Fractional Bagley-Torvik Problem Analysis with Hilfer Fractional Derivatives and Pettis Function Space

Abstract

This paper analyzes the Bagley–Torvik fractional-order equation with generalized fractional Hilfer derivatives of two orders for functions in Banach spaces under conditions expressed in the language of weak topology. We develop a comprehensive theory of fractional-order differential equations of various orders. Our focus is on the equivalence results (or the lack thereof) of this new class of fractional-order Hilfer operators and on maximizing the regularity of the solution. To this end, we examine the equivalence of differential problems involving pseudo-derivatives and integral problems involving Pettis integrals. Our results are novel, even within the context of integer-order differential equations. Another objective is to incorporate fractional-order problems into the growing research field that uses weak topology and function spaces to study vector-valued functions. The auxiliary results obtained in this article are general and applicable beyond its scope.

1. Background and Motivations

In this paper, we examine the Bagley–Torvik equation in the context of generalized Hilfer derivatives of fractional order. Specifically, we will analyze the equation in the context of vector-valued functions. This is an interesting example of an equation containing derivatives of two different orders, which have important practical applications. In connection with the study of vector functions, where the assumptions about the problem are expressed in terms of weak topology on the target space, the equation illustrates the role of fractional calculus in the widely researched topic of applications involving such functions (differential and integral equations). The main tool used in this study is the operator form of the problems. This form is used to investigate the properties of differential and integral operators, as well as to maximize the regularity of solutions by examining their domains. Unlike previous studies, which focused on classical operators (primarily Volterra) or typical Caputo or Riemann–Liouville fractional-order operators, the Bagley–Torvik equation introduces an intriguing new class of operators. While the results are motivated by solving this equation, they also have interesting connections to operator theory and function spaces. We will examine vector-valued functions and assumptions about functions in terms of weak topology because we believe this yields the most mathematically interesting cases of generalized solutions. This approach enables us to study these problems under very general assumptions.

The four main goals and steps of the research are described and explained below.

Bagley–Torvik equation. First, here are some comments on the fractional-order equation under investigation. The Bagley–Torvik equation was not an equation that was later “fractionalized” by others. Rather, Bagley and Torvik conceived it as a fractional differential equation from the beginning to solve a specific physical problem [1,2]:

$$u^{″}\left(t\right)+a{D}_t^{3/2}u\left(t\right)+bu\left(t\right)=f\left(t\right),$$

where a and b are constants, and it was originally derived by Bagley and Torvik to model the motion of a rigid plate in a Newtonian fluid. The step of replacing the scalar coefficient b with the operator $A=-\Delta$ is a classical method (e.g., for the wave equation $u^{″}+Au=f$). Extending this to include a fractional derivative term $A{D}_t^{\alpha }u$ is a natural generalization studied in the late 20th and early 21st centuries.

The work by Bagley and Torvik was a major milestone in demonstrating the practical utility of fractional calculus in engineering. An original Bagley–Torvik problem is a differential equation that includes both a standard integer-order derivative, such as acceleration (${\displaystyle \frac{d^{2}y}{d{t}^2}}$), and a fractional-order derivative. The canonical form of the Bagley–Torvik equation is

$$A·{\displaystyle \frac{d^{2}y\left(t\right)}{d{t}^2}}+B·D_t^{\alpha }y\left(t\right)+C·y\left(t\right)=f\left(t\right),\mathrm{with}\alpha ={\displaystyle \frac{3}{2}},$$

where A, B, and C are constants, $D_t^{\alpha }y\left(t\right)$ is the Caputo or Riemann–Liouville fractional derivative of order $\alpha$, and $f\left(t\right)$ is a forcing function. The key feature is the presence of this ${\displaystyle \frac{3}{2}}$-order derivative, which sits conceptually and mathematically between the first derivative (velocity) and the second derivative (acceleration). The issues they examine naturally lead to generalized problems concerning two different orders of derivatives. Furthermore, they point to the need for research on operators between fractional-order (e.g., $\alpha ={\displaystyle \frac{1}{2}}$) and integer-order (e.g., 1 or 2) operators. The original and primary motivation for studying the Bagley–Torvik equation was to model the damping behavior of viscoelastic materials [1]. This is why fractional derivatives were applied ([2,3], for instance): Viscoelastic materials (e.g., polymers, rubber, biological tissues, and asphalt) exhibit a blend of elastic solid and viscous fluid behaviors. Their response depends not only on the current deformation, but also on the history of deformation. A first-derivative term (velocity) is insufficient to accurately capture this “memory” effect. However, this property is common among many fractional derivatives. This type of fractional model has many applications in situations where viscoelastic damping is significant. It is particularly useful for modeling and designing damping systems, such as vehicle suspension systems, shock absorbers, and vibration isolators that use polymer-based dampers.

Bagley–Torvik-type problems have been extensively studied. One can find detailed accounts of recent work involving these problems and the following operators in [3,4]: Caputo, Riemann–Liouville, Hadamard, Hadamard–Caputo, and generalized fractional derivatives. These accounts also cover various boundary conditions. We propose considering the Hilfer fractional derivative operator, which encompasses the Caputo and the Riemann–Liouville operators and has received significant attention because it is involved in the equations under investigation.

Vector-valued fractional calculus. The second goal of the paper is to incorporate fractional calculus into the study of vector-valued differential equations. The transition from scalar to abstract cases is motivated by the need to model infinite systems of coupled oscillators or systems with continuous state distributions. This naturally leads to equations in infinite-dimensional spaces. Many time-dependent differential equations, particularly partial differential equations (PDEs), can be rewritten as integral equations in infinite-dimensional function spaces, such as Banach or Hilbert spaces [5,6]. More about vector-valued fractional problems can be found in [7].

Simple motivation: Consider a viscoelastic plate whose deformation $w(x,t)$ is governed by a equation like

$${\displaystyle \frac{{\partial }^{2}w}{\partial t^2}}-\Delta \left(a{D}_t^{\alpha }w+bw\right)=g(x,t).$$

For a plate element, the Newton second law states

$$\rho h{\displaystyle \frac{{\partial }^{2}w}{\partial t^2}}=\Delta M+g(x,t),$$

where $\rho h$ is the mass per unit area. The reasoning behind converting partial differential equations into abstract evolution equations is quite classic, and it is used to solve various types of equations. We can usually start with a PDE for a viscoelastic structure, then by defining the operator $A=-\Delta$ on a function space $X=L^p(\Omega )$ with the appropriate domain, this PDE becomes a Banach-space valued the Bagley–Torvik equation. In the form of the abstract equation we can prove well-possessedness and solution properties using the operator theory:

$$u^{″}\left(t\right)+aA{D}_t^{\alpha }u\left(t\right)+bAu\left(t\right)=f\left(t\right),$$

where $u\left(t\right):=w(·,t)$ is a function-valued trajectory (see [8,9]). See also [10,11] for evolution equations with fractional derivatives:

$$u^{\prime }\left(t\right)=Au\left(t\right)+D_t^{1-\alpha }f\left(t\right),$$

or similar forms. For a general, abstract form of these equations and their background, see [3]. Note that the regularity of solutions to this equation is one of the paper’s central topics. Building on this idea, we examine improved Hölder regularity. We investigate the regularity (smoothness) of the solution $u\left(t\right)$ based on the regularity of the initial data and the properties of differential or integral fractional operators. Unfortunately, when considering the weak topology on Banach spaces, the equivalence problem between differential and integral equations is very delicate. Surprisingly, this is generally only possible for certain spaces [12,13].

Furthermore, problems involving integrals where the kernel is an operator-valued function are prevalent in control theory, non-local phenomena, and quantum mechanics. This also leads to differential problems in abstract spaces (see also [14,15,16]).

Finally, fractional differential equations are useful for modeling memory and hereditary properties. Extending these models to infinite-dimensional settings, such as fractional partial differential equations (PDEs), is a major research area that requires further study of new operators and function spaces (see [17] for the fractional partial differential equations rewritten in the operator form). The fundamental paper to be recommended in this topic is [6]. This paper concerns the equation

$$D_t^{\alpha }{u}^{\prime }\left(t\right)+\mu D_t^{\alpha }u\left(t\right)=Au\left(t\right)+{\displaystyle \frac{t^{-\alpha }}{\Gamma (1-\alpha )}}(u^{\prime }\left(0\right)+\mu u\left(0\right))+f\left(t\right),$$

for $t>0$, $0<\alpha \le 1$, $\mu \ge 0$, where A is an unbounded closed operator defined on a Banach space X and f is an X-valued function. Throughout this paper, recall that the constant $\Gamma (1-\alpha )$ is the normalization constant that arises directly from the definition of a fractional derivative. In the definition of the Riemann–Liouville integral, it guarantees that $I^{\alpha }$ is the proper antiderivative. This equation is then studied in abstract Banach spaces that motivates our research. However, we are dealing with an equation that has two orders of fractional derivatives, and in this case, we have $\beta =1+\alpha$. Most importantly, we increase the regularity of the solutions and use the weak topology.

Let us mention the paper [18], in which the study of abstract fractional differential equations, including versions with damped terms, leads to fractional evolution problems similar to the Bagley–Torvik equation. The abstract formulation of the problem (a prototype for our vector-valued problem) is typically of the form:

$$\begin{array}{ccc}& & {\displaystyle \frac{d^{2}u\left(t\right)}{d{t}^2}}+B{D}_t^{\alpha }u\left(t\right)+Cu\left(t\right)=f\left(t\right),\hfill \\ & & u\left(0\right)=u_0,u^{\prime }\left(0\right)=u_1,\hfill \end{array}$$

where $t\in [0,T]$, $u:[0,T]\to X$ is the unknown function, taking values in the Banach space X, and B and C are operators on X. Importantly, B is often a sectorial operator or closed, densely defined operator (e.g., the Laplacian, $\Delta$), enabling the development of a solution theory analogous to the scalar case. Clearly, $D_t^{\alpha }$ is the fractional derivative of order $\alpha$, most commonly $\alpha =1/2$ or $\alpha =3/2$. Until now, it was usually defined as the Caputo or the Riemann–Liouville derivative. Moreover, $f:[0,T]\to X$ is a forcing term, and $u_0,u_1\in X$ are the initial conditions. The Bagley–Torvik equation is a perfect example of this problem (see also [19]).

Weak topology. The third motivation for researching this problem based on assumptions expressed in terms of weak topology remains to be indicated. This leads to new definitions of the “weak” fractional derivative and the use of the Pettis integral instead of the Lebesgue–Bochner integral. This approach to the subject has several advantages when it comes to studying vector functions and developing integral theory and function spaces. Due to the broad scope of the results and their applications, many problems have been examined in terms of weak topology (cf. [20], for the role of weak topologies in stochastic mechanics). However, this has primarily been done in the context of integer derivatives (see, for example, [21,22,23,24,25], in different applications). These results are inspiring, even with purely mathematical motivations, and they have interesting implications for fractional-order equations, particularly those with two distinct orders of derivatives, such as the Bagley–Torvik equation.

The main reason to use the Pettis integral is when our function is not Bochner-integrable, but still “integrable” in a weaker sense. The definition of a Pettis integral seems like a natural generalization of the Bochner one. However, the critical challenge is loss of compactness and convergence. In infinite dimensions, a bounded sequence does not necessarily have a convergent subsequence. For the convenience of our readers, we will review all the necessary concepts and results. Then, we will extend these concepts to the framework of fractional operators. In practice, the key point when using weak Pettis integrals is to use them with additional structures, such as the geometric properties of the space. This leads to the problem of differential and integral operators defined by the investigated problems, as well as the function spaces in which they are defined. One central problem is the equivalence between solutions of fractional differential equations and their integral forms [12,13]. We will address this problem in our paper. The equivalence between the differential and integral forms of a problem is not just a mathematical curiosity. It signifies a fundamental shift in how we analyze problems. This equivalence plays a multifaceted role, particularly when dealing with concepts such as pseudo-solutions and the Pettis integral. Proving the existence of a solution to the integral formulation using a fixed point theorem also proves the existence of a generalized solution to the original differential problem. This method is primarily used to prove the existence of solutions for ordinary and partial evolution equations.

Nevertheless, the application of this integral and topology has many advantages and has long been used in research [26,27,28], for instance. In the case of fractional problems related to, we should mention [29,30,31,32,33].

Regularity of solutions. Additionally, it should be emphasized that the research is conducted with maximal (Hölder) regularity of solutions when solving the examined equations. This aligns with the current research trend ([3,34,35,36]).

2. Preliminaries

To understand the purpose of this paper and the mathematical difficulties concerning operators, including integral and pseudo-derivatives as well as function spaces and their relations, we will review the necessary results. Then, we will extend this scope to the fractional and non-fractional order operators under investigation.

In what follows, let $\left[X,∥·∥\right]$ denote a Banach space with dual $X^{\ast }$. Let $X_w$ denote the space X when it is endowed with the weak topology. Moreover, let $C([a,b],X_w)$ denote the Banach space of continuous functions from $[a,b]$ to X, with its weak topology (see also [27]).

A function $f:[a,b]\to X$ is said to be weakly measurable on $[a,b]$ if for every ${\phi }^{\ast }\in X^{\ast }$ we have real function ${\phi }^{\ast }f\in L_1[a,b]$. Also, the pair $\left[{\mathcal{H}}^{\lambda }\left[[a,b],X\right],\parallel |·{|\parallel }_{\lambda }\right],\lambda \in (0,1)$ denotes the Hölder space endowed by the norm

$${∥\left|f\right|∥}_{\lambda }=∥f\left(a\right)∥+{\left[\left[f\right]\right]}_{\lambda },\mathrm{where}{\left[\left[f\right]\right]}_{\lambda }:=\underset{t,s\in [a,b],t\ne s}{sup}\left\{{\displaystyle \frac{∥f\left(t\right)-f\left(s\right)∥}{{|t-s|}^{\lambda }}}\right\}.$$

In addition, if $f\left(a\right)\equiv 0$ (where 0 denotes the zero element of X), then we write $f\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$. However, the pair $\left[{\mathcal{H}}_0^{\lambda }\left[[a,b],X\right],{\left[[·]\right]}_{\lambda }\right]$ becomes a Banach space. We need the following well-known fact: For $0<{\lambda }_1<{\lambda }_2<1$, we have

$${\mathcal{H}}^1\left[[a,b],X\right]\subset {\mathcal{H}}^{{\lambda }_2}\left[[a,b],X\right]\subset {\mathcal{H}}^{{\lambda }_1}\left[[a,b],X\right]\subset C([a,b],X).$$

Now, we need the following important definitions [37]:

Definition 1. 

A weakly measurable function $f:[a,b]\to X$ is said to be Pettis-integrable on $[a,b]$ if $\phi f\in L_1[a,b]$ for every $\phi \in X^{\ast }$ and for any measurable $A\subset [a,b]$ there exists an element in X denoted by ${\int }_{A}f$ such that

$$\phi \left({\int }_{A}f\right)={\int }_A\phi f\left(s\right)ds\mathrm{for}\mathrm{every}\phi \in X^{\ast }.$$

By $\mathfrak{P}\left[[a,b],X\right]$, we denote the space of all X-valued Pettis-integrable functions on $[a,b]$.

The fundamental reason to use the Pettis integral in studying differential or integral equations is when the function is not Bochner integrable (i.e., not strongly measurable and whose norm is not Lebesgue integrable) but is still “integrable” in a weaker sense, described above. This seems like a natural generalization. However, the critical challenge is that the weak topology does not control the norm topology.

In practice, when using weak Pettis integrals, the key point is to employ them in conjunction with additional structure, such as weak compactness (the unit ball of X is weakly compact if X is reflexive) or if the function f takes values in a weakly compact set. Another approach is based on geometry of the space. The theory is much more manageable if the Banach space X has the Radon-Nikodým Property (RNP) (e.g., reflexive spaces, separable dual spaces). For spaces with RNP, a function f is Pettis-integrable if and only if it is Dunford integrable (i.e., there exist Lebesgue integrals of $x^{\ast }f$ for every fixed functional $x^{\ast }\in X^{\ast }$) and its range is essentially separable. In connection with the concept of pseudo-derivative, we will study certain other types of spaces.

Definition 2 

([27]). A function $f:[a,b]\to X$ is said to be pseudo-differentiable if there exists a function $g:[a,b]\to X$ such that, for every $\phi \in X^{\ast }$, the real-valued function $\phi g$ is differentiable a.e., to the value $\phi g$. In this case, the function g is called the pseudo-derivative of f will be denoted by ${\mathfrak{D}}_{p}f$.

If the null set independent on $\phi$, then f is said to be a.e. weakly differentiable on $[a,b]$ and g (in this case) is called weak derivative of f (denoted by ${\mathfrak{D}}_{w}f$) and exists almost everywhere on $[a,b]$. In particular, when $X=\mathbb{R}$ it is clear that the pseudo and a.e. weak derivatives coincide with the classical derivatives of real-valued functions.

Remark 1. 

1. 

Unfortunately (unless X has a total dual $X^{\ast }$, i.e., with the property that for every non-zero element $x\in X$, there exists a bounded linear functional $\phi \in X^{\ast }$ such that $\phi x\ne 0$), the pseudo-derivative of the pseudo-differentiable function $f:[a,b]\to X$ is not uniquely determined and two pseudo-derivatives need not be a.e. equal [38] (see Example 9.1 in [37] or [39]). Indeed, the pseudo-derivatives of $f:[a,b]\to X$ are weakly equivalent ([40], Theorem VI.8) (cf. also [38]).

2. 

It is well known that, in a finite dimension space X, then the weak absolute continuity of $f:[a,b]\to X$ already implies that ${\mathfrak{D}}_{p}f$ exists almost everywhere and it is in $\mathfrak{P}\left[[a,b],X\right]$. This one of the few properties of the finite dimension-valued functions that is not not carry over in arbitrary infinite dimensions: In fact, even in separable spaces, there is an weakly absolutely continuous function having no pseudo-derivative [38,40]. The question of whether weak absolute continuity of a X-valued function implies the existence of a Pettis integrable pseudo-derivative is related to the geometry of X. We also remark that, if X is finite dimensional, then the indefinite integral of a Pettis integrable function is a.e. weakly differentiable. In 1995 in [41] it was proved that the indefinite Pettis integral of a Pettis-integrable function $f:[a,b]\to X$ does not enjoy the property of being a.e. weakly differentiable. Therefore, it is a different notion from the weak derivative.

We will consider here pseudo-solutions of considered differential problems. The term “pseudo-solution” typically refers to a function that satisfies the integral form of a problem but not necessarily the original differential form. Here the integral form is based on the Pettis integral, and the differential one on pseudo-derivatives. For well-posed problems in appropriate spaces, the equivalence tells us that the concept of a pseudo-solution is a correct and natural generalization of a classical solution. It is not a “fake” solution but rather the correct type of solution for a broader context. However, this approach raises many problems, particularly with equivalence problems and, above all, with correctly defining the integral and differential fractional operators. We will discuss these issues in the paper.

First, let us start with the basics. The following proposition tells us weakly that an absolutely continuous function is pseudo-differentiable if, and only if, it is an indefinite Pettis integral of a Pettis-integrable function:

Proposition 1 

(([37,38]), ([40], Theorem VIII.3)). A function $f:[a,b]\to X$ is an indefinite Pettis integral if and only if f is weakly absolutely continuous having a pseudo-derivative ${\mathfrak{D}}_{p}f$ on $[a,b]$. In this case, ${\mathfrak{D}}_{p}f\in \mathfrak{P}\left[[a,b],X\right]$ and

$${\int }_a^t{\mathfrak{D}}_{p}f\left(s\right)ds=f\left(t\right)-f\left(a\right),t\in [a,b].$$

The following result can be obtained by assuming weak absolute continuity of the function:

Proposition 2 

([42,43], Theorem 5.1). The indefinite integral of Pettis-integrable (resp. weakly continuous) function is weakly absolutely continuous. It is pseudo (resp. weakly) differentiable with respect to the right endpoint of the integration interval and its pseudo (resp. weak) derivative equals the integrand at that point.

Throughout this paper, we let the functions ${\mathsf{\ell }}_1,{\mathsf{\ell }}_2:[0,1]\times [a,b]\to [0,\infty )$ be continuous such that

$$\left\{\begin{array}{c}{\displaystyle \underset{\alpha \to 0}{lim}{\mathsf{\ell }}_2(\alpha ,t)=1,\underset{\alpha \to 0}{lim}{\mathsf{\ell }}_1(\alpha ,t)=0,\mathrm{for} \mathrm{all}t\in [a,b],}\hfill \\ {\displaystyle \underset{\alpha \to 1}{lim}{\mathsf{\ell }}_2(\alpha ,t)=0,\underset{\alpha \to 1}{lim}{\mathsf{\ell }}_1(\alpha ,t)=1,\mathrm{for} \mathrm{all}t\in [a,b],}\hfill \\ {\displaystyle {\mathsf{\ell }}_2(\alpha ,t)\ne 0,\alpha \in [0,1),{\mathsf{\ell }}_1(\alpha ,t)\ne 0,\alpha \in (0,1],\mathrm{for} \mathrm{all}t\in [a,b].}\hfill \end{array}\right.$$

(1)

Throughout this pages, we assume that, for any $\rho \in [0,1]$

$${\left({\mathsf{\ell }}_1(\rho ,·)\right)}^{-1}\in L_q[a,b]\mathrm{and}{\left({\mathsf{\ell }}_2(\rho ,·)\right)}^{-1}\in L_q[a,b]\mathrm{for} \mathrm{some}q\ge 1.$$

(2)

A simple example: ${\mathsf{\ell }}_1(\rho ,t)=\rho t^{1-\rho },{\mathsf{\ell }}_2(\rho ,t)=(1-\rho )t^{\rho },\rho \in [0,1]$ and $t\in [0,1]$. Define $\mathsf{\ell }(·,·):={\mathsf{\ell }}_2(·,·)/{\mathsf{\ell }}_1(·,·)$.

Next, we will introduce the (non-fractional) differential operators that are useful for studying Bagley–Torvik type problems. Let $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ be a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$.

Define the non-fractional pseudo (in particular, weakly) differentiation operators (see e.g., [44,45,46] and the references therein for background on these topics):

$${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f:=\left({\mathsf{\ell }}_2(\rho ,t)+{\mathsf{\ell }}_1(\rho ,t)\left[\mu +{\displaystyle \frac{1}{{\psi }^{\prime }}}{\mathfrak{D}}_p\right]\right)f,$$

$${\Delta }_{w,\psi }^{\rho ,\mathsf{\ell }}f:=\left({\mathsf{\ell }}_2(\rho ,t)+{\mathsf{\ell }}_1\rho ,t)\left[\mu +{\displaystyle \frac{1}{{\psi }^{\prime }}}{\mathfrak{D}}_w\right]\right)f,$$

where f is a pseudo (in particular, weakly) differentiable function of the argument $t\in \mathbb{R}$.

The following general definition of fractional integral operator has been established for the first time by the authors in [47] for real-valued functions.

Definition 3 

([47]). Let $\alpha >0$, $\mu \ge 0$, $\rho \in (0,1)$. Assume that the assumptions (1) and (2) are satisfied. We define the combination of the fractional and non-fractional Pettis integral operator acting on $f\in \mathfrak{P}\left[[a,b],X\right]$ by

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t\right):={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right)f\left(s\right)ds,$$

$$\omega (·):={\int }_a^{(·)}{\displaystyle \frac{{\psi }^{\prime }\left(\theta \right)d\theta }{{\mathsf{\ell }}_1(\rho ,\theta )}}.$$

For completeness, we define $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(a\right):=0$ and $K_{\rho ,\psi }^{0,\mathsf{\ell },\mu }f:=f$. In the above definition the sign ∫ denotes the Pettis integral.

These operators and their corresponding differential operators will be described later. Together, they encompass all standard fractional operators for real-valued and vector-valued functions.

In particular, when ${\mathsf{\ell }}_1(\rho ,t)\equiv \rho ,{\mathsf{\ell }}_2(\rho ,t)\equiv (1-\rho )$,

$$K_{\rho ,\psi }^{\alpha ,{\displaystyle \frac{1-\rho }{\rho }},\mu }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right){\rho }^{\alpha }}}{\int }_a^t{e}^{-\left({\displaystyle \frac{1-\rho }{\rho }}+\mu \right)(\psi \left(t\right)-\psi \left(s\right))}{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}f\left(s\right){\psi }^{\prime }\left(s\right)ds.$$

Furthermore, when $\rho =1$

$$K_{\rho ,\psi }^{\alpha ,0,\mu }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{e}^{-\mu \left(\psi \right(t)-\psi (s\left)\right)}{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}f\left(s\right){\psi }^{\prime }\left(s\right)ds.$$

Also

$${\displaystyle K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f\left(t\right):={\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\omega }^{\prime }\left(s\right)f\left(s\right)ds.}$$

Clearly, for any $\rho \in [0,1]$,

$${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}\left\{exp{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\right\}=0.$$

(3)

Remark 2. 

Our definition that $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(a\right):=0$ makes sense when $f\in C\left([a,b],X_w\right)$.

Let $f\in C\left([a,b],X_w\right)$. Then, for any $\phi \in X^{\ast }$, we have

$$\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t\right)\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{e}^{\left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right)\phi \left(f\left(s\right)\right)ds,$$

for $t\in [a,b]$. Therefore,

$$\left|\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t\right)\right)\right|\le \underset{t\in [a,b]}{max}\left|\phi \left(f\right(t\left)\right)\right|{\displaystyle \frac{{\left(\omega \left(t\right)\right)}^{\alpha }}{\Gamma (1+\alpha )}}\to 0ast\to a^{+}.$$

Hence

$$\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(a\right)\right)=0,\mathit{for} \mathit{any}\phi \in X^{\ast }.$$

Thus, $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(a\right)=0$, as required.

We will use the following

Remark 3. 

Let $0\ne f\in C\left([a,b],X_w\right)$. An immediate consequence of the Hahn–Banach theorem is that there exists $\phi \in X^{\ast }$ with ${∥\phi ∥}_{X^{\ast }}=1$ such that $∥K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f\left(t\right)∥=\left|\phi \left(K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f\left(t\right)\right)\right|=\left|K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\phi \left(f\left(t\right)\right)\right|$. Hence

$$∥K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f\left(t\right)∥=\left|{\int }_a^t{e}^{\left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\omega }^{\prime }\left(s\right)\phi \left(f\left(s\right)\right)ds\right|\le ∥\omega ∥·{∥\left|f\right|∥}_C,$$

where ${∥\left|f\right|∥}_C:={max}_{t\in [a,b]}∥f\left(t\right)∥$.

In order to examine the existence of the operator $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }$ acting on functions from the space $\mathfrak{P}\left[[a,b],X\right]$, we recall the following function spaces:

Definition 4 

([37]). Let $\mathfrak{P}\left[[a,b],X\right]$ denote the space of X-valued Pettis-integrable functions on $[a,b]$. For any $p\in [1,\infty ]$, we define its subspace

$${\mathbb{H}}^p\left(X\right):=\left\{f:[a,b]\to X:fweakly measurable such that\phi f\in L_p[a,b]for every\phi \in X^{\ast }\right\}.$$

Moreover, we define the subspace ${\mathbb{H}}_0^p\left(X\right)$ of ${\mathbb{H}}^p\left(X\right)$ composed of Pettis-integrable functions on $[a,b]$.

Since the weak continuity implies the strong measurability ([42], p. 73), it is not hard to be seen that

$$C\left([a,b],X_w\right)\subset {\mathcal{H}}_0^p\left(X\right)\subseteq {\mathcal{H}}_0^q\left(X\right).$$

To make it easier for readers to understand, we will present a summary of our considerations in the form of the following results:

Proposition 3. 

Let $p,q\in [1,\infty ]$ such that ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$ (with the convention that ${\displaystyle \frac{1}{\infty }}=0$).

The following implications hold:

1. 

$f\in {\mathbb{H}}_0^p\left(X\right),p>1,g\in L_q[a,b]⟹fg\in \mathfrak{P}\left[[a,b],X\right]$,

2. 

$f\in \mathfrak{P}\left[[a,b],X\right],g\in L_{\infty }[a,b]⟹fg\in \mathfrak{P}\left[[a,b],X\right]$,

3. 

For any $p>1$, we have $\{f\in {\mathbb{H}}^p\left(X\right):fstrongly measurable\}\subseteq {\mathbb{H}}_0^p\left(X\right)$.

If E weakly sequentially complete, this is also true for $p=1$,

4. 

$f\in {\mathbb{H}}^1\left(X\right),Xis reflexive⟹f\in \mathfrak{P}\left[[a,b],X\right]$.

As a direct consequence of Proposition 3, we have the following existence lemmas

Lemma 1. 

For every weakly measurable $f:[a,b]\to X$ and $\alpha >0$, $\mu \ge 0$, $\rho \in (0,1)$, the function $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f$ makes sense almost everywhere on $[a,b]$ if at least one of the following cases holds:

(a) 

$f\in {\mathbb{H}}_0^p\left(X\right)$ with $p>{\displaystyle \frac{1}{\alpha }}$, where $\alpha \in (0,1)$,

(b) 

$f\in {\mathbb{H}}_o^1\left(X\right)\equiv \mathfrak{P}\left[[a,b],X\right]$ and $\alpha \ge 1$,

(c) 

If X is weakly complete and $f\in {\mathbb{H}}^p\left(X\right)$ is strongly measurable where $p\in [1,\infty ]$.

If X is reflexive, this is also true for any $f\in {\mathbb{H}}^p$ with $p\in [1,\infty ]$. In all cases, for every $\phi \in X^{\ast }$ we have $\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right)=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi f$.

Proof. 

Define the real-valued continuous function $\chi (·):={\int }_0^{\psi (·)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du$. To prove assertion $\left(a\right)$, let $p>{\displaystyle \frac{1}{\alpha }}$ and note that $q(\alpha -1)+1>0$ (${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$). Since $q(\alpha -1)+1>0$, we therefore have

$$g(·):=e^{-\left(\chi \right(t)-\chi (·\left)\right)}{(\omega \left(t\right)-\omega (·))}^{\alpha -1}{\omega }^{\prime }(·)\in L_q[a,t],t\in (a,b].$$

Thus, if $f\in {\mathbb{H}}_0^p\left(X\right)$ with $p>max\{1,{\displaystyle \frac{1}{\alpha }}\}$, the assertion $\left(a\right)$ follows by the first part of Proposition 3. Secondly, if $\alpha \ge 1$, then $g\in L_{\infty }[a,t]$. Therefore, if $f\in \mathfrak{P}\left[[a,b],X\right]$, assertion $\left(b\right)$ follows from the second part of Proposition 3.

Next, to prove $\left(c\right)$, we let X be weakly complete and assume that $f\in {\mathbb{H}}^p\left(X\right)$, $p\ge 1$ is strongly measurable on $[a,b]$. Since the product $g(·)f(·):[a,t]\to X$ is almost everywhere strongly measurable on $[a,t]$, $t\in (a,b]$, it follows from the Young inequality that, for every $\phi \in X^{\ast }$, $\phi \left(g(·)f(·)\right)=g(·)\phi f(·)\in L_1[a,t]$, $t\in (a,b]$. Therefore, the assertion $\left(c\right)$ follows directly from the third part of Proposition 3. Finally, when X is reflexive, the result follows from Proposition 3.

In all cases, however, $e^{-\left(\chi \right(t)-\chi (·\left)\right)}{(\omega \left(t\right)-\omega (·))}^{\alpha -1}{\omega }^{\prime }(·)f(·)\in f\in \mathfrak{P}\left[[a,t],X\right]$ for almost every $t\in [a,b]$. By the definition of the Pettis integral, there exists an element from X denoted by $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t\right)$, $t\in [a,b]$ such that

$$\begin{array}{ccc}\hfill \phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t\right)\right)& =& {\int }_a^t\phi \left({\displaystyle \frac{e^{-\left(\chi \right(t)-\chi (s\left)\right)}{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right)}{\Gamma \left(\alpha \right)}}f\left(s\right)\right)ds\hfill \\ & =& {\int }_a^t{\displaystyle \frac{e^{-\left(\chi \right(t)-\chi (s\left)\right)}{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right)}{\Gamma \left(\alpha \right)}}\phi f\left(s\right)ds\hfill \\ & =& K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi f\left(t\right),\hfill \end{array}$$

holds for every $\phi \in X^{\ast }$. This completes the proof. □

Let us prove that the operator ${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}$ is the left inverse of the operator $K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }$ on the space $C\left([a,b],X_w\right)$ of weakly continuous functions $f:[a,b]\to X$. In fact, based on Proposition 2, it is easy to prove that

Lemma 2. 

(a) For any $f\in C\left([a,b],X_w\right)$ and $\rho \in (0,1)$, and $t\in [a,b]$ we have

$${\Delta }_{w,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f\left(t\right)={\Delta }_{w,\psi }^{\rho ,\mathsf{\ell }}\left({\int }_a^{t}exp{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\omega }^{\prime }\left(s\right)f\left(s\right)ds\right)=f\left(t\right).$$

(b) For any $f\in \mathfrak{P}\left([a,b],X\right)$ and $\rho \in (0,1)$, and $t\in [a,b]$ we have

$${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f\left(t\right)={\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}\left({\int }_a^{t}exp{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\omega }^{\prime }\left(s\right)f\left(s\right)ds\right)=f\left(t\right).$$

Remark 4. 

The second assertion of Lemma is not uniquely determined unless X has total dual $X^{\ast }$. According to (e.g., ([38], p. 2)), it may happen that ${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f=g$, with g being weakly equivalent to f. However, they do not necessarily have to be equal.

The situation is simple in the case of absolutely continuous functions. We can prove the following:

Proposition 4. 

Let $\rho \in [0,1]$ and assume that $f:[a,b]\to X$ be a weakly absolutely continuous with a pseudo-derivative ${\mathfrak{D}}_{p}f\in \mathfrak{P}\left[[a,b],X\right]$ uniquely determined up to a negligible set. Then

$$K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\left(t\right)=f\left(t\right)-exp{\displaystyle \left[-{\int }_0^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f\left(a\right),t\in [a,b].$$

Proof. 

By the definition of the pseudo-derivative of f on $[a,b]$, for any $\phi \in X^{\ast }$, we have ${\left(\phi f\left(s\right)\right)}^{\prime }=\phi \left({\mathfrak{D}}_{p}f\right)$. Therefore, for any $\phi \in X^{\ast }$, we have

$$\begin{array}{ccc}& & \phi \left(K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\left(t\right)\right)=K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}\phi \left(f\left(t\right)\right)\hfill \\ & =& {\int }_a^t{e}^{\left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\left[\left({\mathsf{\ell }}_2(\rho ,t)+\mu {\mathsf{\ell }}_1(\rho ,t)\right)\phi \left(f\left(s\right)\right){\omega }^{\prime }\left(s\right)\right.\hfill \\ & & +\left.{\left(\phi \left(f\right(s\left)\right)\right)}^{\prime }\right]ds\hfill \\ & =& {\int }_a^t{e}^{\left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\left[\phi \left(f\left(s\right)\right){\displaystyle \frac{d}{ds}}\left(-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right)\right]ds\hfill \\ & & +{\int }_a^t{e}^{\left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\left(\phi \left(f\right(s\left)\right)\right)}^{\prime }ds.\hfill \end{array}$$

Therefore, integration by parts provides

$$\begin{array}{ccc}\hfill \phi \left(K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\left(t\right)\right)& =& {\left[\phi \left(f\left(s\right)\right)e^{\left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\right]}_a^t\hfill \\ & & -{\int }_a^t{e}^{\left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\left(\phi \left(f\right(s\left)\right)\right)}^{\prime }ds\hfill \\ & & +{\int }_a^t{e}^{\left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\left(\phi \left(f\right(s\left)\right)\right)}^{\prime }ds\hfill \\ & =& \phi \left(f\left(t\right)\right)-e^{-{\int }_0^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du}\phi \left(f\left(a\right)\right)\hfill \\ & =& \phi \left(f\left(t\right))-e^{-{\int }_0^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du}f\left(a\right)\right).\hfill \end{array}$$

As a result,

$$K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\left(t\right)=f\left(t\right)-e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f\left(a\right),$$

(4)

for $t\in [a,b]$, as required. □

Lemma 3. 

Let $\alpha \in (0,1),\rho \in [0,1]$, $\mu \ge 0$ and assume that $\psi \in C^1([a,b],\mathbb{R})$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0.$ If $p>1/\alpha$, then

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }:{\mathbb{H}}_0^p\left(X\right)\to C([a,b],X).$$

In particular, if X is a reflexive space, then for all $p\in [1,\infty ]$,

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }:{\mathbb{H}}^p\left(X\right)\to {\mathbb{H}}_0^p\left(X\right).$$

Proof. 

Let $f\in {\mathbb{H}}_0^p\left(X\right)$, where $p>1/\alpha$, and let $a\le t_1<t_2\le b$. Without loss of generality, assume that $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_2\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_1\right)\ne 0$. As we observed, the Hahn–Banach theorem implies the existence of a functional $\phi \in X^{\ast }$ with $∥\phi ∥=1$, such that

$$∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_2\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_1\right)∥=\left|\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_2\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_1\right)\right)\right|.$$

Therefore,

$$\begin{array}{ccc}& & \Gamma \left(\alpha \right)∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_2\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_1\right)∥=\Gamma \left(\alpha \right)\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi f\left(t_2\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi f\left(t_1\right)\right|\hfill \\ & & ={\int }_a^{t_1}\left|e^{-(\chi \left(t_2\right)-\chi \left(s\right))}{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}-e^{-(\chi \left(t_1\right)-\chi \left(s\right))}{(\omega \left(t_1\right)-\omega \left(s\right))}^{\alpha -1}\right|\left|\phi f\left(s\right)\right|{\omega }^{\prime }\left(s\right)ds\hfill \\ & & +{\int }_{t_1}^{t_2}\left|e^{-(\chi \left(t_2\right)-\chi \left(s\right))}{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}\right|\left|\phi f\left(s\right)\right|{\omega }^{\prime }\left(s\right)ds\hfill \\ & & \le {\int }_a^{t_1}{e}^{-(\chi \left(t_2\right)-\chi \left(s\right))}\left|{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}-{(\omega \left(t_1\right)-\omega \left(s\right))}^{\alpha -1}\right|\left|\phi f\left(s\right)\right|{\omega }^{\prime }\left(s\right)ds\hfill \\ & & +{\int }_a^{t_1}\left|e^{-(\chi \left(t_2\right)-\chi \left(s\right))}-e^{-(\chi \left(t_1\right)-\chi \left(s\right))}\right|{(\omega \left(t_1\right)-\omega \left(s\right))}^{\alpha -1}\left|\phi f\left(s\right)\right|{\omega }^{\prime }\left(s\right)ds\hfill \\ & & +{\int }_{t_1}^{t_2}{e}^{-(\chi \left(t_2\right)-\chi \left(s\right))}{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}\left|\phi f\left(s\right)\right|{\omega }^{\prime }\left(s\right)ds.\hfill \end{array}$$

Since $e^{-\left(\chi \right(t)-\chi (s\left)\right)}\le 1,$ for all $t,s\in [a,b],s\le t$, the Hölder inequality implies that

$$\begin{array}{ccc}& & \Gamma \left(\alpha \right)∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_2\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_1\right)∥\hfill \\ & & \le {\left[{\int }_a^{t_1}{\left|{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}-{(\omega \left(t_1\right)-\omega \left(s\right))}^{\alpha -1}\right|}^q{\left|{\omega }^{\prime }\left(s\right)\right|}^{q}ds\right]}^{{\displaystyle \frac{1}{q}}}{∥\phi f∥}_p\hfill \\ & & +\underset{t\in [a,b]}{max}\left|e^{-(\chi \left(t_2\right)-\chi \left(t\right))}-e^{-(\chi \left(t_1\right)-\chi \left(t\right))}\right|{\left[{\int }_a^{t_1}{(\omega \left(t_1\right)-\omega \left(s\right))}^{q(\alpha -1)}{\left|{\omega }^{\prime }\left(s\right)\right|}^{q}ds\right]}^{{\displaystyle \frac{1}{q}}}{∥\phi f∥}_p\hfill \\ & & +{\left[{\int }_{t_1}^{t_2}{(\omega \left(t_2\right)-\omega \left(s\right))}^{q(\alpha -1)}{\left|{\omega }^{\prime }\left(s\right)\right|}^{q}ds\right]}^{{\displaystyle \frac{1}{q}}}{∥\phi f∥}_p.\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}& & {\displaystyle \frac{\Gamma \left(\alpha \right)}{{∥\phi f∥}_p{∥{\omega }^{\prime }∥}^{\frac{1}{p}}}}∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_2\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_1\right)∥\hfill \\ & & \le {\left[{\int }_a^{t_1}\left|{(\omega \left(t_2\right)-\omega \left(s\right))}^{q(\alpha -1)}-{(\omega \left(t_1\right)-\omega \left(s\right))}^{q(\alpha -1)}\right|{\omega }^{\prime }\left(s\right)ds\right]}^{{\displaystyle \frac{1}{q}}}\hfill \\ & & +∥e^{-(\chi \left(t_2\right)-\chi (·))}-e^{-(\chi \left(t_1\right)-\chi (·))}∥{\left[{\int }_a^{t_1}{(\omega \left(t_1\right)-\omega \left(s\right))}^{q(\alpha -1)}{\omega }^{\prime }\left(s\right)ds\right]}^{{\displaystyle \frac{1}{q}}}\hfill \\ & & +{\left[{\int }_{t_1}^{t_2}{(\omega \left(t_2\right)-\omega \left(s\right))}^{q(\alpha -1)}{\omega }^{\prime }\left(s\right)ds\right]}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

Since $p>1/\alpha$, it follows that $q(\alpha -1)+1>0$ and then

$$\begin{array}{ccc}& & {\displaystyle \frac{\Gamma \left(\alpha \right)\sqrt[q]{q(\alpha -1)+1}}{{∥\phi f∥}_p{∥{\omega }^{\prime }∥}^{\frac{1}{p}}}}∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_2\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_1\right)∥\hfill \\ & \le & {\left|{(\omega \left(t_2\right)-\omega \left(t_1\right))}^{q(\alpha -1)+1}+\underset{<0}{\underbrace{{\left(\omega \left(t_1\right)\right)}^{q(\alpha -1)+1}-{\left(\omega \left(t_2\right)\right)}^{q(\alpha -1)+1}}}\right|}^{{\displaystyle \frac{1}{q}}}\hfill \\ & & +∥e^{-(\chi \left(t_2\right)-\chi (·))}-e^{-(\chi \left(t_1\right)-\chi (·))}∥{\left[{\left(\omega \left(t_1\right)\right)}^{q(\alpha -1)+1}\right]}^{{\displaystyle \frac{1}{q}}}+{\left[{(\omega \left(t_2\right)-\omega \left(t_1\right))}^{q(\alpha -1)+1}\right]}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le & |\omega \left(t_2\right)-\omega \left(t_1\right){|}^{\alpha -{\displaystyle \frac{1}{p}}}+∥e^{-(\chi \left(t_2\right)-\chi (·))}-e^{-(\chi \left(t_1\right)-\chi (·))}∥{∥\omega ∥}^{\alpha -{\displaystyle \frac{1}{p}}}+{|\omega \left(t_2\right)-\omega \left(t_1\right)|}^{\alpha -{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}& & ∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_2\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t_1\right)∥\hfill \\ & \le & {\displaystyle \frac{{∥\phi f∥}_p{∥{\omega }^{\prime }∥}^{\frac{1}{p}}}{\Gamma \left(\alpha \right)\sqrt[q]{q(\alpha -1)+1}}}\left[2|\omega \left(t_2\right)-\omega \left(t_1\right){|}^{\alpha -{\displaystyle \frac{1}{p}}}+∥e^{-(\chi \left(t_2\right)-\chi (·))}-e^{-(\chi \left(t_1\right)-\chi (·))}∥{∥\omega ∥}^{\alpha -{\displaystyle \frac{1}{p}}}\right].\hfill \end{array}$$

These estimations show that $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f$ is norm continuous on $[a,b]$, as required.

Now, if X is a reflexive space and $p\in [1,\infty ]$, then for any $f\in {\mathbb{H}}^p\left(X\right)$ and $\phi \in X^{\ast }$,

$$\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right)=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi f\in L_1[a,b].$$

In view of Lemma 4, the reflexivity of X implies that $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\in {\mathbb{H}}_0^p\left(X\right)$ for all $f\in {\mathbb{H}}^p\left(X\right)$. □

It should be noted that, Lemma 3 is false for $p=1$ and for $p=1/\alpha$ (see e.g., ([48], Example 3.3)).

Now we are in the position to state and prove the following

Lemma 4. 

(semi-group property) Let $\alpha ,\beta >0,\rho \in [0,1]$, $\mu \ge 0$ and assume that $\psi \in C^1([a,b],\mathbb{R})$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. For any $f\in {\mathbb{H}}_0^p\left(X\right)$, $p>max\{1,{\displaystyle \frac{1}{\beta }},{\displaystyle \frac{1}{\alpha }}\}$, we have

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\beta ,\mathsf{\ell },\mu }f=K_{\rho ,\psi }^{\beta ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{\Im }_{a,g}^{\alpha ,\mu }f=K_{\rho ,\psi }^{\alpha +\beta ,\mathsf{\ell },\mu }f.$$

In particular, if X is a reflexive space, it holds for all $p\in [1,\infty ]$.

Proof. 

Let $f\in {\mathbb{H}}_0^p\left(X\right),p>max\{1,1/\beta ,1/\alpha \}$. First, note that since $p>max\{1,1/\beta ,1/\alpha \},$ it follows that $p>max\{1,1/(\beta +\alpha \left)\right\}$. Therefore, by Lemma 4, we deduce that $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f,K_{\rho ,\psi }^{\beta ,\mathsf{\ell },\mu }f$ and $K_{\rho ,\psi }^{\alpha +\beta ,\mathsf{\ell },\mu }f$ exist for all $f\in {\mathbb{H}}_0^p\left(X\right)$. This also true if X is reflexive and $p\in [1,\infty ]$. The result now follows from the the semi-group property of the real-valued functions $\phi f\in L_p[a,b],\phi \in X^{\ast }$:

$$\begin{array}{ccc}\hfill \phi \left(K_{\rho ,\psi }^{\alpha +\beta ,\mathsf{\ell },\mu }f\right)& =& K_{\rho ,\psi }^{\alpha +\beta ,\mathsf{\ell },\mu }\phi f=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\beta ,\mathsf{\ell },\mu }\phi \left(f\right)=K_{\rho ,\psi }^{\beta ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi f\hfill \\ & =& K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi \left(K_{\rho ,\psi }^{\beta ,\mathsf{\ell },\mu }f\right)=\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\beta ,\mathsf{\ell },\mu }f\right).\hfill \end{array}$$

□

The next lemma ensures that the operator $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }$ maps the space of weakly absolutely continuous functions $AC\left[[a,b],X_w\right]$ into itself. Define

$$A{C}^p\left[[a,b],X_w\right]:=\left\{f\in AC\left[[a,b],X_w\right]:f\mathrm{has} a \mathrm{pseudo-derivative}{\mathfrak{D}}_{p}f\in {\mathbb{H}}_0^p\left(X\right)\right\}.$$

Lemma 5. 

Let $\alpha \in (0,1),\rho \in [0,1]$, $\mu \ge 0$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. If $p>{\displaystyle \frac{1}{\alpha }}$, then

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }:A{C}^p\left[[a,b],X_w\right]\to A{C}^1\left[[a,b],X_w\right].$$

In particular, if X is a reflexive space, then this holds for all $p\in [1,\infty ]$.

Proof. 

First, as claimed in the proof of Lemma 4,

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t\right)=e^{-\chi \left(t\right)}{K}_{\rho ,\psi }^{\alpha ,0,0}\left(e^{\chi \left(t\right)}f\left(t\right)\right),\mathrm{for} \mathrm{any}f\in \mathfrak{P}\left[[a,b],X\right].$$

Let $f\in A{C}^p\left[[a,b],X_w\right]$. Note that

$${\mathfrak{D}}_p\left(e^{\chi \left(t\right)}f\left(t\right)\right)=e^{\chi \left(t\right)}{\mathfrak{D}}_{p}f\left(t\right)+f\left(t\right)e^{\chi \left(t\right)}{\chi }^{\prime }\left(t\right),$$

for $t\in [a,b]$. Since $\phi \left(f(·)e^{\chi (·)}{\chi }^{\prime }(·)\right)=e^{\chi (·)}{\chi }^{\prime }(·)\phi \left(f(·)\right)\in L_{\infty }[a,b]$, for all $\phi \in X^{\ast }$, and

$$\phi \left(e^{\chi (·)}{\mathfrak{D}}_{p}f(·)\right)=e^{\chi (·)}\phi \left({\mathfrak{D}}_{p}f(·)\right)=e^{\chi (·)}{\left(\phi f(·)\right)}^{\prime }\in L_p[a,b],$$

for all $\phi \in X^{\ast }$, it follows from Proposition 3, that ${\mathfrak{D}}_p\left(e^{\chi (·)}f(·)\right)\in \mathfrak{P}\left[[a,b],X\right]$. Consequently, by Proposition 1, we have, in view of the fact that $\chi \left(a\right)=0$,

$$f\left(t\right)e^{\chi \left(t\right)}=f\left(a\right)+{\int }_a^t{\mathfrak{D}}_p\left(f\left(s\right)e^{\chi \left(s\right)}\right)ds,t\in [a,b].$$

Therefore,

$$\begin{array}{ccc}& & K_{\rho ,\psi }^{\alpha ,0,0}\left(e^{\chi \left(t\right)}f\left(t\right)\right)\hfill \\ & =& K_{\rho ,\psi }^{\alpha ,0,0}f\left(a\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha -1}\left[{\int }_a^s{\mathfrak{D}}_p\left(f\left(\vartheta \right)e^{\chi \left(\vartheta \right)}\right)d\vartheta \right]{\omega }^{\prime }\left(s\right)ds\hfill \\ & =& {\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{\alpha }}{\Gamma (1+\alpha )}}+K_{\rho ,\psi }^{\alpha ,0,0}{K}_{\rho ,\psi }^{1,0,0}\left({\displaystyle \frac{{\mathfrak{D}}_p\left(f\left(t\right)e^{\chi \left(t\right)}\right)}{{\omega }^{\prime }\left(t\right)}}\right).\hfill \end{array}$$

Remembering that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{{\omega }^{\prime }\left(t\right)}}{\mathfrak{D}}_p\left(e^{\chi \left(t\right)}f\left(t\right)\right)& =& {\displaystyle \frac{e^{\chi \left(t\right)}}{{\omega }^{\prime }\left(t\right)}}\left[{\mathfrak{D}}_{p}f\left(t\right)+f\left(t\right)\left(\mathsf{\ell }(\rho ,t)+\mu \right){\psi }^{\prime }\left(t\right)\right]\hfill \\ & =& {\displaystyle \frac{e^{\chi \left(t\right)}}{{\psi }^{\prime }\left(t\right)}}\left[{\mathsf{\ell }}_1(\rho ,t){\mathfrak{D}}_{p}f\left(t\right)+f\left(t\right)\left({\mathsf{\ell }}_2(\rho ,t)+\mu {\mathsf{\ell }}_1(\rho ,t)\right){\psi }^{\prime }\left(t\right)\right]\hfill \\ & =& e^{\chi \left(t\right)}{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\left(t\right)\in {\mathbb{H}}_0^p\left(X\right),\hfill \end{array}$$

and $p>{\displaystyle \frac{1}{\alpha }}$, we deduce from the semi-group property (see Lemma 4)

$$\begin{array}{ccc}\hfill K_{\rho ,\psi }^{\alpha ,0,0}\left(e^{\chi \left(t\right)}f\left(t\right)\right)& =& {\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{\alpha }}{\Gamma (1+\alpha )}}+K_{\rho ,\psi }^{1,0,0}{K}_{\rho ,\psi }^{\alpha ,0,0}\left({\displaystyle \frac{{\mathfrak{D}}_p\left(f\left(t\right)e^{\chi \left(t\right)}\right)}{{\omega }^{\prime }\left(t\right)}}\right)\hfill \\ & =& {\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{\alpha }}{\Gamma (1+\alpha )}}+K_{\rho ,\psi }^{1,0,0}{K}_{\rho ,\psi }^{\alpha ,0,0}\left(e^{\chi \left(t\right)}{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\left(t\right)\right)\hfill \\ & =& {\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{\alpha }}{\Gamma (1+\alpha )}}+K_{\rho ,\psi }^{1+\alpha ,0,0}\left(e^{\chi \left(t\right)}{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\left(t\right)\right).\hfill \end{array}$$

Hence

$$\begin{array}{ccc}& & K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t\right)=e^{-\chi \left(t\right)}{K}_{\rho ,\psi }^{\alpha ,0,0}\left(e^{\chi \left(t\right)}f\left(t\right)\right)\hfill \\ & =& e^{-\chi \left(t\right)}{\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{\alpha }}{\Gamma (1+\alpha )}}+{\displaystyle \frac{1}{\Gamma (1+\alpha )}}{\int }_a^t{e}^{-\left(\chi \right(t)-\chi (s\left)\right)}{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\left(s\right){\omega }^{\prime }\left(s\right)ds\hfill \\ & =& e^{-\chi \left(t\right)}{\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{\alpha }}{\Gamma (1+\alpha )}}+K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\left(t\right).\hfill \end{array}$$

Consequently, $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f$ is weakly absolutely continuous and has a Pettis-integrable pseudo-derivative on $[a,b]$. Also,

$${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t\right)=e^{-\chi \left(t\right)}{\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{\alpha -1}}{\Gamma \left(\alpha \right)}}+K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\left(t\right).$$

(5)

Furthermore, this is also true for all $p\ge 1$ when X is reflexive. □

Reasoning, now, as in ([47], Theorem 2), we know that

Lemma 6. 

Let $\alpha ,\lambda \in (0,1)$ such that $\alpha +\lambda <1$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. If $f\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$, then $\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right)=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi \left(f\right)\in {\mathcal{H}}_0^{\alpha +\lambda }\left[[a,b],X_w\right]$, for every functional $\phi \in X^{\ast }$ and we can estimate the Hölder constant for the operator $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }$, i.e., there exists a Hölder constant $C>0$ for this operator such that

$${\left[K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi \left(f\right)\right]}_{\lambda +\alpha }={\left[\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right)\right]}_{\lambda +\alpha }\le C{\left[\phi \left(f\right)\right]}_{\lambda },$$

(6)

for every $\phi \in X^{\ast }$, $\alpha +\lambda <1$, where

$${\left[\phi f\right]}_{\lambda }:=\underset{t,s\in [a,b],t\ne s}{sup}{\displaystyle \frac{\left|\phi f\left(t\right)-\phi f\left(s\right)\right|}{{|t-s|}^{\lambda }}}.$$

We remark that, the precise estimate of the constant C for real-valued functions can be found in [47]. Since the function $\phi f$ is also real-valued, that same calculation applies to our case. Due to its length, we will not repeat the analysis here. Consequently, we can prove the next

Lemma 7. 

Let $\lambda \in (0,1),\alpha \in [1,2)$ such that $\alpha +\lambda \in (1,2)$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. If $f\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$, then $\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right)=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi \left(f\right)\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$, for every functional $\phi \in X^{\ast }$ and there is finite $C>0$ such that

$${\left[\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right)\right]}_{\lambda }\le C{\left[\phi \left(f\right)\right]}_{\lambda },$$

(7)

for every $\lambda \in (0,1)$, $\alpha \in [1,2)$, such that $\alpha +\lambda \in (1,2)$.

Proof. 

Let $f\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$, $\phi \in X^{\ast }$, and define $g:=K_{\rho ,\psi }^{\alpha -1,\mathsf{\ell },\mu }f\in {\mathcal{H}}_0^{\lambda +\alpha -1}\left[[a,b],X_w\right]$. Arguing similarly as in ([47], Theorem 2), we can show for every $t,s\in [a,b]$, that

$$\left|\phi \left(K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }g\left(t\right)-K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }g\left(s\right)\right)\right|=\left|K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\phi g\left(t\right)-K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\phi g\left(s\right)\right|\le C^{\prime }{|t-s|}^{\lambda +\alpha }{\left[\phi \left(g\right)\right]}_{\lambda +\alpha -1},$$

(8)

where $C^{\prime }>0$.

By the aid of (6), we have

$${\left[\phi \left(g\right)\right]}_{\lambda +\alpha -1}={\left[\phi \left(K_{\rho ,\psi }^{\alpha -1,\mathsf{\ell },\mu }f\right)\right]}_{\lambda +\alpha -1}\le C{\left[\phi \left(f\right)\right]}_{\lambda },$$

for every $\phi \in X^{\ast }$, $\alpha +\lambda \in (1,2)$. Hence

$$\begin{array}{ccc}\hfill {\left[\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right)\right]}_{\lambda }& =& {\left[K_{1,\psi }^{1,\mathsf{\ell },\mu }\phi \left(g\right)\right]}_{\lambda }:=\underset{t,s\in [a,b],t\ne s}{sup}{\displaystyle \frac{\left|K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\phi g\left(t\right)-K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\phi g\left(s\right)\right|}{{|t-s|}^{\lambda }}}\hfill \\ & & \le C^{\prime }{|b-a|}^{\alpha }{\left[\phi \left(g\right)\right]}_{\lambda +\alpha -1}.\hfill \end{array}$$

Consequently, $\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right)=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi \left(f\right)\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$, for every functional $\phi \in X^{\ast }$ and there is a finite constant $C>0$ such that

$${\left[\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right)\right]}_{\lambda }\le C{\left[\phi \left(f\right)\right]}_{\lambda },$$

for every $\phi \in X^{\ast }$, $\alpha +\lambda \in (1,2)$. □

Lemma 8. 

Let $\alpha ,\lambda \in (0,1)$ such that $\alpha +\lambda <1$ and assume that $\psi \in C^1([a,b],\mathbb{R})$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Then

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }:{\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]\to {\mathcal{H}}_0^{\alpha +\lambda }\left[[a,b],X_w\right],$$

is injective. In particular,

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }:{\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]\to {\mathcal{H}}_0^{\alpha +\lambda }\left[[a,b],X\right],$$

is injective, and

$${∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right|∥}_{\alpha +\lambda }={\left[\left[K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right]\right]}_{\alpha +\lambda }\le C{\left[\left[f\right]\right]}_{\lambda },$$

(9)

for some $C=C\left(\rho ,\psi ,\alpha ,\mathsf{\ell },\mu \right)>0$.

Proof. 

The first part, namely for $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }:{\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]\to {\mathcal{H}}_0^{\alpha +\lambda }\left[[a,b],X_w\right]$, follows from Lemma 6. Also, since ${\left[\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right)\right]}_{\lambda +\alpha }\le C{\left[\phi \left(f\right)\right]}_{\lambda }$, then for any $t,t+h\in [a,b]$ and $\phi \in X^{\ast }$, we deduce

$$\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi f(t+h)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi f\left(t\right)\right|=\left|\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f(t+h)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t\right)\right)\right|\le C{h}^{\lambda +\alpha }{\left[\phi f\right]}_{\lambda },$$

with $C>0$. Next, without loss of generality, assume that $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f(t+h)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t\right)\ne 0$, $f\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$. Thus, as an immediate consequence of the Hahn–Banach theorem, there exists a functional $\phi \in X^{\ast }$ with ${∥\phi ∥}_{X^{\ast }}=1$ such that

$$∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f(t+h)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t\right)∥=\left|\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f(t+h)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t\right)\right)\right|\le C{h}^{\lambda +\alpha }{\left[\phi f\right]}_{\lambda },$$

with $C>0$. Since

$${\left[\phi f\right]}_{\lambda }=\underset{t,s\in [a,b],t\ne s}{sup}{\displaystyle \frac{\left|\phi \left(f\right(t)-f(s\left)\right)\right|}{{|t-s|}^{\lambda }}}\le {∥\phi ∥}_{X^{\ast }}\underset{t,s\in [a,b],t\ne s}{sup}{\displaystyle \frac{∥f\left(t\right)-f\left(s\right)∥}{{|t-s|}^{\lambda }}}\le {\left[\left[f\right]\right]}_{\lambda },$$

and we therefore deduce

$$∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f(t+h)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(t\right)∥\le C{h}^{\alpha +\lambda }{\left[\left[f\right]\right]}_{\lambda }.$$

Hence, ${∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right|∥}_{\alpha +\lambda }={\left[\left[K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right]\right]}_{\alpha +\lambda }\le C{\left[\left[f\right]\right]}_{\lambda }<\infty$ and so $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\in {\mathcal{H}}_0^{\alpha +\lambda }\left[[a,b],X\right]$, as required.

In order to show that the map $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }:{\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]\to {\mathcal{H}}_0^{\alpha +\lambda }\left[[a,b],X\right]$ is injective, we assume that $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }g$ for some $f,g\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right].$ So, for any $\phi \in X^{\ast }$, it follows in view of the definition of the Pettis integral $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi f=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi g$. Then, for every $\phi \in X^{\ast }$ and almost every $t\in [a,b]$ we obtain $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi \left(f\left(t\right)-g\left(t\right)\right)=0$. Therefore, for every $\phi \in X^{\ast }$ and almost every $t\in [a,b]$ we have

$$K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\phi \left(f\left(t\right)-g\left(t\right)\right)=K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi \left(f\left(t\right)-g\left(t\right)\right)=0.$$

Due to the continuity of the function $\phi \left(f(·)-g(·)\right)$, it follows that $\phi \left(f(·)-g(·)\right)=0$ for every $\phi \in X^{\ast }$ and every $t\in [a,b]$. Namely, $f\equiv g$ on $[a,b]$, and the result is now established. □

Lemma 8 may be combined with Lemma 7 in order to assure the following

Lemma 9. 

Let $\lambda \in (0,1),\alpha \in [1,2)$ such that $\alpha +\lambda \in (1,2)$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Then

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }:{\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]\to {\mathcal{H}}_0^{\alpha +\lambda -1}\left[[a,b],X\right],$$

is injective and

$${∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right|∥}_{\alpha +\lambda -1}={\left[\left[K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right]\right]}_{\alpha +\lambda -1}\le C{\left[\left[f\right]\right]}_{\lambda }.$$

(10)

Proof. 

Let $f\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$, $\phi \in X^{\ast }$ and define $g:=K_{\rho ,\psi }^{\alpha -1,\mathsf{\ell },\mu }f\in {\mathcal{H}}_0^{\lambda +\alpha -1}\left[[a,b],X_w\right]$. By Lemma 8, we know that

$${∥\left|g\right|∥}_{\alpha +\lambda -1}={∥\left|K_{\rho ,\psi }^{\alpha -1,\mathsf{\ell },\mu }f\right|∥}_{\alpha +\lambda -1}={\left[\left[K_{\rho ,\psi }^{\alpha -1,\mathsf{\ell },\mu }f\right]\right]}_{\alpha +\lambda -1}\le C{\left[\left[f\right]\right]}_{\lambda }.$$

Without loss of generality, assume that $K_{\rho ,\psi }^{\alpha -1,\mathsf{\ell },\mu }f\left(t\right)-K_{\rho ,\psi }^{\alpha -1,\mathsf{\ell },\mu }f\left(s\right)\ne 0$, $f\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right],t,s\in [a,b]$. Thus, there exists a functional $\phi \in X^{\ast }$ with ${∥\phi ∥}_{X^{\ast }}=1$ such that (cf. (8))

$$\begin{array}{ccc}\hfill ∥K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }g\left(t\right)-K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }g\left(s\right)∥& =& \left|\phi \left(K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }g\left(t\right)-K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }g\left(s\right)\right)\right|\hfill \\ & & \le C^{\prime }{|t-s|}^{\lambda +\alpha }{\left[\phi g\right]}_{\alpha +\lambda -1}\le C^{\prime }{|t-s|}^{\lambda +\alpha }{∥\left|g\right|∥}_{\alpha +\lambda -1},\hfill \end{array}$$

where $C^{\prime }>0$. Therefore

$${∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right|∥}_{\alpha +\lambda -1}:=\underset{t,s\in [a,b],t\ne s}{sup}{\displaystyle \frac{∥K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }g\left(t\right)-K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }g\left(s\right)∥}{{|t-s|}^{\alpha +\lambda -1}}}\le C{\left[\left[f\right]\right]}_{\lambda }.$$

We can now complete our proof exactly as in Lemma 8. □

Now, we will finish this part of our discussion with the following theorem on linear fractional integral equations.

Theorem 1. 

Let $\alpha ,{\alpha }_1,\lambda \in (0,1),\rho \in [0,1]$, $\mu \ge 0$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Consider the linear fractional integral equation

$$y\left(t\right)+A{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)=F\left(t,K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\left(t\right),y\left(t\right)\right),$$

(11)

for $A\in \mathbb{R}$ and $t\in [a,b]$. If $\lambda +max\{{\alpha }_1,\alpha \}<1$ and $F(·,y(·),x(·))\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ for every $x,y\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$, such that for any $t\in [a,b]$ and any $x_1,x_2,y_1,y_2\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$, we have

$${∥\left|F\left(·,y_1,x_1\right)-F\left(·,y_2,x_2\right)\right|∥}_{\lambda }\le A_1{∥\left|y_1-y_2\right|∥}_{\lambda }+A_2{∥\left|x_1-x_2\right|∥}_{\lambda },$$

(12)

for $A_1,A_2>0$. Then, for sufficiently small values of the numbers $A,A_1,A_2$, the integral Equation (11) has a unique Hölder continuous solution $y\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$.

First, let us recall that a simple sufficient condition needed for (12) is that, for any $x_1,x_2,y_1,y_2\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$, we have

$$∥F(t,y_1\left(t\right),x_1\left(t\right))-F(t,y_2\left(t\right),x_2\left(t\right))∥\le M_1∥y_1\left(t\right)-y_2\left(t\right)∥+M_2∥x_1\left(t\right)-x_2\left(t\right)∥,$$

where $M_1,M_2>0$. Clearly, for any $x_1,x_2,y_1,y_2\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ we have

$$\begin{array}{ccc}& & {∥\left|F\left(·,y_1,x_1\right)-F\left(·,y_2,x_2\right)\right|∥}_{\lambda }\hfill \\ & =& \underset{t,s\in [a,b],t\ne s}{sup}\left\{{\displaystyle \frac{∥F\left(t,y_1\left(t\right),x_1\left(t\right)\right)-F\left(t,y_2\left(t\right),x_2\left(t\right)\right)∥}{{|t-s|}^{\lambda }}}\right\}\hfill \\ & & +\underset{t,s\in [a,b],t\ne s}{sup}\left\{{\displaystyle \frac{∥F\left(s,y_1\left(s\right),x_1\left(s\right)\right)-F\left(s,y_2\left(s\right),x_2\left(s\right)\right)∥}{{|t-s|}^{\lambda }}}\right\}\hfill \\ & \le & 2\underset{t,s\in [a,b],t\ne s}{sup}\left\{M_1{\displaystyle \frac{∥y_1\left(t\right)-y_2\left(t\right)∥+M_2∥x_1\left(t\right)-x_2\left(t\right)∥}{{|t-s|}^{\lambda }}}\right\}\hfill \\ & \le & 2M_1{∥\left|y_1-y_2\right|∥}_{\lambda }+2M_2{∥\left|x_1-x_2\right|∥}_{\lambda }.\hfill \end{array}$$

Proof of Theorem 1. 

Define the non-linear operator $\mathbf{Q}:{\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]\to {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ by

$$\mathbf{Q}y\left(t\right)=F\left(t,K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\left(t\right),y\left(t\right)\right)-A{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right),$$

for $t\in [a,b]$, $A\in \mathbb{R}$. Since for any $y\in {\mathcal{H}}_0^{\lambda }[a,b]$, we have (in view of Lemma 8) $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\in {\mathcal{H}}_0^{\lambda +\alpha }[a,b]\subset {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ and $K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\in {\mathcal{H}}_0^{\lambda +{\alpha }_1}[a,b]\subset {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$. However,

$$\begin{array}{ccc}\hfill {∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\right|∥}_{\lambda }& =& \underset{t,s\in [a,b],t\ne s}{sup}{\displaystyle \frac{∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(s\right)∥}{{|t-s|}^{\lambda }}}\hfill \\ & \le & \underset{t,s\in [a,b],t\ne s}{sup}{\displaystyle \frac{{(b-a)}^{\alpha }∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(s\right)∥}{{|t-s|}^{\lambda +\alpha }}}={(b-a)}^{\alpha }{∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\right|∥}_{\lambda +\alpha },\hfill \end{array}$$

and (9), implies the existence of constants $A^{\ast },B^{\ast }>0$ such that

$${∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\right|∥}_{\lambda }\le A^{\ast }{∥\left|y\right|∥}_{\lambda },\mathrm{simlarly}{∥\left|K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\right|∥}_{\lambda }\le B^{\ast }{∥\left|y\right|∥}_{\lambda },$$

(13)

for any $y\in {\mathcal{H}}_0^{\lambda }[a,b]$. Also, according to our assumptions, we have $F\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$, so $\mathbf{Q}$ becomes well defined and makes sense.

Now, let $y_1,y_2\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ and note that

$${∥|\mathbf{Q}{y}_1-\mathbf{Q}{y}_2|∥}_{\lambda }\le {∥\left|F\left(·,K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{y}_1,y_1\right)-F\left(·,K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{y}_2,y_2\right)\right|∥}_{\lambda }+\left|A\right|{∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\left(y_1-y_2\right)\right|∥}_{\lambda }.$$

In view of (12), we obtain

$$\begin{array}{ccc}& & {∥\left|F\left(·,K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{y}_1,y_1\right)-F\left(·,K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{y}_2,y_2\right)\right|∥}_{\lambda }\hfill \\ & & \le max\{A_1,A_2\}\left[{∥\left|K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }\left(y_1-y_2\right)\right|∥}_{\lambda }+{∥\left|y_1-y_2\right|∥}_{\lambda }\right].\hfill \end{array}$$

Therefore, we have

$$\begin{array}{ccc}& & {∥|\mathbf{Q}{y}_1-\mathbf{Q}{y}_2|∥}_{\lambda }\hfill \\ & & \le max\{A_1,A_2\}\left[{∥\left|K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }\left(y_1-y_2\right)\right|∥}_{\lambda }+{∥\left|y_1-y_2\right|∥}_{\lambda }\right]+\left|A\right|{∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\left(y_1-y_2\right)\right|∥}_{\lambda }.\hfill \end{array}$$

Thus, in view of (13), there is a constant $M^{\ast }>0$ such that

$${∥|\mathbf{Q}{y}_1-\mathbf{Q}{y}_2|∥}_{\lambda }\le M^{\ast }{∥\left|y_1-y_2\right|∥}_{\lambda }.$$

Consequently, since the pair $\left({\mathcal{H}}_0^{\lambda }\left[[a,b],X\right],{∥|·|∥}_{\lambda }\right)$ forms a complete space, then by Banach fixed point theorem, for sufficiently small $A,A_1,A_2$, the operator $\mathbf{Q}$ has a (unique) fixed point $y\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right].$ □

The result of Theorem 1 still valid for any $\alpha \in [1,2)$. More precisely,

Corollary 1. 

Let ${\alpha }_1,\lambda \in (0,1),\alpha \in [1,2),\rho \in [0,1]$, $\mu \ge 0$ and assume that $\psi \in C^1([a,b],\mathbb{R})$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$.

If $\alpha +\lambda +{\alpha }_1\in (1,2)$ and $F(·,y(·),x(·))\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ for every $x,y\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ satisfying (12), then for sufficiently small $A,A_1,A_2$, the integral Equation (11) has a unique Hölder continuous solution $y\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$.

Proof. 

First, note that since $\alpha \in [1,2)$, it follows (in view of $\alpha +\lambda +{\alpha }_1\in (1,2)$) that $\lambda +{\alpha }_1\in (0,1)$. Let $y\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ and note that

(1)

By Lemma 9, we have $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\in {\mathcal{H}}_0^{\lambda +\alpha -1}\left[[a,b],X\right]\subset {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ and

${∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\right|∥}_{\lambda +\alpha -1}\le C_1{\left[\left[y\right]\right]}_{\lambda }$.

(2)

By Lemma 8, we have $K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\in {\mathcal{H}}_0^{\lambda +{\alpha }_1}\left[[a,b],X\right]\subset {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ and ${∥\left|K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\right|∥}_{{\alpha }_1+\lambda }\le C_2{\left[\left[y\right]\right]}_{\lambda }$.

Since, for any $t,s\in [a,b]$ we have

$$\begin{array}{ccc}\hfill {∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\right|∥}_{\lambda }& =& \underset{t,s\in [a,b],t\ne s}{sup}{\displaystyle \frac{∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(s\right)∥}{{|t-s|}^{\lambda }}}\hfill \\ & \le & \underset{t,s\in [a,b],t\ne s}{sup}{(b-a)}^{\alpha -1}{\displaystyle \frac{∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }g\left(s\right)∥}{{|t-s|}^{\alpha +\lambda -1}}},\hfill \end{array}$$

it follows

$${∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\right|∥}_{\lambda }\le {(b-a)}^{\alpha -1}{∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\right|∥}_{\alpha +\lambda -1}\le C_1{(b-a)}^{\alpha -1}{\left[\left[y\right]\right]}_{\lambda }.$$

Similarly,

$$\begin{array}{ccc}\hfill {∥\left|K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\right|∥}_{\lambda }& =& \underset{t,s\in [a,b],t\ne s}{sup}{\displaystyle \frac{∥K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\left(t\right)-K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\left(s\right)∥}{{|t-s|}^{\lambda }}}\hfill \\ & \le & \underset{t,s\in [a,b],t\ne s}{sup}{(b-a)}^{{\alpha }_1}{\displaystyle \frac{∥K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\left(t\right)-K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\left(s\right)∥}{{|t-s|}^{{\alpha }_1+\lambda }}},\hfill \end{array}$$

hence

$${∥\left|K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\right|∥}_{\lambda }\le {(b-a)}^{{\alpha }_1}{∥\left|K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\right|∥}_{{\alpha }_1+\lambda }\le C_2{(b-a)}^{{\alpha }_1}{\left[\left[y\right]\right]}_{\lambda }.$$

Now, since $F\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$, we can conclude the proof of Theorem 1 and obtain the result. □

Note that the main proof tool used here is the Banach fixed point theorem. This theorem requires that the constant $M^{\ast }$ be less than 1; however, this assumption can be relaxed. Now, we will turn our attention to how to use these operators outside of little Hölder spaces ${\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$.

Note that searching for solutions among functions that satisfy the condition $f\left(a\right)=0$ is essential to the proof. Now, we will demonstrate what this looks like in the space of continuous functions. Assuming the operator is a contraction in the Hölder space can be replaced by assuming the contraction condition with respect to the supremum norm only, with no restriction on $M^{\ast }$.

Proposition 5. 

Let $\alpha ,{\alpha }_1,\lambda \in (0,1)$, $\rho \in [0,1]$, $\mu \ge 0$, and assume that $\psi \in C^1([a,b],\mathbb{R})$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Consider the linear fractional integral equation

$$y\left(t\right)+A{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)=F\left(t,K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\left(t\right),y\left(t\right)\right),$$

(14)

where $t\in [a,b]$, $A\in \mathbb{R}$, $K_{\rho ,\psi }^{\beta ,\mathsf{\ell },\mu }$ denotes the ψ-Hilfer fractional integral operator of order β.

Assume that:

1. 

$\lambda +max\{{\alpha }_1,\alpha \}<1$.

2. 

For every $x,y\in {\mathcal{H}}_0^{\lambda +{\alpha }_1}[[a,b],X]$, we have $F(·,y(·),x(·))\in {\mathcal{H}}_0^{\lambda +{\alpha }_1}[[a,b],X]$.

3. 

There exist constants $A_1,A_2>0$ such that for all $t\in [a,b]$ and all

$x_1,x_2,y_1,y_2\in {\mathcal{H}}_0^{\lambda +{\alpha }_1}[[a,b],X]$,

$$\parallel F(t,y_1\left(t\right),x_1\left(t\right))-F(t,y_2\left(t\right),x_2\left(t\right))\parallel \le A_1\parallel y_1\left(t\right)-y_2\left(t\right)\parallel +A_2\parallel x_1\left(t\right)-x_2\left(t\right)\parallel .$$

4. 

The fractional integral operators $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }$ and $K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }$ are bounded on $C\left(\right[a,b],X)$ with the supremum norm. That is, there exist constants $M_{\alpha },M_{{\alpha }_1}>0$ such that for all $y\in C\left(\right[a,b],X)$,

$$\parallel K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{y\parallel }_{\infty }\le M_{\alpha }{\parallel y\parallel }_{\infty },\parallel K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{y\parallel }_{\infty }\le M_{{\alpha }_1}{\parallel y\parallel }_{\infty }.$$

5. 

The constants $A,A_1,A_2$ satisfy the contraction condition:

$$A_1{M}_{{\alpha }_1}+A_2+\left|A\right|M_{\alpha }<1.$$

Then the operator $\mathbf{Q}:C\left(\right[a,b],X)\to C\left(\right[a,b],X)$ defined by

$$(\mathbf{Q}\left(y\right)\left(t\right)=F\left(t,K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y\left(t\right),y\left(t\right)\right)-A{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)$$

is a contraction in $(C([a,b],X),\parallel ·{\parallel }_{\infty })$.

Proof. 

We will show that $\mathbf{Q}$ is a contraction mapping on the complete metric space $C\left(\right[a,b],X)$ equipped with the supremum norm ${\parallel y\parallel }_{\infty }={sup}_{t\in [a,b]}\parallel y\left(t\right)\parallel$.

First, we need to prove that $\mathbf{Q}$ is well-defined on $C\left(\right[a,b],X)$. Let $y\in C\left(\right[a,b],X)$. Since $C([a,b],X)\subset {\mathcal{H}}_0^{\lambda +{\alpha }_1}[[a,b],X]$ (as continuous functions are Hölder continuous with any exponent less than 1, and $\lambda +{\alpha }_1<1$), the assumptions on F imply that $F(·,K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y(·),y(·))$ is well-defined and continuous. The fractional integrals $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y$ and $K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }y$ are also continuous (by the boundedness assumption). Therefore, $\mathbf{Q}\left(y\right)$ is continuous, so $\mathbf{Q}:C\left(\right[a,b],X)\to C\left(\right[a,b],X)$.

Let $y_1,y_2\in C([a,b],X)$. For any $t\in [a,b]$, we have

$$\begin{array}{cc}& \mathbf{Q}{y}_1\left(t\right)-\mathbf{Q}{y}_2\left(t\right)\hfill \\ & =\left[F\left(t,K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{y}_1\left(t\right),y_1\left(t\right)\right)-F\left(t,K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{y}_2\left(t\right),y_2\left(t\right)\right)\right]-A\left[K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{y}_1\left(t\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{y}_2\left(t\right)\right].\hfill \end{array}$$

Taking norms and using the Lipschitz condition on F, we get

$$\begin{array}{ccc}& & \parallel (\mathbf{Q}{y}_1\left(t\right)-\mathbf{Q}{y}_2\left(t\right)\parallel \hfill \\ & \le & ∥F\left(t,K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{y}_1\left(t\right),y_1\left(t\right)\right)-F\left(t,K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{y}_2\left(t\right),y_2\left(t\right)\right)∥+\left|A\right|∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }(y_1-y_2)\left(t\right)∥\hfill \\ & \le & A_1∥K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }(y_1-y_2)\left(t\right)∥+A_2\parallel y_1\left(t\right)-y_2\left(t\right)\parallel +\left|A\right|∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }(y_1-y_2)\left(t\right)∥.\hfill \end{array}$$

Now take the supremum over $t\in [a,b]$ we obtain

$$\begin{array}{ccc}\hfill \parallel \mathbf{Q}{y}_1-\mathbf{Q}{y}_2{\parallel }_{\infty }& \le & A_1\underset{t\in [a,b]}{sup}∥K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }(y_1-y_2)\left(t\right)∥+A_2\underset{t\in [a,b]}{sup}\parallel y_1\left(t\right)-y_2\left(t\right)\parallel \hfill \\ & +& \left|A\right|\underset{t\in [a,b]}{sup}∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }(y_1-y_2)\left(t\right)∥.\hfill \end{array}$$

Now, using the boundedness of the considered fractional integral operators:

$$\begin{array}{cc}\hfill \parallel \mathbf{Q}{y}_1-\mathbf{Q}{y}_2{\parallel }_{\infty }& \le A_1{M}_{{\alpha }_1}\parallel y_1-y_2{\parallel }_{\infty }+A_2\parallel y_1-y_2{\parallel }_{\infty }+\left|A\right|M_{\alpha }{\parallel y_1-y_2\parallel }_{\infty }\hfill \\ & =\left(A_1{M}_{{\alpha }_1}+A_2+\left|A\right|M_{\alpha }\right){\parallel y_1-y_2\parallel }_{\infty }.\hfill \end{array}$$

By assumption, $A_1{M}_{{\alpha }_1}+A_2+\left|A\right|M_{\alpha }<1$, so $\mathbf{Q}$ is a contraction on $C\left(\right[a,b],X)$. □

Importantly, we can guarantee the uniqueness of the solution. Due to the results from [49], the operator does not have to be a contraction in the expected Hölder space. Note, that the norm ${∥\left|f\right|∥}_{\lambda }=∥f\left(a\right)∥+{\left[\left[f\right]\right]}_{\lambda }$ considered on Hölder spaces is equivalent to the following one

$${\left|\right|\left|f\right|\left|\right|}_{\infty ,\lambda }={\parallel f\parallel }_{\infty }+{\left[\left[f\right]\right]}_{\lambda }.$$

However, according to [49], to ensure the regularity of solutions and obtain the uniqueness property, it is sufficient to verify the assumptions of the operator $\mathbf{Q}$ with respect to the norm ${\parallel ·\parallel }_{\infty }$, which is done in Proposition 5. Therefore, by applying the fixed point results from [49], we immediately obtain the following:

Corollary 2. 

Under the assumptions of Proposition 5, where we replace the little Hölder space ${\mathcal{H}}_0^{\lambda +{\alpha }_1}[[a,b],X]$ by the Hölder space ${\mathcal{H}}^{\lambda +{\alpha }_1}\left[[a,b],X\right]$, the integral Equation (14) has a unique solution in ${\mathcal{H}}^{\lambda +{\alpha }_1}\left[[a,b],X\right]$.

3. Generalized Hilfer-Type Fractional Derivatives

After recalling the definition of combinations of fractional and non-fractional Pettis integral operators, the following definition of combinations of fractional and non-fractional differential operators is a logical next step:

Definition 5. 

Let $\psi \in C^1[a,b]$ be a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in [a,b]$, and let $f\in \mathfrak{P}\left[[a,b],X\right]$. For $\alpha \in (0,1)$ and $n\in \mathbb{N},$ we define the Hilfer-type combination of the fractional and non-fractional pseudo derivatives of order $n+\alpha$, with parameters $\rho \in (0,1),\mu \ge 0$ and type $\beta \in [0,1]$ by

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{n+\alpha ,\beta ,\mathsf{\ell }}f\left(t\right):={\left({\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}\right)}^n{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f\left(t\right),$$

(15)

where

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}:=K_{\rho ,\psi }^{\beta (1-\alpha ),\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }.$$

Similarly, we can define ${}_w{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{n+\alpha ,\beta ,\mathsf{\ell }}$ “the Hilfer-type combination between fractional and non-fractional weak derivatives”. In particular, we define

1. 

The Riemann–Liouville-type combination of the fractional and non-fractional pseudo (resp. weak) derivative of order $n+\alpha$, with parameters $\rho \in (0,1)$, $\mu \ge 0$:

$${\mathfrak{D}}_{p,\psi ,\mu }^{n+\alpha ,\rho ,\mathsf{\ell }}f\left(t\right):={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{n+\alpha ,0,\mathsf{\ell }}f\left(t\right)$$

(16)

respectively,

$${\mathfrak{D}}_{w,\psi ,\mu }^{n+\alpha ,\rho ,\mathsf{\ell }}f\left(t\right):={}_w{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{n+\alpha ,0,\mathsf{\ell }}f\left(t\right).$$

2. 

The Caputo-type combination of the fractional and non-fractional pseudo (resp. weak) derivative of order $n+\alpha$, with parameters $\rho \in (0,1)$, $\mu \ge 0$ defined as

$${}^C\mathfrak{D}_{p,\psi ,\mu }^{n+\alpha ,\rho ,\mathsf{\ell }}:={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{n+\alpha ,1,\mathsf{\ell }}f\left(t\right)$$

(17)

respectively,

$${}^C\mathfrak{D}_{w,\psi ,\mu }^{n+\alpha ,\rho ,\mathsf{\ell }}f\left(t\right):={}_w{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{n+\alpha ,1,\mathsf{\ell }}f\left(t\right).$$

Although, the pseudo-derivative is not uniquely determined in general (see Remark 1), we can show that

Lemma 10. 

The Hilfer-type combination between the fractional and non-fractional pseudo derivative ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}$, if it exists, is uniquely determined for any $\beta \in (0,1]$. The same holds true for $\beta \in [0,1]$ whenever X has a total dual $X^{\ast }$.

Proof. 

Let $\beta \in (0,1]$ and assume that $f,g\in \mathfrak{P}\left[[a,b],X\right]$ are two pseudo-derivatives of ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}$. The proof is now a simple consequence of the fact that, for any $\gamma >0$, we have

$$K_{\rho ,\psi }^{\gamma ,\mathsf{\ell },\mu }f\equiv K_{\rho ,\psi }^{\gamma ,\mathsf{\ell },\mu }g,\mathrm{for} \mathrm{any} \mathrm{weakly} \mathrm{equivalent}f,g.$$

(18)

Evidently, since $f,g$ are weakly equivalent on $[a,b]$, then for every $\phi \in X^{\ast }$ there exists a null set N depends on $\phi \in X^{\ast }$ such that

$$e^{-\left(\chi \right(t)-\chi (s\left)\right)}{(\omega \left(t\right)-\omega \left(s\right))}^{\gamma -1}{\omega }^{\prime }\left(s\right)\phi f\left(s\right)=e^{-\left(\chi \right(t)-\chi (s\left)\right)}{(\omega \left(t\right)-\omega \left(s\right))}^{\gamma -1}{\omega }^{\prime }\left(s\right)\phi g\left(s\right)$$

for every $s\in [a,t]\setminus N$. Consequently, for almost every $t\in [a,b]$,

$$\phi \left(K_{\rho ,\psi }^{\gamma ,\mathsf{\ell },\mu }f\left(t\right)\right)=K_{\rho ,\psi }^{\gamma ,\mathsf{\ell },\mu }\phi f\left(t\right)=K_{\rho ,\psi }^{\gamma ,\mathsf{\ell },\mu }\phi g\left(t\right)=\phi \left(K_{\rho ,\psi }^{\gamma ,\mathsf{\ell },\mu }g\left(t\right)\right),$$

holds for every $\phi \in X^{\ast }$. Therefore, (18) holds, and for almost every $t\in [a,b]$,

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f\left(t\right)={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}g.$$

That is what we wanted to show. □

Lemma 11. 

Let $\alpha \in (0,1)$ and $\beta \in [0,1]$. For any $f\in A{C}^p\left[[a,b],X_w\right]$, where

$$p>\left\{\begin{array}{cc}max\left\{1/\alpha ,1/\left(\beta \right(1-\alpha \left)\right),1/\left(\right(1-\beta \left)\right(1-\alpha \left)\right)\right\},\hfill & \beta \in (0,1),\hfill \\ {\displaystyle max\left\{1/\alpha ,1/(1-\alpha )\right\},}\hfill & \beta =0,1,\hfill \end{array}\right.$$

we have mutually inverse operators, i.e., for all $\beta \in [0,1)$

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f=f.$$

In addition,

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f=f,K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}f=f-e^{-\chi \left(t\right)}f\left(a\right).$$

This holds for all $p\in [1,\infty ]$ if X is a reflexive space.

Proof. 

Let $f\in A{C}^p\left[[a,b],X_w\right]$ and let $\beta \in [0,1)$. By the semi-group property (see Lemma 4), we have:

(a)

$$\begin{array}{ccc}\hfill {\displaystyle K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f}& =& K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\beta (1-\alpha ),\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }f\hfill \\ & =& K_{\rho ,\psi }^{\alpha +\beta (1-\alpha ),\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }f.\hfill \end{array}$$

In view of (5), we obtain

$$\begin{array}{ccc}\hfill K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f& =& K_{\rho ,\psi }^{\alpha +\beta (1-\alpha ),\mathsf{\ell },\mu }\left[e^{-\chi \left(t\right)}{\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{(1-\beta )(1-\alpha )-1}}{\Gamma \left(\right(1-\beta \left)\right(1-\alpha \left)\right)}}+K_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\right]\hfill \\ & =& e^{-\chi \left(t\right)}f\left(a\right)+K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f.\hfill \end{array}$$

Hence, by Proposition 4, $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f=f.$

(b) Conversely,

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f=K_{\rho ,\psi }^{\beta (1-\alpha ),\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha )+\alpha ,\mathsf{\ell },\mu }f=K_{\rho ,\psi }^{\beta (1-\alpha ),\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{1-\beta (1-\alpha ),\mathsf{\ell },\mu }f.$$

In view of (5), we get

$$\begin{array}{ccc}\hfill {}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f& =& K_{\rho ,\psi }^{\beta (1-\alpha ),\mathsf{\ell },\mu }\left[e^{-\chi \left(t\right)}{\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{-\beta (1-\alpha )}}{\Gamma (1-\beta (1-\alpha \left)\right)}}+K_{\rho ,\psi }^{1-\beta (1-\alpha ),\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\right]\hfill \\ & =& e^{-\chi \left(t\right)}f\left(a\right)+K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f.\hfill \end{array}$$

Hence, by Proposition 4, ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f=f$. Next, in view of (5), applying Lemma 4 and Proposition 4, yield

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}f=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f=K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f=f-e^{-\chi \left(t\right)}f\left(a\right).$$

Conversely,

$$\begin{array}{ccc}\hfill {}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f& =& K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f=K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }\left[e^{-\chi \left(t\right)}{\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{\alpha -1}}{\Gamma \left(\alpha \right)}}+K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\right]\hfill \\ & =& e^{-\chi \left(t\right)}f\left(a\right)+K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f=f.\hfill \end{array}$$

□

Now, we will examine the maximum regularity of differential operators by defining the appropriate domains and codomains. Interestingly, the operators we study behave similarly to fractional-order operators. They change the order of the Hölder space according to the order of the operator $\alpha$. The same is true for differential operators of integer order in $C^n$ spaces. This interesting symmetry suggests interpolation relations for these spaces. This research direction was recently presented in [34,35].

Lemma 12. 

Let $\alpha ,\lambda \in (0,1),\beta \in (0,1),\rho \in [0,1]$, $\mu \ge 0$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. If $\lambda \in \left(\alpha +\beta (1-\alpha ),1\right)$, then

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]\to {\mathcal{H}}_0^{\lambda -\alpha }\left[[a,b],X_w\right],$$

in particular

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]\to {\mathcal{H}}_0^{\lambda -\alpha }\left[[a,b],X\right].$$

Proof. 

Let $f\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$ and define $\delta :=\alpha +\beta (1-\alpha )$. Recall that

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f=K_{\rho ,\psi }^{\beta (1-\alpha ),\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,0,\mathsf{\ell }}f=K_{\rho ,\psi }^{\beta (1-\alpha ),\mathsf{\ell },\mu }{\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}f,$$

for $\beta \in [0,1)$. Let $h>0$, be such that $t,t+h\in [a,b]$. Similarly to the proof of Lemma 3 in [50], it can be easily verified that $K_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }f$ is pseudo-differentiable on $[a,b]$ and that there is a Hölder constant $L>0$ for this operator such that

$$\left|\phi \left({\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}f\right)(t+h)-\phi \left({\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}f\right)\left(t\right)\right|=\left|{\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}\phi f(t+h)-{\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}\phi f\left(t\right)\right|\le L{h}^{\lambda -\delta }{\left[\phi f\right]}_{\lambda },$$

holds true for any $\phi \in X^{\ast }$. Hence ${\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}f={\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }f\in {\mathcal{H}}_0^{\lambda -\delta }\left[[a,b],X_w\right]$. Therefore, in view of Lemma 8, we deduce that

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f=K_{\rho ,\psi }^{\beta (1-\alpha ),\mathsf{\ell },\mu }{\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}f\in {\mathcal{H}}_0^{\lambda -\alpha }\left[[a,b],X_w\right].$$

That is, ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]\to {\mathcal{H}}_0^{\lambda -\alpha }\left[[a,b],X_w\right]$ as required.

Now, let $f\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ and, without loss of generality, assume that ${\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}f(t+h)-{\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}f\left(t\right)\ne 0$. An immediate consequence of the Hahn–Banach theorem, there exists $\phi \in X^{\ast }$ with ${∥\phi ∥}_{X^{\ast }}=1$ such that

$$\begin{array}{ccc}\hfill & & ∥{\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}f(t+h)-{\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}f\left(t\right)∥=\left|\phi \left({\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}(t+h)-{\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}f\left(t\right)\right)\right|\hfill \\ & =& \left|{\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}\phi f(t+h)-{\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}\phi f\left(t\right)\right|\le L{h}^{\lambda -\delta }{\left[\phi f\right]}_{\lambda }.\hfill \end{array}$$

(19)

Since ${\left[\phi f\right]}_{\lambda }\le {\left[\left[f\right]\right]}_{\lambda }$, it follows ${\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}f={\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }f\in {\mathcal{H}}_0^{\lambda -\delta }\left[[a,b],X\right]$. Moreover, by Lemma 8,

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f=K_{\rho ,\psi }^{\beta (1-\alpha ),\mathsf{\ell },\mu }{\mathfrak{D}}_{p,\psi ,\mu }^{\delta ,\rho ,\mathsf{\ell }}f\in {\mathcal{H}}_0^{\lambda -\alpha }\left[[a,b],X\right].$$

That is ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]\to {\mathcal{H}}_0^{\lambda -\alpha }\left[[a,b],X\right]$ as required. □

Recall an important goal of this work: the introduced operators are intended to establish equivalence between differential and integral problems, at least in certain cases. The first step is to investigate whether fractional differential and integral operators are inverses of each other, and if so, to what extent. Typically, a generalization and weakening of solutions is required. Recall the procedure for ordinary equations.

An abstract differential form (e.g., $du/dt=A\left(t\right)u\left(t\right)+f\left(t\right)$) requires a strong notion of differentiability. The solution $u\left(t\right)$ must be differentiable in the norm topology of the Banach space X and the equation must hold pointwise for (almost) every t. The integral form (e.g., $u\left(t\right)=u_0+{\int }_0^t[A\left(s\right)u\left(s\right)+f\left(s\right)]ds$) only requires that the integrand be strongly (Bochner) integrable. This allows us to define generalized solutions for problems that may not have a classical, differentiable solution, even in the weak sense. The integral equation generalizes the differential equation.

Our earlier reasoning allows us to achieve this goal:

Lemma 13. 

Let $\alpha ,\lambda \in (0,1),\beta \in (0,1),\rho \in [0,1]$, $\mu \ge 0$ and assume that $\psi \in C^1([a,b],\mathbb{R})$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. If $\lambda +\alpha +\beta (1-\alpha )<1$, then the operator $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }:{\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]\to {\mathcal{H}}_0^{\lambda +\alpha }\left[[a,b],X_w\right]$ is bijective with the continuous inverse ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}$. That is

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f=fon{\mathcal{H}}_0^{\lambda +\alpha }\left[[a,b],X_w\right].$$

(20)

This holds true for all $\beta \in [0,1)$ whenever X has a total dual $X^{\ast }$.

Proof. 

1.

The operator $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }$ is injective by Lemma 8.

2.

The operator $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }$ is surjective with the right inverse ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}$. That is, for all $f\in {\mathcal{H}}_0^{\alpha +\lambda }\left[[a,b],X_w\right]$, there is $g\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$ such that we have $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }g=f$ with $g:={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f$. By Lemma 12, we know that $g\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$ is defined, and uniquely determined for all $\beta \in (0,1)$ by Lemma 10 and for $\beta =0$ if X has a total dual (by Remark 1). Therefore, due to Lemma 12 and by using the definition of generalized Hilfer-type fractional derivatives (Definition 5), for all $t\in [a,b]$ we have:

$$K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }g=K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f=K_{\rho ,\psi }^{1+\beta (1-\alpha ),\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}\left[K_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }f\right].$$

Using the definition of non-fractional pseudo-differential operators ${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}$, we can derive

$$\begin{array}{ccc}& & K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }g\hfill \\ & =& K_{\rho ,\psi }^{1+\beta (1-\alpha ),\mathsf{\ell },\mu }\left[\left(\left[{\mathsf{\ell }}_2(\rho ,t)+\mu {\mathsf{\ell }}_1(\rho ,t)\right]K_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }f\right.\right.\hfill \\ & & \left.\left.+{\displaystyle \frac{1}{{\omega }^{\prime }\left(t\right)}}{\mathfrak{D}}_p{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }f\right)\right]\hfill \\ & =& K_{\rho ,\psi }^{\beta (1-\alpha ),\mathsf{\ell },\mu }\left[{\int }_a^t{e}^{-\left(\chi \right(t)-\chi (s\left)\right)}\left(\left[{\mathsf{\ell }}_2(\rho ,s)+\mu {\mathsf{\ell }}_1(\rho ,s)\right]K_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }f\left(s\right)\right.\right.\hfill \\ & & +\left.\left.{\displaystyle \frac{1}{{\omega }^{\prime }\left(s\right)}}{\mathfrak{D}}_p{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }f\left(s\right)\right){\omega }^{\prime }\left(s\right)ds\right].\hfill \end{array}$$

By definitions of the Pettis integral and the pseudo-derivative, for every $\phi \in X^{\ast }$, we have

$$\begin{array}{ccc}& & \phi \left(K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }g\left(t\right)\right)\hfill \\ & & =K_{\rho ,\psi }^{\beta (1-\alpha ),\mathsf{\ell },\mu }\left[{\int }_a^t{e}^{-\left(\chi \right(t)-\chi (s\left)\right)}\left[{\mathsf{\ell }}_2(\rho ,s)+\mu {\mathsf{\ell }}_1(\rho ,s)\right]K_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }\phi f\left(s\right){\omega }^{\prime }\left(s\right)ds\right.\hfill \\ & & +\left.{\int }_a^t{e}^{-\left(\chi \right(t)-\chi (s\left)\right)}{\displaystyle \frac{d}{ds}}\left[K_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }\phi f\left(s\right)\right]ds\right].\hfill \end{array}$$

Using the integration by parts formula, we can conclude that

$$\begin{array}{ccc}& & \phi \left(K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }g\left(t\right)\right)\hfill \\ & & =K_{\rho ,\psi }^{\beta (1-\alpha ),\mathsf{\ell },\mu }\left[{\int }_a^t{e}^{-\left(\chi \right(t)-\chi (s\left)\right)}\left[{\mathsf{\ell }}_2(\rho ,s)+\mu {\mathsf{\ell }}_1(\rho ,s)\right]K_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }\phi f\left(s\right){\omega }^{\prime }\left(s\right)ds\right.\hfill \\ & & +\left.K_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }\phi f\left(t\right)-{\int }_a^t{e}^{-\left(\chi \right(t)-\chi (s\left)\right)}{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }\phi f\left(s\right){\chi }^{\prime }\left(s\right)ds\right].\hfill \end{array}$$

Since ${\chi }^{\prime }(·)={\psi }^{\prime }(·)[\mathsf{\ell }(\rho ,·)+\mu ]={\mathsf{\ell }}_2(\rho ,·){\omega }^{\prime }(·)+\mu {\mathsf{\ell }}_1(\rho ,·){\omega }^{\prime }(·)$, it follows

$$\phi \left(K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }g\left(t\right)\right)=K_{\rho ,\psi }^{\beta (1-\alpha ),\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\mathsf{\ell },\mu }\phi f\left(t\right)=K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }\phi f\left(t\right).$$

Therefore, for each $\phi \in X^{\ast }$, we obtain

$$K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }g\left(t\right)-f\left(t\right)\right)=0.$$

Since $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }g-f$ lies in ${\mathcal{H}}_0^{\lambda +\alpha }\left[[a,b],X_w\right]$, is therefore weakly continuous. Consequently, for every $\phi \in X^{\ast }$, $\phi \left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }g\left(t\right)\right)=\phi f\left(t\right)$, for every $t\in [a,b]$. Hence

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }g=f$$

for any $f\in {\mathcal{H}}_0^{\lambda +\alpha }\left[[a,b],X_w\right]$.

3.

Finally, we will show that $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }$ with the left inverse of ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}$. Evidently, since $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }$ is bijective, the right inverse is equal to the left inverse (and is equal to ${\left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\right)}^{-1}$):

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }={\left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\right)}^{-1}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }={\left(K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }\right)}^{-1}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }=Id.$$

Since ${\mathcal{H}}_0^{\lambda +\alpha }\left[[a,b],X_w\right]$ and ${\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$ are Banach spaces, the continuity of ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}={\left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\right)}^{-1}$ follows from Lemma 8 and by by the continuous inverse theorem. Therefore, ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f=f$, for any $f\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$. Since ${\mathcal{H}}_0^{\lambda +\alpha }\left[[a,b],X_w\right]\subset {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$, the Equation (20) holds true, as required.

□

Corollary 3. 

If the Assumptions of Lemma 13 are satisfied, then

$$K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f=K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }\left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}f\right)=K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }fon{\mathcal{H}}_0^{\lambda +\alpha }\left[[a,b],X_w\right].$$

This holds true for all $\beta \in [0,1)$ whenever X has a total dual $X^{\ast }$.

Theorem 2. 

Let $\alpha ,{\alpha }_1,\lambda \in (0,1),\rho \in [0,1]$, $\mu \ge 0$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Consider the linear fractional integral equation

$$y\left(t\right)+A{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)=F\left(t,K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}y\left(t\right),y\left(t\right)\right),$$

(21)

for $\alpha <{\alpha }_1$, $t\in [a,b]$, $A\in \mathbb{R}$. If $0<\lambda <{\alpha }_1-\alpha$ and $F(·,y(·),x(·))\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$ for every $x,y\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$ such that for any $t\in [a,b]$ and any $x_1,x_2,y_1,y_2\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$, we have

$${∥\left|F\left(·,y_1,x_1\right)-F\left(·,y_2,x_2\right)\right|∥}_{1+\lambda -{\alpha }_1}\le A_1{∥\left|y_1-y_2\right|∥}_{1+\lambda -{\alpha }_1}+A_2{∥\left|x_1-x_2\right|∥}_{1+\lambda -{\alpha }_1},$$

(22)

where $A_1,A_2>0$. Then for sufficiently small $A,A_1,A_2$, the integral Equation (21) has a Hölder continuous solution $y\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$.

Proof. 

Define the non-linear operator $\mathbf{U}:{\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]\to {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$ by

$$\mathbf{U}y\left(t\right)=F\left(t,K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}y\left(t\right),y\left(t\right)\right)-A{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right),$$

for $\alpha <{\alpha }_1$, $t\in [a,b]$, $A\in \mathbb{R}$. Let $y\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$. Note that

(1)

Since $\lambda <{\alpha }_1$, it follows from Lemma 12, that $g:={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}y\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$, and by (19) we have ${∥g∥}_{\lambda }={∥{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}y∥}_{\lambda }\le L{∥y∥}_{1+\lambda -{\alpha }_1}$ for some $L>0$. Also, for any $t,s\in [a,b]$, we have

$$∥K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }g\left(t\right)-K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }g\left(s\right)∥\le C_1{|t-s|}^{\lambda +1}{∥\left|g\right|∥}_{\lambda },$$

where $C_1>0$. Therefore,

$$\begin{array}{ccc}\hfill {∥\left|K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}y\right|∥}_{1+\lambda -{\alpha }_1}& =& {∥\left|K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }g\right|∥}_{1+\lambda -{\alpha }_1}\hfill \\ & :=& \underset{t,s\in [a,b],t\ne s}{sup}{\displaystyle \frac{∥K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }g\left(t\right)-K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }g\left(s\right)∥}{{|t-s|}^{1+\lambda -{\alpha }_1}}}\hfill \\ & \le & C_1{|b-a|}^{{\alpha }_1}{∥\left|g\right|∥}_{\lambda }\le C_{1}L{|b-a|}^{{\alpha }_1}{∥y∥}_{1+\lambda -{\alpha }_1}.\hfill \end{array}$$

Hence $K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}y\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right].$

(2)

Since $\lambda <{\alpha }_1-\alpha$, it follows from Lemma 8, that $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1+\alpha }\left[[a,b],X\right]\subset {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$ and ${∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\right|∥}_{1+\lambda -{\alpha }_1+\alpha }\le C_2{\left[\left[y\right]\right]}_{1+\lambda -{\alpha }_1}$. Therefore,

$$\begin{array}{ccc}\hfill {∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\right|∥}_{1+\lambda -{\alpha }_1}& =& \underset{t,s\in [a,b],t\ne s}{sup}{\displaystyle \frac{∥K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)-K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(s\right)∥}{{|t-s|}^{1+\lambda -{\alpha }_1}}}\hfill \\ & \le & {(b-a)}^{\alpha }{∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\right|∥}_{1+\lambda -{\alpha }_1+\alpha }\hfill \\ & \le & C_2{(b-a)}^{\alpha }{\left[\left[y\right]\right]}_{1+\lambda -{\alpha }_1}.\hfill \end{array}$$

Since $F\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$, it is clear that $\mathbf{U}$ becomes well defined and makes sense. Now, let $y_1,y_2\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$. Therefore,

$$\begin{array}{ccc}& & {∥|\mathbf{U}{y}_1-\mathbf{U}{y}_2|∥}_{1+\lambda -{\alpha }_1}\hfill \\ & \le & {∥\left|F\left(·,K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}{y}_1,y_1\right)-F\left(·,K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}{y}_2,y_2\right)\right|∥}_{1+\lambda -{\alpha }_1}\hfill \\ & & +\left|A\right|{∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\left(y_1-y_2\right)\right|∥}_{1+\lambda -{\alpha }_1}.\hfill \end{array}$$

In view of (22), we obtain

$$\begin{array}{ccc}& & {∥\left|F\left(·,K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}{y}_1,y_1\right)-F\left(·,K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}{y}_2,y_2\right)\right|∥}_{1+\lambda -{\alpha }_1}\hfill \\ & \le & \mathbf{A}\left[{∥\left|K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}\left(y_1-y_2\right)\right|∥}_{1+\lambda -{\alpha }_1}\right.\hfill \\ & & +\left.{∥\left|y_1-y_2\right|∥}_{1+\lambda -{\alpha }_1}\right],\mathbf{A}=max\{A_1,A_2\}\hfill \\ & \le & \mathbf{A}\left[C_{2}L{|b-a|}^{{\alpha }_1}{∥\left|y_1-y_2\right|∥}_{1+\lambda -{\alpha }_1}\right.\hfill \\ & & +\left.{∥\left|y_1-y_2\right|∥}_{1+\lambda -{\alpha }_1}\right].\hfill \end{array}$$

Therefore, there is a constant $\mathbf{M}>0$ such that

$${∥|\mathbf{U}{y}_1-\mathbf{U}{y}_2|∥}_{1+\lambda -{\alpha }_1}\le \mathbf{M}{∥\left|y_1-y_2\right|∥}_{1+\lambda -{\alpha }_1}+\left|A\right|{∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\left(y_1-y_2\right)\right|∥}_{1+\lambda -{\alpha }_1}.$$

Consequently, there is a constant $M^{\ast }>0$ such that

$${∥|\mathbf{U}{y}_1-\mathbf{U}{y}_2|∥}_{1+\lambda -{\alpha }_1}\le M^{\ast }{∥\left|y_1-y_2\right|∥}_{1+\lambda -{\alpha }_1}.$$

Since the pair $\left({\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right],{∥|·|∥}_{1+\lambda -{\alpha }_1}\right)$ forms a complete space, by the Banach fixed point theorem, for sufficiently small constants $A,A_2$, the operator $\mathbf{U}$ has a fixed point $y\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$. □

Recall, that in the space ${\mathcal{H}}^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$, to obtain the norm contraction, we must also control the supremum norm (see Proposition 5). Furthermore, the result of Theorem 2 is still valid for any $\alpha \in [1,2)$. Namely,

Corollary 4. 

Let ${\alpha }_1,\lambda \in (0,1),\alpha \in [1,2),\rho \in [0,1]$, $\mu \ge 0$ and assume that $\psi \in C^1([a,b],\mathbb{R})$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Consider the linear fractional integral equation

$$y\left(t\right)+A{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)=F\left(t,K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}y\left(t\right),y\left(t\right)\right),$$

(23)

for $t\in [a,b]$, $A\in \mathbb{R}$. If $\lambda -{\alpha }_1+\alpha \in (0,1)$ and $F(·,y(·),x(·))\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$ for every $x,y\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$ satisfying (22), then for sufficiently small $A,A_1,A_2$, the integral Equation (23) admits a Hölder continuous solution $y\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$.

Proof. 

Let $y\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$. As in the proof of Theorem 2, it can be easily seen that

(1)

$K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}y\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right],$ and there is a constant $C_1>0$ such that

$${∥\left|K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_1,0,\mathsf{\ell }}y\right|∥}_{1+\lambda -{\alpha }_1}\le C_1{∥y∥}_{1+\lambda -{\alpha }_1}.$$

(2)

Since $1+\lambda -{\alpha }_1+\alpha \in (1,2)$, it follows by Lemma 9, $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\in {\mathcal{H}}_0^{\lambda -{\alpha }_1+\alpha }\left[[a,b],X\right]$ and ${∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\right|∥}_{\lambda -{\alpha }_1+\alpha }\le C_2{\left[\left[y\right]\right]}_{1+\lambda -{\alpha }_1}$.

Now, since $F\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$, we are able to complete the rest of the proof of Theorem 2 and derive the result. □

As a consequence of Lemma 13, we have the following characterization of the one-sided invertibility of differential and integral operators.

Lemma 14. 

Let ${\alpha }_1,{\alpha }_2\in (0,1)$, $\beta \in (0,1)$, $\rho \in [0,1]$, $\mu \ge 0$. Also, assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Let $\lambda >0$ be such that $(1-\beta ){\alpha }_2<\lambda +{\alpha }_1+\beta (1-{\alpha }_1)<1$. Then

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{{\alpha }_2,\mathsf{\ell },\mu }f=\left\{\begin{array}{cc}{\displaystyle {}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-{\alpha }_2,\beta ,\mathsf{\ell }}f,}\hfill & {\alpha }_1>{\alpha }_2,\hfill \\ {\displaystyle K_{\rho ,\psi }^{{\alpha }_2-{\alpha }_1,\mathsf{\ell },\mu }f,}\hfill & {\alpha }_1\le {\alpha }_2,\hfill \end{array}\right.fin{\mathcal{H}}_0^{\lambda +{\alpha }_1-{\alpha }_2}\left[[a,b],X_w\right].$$

This holds true for all $\beta \in [0,1)$ whenever X has a total dual $X^{\ast }$.

Proof. 

Let $f\in {\mathcal{H}}_0^{\lambda +{\alpha }_1-{\alpha }_2}\left[[a,b],X_w\right]$.

The case when ${\alpha }_1>{\alpha }_2$.

Since $0<\lambda +{\alpha }_1-{\alpha }_2+\beta (1-{\alpha }_1+{\alpha }_2)<1$, it follows from Lemma 13 that

$$u:=K_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-{\alpha }_2,\beta ,\mathsf{\ell }}f=f\in {\mathcal{H}}_0^{\lambda +{\alpha }_1-{\alpha }_2}\left[[a,b],X_w\right],$$

and by Lemma 13 (in view of $0<\lambda +{\alpha }_1+\beta (1-{\alpha }_1)<1$), we obtain

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }u=u.$$

Therefore,

$$\begin{array}{ccc}\hfill {}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{{\alpha }_2,\mathsf{\ell },\mu }f& =& {}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{{\alpha }_2,\mathsf{\ell },\mu }\left(K_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-{\alpha }_2,\beta ,\mathsf{\ell }}f\right)\hfill \\ & =& {}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }\left({}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-{\alpha }_2,\beta ,\mathsf{\ell }}f\right)={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-{\alpha }_2,\beta ,\mathsf{\ell }}f.\hfill \end{array}$$

The case when ${\alpha }_1\le {\alpha }_2$.

Since $0<\lambda +{\alpha }_1-{\alpha }_2<1$, it follows $v:=K_{\rho ,\psi }^{{\alpha }_2-{\alpha }_1,\mathsf{\ell },\mu }f\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right],\lambda +{\alpha }_1+\beta (1-{\alpha }_1)<1$. According to the proof of part (3) of Lemma 13, we have that ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }v=v$. Therefore,

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{{\alpha }_2,\mathsf{\ell },\mu }f={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }\left(K_{\rho ,\psi }^{{\alpha }_2-{\alpha }_1,\mathsf{\ell },\mu }f\right)=K_{\rho ,\psi }^{{\alpha }_2-{\alpha }_1,\mathsf{\ell },\mu }f.$$

□

The following example shows why Lemmas 13 and 14 should not be applied to the critical case $\beta =0$. Indeed, we will show that the hypothesis that the target space X has a total dual, in these Lemmas, is crucial and cannot be omitted.

Example 1. 

Let $\mathbb{B}[0,1]$ be the Banach space of bounded real-valued functions on $[0,1]$. It has no total dual. Let $\mathfrak{J}\subset [0,1]$ be of positive measure, and define $f:[0,1]\to \mathbb{B}[0,1]$ by

$$f\left(t\right):=\left\{\begin{array}{c}{\chi }_t(·),t\in \mathfrak{J},\hfill \\ 0,t\notin \mathfrak{J}.\hfill \end{array}\right.$$

(24)

According to Example 9.1 in [37], we know that $f\in \mathfrak{P}\left[[0,1],\mathbb{B}\right]$ with $\phi f\equiv 0$ for each $\phi \in {\mathbb{B}}^{\ast }[0,1]$. Clearly, $f\in {\mathcal{H}}_0^{\lambda -\alpha +1}\left[[0,1],{\mathbb{B}}_w\right]$ for any $\lambda <\alpha .$ Also, $t\to K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f$ has two pseudo-derivatives $0$ and $f$, on $[0,1]$ that differ on the set of positive measure $\mathfrak{J}$. Therefore, ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,0,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f={\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f$ is equal either f (by Lemma 4) or 0. Consequently, we conclude that on J,

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,0,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\ne f.$$

Now, consider an analogous result to that in Lemma 13 with $\beta =1$.

Lemma 15. 

Let $\alpha \in (0,1)$, $\rho \in [0,1]$, $\mu \ge 0$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. If we define ${\mathbb{K}}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }=K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}$, then for any $\lambda \in (0,\alpha )$,

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}{\mathbb{K}}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f=f,f\in {\mathcal{H}}_0^{1+\lambda -\alpha }\left[[a,b],X_w\right].$$

Remark 5. 

Before proving this theorem, one remark should be added. In view of Lemma 12, a pseudo-derivative ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}f$ exists for any Hölder continuous function f of some critical order less than one, and moreover ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}f\in C\left([a,b],X_w\right)$. This implies the existence of ${\mathbb{K}}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f$. In this case, Proposition 2 states that ${\mathbb{K}}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }x$ is weakly- (trivially pseudo-) differentiable on $[a,b]$, and ${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{\mathbb{K}}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }x={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}f$.

Proof of Lemma 15.  

Let $f\in {\mathcal{H}}_0^{1+\lambda -\alpha }\left[[a,b],X_w\right]$. By Lemma 12, we know that $g:{=}_p^H{\mathrm{D}}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}f={\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$. Consequently,

$$\begin{array}{ccc}& & K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }g=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}f=K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\hfill \\ & =& K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\left[\left(\left[{\mathsf{\ell }}_2(\rho ,t)+\mu {\mathsf{\ell }}_1(\rho ,t)\right]K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f+{\displaystyle \frac{1}{{\omega }^{\prime }\left(t\right)}}{\mathfrak{D}}_p{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right)\right]\hfill \\ & =& {\int }_a^t{e}^{-\left(\chi \right(t)-\chi (s\left)\right)}\left(\left[{\mathsf{\ell }}_2(\rho ,s)+\mu {\mathsf{\ell }}_1(\rho ,s)\right]K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(s\right)+{\displaystyle \frac{1}{{\omega }^{\prime }\left(s\right)}}{\mathfrak{D}}_p{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\left(s\right)\right){\omega }^{\prime }\left(s\right)ds\hfill \\ & =& K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f.\hfill \end{array}$$

Therefore, for every $\phi \in X^{\ast }$ and for a.e. $t\in [a,b]$, we obtain

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }\phi \left(K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }g\left(t\right)-f\left(t\right)\right)=0.$$

Consequently,

$$K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}f=K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }g=f,f\in {\mathcal{H}}_0^{1+\lambda -\alpha }\left[[a,b],X_w\right].$$

(25)

By Lemmas 4 and 13, since $1+\lambda -\alpha >1-\alpha$, it follows that

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}{\mathbb{K}}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f=K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }\left({\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\right){}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}f=K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}f=f.$$

This completes the proof. □

As a generalization of Lemma 15 for two different orders, which is of special interest for the Bagley–Torvik equation, we have the following:

Lemma 16. 

Let ${\alpha }_1<{\alpha }_2\in (0,1)$, $\rho \in [0,1]$, and $\mu \ge 0$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. If $\lambda \in (0,{\alpha }_1)$, then

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,1,\mathsf{\ell }}{\mathbb{K}}_{\rho ,\psi }^{{\alpha }_2,\mathsf{\ell },\mu }f=K_{\rho ,\psi }^{{\alpha }_2-{\alpha }_1,\mathsf{\ell },\mu }f,$$

where $f\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_2}\left[[a,b],X_w\right]$.

Proof. 

Let ${\alpha }_1<{\alpha }_2$ and $f\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_2}\left[[a,b],X_w\right]$. By Lemma 12, we know that ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_2,0,\mathsf{\ell }}f\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X_w\right]$. Therefore, $K_{\rho ,\psi }^{1-{\alpha }_1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_2,0,\mathsf{\ell }}f\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X_w\right]$. Consequently, in view of Lemma 4), it follows that

$$\begin{array}{ccc}\hfill {}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,1,\mathsf{\ell }}{\mathbb{K}}_{\rho ,\psi }^{{\alpha }_2,\mathsf{\ell },\mu }f& =& K_{\rho ,\psi }^{1-{\alpha }_1,\mathsf{\ell },\mu }\left({\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\right){}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_2,0,\mathsf{\ell }}f=K_{\rho ,\psi }^{1-{\alpha }_1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_2,0,\mathsf{\ell }}f\hfill \\ & =& K_{\rho ,\psi }^{{\alpha }_2-{\alpha }_1,\mathsf{\ell },\mu }\left(K_{\rho ,\psi }^{1-{\alpha }_2,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_2,0,\mathsf{\ell }}f\right),\hfill \end{array}$$

Since $1+\lambda -{\alpha }_2>1-{\alpha }_2$, from Lemma 14, it follows that $K_{\rho ,\psi }^{1-{\alpha }_2,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-{\alpha }_2,0,\mathsf{\ell }}f=f$, and so ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,1,\mathsf{\ell }}{\mathbb{K}}_{\rho ,\psi }^{{\alpha }_2,\mathsf{\ell },\mu }f=f$, as required. □

Lemma 17. 

Let ${\alpha }_1>{\alpha }_2\in (0,1)$, $\rho \in [0,1]$, and $\mu \ge 0$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Then

$${\mathbb{K}}_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\mathsf{\ell }}f=K_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }f\left(t\right)-{\displaystyle \frac{f\left(a\right)e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\Gamma (1+{\alpha }_1-{\alpha }_2){\left(\omega \left(t\right)\right)}^{{\alpha }_2-{\alpha }_1}}},$$

where $f\in A{C}^1\left[[a,b],X_w\right]$.

Proof. 

Let $f\in A{C}^1\left[[a,b],X_w\right]$. The proof is derived directly from Proposition 4, since

$$\begin{array}{ccc}\hfill {\mathbb{K}}_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\mathsf{\ell }}f& =& K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}\left(K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{1-{\alpha }_2,\mathsf{\ell },\mu }\right){\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\hfill \\ & =& K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\left({\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\right)K_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\hfill \\ & =& K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f=K_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }\left(K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}f\right)\hfill \\ & =& K_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }\left[f-f\left(a\right)e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\right]\hfill \\ & =& K_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }f\left(t\right)-{\displaystyle \frac{f\left(a\right)e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\Gamma (1+{\alpha }_1-{\alpha }_2){\left(\omega \left(t\right)\right)}^{{\alpha }_2-{\alpha }_1}}}.\hfill \end{array}$$

□

4. Bagley–Torvik-Type Problem with Generalized Hilfer Fractional Derivatives

Combining the constitutive law of a material with classical equations of motion (e.g., Newton’s second law or Euler-Bernoulli beam theory) for a viscoelastic system results in an equation of motion containing a fractional derivative of order ${\displaystyle \frac{3}{2}}$. Acting upon ${\displaystyle \frac{1}{2}}$-order derivative in the stress-strain law with the second derivative from the equation of motion, leads to the ${\displaystyle \frac{1}{2}}+1={\displaystyle \frac{3}{2}}$ order term:

$${\displaystyle \frac{d^2}{d{t}^2}}\left(D_t^{1/2}y\left(t\right)\right)=D_t^{3/2}y\left(t\right).$$

Thus, the Bagley–Torvik equation emerged from the need to model the dynamics of mechanical systems with physically realistic viscoelastic material behavior constraints. The fractional version of this equation is named after the two researchers who first derived and solved it: P. J. Torvik and R. L. Bagley.

In their paper [1] and others from that period, Torvik and Bagley derived the fractional differential equation from fundamental thermodynamic and mechanical principles in order to describe viscoelastic material behavior. They provided methods for solving the equation. They are universally credited with creating and popularizing this fractional model. Subsequent research expanded to include other types of fractional derivatives and their benefits. As previously mentioned, additional issues arose that can be reduced to the vector Bagley–Torvik equation. Here, we will focus on the fractional Hilfer derivative.

Drawing inspiration from the work of numerous authors, our goal to contribute to the literature by introducing and analyzing a Bagley–Torvik type problem involving Hilfer fractional derivatives of two different orders:

$$\left\{\begin{array}{c}{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathsf{\ell }}x\left(t\right)+A{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathsf{\ell }}x\left(t\right)=f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right),\hfill \\ x\left(a\right)={\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x\left(a\right)=0,\hfill \end{array}\right.$$

(26)

where $t\in [a,b]$, $A\in \mathbb{R}$, $\beta \in [0,1]$, for ${\alpha }_1\in [1,2)$, ${\alpha }_2,\alpha \in (0,1)$ such that $\alpha \le {\alpha }_2$.

When researching maximal regularity and equivalence problems of solutions, the choice of maximal domain and codomain spaces is paramount. Equivalence may or may not hold depending on this choice.

The integral in the equation must be defined. If we choose a codomain space X that lacks the necessary properties, the integral may not exist. To use the Bochner integral, the space must be such that the integrand is strongly measurable and its norm is integrable. The space must be a Banach space (i.e., complete), otherwise the integral limits might not exist. If we must use the Pettis integral, the space X must have some extra properties that ensures the Pettis integral behaves well; otherwise, we may be unable to prove the necessary compactness theorems for existence.

Since we just reviewed the Bagley–Torvik problem, the expected integral operator cannot simply be a Volterra operator with a Pettis integral. Therefore, selecting a domain for the operators requires a thorough analysis. For the differential and integral forms of the problem to be equivalent, we need a function space, E, such that the solution, x, and its derivative $Dx$ (appropriate to the chosen problem) are both in E. However, such a space would be enormous. Thus, we typically consider much smaller spaces that maximize additional properties of solutions. The “maximal” space yields the sharpest well-posedness result and the largest set of forcing functions, f, for which the solution is as regular as possible. Finding the maximal space reveals the exact “price” of a solution. It demonstrates how irregular the input data can be while still yielding controlled behavior. This is of immense practical importance.

Let us formally convert the problem (26) into its integral form. Clearly, for sufficiently smooth f and for any $\beta \in [0,1]$, we have

$$\left({\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}\right){}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-1,\beta ,\mathsf{\ell }}x\left(t\right)+A{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathsf{\ell }}x\left(t\right)=f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right),$$

for $t\in [a,b]$, $A\in \mathbb{R}$, $\beta \in [0,1]$. In view of Proposition 4, we have that

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-1,\beta ,\mathsf{\ell }}x\left(t\right)+A{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathsf{\ell }}x\left(t\right)=e^{-\chi \left(t\right)}{C}_0+K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right),$$

(27)

for $\beta \in [0,1]$. Let us investigate the following two different cases:

I

For $\beta \in [0,1)$. Because of (cf. also [47])

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-1,\beta ,\mathsf{\ell }}\left\{e^{{\displaystyle \left[-{\int }_0^{\psi \left(t\right)}\left(\mathsf{\ell }\left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\left(\omega \left(t\right)\right)}^{({\alpha }_1-2)(1-\beta )}\right\}=0,$$

in view of Lemmas 4 and 11, it follows that

$$\begin{array}{ccc}\hfill x\left(t\right)& +& A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }\left(K_{\rho ,\psi }^{{\alpha }_2,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\mathsf{\ell }}x\left(t\right)\right)\hfill \\ & =& e^{-\chi \left(t\right)}\left[C_0+{\displaystyle \frac{C_1}{{\left(\omega \left(t\right)\right)}^{(2-{\alpha }_1)(1-\beta )}}}\right]+K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right).\hfill \end{array}$$

Then (still formally) we obtain

$$\begin{array}{ccc}\hfill x\left(t\right)& +& A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }x\left(t\right)\hfill \\ & =& e^{-\chi \left(t\right)}\left[C_0+{\displaystyle \frac{C_1}{{\left(\omega \left(t\right)\right)}^{(2-{\alpha }_1)(1-\beta )}}}\right]+K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right).\hfill \end{array}$$

Assuming that we look for a weakly continuous solution x that satisfies $x\left(a\right)=0$, it follows from Remark 2 that $C_0=C_1=0$. Thus, we arrive to the following integral form corresponding to the problem (26)

$$x\left(t\right)+A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }x\left(t\right)=K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right),\beta \in [0,1).$$

(28)

Consequently,

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)+A{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }x\left(t\right)={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right),$$

for $\beta \in [0,1)$. This reads as

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)+A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2-\alpha ,\mathsf{\ell },\mu }x\left(t\right)=K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right),$$

(29)

$\beta \in [0,1)$. Define $y:={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x$, $\beta \in [0,1)$. By applying Lemma 13, we obtain that $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y=x$. Therefore,

$$K_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2-\alpha ,\mathsf{\ell },\mu }x=K_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2-\alpha ,\mathsf{\ell },\mu }\left(K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\right)=K_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }y.$$

Consequently, we arrive at the following integral equation

$$y\left(t\right)+A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }y\left(t\right)=K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f\left(t,K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right),y\left(t\right)\right),$$

(30)

for ${\alpha }_1\in [1,2)$, ${\alpha }_2,\alpha \in (0,1)$, $\alpha \le {\alpha }_2$.

II

For $\beta =1$. In view of Lemma 17 and our assumption that $x\left(a\right)=0$, acting on both sides of (27) with the operator ${\mathbb{K}}_{\rho ,\psi }^{{\alpha }_1-1,\mathsf{\ell },\mu }$ yields that

$$\begin{array}{ccc}\hfill x\left(t\right)& +& A{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{2-{\alpha }_1,0,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\mathsf{\ell }}x\left(t\right)\hfill \\ & & ={\displaystyle \frac{C_0{e}^{-\chi \left(t\right)}{\left(\omega \left(t\right)\right)}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}}+K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{2-{\alpha }_1,0,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\left(t\right)\right).\hfill \end{array}$$

Since

$$\begin{array}{ccc}& & K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{2-{\alpha }_1,0,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\mathsf{\ell }}x\hfill \\ & & =K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\mathsf{\ell }}x=K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\left({\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\right)K_{\rho ,\psi }^{{\alpha }_1-1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\mathsf{\ell }}x\hfill \\ & & =K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\mathsf{\ell }}x,\hfill \end{array}$$

and

$$K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{2-{\alpha }_1,0,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f=K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }f,$$

we arrive at

$$x\left(t\right)+A{K}_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\mathsf{\ell }}x\left(t\right)={\displaystyle \frac{C_0{e}^{-\chi \left(t\right)}{\left(\omega \left(t\right)\right)}^{{\alpha }_1-1}}{\Gamma \left({\alpha }_1\right)}}+K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\left(t\right)\right).$$

Therefore, by Lemma 4, we obtain

$$\begin{array}{ccc}\hfill {\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x\left(t\right)& +& A{K}_{\rho ,\psi }^{{\alpha }_1-1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\mathsf{\ell }}x\left(t\right)\hfill \\ & & ={\displaystyle \frac{C_0{e}^{-\chi \left(t\right)}{\left(\omega \left(t\right)\right)}^{{\alpha }_1-2}}{\Gamma ({\alpha }_1-1)}}+K_{\rho ,\psi }^{{\alpha }_1-1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\left(t\right)\right).\hfill \end{array}$$

Thus, since ${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x\left(a\right)=0,$ it follows in view of Remark 2 that $C_0=0$. Operating by $K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }$ on both sides of the last equation leads to

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\left(t\right)+A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\left(t\right)=K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\left(t\right)\right).$$

Thus, if we define $y:={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x=K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x$, it follows by Lemma 17 in view of $x\left(a\right)=0$, that

$${\mathbb{K}}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y={\mathbb{K}}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x=x.$$

Therefore, we arrive at the following integral equation:

$$y\left(t\right)+A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }y\left(t\right)=K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f\left(t,{\mathbb{K}}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right),y\left(t\right)\right),$$

(31)

for ${\alpha }_1\in [1,2),{\alpha }_2,\alpha \in (0,1),\alpha \le {\alpha }_2$.

Lemma 18. 

Let $\rho \in [0,1]$, $\beta \in [0,1)$, $\mu \ge 0$ and ${\alpha }_1\in [1,2)$, ${\alpha }_2,\alpha \in (0,1)$ such that $\alpha \le {\alpha }_2$, ${\alpha }_1-\alpha <1$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Suppose that for all $x,y\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$, we have that the superposition $f(·,y(·),x(·))\in {\mathcal{H}}_0^{\lambda +\alpha }\left[[a,b],X\right]$, for $\lambda +\beta (1-\alpha )+max\{{\alpha }_2,{\alpha }_1-{\alpha }_2\}<1$. Then, for any $x_1,x_2,y_1,y_2\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$, we have

$${∥\left|f\left(·,y_1,x_1\right)-f\left(·,y_2,x_2\right)\right|∥}_{\lambda +\alpha }\le A_1{∥\left|y_1-y_2\right|∥}_{\lambda }+A_2{∥\left|x_1-x_2\right|∥}_{\lambda }.$$

(32)

Then the problem (26) with ${\alpha }_1-\alpha <1$ has a unique solution $x\in {\mathcal{H}}_0^{\lambda +\alpha }\left[[a,b],X\right]$ for any $\beta \in (0,1)$ and for all $\beta \in [0,1)$, whenever X has a total dual $X^{\ast }$.

Proof. 

Define the operator $F:=K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f$, for $f\in {\mathcal{H}}_0^{\lambda +\alpha }\left[[a,b],X\right]$. Note that $K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f=K_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{{\alpha }_2-\alpha ,\mathsf{\ell },\mu }f$. Since (cf. Lemma 8) $K_{\rho ,\psi }^{{\alpha }_2-\alpha ,\mathsf{\ell },\mu }f\in {\mathcal{H}}_0^{\lambda +{\alpha }_2}\left[[a,b],X\right]\subset {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$, again by Lemma 8), it follows that $F\in {\mathcal{H}}_0^{\lambda +{\alpha }_1-{\alpha }_2}\left[[a,b],X\right]\subset {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$. Also, in view of (9), for any $x_1,x_2,y_1,y_2\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$, there exists $M>0$ such that

$${∥\left|K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f\left(·,y_1,x_1\right)-K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f\left(·,y_2,x_2\right)\right|∥}_{\lambda }\le M{∥\left|f\left(·,y_1,x_1\right)-f\left(·,y_2,x_2\right)\right|∥}_{\lambda +\alpha }.$$

Thus, by (32), the operator F satisfies the assumptions of Theorem 1, and so, for sufficiently small $A,A_1$, and $A_2$, the integral Equation (30) has a unique solution $y\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$.

Suppose that $y\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ satisfies the Equation (30). We will show that the function $x\in {\mathcal{H}}_0^{\lambda +\alpha }\left[[a,b],X\right]$ defined by $x:=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y$ satisfies the problem (26) with ${\alpha }_1-\alpha <1$ for any $\beta \in [0,1)$. First, note that in view of Remark 2, we obtain $x\left(a\right)=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(a\right)=0$. Also, we have that ${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x\left(a\right)={\mathfrak{D}}_{p,\psi ,\mu }^{1-\alpha ,\rho ,\mathsf{\ell }}y\left(a\right)={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}y\left(a\right)=0$.

Now, consider the case when $\beta \in (0,1)$. Note that, the following also holds for $\beta =0$ whenever X has a total dual $X^{\ast }$). In view of the semi-group property, operating with $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }$ on both sides of (30) yields that

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)+A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)=K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }f\left(t,K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right),y\left(t\right)\right).$$

Since $\lambda +\beta (1-\alpha )<1$, we have that (cf. Lemma 13) ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y=y$, and then

$$x\left(t\right)+A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }x\left(t\right)=K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right).$$

Next, applying operator ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-1,\beta ,\mathsf{\ell }}$, by Lemma 14, it follows that

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-1,\beta ,\mathsf{\ell }}x\left(t\right)+A{K}_{\rho ,\psi }^{1-{\alpha }_2,\mathsf{\ell },\mu }x\left(t\right)=K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right).$$

In view of Corollary 3, we obtain

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-1,\beta ,\mathsf{\ell }}x\left(t\right)+A{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathsf{\ell }}x\left(t\right)=K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right).$$

Finally, by Lemma 2, we arrive at

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,\beta ,\mathsf{\ell }}x\left(t\right)+A{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathsf{\ell }}x\left(t\right)=f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right),x\left(a\right)={\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x\left(a\right)=0.$$

Therefore, the function $x:=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y$ solves problem (26) with $\beta \in [0,1)$, as required. □

The results of Lemma 18 still hold whenever ${\alpha }_1-{\alpha }_2\ge 1$.

Corollary 5. 

Let $\rho \in [0,1]$, $\beta \in [0,1)$, $\mu \ge 0$. Also, let ${\alpha }_1\in [1,2)$, ${\alpha }_2,\alpha \in (0,1)$ be such that $\alpha \le {\alpha }_2,{\alpha }_1-{\alpha }_2\ge 1$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. For all $x,y\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ we have $f(·,y(·),x(·))\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1+\alpha }\left[[a,b],X\right]$, where $1+\lambda +\beta (1-\alpha )-({\alpha }_1-\alpha )\in (0,1)$, such that for any $x_1,x_2,y_1,y_2\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$. Consequently, we have

$${∥\left|f\left(·,y_1,x_1\right)-f\left(·,y_2,x_2\right)\right|∥}_{1+\lambda -({\alpha }_1-\alpha )}\le A_1{∥\left|y_1-y_2\right|∥}_{\lambda }+A_2{∥\left|x_1-x_2\right|∥}_{\lambda }.$$

(33)

Then, for any $\beta \in [0,1)$, the problem (26) with ${\alpha }_1-{\alpha }_2\ge 1$ has a unique solution $x\in {\mathcal{H}}_0^{\lambda +\alpha }\left[[a,b],X\right]$ whenever X has a total dual $X^{\ast }$.

Proof. 

Define the operator $F=K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f$. Note that

(1)

Since $1+\lambda -({\alpha }_1-\alpha )\in (0,1)$ we have, by the aid of Lemma 9, $F\in {\mathcal{H}}_0^{\alpha +\lambda -1}\left[[a,b],X\right])$, and

$${∥\left|F\right|∥}_{\lambda }={∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right|∥}_{\lambda }\le C{\left[\left[f\right]\right]}_{1+\lambda -({\alpha }_1-\alpha )}$$

with $C>0$.

(2)

For any $x_1,x_2,y_1,y_2\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$, we have

$${∥\left|F\left(·,y_1,x_1\right)-F\left(·,y_2,x_2\right)\right|∥}_{\lambda }\le C{∥\left|f\left(·,y_1,x_1\right)-f\left(·,y_2,x_2\right)\right|∥}_{1+\lambda -({\alpha }_1-\alpha )},$$

with $C>0$.

This, along with (33) guarantees the satisfaction of the assumptions of Corollary 1. Thus, for sufficiently small $A,A_1,A_2$, the integral Equation (30) with ${\alpha }_1-{\alpha }_2\ge 1$ has a unique solution $y\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$.

Let $y\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$ solves (30) and define $x\in {\mathcal{H}}_0^{\lambda +\alpha }\left[[a,b],X\right]$ by $x:=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y$. In view of Remark 2, we obtain that $x\left(a\right)=K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(a\right)=0$.

Now consider the case when $\beta \in (0,1)$. Note that the following also holds for $\beta =0$ whenever X has a total dual $X^{\ast }$). Applying $K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }$ to both sides of (30), in view of the semi-group property, we have that

$$K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)+A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)=K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }f\left(t,K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right),y\left(t\right)\right).$$

In view of Lemma 13, since $\lambda +\beta (1-\alpha )<1$, we have that ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}{K}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y=y$. It follows that

$$x\left(t\right)+A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2,\mathsf{\ell },\mu }x\left(t\right)=K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right).$$

In view of Proposition 2, since ${\alpha }_1-{\alpha }_2\ge 1$, we obtain

$${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x\left(t\right)+A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2-1,\mathsf{\ell },\mu }x\left(t\right)=K_{\rho ,\psi }^{{\alpha }_1-1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right).$$

In view of Remark 2, we obtain ${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x\left(a\right)=0$. The result now follows by completing the rest of the proof of Lemma 18 with minor necessary changes. □

The following result complements the results of Lemma 18 and Corollary 5, which deal with the case when $\beta =1$.

Lemma 19. 

Let $\rho \in [0,1]$, $\beta \in [0,1)$, $\mu \ge 0$. Also, let ${\alpha }_1\in [1,2),{\alpha }_2,\alpha \in (0,1)$ be such that $\alpha \le {\alpha }_2,{\alpha }_1-{\alpha }_2<\alpha$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Let $f(·,y(·),x(·))\in {\mathcal{H}}_0^{1+\lambda -({\alpha }_1-{\alpha }_2)}\left[[a,b],X\right]$, where $0<\lambda <{\alpha }_1+\alpha -{\alpha }_2<1$ for all $x,y\in {\mathcal{H}}_0^{1+\lambda -\alpha }\left[[a,b],X\right]$, such that for any $t\in [a,b]$ and any $x_1,x_2,y_1,y_2\in {\mathcal{H}}_0^{1+\lambda -\alpha }\left[[a,b],X\right]$, we have

$$\begin{array}{ccc}\hfill & & {∥\left|f\left(·,y_1,x_1\right)-f\left(·,y_2,x_2\right)\right|∥}_{1+\lambda -({\alpha }_1-{\alpha }_2)}\hfill \\ & & \le A_1{∥\left|y_1-y_2\right|∥}_{1+\lambda -\alpha }+A_2{∥\left|x_1-x_2\right|∥}_{1+\lambda -\alpha }.\hfill \end{array}$$

(34)

Then the problem (26) with $\beta =1$, ${\alpha }_1-{\alpha }_2<1$ has a unique solution $x\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$.

Proof. 

Define the operator $F\in {\mathcal{H}}_0^{1+\lambda -\alpha }\left[[a,b],X\right]$ by $F=K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f$. As in the the proof of Lemma 8, it can easily be seen that for any $t,s\in [a,b]$, there exists $M>0$ such that

$$∥K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f\left(t\right)-K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f\left(s\right)∥\le M{|t-s|}^{2+\lambda -({\alpha }_1-{\alpha }_2)}{\left[\left[f\right]\right]}_{1+\lambda -({\alpha }_1-{\alpha }_2)},$$

where $M>0$. Hence

$$\begin{array}{ccc}\hfill {∥\left|F\right|∥}_{1+\lambda -\alpha }& =& \underset{t,s\in [a,b],t\ne s}{sup}{\displaystyle \frac{∥K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f\left(t\right)-K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f\left(s\right)∥}{{|t-s|}^{1+\lambda -\alpha }}}\hfill \\ & \le & {M|b-a|}^{1+\alpha -({\alpha }_1-{\alpha }_2)}{\left[\left[f\right]\right]}_{1+\lambda -({\alpha }_1-{\alpha }_2)},\hfill \end{array}$$

where $M>0$. In view of (9), for any $x_1,x_2,y_1,y_2\in {\mathcal{H}}_0^{1+\lambda -\alpha }\left[[a,b],X\right]$, we see that $F\left(·,y_1,x_1\right)\in {\mathcal{H}}_0^{1+\lambda -\alpha }\left[[a,b],X\right]$, and

$${∥\left|K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f\left(·,y_1,x_1\right)-K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f\left(·,y_2,x_2\right)\right|∥}_{1+\lambda -\alpha }\le M{|b-a|}^{1+\alpha -({\alpha }_1-{\alpha }_2)}{\left[\left[f\right]\right]}_{1+\lambda -({\alpha }_1-{\alpha }_2)}.$$

In view of (34), F satisfies the assumptions of Theorem 2, so for sufficiently small $A,A_1,A_2$, the integral Equation (31) has a unique solution $y\in {\mathcal{H}}_0^{1+\lambda -\alpha }\left[[a,b],X\right]$.

Let $y\in {\mathcal{H}}_0^{1+\lambda -\alpha }\left[[a,b],X\right]$ solves Equation (31). Define $x:={\mathbb{K}}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y=K_{\rho ,\psi }^{1,\mathsf{\ell },\mu } {}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}y$. According to Proposition 2, x is weakly absolutely continuous and has a Pettis-integrable pseudo-derivative on $[a,b]$. Clearly, $x\left(a\right)=0$. In view of Lemma 13, it follows that

$${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x={}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}y⟹{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x=K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x=K_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}y=y,$$

and ${\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x\left(a\right)=0$. With ${\mathbb{K}}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }$ now operating on both sides of (31), it follows that

$$\begin{array}{ccc}\hfill {\mathbb{K}}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)& +& A{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\left({}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}{K}_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }\right)K_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2-1+\alpha ,\mathsf{\ell },\mu }y\left(t\right)\hfill \\ & & =K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\left({}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{1-\alpha ,0,\mathsf{\ell }}{K}_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }\right)K_{\rho ,\psi }^{{\alpha }_1-1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\left(t\right)\right).\hfill \end{array}$$

Thus, by the semi-group property of considered operators, and by Lemma 13, we arrive at

$${\mathbb{K}}_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }y\left(t\right)+A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2+\alpha ,\mathsf{\ell },\mu }y\left(t\right)=K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\left(t\right)\right).$$

This reads as

$$x\left(t\right)+A{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2+\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\left(t\right)=K_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\left(t\right)\right).$$

Operating with ${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-1,1,\mathsf{\ell }}\equiv K_{\rho ,\psi }^{2-{\alpha }_1,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}$, and keeping in mind that

$$\begin{array}{ccc}\hfill {}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-1,1,\mathsf{\ell }}{K}_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2+\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x& =& K_{\rho ,\psi }^{2-{\alpha }_1,\mathsf{\ell },\mu }\left({\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\right)K_{\rho ,\psi }^{{\alpha }_1-{\alpha }_2+\alpha -1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\hfill \\ & =& K_{\rho ,\psi }^{1-{\alpha }_2+\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x,\hfill \end{array}$$

and that

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-1,1,\mathsf{\ell }}{K}_{\rho ,\psi }^{{\alpha }_1,\mathsf{\ell },\mu }f=K_{\rho ,\psi }^{2-{\alpha }_1,\mathsf{\ell },\mu }\left({\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }\right)K_{\rho ,\psi }^{{\alpha }_1-1,\mathsf{\ell },\mu }f=K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f.$$

It follows that

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-1,1,\mathsf{\ell }}x\left(t\right)+A{K}_{\rho ,\psi }^{1-{\alpha }_2+\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\left(t\right)=K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\left(t\right)\right).$$

Since

$$\begin{array}{ccc}\hfill K_{\rho ,\psi }^{1-{\alpha }_2+\alpha ,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x& =& K_{\rho ,\psi }^{1-{\alpha }_2+\alpha ,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{1-\alpha ,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x=K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{K}_{\rho ,\psi }^{1-{\alpha }_2,\mathsf{\ell },\mu }{\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x\hfill \\ & =& K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\mathsf{\ell }}x,\hfill \end{array}$$

we arrive at

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1-1,1,\mathsf{\ell }}x\left(t\right)+A{K}_{\rho ,\psi }^{1,\mathsf{\ell },\mu }{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\mathsf{\ell }}x\left(t\right)=K_{\rho ,\psi }^{1,\mathsf{\ell },\mu }f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\mathsf{\ell }}x\left(t\right)\right).$$

Therefore, according to Proposition 2, we have

$${}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,1,\mathsf{\ell }}x\left(t\right)+A{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,\beta ,\mathsf{\ell }}x\left(t\right)=f\left(t,x\left(t\right),{}_p{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\mathsf{\ell }}x\left(t\right)\right),x\left(a\right)={\Delta }_{p,\psi }^{\rho ,\mathsf{\ell }}x\left(a\right)=0,$$

as required. □

The following result complements the result of Lemma 19, dealing with the case when $\beta =1$, ${\alpha }_1-{\alpha }_2\ge 1$.

Corollary 6. 

Let $\rho \in [0,1]$, $\beta \in [0,1)$, $\mu \ge 0$. Also, let ${\alpha }_1\in [1,2),{\alpha }_2,\alpha \in (0,1)$ be such that $\alpha \le {\alpha }_2,{\alpha }_1-{\alpha }_2\ge 1$. Assume that $\psi \in C^1\left[[a,b],\mathbb{R}\right]$ is a positive, increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Let $f(·,y(·),x(·))\in {\mathcal{H}}_0^{2+\lambda -({\alpha }_1+\alpha )}\left[[a,b],X\right]$, where $1+\lambda <{\alpha }_1+\alpha$ for all $x,y\in {\mathcal{H}}_0^{1+\lambda -\alpha }\left[[a,b],X\right]$, such that for any $t\in [a,b]$ and any $x_1,x_2,y_1,y_2\in {\mathcal{H}}_0^{1+\lambda -\alpha }\left[[a,b],X\right]$, we have

$$\begin{array}{ccc}\hfill & & {∥\left|f\left(·,y_1,x_1\right)-f\left(·,y_2,x_2\right)\right|∥}_{2+\lambda -({\alpha }_1+\alpha )}\hfill \\ & & \le A_1{∥\left|y_1-y_2\right|∥}_{1+\lambda -\alpha }+A_2{∥\left|x_1-x_2\right|∥}_{1+\lambda -\alpha }.\hfill \end{array}$$

(35)

Then the problem (26) with $\beta =1,{\alpha }_1-{\alpha }_2\ge 1$ has a unique solution $x\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right]$.

Proof. 

Define the operator $F=K_{\rho ,\psi }^{{\alpha }_1-\alpha ,\mathsf{\ell },\mu }f$. Note that

(1)

Since $1+\lambda -({\alpha }_1+\alpha )\in (1,2)$, we have, by the aid of Lemma 9, that $F\in {\mathcal{H}}_0^{1+\lambda -{\alpha }_1}\left[[a,b],X\right])$ and

$${∥\left|F\right|∥}_{1+\lambda -{\alpha }_1}={∥\left|K_{\rho ,\psi }^{\alpha ,\mathsf{\ell },\mu }f\right|∥}_{1+\lambda -\alpha }\le C{\left[\left[f\right]\right]}_{2+\lambda -({\alpha }_1+\alpha )}$$

with $C>0$.

(2)

For any $x_1,x_2,y_1,y_2\in {\mathcal{H}}_0^{\lambda }\left[[a,b],X\right]$, we have

$${∥\left|F\left(·,y_1,x_1\right)-F\left(·,y_2,x_2\right)\right|∥}_{1+\lambda -{\alpha }_1}\le C{∥\left|f\left(·,y_1,x_1\right)-f\left(·,y_2,x_2\right)\right|∥}_{2+\lambda -({\alpha }_1+\alpha )}$$

with $C>0$.

This, along with (35), ensures that the assumptions of Corollary 4 are met. Thus, for sufficiently small $A,A_1,A_2$, the integral Equation (30) with ${\alpha }_1-{\alpha }_2\ge 1$ has a unique solution $y\in {\mathcal{H}}_0^{1+\lambda -\alpha }\left[[a,b],X\right]$. The result now follows by completing the rest of the proof as in the proof of Lemma 19 with minor changes. □

To avoid making the paper too long, we will briefly present two additional aspects of our research in the form of comments and recommendations for readers.

Remark 6. 

Developing numerical schemes for Hilfer-type fractional derivatives in Banach spaces that preserve the analytical properties of the solution poses a significant challenge. However, methods and theoretical frameworks do exist. The Hilfer derivative is a non-local operator, meaning its value at any time t depends on the entire history of the function. This non-locality must be accurately approximated. Moreover, standard discretizations (like simple finite difference or finite element methods) can often violate key analytical properties of the continuous solution, such as positivity, monotonicity, or the conservation of energy or mass (new, interesting methods can be found in [51,52], for instance).

A significant amount of theoretical work has been dedicated to proving that Hilfer fractional differential equations are "Ulam-Hyers stable." This means that an approximate solution, which is what a numerical method produces, stays close to the true solution. Researchers then develop discrete numerical schemes, i.e., Hilfer fractional difference equations, that possess this stability as well, ensuring the numerical approximation is robust.

Numerical methods for abstract problems often use finite-dimensional approximations that are consistent in a weak or dual sense. So, are numerical methods ineffective? Even though standard algorithms generate approximations that only weakly converge to the true solution, it’s still worth studying them.

Remark 7. 

Are the results obtained here merely theoretical constructs, or do they have practical significance? Similar to Hilfer’s original construction, our modifications have many applications that significantly expand the range of models that can be studied without these operators. Below is a brief comment related to the Bagley-Torvik equation studied in the paper.

The Hilfer fractional operator interpolates between the Riemann–Liouville and Caputo types, solving theoretical and applied problems in fractional differential equations. This includes certain formulations of the Bagley–Torvik problem, especially when addressing initial conditions for intermediate derivatives and ensuring well-posedness in multi-term equations where the derivative orders range from 1 to 2.

Example 2. 

The central challenge with mixed-order fractional differential equations is establishing a well-posed initial value problem. The difficulty lies in formulating a well-posed initial value problem for an equation containing fractional derivatives of different orders. The Hilfer derivative elegantly unifies the required initial conditions.

Suppose we consider the Bagley-Torvik equation:

$$A·D^{\alpha }y\left(t\right)+B·D^{\beta }y\left(t\right)+C·y\left(t\right)=f\left(t\right),$$

where $1<\alpha \le 2$, $0<\beta \le 1$, but not necessarily $\alpha =2$ and $\beta =3/2$.

Using pure Riemann-Liouville derivatives for $D^{\alpha }$ (order $1<\alpha \le 2$) and $D^{\beta }$ (order $0<\beta \le 1$) requires two different types of incompatible, fractional-order initial conditions (e.g., $D^{\alpha -k}y\left(0\right)=b_k$, ($k=1,2$) (for $1<\alpha \le 2$), which lack clear physical meaning.

If we consider Caputo derivatives, the initial conditions are natural: $y\left(0\right),y^{\prime }\left(0\right)$. However, the derivative $D^{\beta }$ in the equation is Caputo, if β is small. If the other term $D^{\alpha }$ is of order between 1 and 2, the composition rules can be unclear when α and β are not integers.

For the Hilfer derivative of order α and type μ, the initial conditions for $D^{\alpha ,\mu }y$ with $1<\alpha \le 2$ are as follows:

$$I^{(1-\mu )(2-\alpha )}y\left(0\right)=c_1,{\displaystyle \frac{d}{dt}}{I}^{(1-\mu )(2-\alpha )}y\left(0\right)=c_2.$$

By choosing μ and ν appropriately, i.e.,

$$(1-\mu )(2-\alpha )=(1-\nu )(1-\beta )=k$$

we can match the same type of initial conditions for both derivatives. This allows you to formulate a well-posed IVP unified by a single fractional-type initial condition (e.g., $I^{k}y\left(0\right)$), a flexibility that Riemann–Liouville and Caputo derivatives alone do not offer (unless μ and ν are 0 or 1).

5. Conclusions

The results obtained in this paper are an important step toward developing a comprehensive theory of fractional-order differential equations of various orders. Equivalence results (or the lack thereof) were obtained for operators connecting classes of fractional and non-fractional operators in Hölder spaces. These spaces are a natural class in fractional-order calculus, and the operators under investigation act on vector-valued function spaces.

Our proof methods are based on operator properties expressed in terms of weak topology, providing strong links to earlier studies of total order operators. Furthermore, we solved problems concerning the existence of certain Hilfer-type fractional derivatives and their corresponding fractional Pettis-integrals. These results are motivated and complemented by applications to Bagley-Torvik problems.