A Unified Framework for Fractional and Non-Fractional Operators in Some Function Spaces

Abstract

This paper examines a family of operators that combine the features of fractional-order and classical operators. Our goal is to obtain results on their invertibility in function spaces, based on their inherent improving properties. The class of proportional operators we study is extensive and includes both fractional-order and classical operators. This leads to interesting function spaces in which we obtain the right- and left-handed properties of invertibility. Thus, we extend and unify results concerning fractional-order and proportional operators. To confirm the relevance of our results, we have supplemented the paper with a series of results on the equivalence of differential and integral forms for various problems, including terminal value problems.

1. Introduction and Preliminaries

Fractional-order calculus has evolved—and continues to evolve—for reasons different from those that drove the development of classical integer-order calculus ([1,2,3,4]). In fact, integer-order calculus is increasingly being replaced in many areas of physics, with fractional-order derivatives now serving as the preferred tool for modeling and analysis ([5,6,7,8,9], among others).

This paper is not just another application of fractional-order derivatives or integrals to solve specific differential or integral equations. Instead, our goal is to systematically study a broad class of operators within their natural domains (i.e., function spaces). A critical aspect of this analysis involves investigating their inverses, which requires identifying the appropriate parameter sets for these operators. The existence and form of an inverse operator fundamentally depend on this choice—an area in which we propose that significant advancements should be made. The recent results from studying this type can be found in [2,10,11,12,13,14].

The study of operators and their properties is central to many differential calculus methods. In this paper, we focus specifically on integral and differential operators. Regardless of their classification, a fundamental consideration is their reciprocal inverse, which depends crucially on the function spaces in which they are defined, among other factors. For certain classical integral operators (such as Volterra-type operators), the action of the operator leads to increased smoothness of the function as the total order increases (e.g., mapping from $C[a,b]$ to $C^1[a,b]$ for Volterra-type operators; see [15]). In the case of fractional-order operators (e.g., the Riemann–Liouville operator of order $\alpha$), this phenomenon is known as fractional-order smoothing (for instance, transforming Hölder functions of order $\lambda$ to those of order $\lambda +\alpha$).

This paper focuses on this critical aspect of operator analysis, particularly the rigorous definition of inverse operators in appropriate function spaces. Furthermore, we present a unified framework for systematically studying combinations of fractional and non-fractional operators, thereby bridging the gap between classical and fractional calculus.

We examine operators that, in a certain sense, blend classical fractional-order integral operators with the identity operator. These “interpolated” operators, which are frequently employed in physics, cannot be categorized as standard fractional-order or classical differentiation/integration operators. Often termed “proportional” operators, they possess a unique property: operators in proportional calculus cannot exhibit weak singularity because of the proportional relationship between the singular operator and the identity operator. This characteristic leads us to classify them as “non-fractional”. It is important to note that proportional calculus does not align completely with fractional calculus, as it fails to satisfy certain fundamental criteria established for fractional operators (see [16,17,18,19,20]). Nevertheless, our approach provides a unified framework for analyzing these operators alongside standard and fractional ones.

As previously noted, these operators are of particular interest due to their analogous properties to those of classical fractional-order operators. This is a key motivation for our paper. Additionally, they exhibit notable “regularizing” properties. While many studies focus on applications of proportional calculus in physics, we are not aiming to evaluate these applications, despite their empirical validity. Rather, we argue that proportional operators warrant theoretical investigation in their own right. For instance, these operators naturally arise in the study of fractional-order Langevin equations ([4,12]), where maintaining solution regularity is essential. This is a direct consequence of our research. The proportional definition of the fractional gradient also yields insightful results in modeling physical phenomena, even though the fractional gradient operation lacks rotational invariance in the coordinate system ([21]). Consequently, the Caputo fractional gradient cannot be treated as a true vector operator, precluding its use in defining tensor fields for deformation analysis. Despite these limitations, the proportional approach remains valuable, and we seek to systematically integrate it into broader operator theory.

The central challenge lies in identifying and analyzing function spaces X for which operators T map into a subsequent space Y, ensuring that their inverses $T^{-1}$ (whether right or left) preserve structural properties when acting from Y back to X. This is the core goal of our investigation. We focus on proportional operators within generalized Hölder spaces, which extend beyond classical continuous function frameworks. We also examine their behavior in Orlicz spaces that generalize Hardy–Littlewood-type results (see, for example, [22,23]). Our study naturally encompasses Riemann–Liouville operators in Lebesgue spaces, creating a unified analytical framework.

A fundamental aspect of operator theory, particularly for fractional-order operators, is characterizing the image of function spaces under their action—a crucial step for determining solution regularity. This characterization enables two critical investigations: first, identifying the precise function spaces where these operators admit inverses, and second, establishing equivalences between differential and integral problem formulations. These equivalences are indispensable when studying fixed points or approximation sequences, repeatedly emphasizing the foundational role of the underlying function spaces.

Our paper builds upon existing results by generalizing the classes of operators and their associated function spaces considered, paying particular attention to maximizing operator regularity relative to their parameters. Although foundational studies of Riemann–Liouville operators in $L_p$ spaces [22]—particularly in the context of Abel equations—have been established, important equivalence questions with Caputo derivatives remained unresolved. Subsequent research consequently shifted its focus to continuous function spaces and their weighted subspaces. This shift revealed richer structural properties that we now incorporate into our broader analytical framework. For an interesting summary of these results, see [23]. This progression underscores the necessity of our unified approach to conducting operator analysis across diverse function space settings.

The transition from Lebesgue spaces to Orlicz spaces in [24,25] demonstrated an improved regularity of operator images. While fractional-order equations find numerous applications in physics, particularly when examined through the lens of Hölder and Orlicz spaces (cf. [26,27]), a fundamental challenge remains: ensuring that fractional-order operators acting on continuous functions maintain their values in Hölder spaces ([28]). This naturally leads us to the core questions of operator invertibility and regularity preservation. Our work addresses these questions by investigating a broad class of operators in Orlicz-type spaces—where we establish boundary conditions on the generating functions (analogous to the role of exponentials in Lebesgue spaces [23]) —alongside Hölder spaces.

The study of invertibility for differential and integral operators, especially fractional-order ones, has deep historical roots spanning from foundational papers [11,22,23] to recent research [29,30]. These investigations have proven particularly valuable when applied to concrete problems ([31,32]).

Our analysis of the underlying function spaces builds upon earlier approaches focused on $L^1[a,b]$ and $AC[a,b]$, while incorporating both fractional and classical operators. Notably, classical integral operators like Volterra exhibit regularity-improving properties, a characteristic that extends to fractional operators in Hölder spaces (cf. [23,28,33,34,35]), where they increase the Hölder continuity order by their operator order.

This paper begins by rigorously defining the key function spaces central to our investigation.

First, we briefly discuss the class of Orlicz spaces. Recall that a function $\phi :[0,\infty )\to [0,\infty )$ is called a Young function if $\phi$ is a continuous, even, convex nondecreasing function with $\phi \left(0\right)=0$ and $\phi \left(t\right)\to \infty$ as $t\to \infty .$ For any Young function $\phi$, the convex function $\tilde{\phi }:[0,\infty )\to [0,\infty )$ defined by $\tilde{\phi }\left(t\right):={sup}_{s\ge 0}\left\{\right|t|s-\phi \left(s\right)\}$ is called the Young complement of $\phi$, and it is known that $\tilde{\phi }$ is also a Young function.

Example 1.

(Fundamental examples)

1.

${\phi }_p\left(s\right)={\displaystyle \frac{1}{p}}{\left|s\right|}^p,p\in [1,\infty )$ and for any $p,q\in (1,\infty )$, we have ${\tilde{\phi }}_p\equiv {\phi }_q$.

2.

${\phi }_{\infty }\left(s\right)=\left\{\begin{array}{cc}0\hfill & {\displaystyle for\left|s\right|\le 1,}\hfill \\ \infty \hfill & otherwise.\hfill \end{array}\right.$ and ${\tilde{\phi }}_{\infty }\equiv {\phi }_1.$

3.

${\phi }_{exp}\left(s\right)=e^{\left|s\right|}-\left|s\right|-1$ and ${\tilde{\phi }}_{exp}\left(s\right)=\left(\right|s|+1)ln(1+\left|s\right|)-\left|s\right|.$

The Orlicz space $L_{\phi }[a,b]$ consists of all measurable functions $f:[a,b]\to \mathbb{R}$ such that

$${\int }_{[a,b]}\phi \left(\left|f\right(t\left)\right|\right)dt<\infty .$$

Equivalently,

$$L_{\phi }[a,b]:=\left\{f:f\mathrm{measurable},{\int }_{[a,b]}\phi \left({\displaystyle \frac{\left|f\right(t\left)\right|}{k}}\right)dt\le 1\mathrm{for}\mathrm{some}k>0\right\}.$$

The pair $\left[L_{\phi }[a,b],{∥·∥}_{\phi }\right]$, where

$${∥f∥}_{\phi }:=inf\left\{k>0:{\int }_{[a,b]}\phi \left({\displaystyle \frac{\left|f\right(t\left)\right|}{k}}\right)dt\le 1\right\},$$

forms a Banach space. Whenever $[a,b]$ is finite interval, then for any non-trivial Young function $\phi$, it can be easily seen that $L_{\infty }[a,b]\subset L_{\phi }[a,b]\subseteq L_1[a,b]$. It is worth noting that the special choice of the Young function, defined as $\phi \left(f\right)={\phi }_p\left(f\right)$, $p\in [1,\infty )$ leads to the Lebesgue space $L_p[a,b]$ while $L_{{\phi }_{\infty }}[a,b]\equiv L_{\infty }[a,b]$.

As their definition implies, spaces are the natural domain of operators when studying differential problems involving functions that grow faster than the polynomial rate. The problem of admissible function growth in fractional-order problems leads to intriguing blow-up problems and is a promising area of research, provided that the existence and equivalence of differential and integral problems are established ([36,37]).

Our goal is to study the broadest possible group of operators. Therefore, we are interested in Stieltjes-type integration and differentiation operators, which are defined by a different function (cf. [38]). This type of modeling ability, involving derivatives and integrals depending on a specific function, is important in physics for describing subdiffusion processes, among others (see, for instance, [5]).

This is a well-known case for both classical and fractional-order operators. Another aspect of interest is studying such operators in generalized Hölder spaces. Now, let us discuss these spaces.

Throughout the following pages, we assume that $\psi \in C^1[a,b]$ 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 a continuous increasing function $\vartheta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ having $\vartheta \left(0\right)=0$, we say that a function $f\in C[a,b]$ satisfies the $\vartheta$-Hölder condition on the interval $[a,b]$ (or f belongs to the (generalized) Hölder space ${\mathbb{H}}_{\psi }^{\vartheta }[a,b]$) if

$$\left|f\left(t\right)-f\left(s\right)\right|\le A\vartheta \left(\left|\psi \left(t\right)-\psi \left(s\right)\right|\right),A>0,t,s\in [a,b].$$

The pair $\left({\mathbb{H}}_{\psi }^{\vartheta }[a,b],{∥·∥}_{\vartheta }\right)$, where the norm is defined by

$${∥f∥}_{\vartheta }:=\left|f\left(a\right)\right|+{\left[f\right]}_{\vartheta },\mathrm{with}{\left[f\right]}_{\vartheta }:={\displaystyle \frac{\left|f\left(t\right)-f\left(s\right)\right|}{\vartheta \left(\left|\psi \left(t\right)-\psi \left(s\right)\right|\right)}},$$

forms a Banach space. The particular choice $\vartheta \left(t\right)=t^{\lambda },\lambda \in (0,1],\psi \left(t\right)=t$ naturally leads to the classical Hölder space ${\mathbb{H}}^{\lambda }[a,b]$.

In constructing the operators under consideration, the parameters responsible for determining their scale are not numbers, but rather, appropriate functions. We will now define these functions and establish their necessary properties. Throughout this paper, we define the functions ${\ell }_1,{\ell }_2:[0,1]\times [a,b]\to [0,\infty )$ as follows:

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

(1)

Regarding the function spaces under consideration, let us establish the weakest possible set of assumptions about these functions. Assume that

$${\ell }_1(·,t),{\ell }_2(·,·)\mathrm{are}\mathrm{continuous}\mathrm{and}\forall \rho \in (0,1),{\left({\ell }_1(\rho ,·)\right)}^{-1},{\left({\ell }_2(\rho ,·)\right)}^{-1}\in L_{\widehat{\phi }}[a,b].$$

(2)

Also, define the constants

$$L_0:=\underset{t\in [a,b]}{max}\left|{\ell }_2(\rho ,t)\right|,L_1:={∥{\left({\ell }_1(\rho ,·)\right)}^{-1}∥}_{\tilde{\phi }}and{L}_2:={∥{\ell }_2{(\rho ,·))}^{-1}∥}_{\tilde{\phi }}.$$

Similarly, let $\ell (·,·)={\ell }_2(·,·)/{\ell }_1(·,·)\in L_1[a,b]$. Using the generalized Hölder inequality between different Orlicz spaces, we obtain

$$∥\ell ∥:={\int }_a^b|\ell \left(\rho ,s\right)|ds\le 2{∥{\ell }_1(\rho ,·)∥}_{\phi }{∥{\ell }_2{(\rho ,·))}^{-1}∥}_{\tilde{\phi }}=2L_2{∥{\ell }_1(\rho ,·)∥}_{\phi }.$$

(3)

These are very typical assumptions about such functions. One simple example of such functions might be ${\ell }_1(\rho ,t)=\rho t^{1-\rho }$, ${\ell }_2(\rho ,t)=(1-\rho )t^{\rho }$, $\rho \in (0,1)$ and $t\in [0,1]$. In such a case, the assumption (2) holds with $\tilde{\phi }={\phi }_q,$ for $q\in \left[1,min\{1/\rho ,1/(1-\rho \left)\right\}\right]$. Meanwhile, (2) holds with $\tilde{\phi }={\phi }_{\infty }$ when, e.g., ${\ell }_1(\rho ,t)=\rho e^{(1-\rho )t}$ and ${\ell }_2(\rho ,t)=(1-\rho )e^{\rho t}$, $\rho \in (0,1)$.

Remark 1.

Let $\alpha \in (0,1]$ and φ be a Young function φ with its complementary function $\tilde{\phi }$ satisfying

$${\int }_0^t\tilde{\phi }\left(s^{\alpha -1}\right)ds<\infty ,t>0.$$

Similarly to the proof of Proposition 2 in [39], we can show that the function ${\tilde{\Phi }}_{\alpha }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is defined by

$${\tilde{\Phi }}_{\alpha }\left(\xi \right):=inf\left\{k>0:{\displaystyle \frac{∥{\ell }_1(\rho ,·)∥}{∥{\psi }^{\prime }∥}}{\int }_0^{k^{{\displaystyle \frac{1}{1-\alpha }}}\xi }\tilde{\phi }\left(s^{\alpha -1}\right)ds\le k^{{\displaystyle \frac{1}{1-\alpha }}}\right\},$$

(4)

which is increasing and continuous with ${\tilde{\Phi }}_{\alpha }\left(0\right)=0.$ That is, for any $\alpha \in (0,1]$, the space ${\mathbb{H}}_{\psi }^{{\tilde{\Phi }}_{\alpha }}[a,b]$ is a (generalized) Hölder space.

We are now ready to present the following generalizations of the standard differentiation operator. For an overview of these topics, see, for example, refs. [40,41] as well as the references therein. This treatment generalizes the classical fractional-order proportional derivative and a fractional-order calculus with derivatives depending on another function.

However, we must emphasize that there is no universally accepted set of axioms regarding the properties of the derivative and integral in the concept of“fractional calculus”. To avoid disputes beyond the scope of this paper, we will present a generalization that allows for interpolation between fractional and standard operators. In this sense, our results apply to both types of calculus.

Define the following non-fractional differential-type operator of order $\rho \in (0,1)$ and parameter $\mu \ge 0,$

$${\partial }_{\psi ,\mu }^{\rho ,\ell }f:={\ell }_2(\rho ,t)f+{\ell }_1(\rho ,t)\Delta f,\mathrm{where}\Delta :=\mu +{\displaystyle \frac{1}{{\psi }^{\prime }}}{\displaystyle \frac{d}{dt}},$$

where f is a differentiable function of $t\in \mathbb{R}$. In other words,

$${\partial }_{\psi ,\mu }^{\rho ,\ell }f=\left[{\ell }_2(\rho ,t)+\mu {\ell }_1(\rho ,t)\right]f+{\ell }_1(\rho ,t){\displaystyle \frac{1}{{\psi }^{\prime }}}{\displaystyle \frac{d}{dt}}f.$$

This operator relates to the large and expanding theory of conformable derivatives. Clearly, we can interpolate between $Id\left(f\right)$ and $\tilde{\delta }\left(f\right)$:

$$\underset{\rho \to 0}{lim}{\partial }_{p,\psi }^{\rho ,\ell }f=f,\mathrm{and}\underset{\rho \to 1}{lim}{\partial }_{\psi }^{\rho ,\ell }f=\tilde{\delta }f.$$

Because these operators range from fractional order to identity, they cannot fulfill the complete set of requirements for classically conceived fractional-order operators. Instead, they allow us to study issues for which the nature of the operator is determined by parameters.

The inverse operator of ${\partial }_{\psi ,\mu }^{\rho ,\ell }$ can be defined as follows:

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

Formally, we have

$$\underset{\rho \to 1}{lim}{\Im }_{\psi ,0}^{\rho ,\ell }f={\int }_a^{t}f\left(s\right){\psi }^{\prime }\left(s\right)ds.$$

However, we should address the assumptions under which these operations are well defined. This depends heavily on the function spaces in which we study invertibility. We will examine the invertibility conditions for the differential and integral operators under consideration in important function spaces. Clearly, the right and left inverse operators differ. For absolutely continuous and integrable function spaces, we extend known results to our broader class of operators.

Proposition 1.

(cf. [40] for weighted spaces, [41] for $C^1$ space) Let $\rho \in (0,1)$. If the assumptions (1) and (2) are satisfied, then

1. 

${\displaystyle {\partial }_{\psi ,\mu }^{\rho ,\ell }{\Im }_{\psi ,\mu }^{\rho ,\ell }f=fa.e.on[a,b],}$ for any $f\in L_{\phi }[a,b]$.

2. 

${\displaystyle {\Im }_{\psi ,\mu }^{\rho ,\ell }{\partial }_{\psi ,\mu }^{\rho ,\ell }f\left(t\right)=f\left(t\right)-e^{{\displaystyle -\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}f\left(a\right),}$$t\in [a,b]$, for any $f\in AC[a,b].$

Proof. 

The first part involves direct calculations. We will then proceed to prove the second part. Since $f\in AC[a,b]$, the function f has an integrable derivative that is defined almost everywhere on $[a,b]$. Hence,

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

Therefore, integration by parts provides

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

as required. Also, if the assumptions (1) and (2) hold, then for any $\rho \in (0,1),\sigma >-1$, and $t\in [a,b]$,

$$\begin{array}{ccc}\hfill & & {\partial }_{\psi ,\mu }^{\rho ,\ell }\left\{{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{-\sigma }}}\right\}\hfill \\ & & {\displaystyle =\left(\left[{\ell }_2(\rho ,t)+\mu {\ell }_1(\rho ,t)\right]+{\ell }_1(\rho ,t)\frac{1}{{\psi }^{\prime }}\frac{d}{dt}\right)\left\{\frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{-\sigma }}\right\}}\hfill \\ & & =\left[{\ell }_2(\rho ,t)+\mu {\ell }_1(\rho ,t)\right]\left\{{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{-\sigma }}}\right\}\hfill \\ & & +{\displaystyle \frac{1}{{\omega }^{\prime }}}{\displaystyle \frac{d}{dt}}\left\{{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{-\sigma }}}\right\}\hfill \\ & & =\left[{\ell }_2(\rho ,t)+\mu {\ell }_1(\rho ,t)\right]\left\{{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{-\sigma }}}\right\}\hfill \\ & & +{\displaystyle \frac{e^{{\displaystyle \left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}}{{\left(\omega \left(t\right)\right)}^{-\sigma }}}\left\{{\displaystyle \frac{\sigma }{\omega \left(t\right)}}-\left[{\ell }_2(\rho ,t)+\mu {\ell }_1(\rho ,t)\right]\right\}\hfill \\ & & =\sigma {\displaystyle \frac{e^{{\displaystyle \left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}}{{\left(\omega \left(t\right)\right)}^{1-\sigma }}}.\hfill \end{array}$$

(5)

In particular,

$$\begin{array}{ccc}\hfill & & {\partial }_{\psi ,\mu }^{\rho ,\ell }\left\{e^{-{\int }_0^{\psi \left(t\right)}\left[\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right]du}\right\}\hfill \\ & & =e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\left[\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right]du}}\left[{\ell }_2(\rho ,t)-{\ell }_2(\rho ,t)\right]=0.\hfill \end{array}$$

(6)

Now, for any $f\in L_1[a,b]$, we have

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

Therefore,

$${\Im }_{\psi ,\mu }^{\rho ,\ell }{\Im }_{\psi ,\mu }^{\rho ,\ell }f\left(t\right)={\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}\left(\omega \left(t\right)-\omega \left(s\right)\right){\displaystyle \frac{{\psi }^{\prime }\left(s\right)f\left(s\right)ds}{{\ell }_1(\rho ,s)}},$$

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

Similarly, we have (in view of the calculation ${\omega }^{\prime }(·)={\psi }^{\prime }(·)/{\ell }_1(\rho ,·)$)

$$\begin{array}{ccc}& & {\Im }_{\psi ,\mu }^{\rho ,\ell }{\Im }_{\psi ,\mu }^{\rho ,\ell }{\Im }_{\psi ,\mu }^{\rho ,\ell }f\left(t\right)\hfill \\ & & ={\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(\theta \right)}^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}\left({\displaystyle {\int }_{\theta }^t(\omega \left(s\right)-\omega \left(\theta \right))\frac{{\psi }^{\prime }\left(s\right)ds}{{\ell }_1(\rho ,s)}}\right){\displaystyle \frac{{\psi }^{\prime }\left(\theta \right)f\left(\theta \right)d\theta }{{\ell }_1(\rho ,\theta )}}\hfill \\ & =& {\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{(\omega \left(t\right)-\omega \left(s\right))}^2{\omega }^{\prime }\left(s\right)f\left(s\right)ds.\hfill \end{array}$$

This can be extended arbitrarily to obtain

$$\begin{array}{ccc}\hfill K_{\rho ,\psi }^{n,\ell ,\mu }f\left(t\right)& :=& \stackrel{n-\mathrm{times}}{\overbrace{{\Im }_{\psi ,\mu }^{\rho ,\ell }{\Im }_{\psi ,\mu }^{\rho ,\ell }\dots ..{\Im }_{\psi ,\mu }^{\rho ,\ell }}}f\left(t\right)\hfill \\ & =& {\displaystyle \frac{1}{\Gamma \left(n\right)}}{\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{(\omega \left(t\right)-\omega \left(s\right))}^{n-1}{\displaystyle \frac{{\psi }^{\prime }\left(s\right)f\left(s\right)ds}{{\ell }_1(\rho ,s)}}\hfill \\ & =& {\displaystyle \frac{1}{\Gamma \left(n\right)}}{\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{(\omega \left(t\right)-\omega \left(s\right))}^{n-1}{\omega }^{\prime }\left(s\right)f\left(s\right)ds.\hfill \end{array}$$

□

The following general combination (functional combination) of fractional and non-fractional integral operators can be presented.

Definition 1.

Let $\psi \in C^1[a,b]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Let $\alpha >0$ and assume that the assumptions (1) and (2) are satisfied. Then, for any $f\in L_1[a,b]$, we define the combination between fractional and non-fractional integral operators by

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

where $\omega (·):={\int }_a^{(·)}{\displaystyle \frac{{\psi }^{\prime }\left(\theta \right)d\theta }{{\ell }_1(\rho ,\theta )}}$. For completeness, we define $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(a\right):=0.$

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

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

Furthermore, when $\rho =1$,

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

Evidently,

$$\begin{array}{ccc}\hfill & & K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t\right)\hfill \\ & =& e^{{\displaystyle \left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{K}_{\rho ,\psi }^{\alpha ,0,0}\left(e^{{\displaystyle \left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}f\left(t\right)\right)\hfill \\ & =& e^{{\displaystyle \left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)\right)du\right]}}{K}_{\rho ,\psi }^{\beta ,0,\mu }\left(e^{{\displaystyle \left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)\right)du\right]}}f\left(t\right)\right).\hfill \end{array}$$

(7)

Remark 2.

Let us remark that, for any $f\in C[a,b]$, we have

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

Thus, $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(a\right)=0$. Therefore, our definition of $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(a\right)=0$, for any f in $L_1[a,b]$, has important consequences.

The condition that $\psi \left(a\right)=0$ in the above Definition 1 serves several important purposes. First, it is a normalization that ensures consistency in the definition of the operator, particularly with regard to its behavior at the lower limit, $t=a$, and in the construction of the auxiliary function, $\omega$. It also simplifies the handling of the inner integral involving the inverse of the function, or the inverse function. Without this condition, additional constraints or adjustments might be necessary to ensure that the operator has the desired properties.

2. Basic Properties of $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }$

As expected for differential operators, the typical properties can be demonstrated for the operators under consideration. First, consider the semi-group property.

Lemma 1.

(Semi-group property) For any $\alpha ,\beta >0$, $\mu \in {\mathbb{R}}^{+}$ and positive increasing function $\psi \in C^1[a,b]$ such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0,$ we have

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

holds true for any $f\in L_1[a,b]$ almost everywhere on $[a,b]$.

Proof. 

Since $f\in L_1[a,b]$, interchanging the order of integration yields

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

Therefore, the substitution $\omega \left(s\right)=\omega \left(\theta \right)+u\left(\omega \right(t)-\omega (\theta \left)\right)$ with a little experimentation yields

$$\begin{array}{ccc}& & K_{\rho ,\psi }^{\alpha ,\ell ,\mu }{K}_{\rho ,\psi }^{\beta ,\ell ,\mu }f\left(t\right)\hfill \\ & & ={\int }_a^t{\displaystyle \frac{e^{\left[-{\int }_{\psi \left(\theta \right)}^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}}\left({\int }_0^1{u}^{\beta -1}{(1-u)}^{\alpha -1}du\right){(\omega \left(t\right)-\psi \left(\theta \right))}^{\alpha +\beta -1}f\left(\theta \right){\omega }^{\prime }\left(\theta \right)d\theta \hfill \\ & & ={\int }_a^t{\displaystyle \frac{e^{\left[-{\int }_{\psi \left(\theta \right)}^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\Gamma (\alpha +\beta )}}{(\omega \left(t\right)-\psi \left(\theta \right))}^{\alpha +\beta -1}f\left(\theta \right){\omega }^{\prime }\left(\theta \right)d\theta =K_{\rho ,\psi }^{\alpha +\beta ,\ell ,\mu }f\left(t\right).\hfill \end{array}$$

Similarly, we can show that $K_{\rho ,\psi }^{\beta ,\ell ,\mu }{K}_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t\right)=K_{\rho ,\psi }^{\alpha +\beta ,\ell ,\mu }f\left(t\right)$ almost everywhere on $[a,b]$ as required.  □

Lemma 2.

For any $\alpha \in (0,1)$, $\mu \in {\mathbb{R}}^{+}$, and positive increasing function $\psi \in C^1[a,b]$, such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0,$ we have for any $\gamma >-1$,

$$K_{\rho ,\psi }^{\alpha ,\ell ,\mu }\left[{\displaystyle \frac{e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{-\gamma }}}\right]={\displaystyle \frac{\Gamma (1+\gamma )}{\Gamma (1+\gamma +\alpha )}}\left[e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\right]{\left(\omega \left(t\right)\right)}^{\gamma +\alpha }.$$

Proof. 

By the definition of $K_{\rho ,\psi }^{\alpha ,\ell ,\mu },$

$$\begin{array}{ccc}& & K_{\rho ,\psi }^{\alpha ,\ell ,\mu }\left[{\displaystyle \frac{e^{{\displaystyle -\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}}{{\left(\omega \left(t\right)\right)}^{-\gamma }}}\right]\hfill \\ & =& {\int }_a^t{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\Gamma \left(\alpha \right)}}{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha -1}{\left(\omega \left(s\right)\right)}^{\gamma }{\omega }^{\prime }\left(s\right)ds\hfill \\ & =& {\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\Gamma \left(\alpha \right)}}{\int }_0^1{\left(1-u\right)}^{\alpha -1}{u}^{\gamma }{\left(\omega \left(t\right)\right)}^{\gamma +\alpha }du\hfill \\ & =& {\displaystyle \frac{\Gamma (1+\gamma )}{\Gamma (1+\gamma +\alpha )}}\left[e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\right]{\left(\omega \left(t\right)\right)}^{\gamma +\alpha }.\hfill \end{array}$$

where $u=\omega \left(s\right)/\omega \left(t\right)$. Hence, the result follows.  □

Lemma 3.

For any $\alpha \in (0,1)$, $\mu \in {\mathbb{R}}^{+}$, and positive increasing function $\psi \in C^1[a,b]$ such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0,$ we have

$$K_{\rho ,\psi }^{\alpha ,\ell ,\mu }:AC[a,b]\to AC[a,b].$$

Proof. 

Let $f\in AC[a,b]$. Define the absolutely continuous function $F:[a,b]\to \mathbb{R}$ by

$$F\left(t\right):=f\left(t\right)e^{{\displaystyle \left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}},t\in [a,b].$$

We notice that, for almost every $t\in [a,b]$, we have

$$\begin{array}{ccc}\hfill F^{\prime }\left(t\right)& =& e^{\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\left[f^{\prime }\left(t\right)+f\left(t\right)\left(\left(\ell \left(\rho ,t\right)+\mu \right){\psi }^{\prime }\left(t\right)\right)\right]\hfill \\ & =& e^{\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\omega }^{\prime }\left(t\right){\partial }_{\psi ,\mu }^{\rho ,\ell }f\left(t\right).\hfill \end{array}$$

Since F is absolutely continuous, by the fundamental theorem of calculus, we get

$$F\left(t\right)=F\left(a\right)+{\int }_a^t{F}^{\prime }\left(s\right)ds=F\left(a\right)+{\int }_a^t{e}^{{\displaystyle \left[{\int }_0^{\psi \left(s\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\omega }^{\prime }\left(s\right){\partial }_{\psi ,\mu }^{\rho ,\ell }f\left(s\right)ds.$$

Consequently, using integration by parts yields

$$\begin{array}{ccc}\hfill K_{\rho ,\psi }^{\alpha ,0,0}F\left(t\right)& =& K_{\rho ,\psi }^{\alpha ,0,0}\left[F\left(a\right)+{\int }_a^t{F}^{\prime }\left(s\right)ds\right]\hfill \\ & =& {\displaystyle \frac{F\left(a\right){\left(\omega \left(t\right)\right)}^{\alpha }}{\Gamma (1+\alpha )}}+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right)\left[{\int }_a^s{F}^{\prime }\left(\theta \right)d\theta \right]ds\hfill \\ & =& {\displaystyle \frac{F\left(a\right){\left(\omega \left(t\right)\right)}^{\alpha }}{\Gamma (1+\alpha )}}+{\displaystyle \frac{1}{\Gamma (1+\alpha )}}{\int }_a^t{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha }{\omega }^{\prime }\left(s\right)F^{\prime }\left(s\right)ds\hfill \\ & =& {\displaystyle \frac{F\left(a\right){\left(\omega \left(t\right)\right)}^{\alpha }}{\Gamma (1+\alpha )}}\hfill \\ & & +{\displaystyle \frac{1}{\Gamma (1+\alpha )}}{\int }_a^t{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha }{e}^{\left[{\int }_0^{\psi \left(s\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\omega }^{\prime }\left(s\right){\partial }_{\psi ,\mu }^{\rho ,\ell }f\left(s\right)ds.\hfill \end{array}$$

Because $F\left(a\right)=f\left(a\right)$, we arrive at

$$\begin{array}{ccc}\hfill K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t\right)& =& e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{K}_{\rho ,\psi }^{\alpha ,0,0}F\left(t\right)\hfill \\ & =& {\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{\alpha }{e}^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\Gamma (1+\alpha )}}\hfill \\ & +& {\displaystyle \frac{1}{\Gamma (1+\alpha )}}{\int }_a^t{e}^{\left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{(\omega \left(t\right)-\omega \left(s\right))}^{\alpha }{\omega }^{\prime }\left(s\right){\partial }_{\psi ,\mu }^{\rho ,\ell }f\left(s\right)ds.\hfill \end{array}$$

Thus, for any $f\in AC[a,b]$, in view of the semi-group property (cf. Lemma 1), we have

$$K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t\right)={\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{\alpha }{e}^{{\displaystyle \left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}}{\Gamma (1+\alpha )}}+K_{\rho ,\psi }^{1,\ell ,\mu }{K}_{\rho ,\psi }^{\alpha ,\ell ,\mu }{\partial }_{\psi ,\mu }^{\rho ,\ell }f\left(t\right).$$

(8)

Since both terms on the right-hand side of (8) are absolutely continuous on $[a,b]$, the left-hand side of (8) is also absolutely continuous on $[a,b]$. The proof is complete.  □

Now, we will turn our attention to the role of the function parameters of the considered family of operators in terms of their properties and the Orlicz space from which they are chosen. The following theorem provides a useful characterization of integral-type operators $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }$ and complements similar results in ([Lemma 1] in [42]) and ([Theorem 2] in [39]) (see also [33,43]).

Theorem 1.

Let $\alpha \in (0,1]$. For any Young function φ with its complementary Young function $\tilde{\phi }$ satisfying

$$\begin{array}{c}\hfill {\int }_0^t\tilde{\phi }\left(s^{\alpha -1}\right)ds<\infty ,\end{array}$$

(9)

for $t>0,$ the operator $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }$ maps the space $L_{\phi }^{\ell }[a,b]:=\left\{f\in L_1[a,b]:{\left({\ell }_1(\rho ,·)\right)}^{-1}f(·)\in L_{\phi }[a,b]\right\}$ into $C[a,b]$. In particular, $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }:C^{\ell }[a,b]:=\left\{f\in L_1[a,b]:{\left({\ell }_1(\rho ,·)\right)}^{-1}f(·)\in C[a,b]\right\}\to C[a,b]$.

Proof. 

Let $a\le t_1\le t_2\le b$ and $f\in L_{\phi }^{\ell }[a,b]$. Then,

$$\begin{array}{ccc}& & K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t_2\right)-K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t_1\right)\hfill \\ & =& e^{\left[{\int }_0^{\psi \left(t_2\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{K}_{\rho ,\psi }^{\alpha ,0,0}\left[e^{\left[{\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f(·)\right]\left(t_2\right)\hfill \\ & & -e^{-\left[{\int }_0^{\psi \left(t_1\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{K}_{\rho ,\psi }^{\alpha ,0,0}\left[e^{\left[{\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f(·)\right]\left(t_1\right)\hfill \\ & =& e^{-\left[{\int }_0^{\psi \left(t_2\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\left(K_{\rho ,\psi }^{\alpha ,0,0}\left[e^{\left[{\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f(·)\right]\left(t_2\right)\right.\hfill \\ & & -\left.K_{\rho ,\psi }^{\alpha ,0,0}\left[e^{\left[{\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f(·)\right]\left(t_1\right)\right)\hfill \\ & & +\left(e^{-\left[{\int }_0^{\psi \left(t_2\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}-e^{-\left[{\int }_0^{\psi \left(t_1\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\right)\hfill \\ & & \times K_{\rho ,\psi }^{\alpha ,0,0}\left[e^{\left[{\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f(·)\right]\left(t_1\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}& & \left|K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t_2\right)\right.-\left.K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t_1\right)\right|\le \left|K_{\rho ,\psi }^{\alpha ,0,0}\left(e^{\left[{\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f(·)\right)\left(t_2\right)\right.\hfill \\ & & -\left.K_{\rho ,\psi }^{\alpha ,0,0}\left(e^{\left[{\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f(·)\right)\left(t_1\right)\right|\hfill \\ & & +\left|e^{-\left[{\int }_0^{\psi \left(t_2\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}-e^{-\left[{\int }_0^{\psi \left(t_1\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\right|\hfill \\ & & \times \left|K_{\rho ,\psi }^{\alpha ,0,0}\left(e^{\left[{\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f(·)\right)\left(t_1\right)\right|\hfill \\ & & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\left({\int }_a^{t_1}{e}^{\left[{\int }_0^{\psi \left(s\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}|{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}-{(\omega \left(t_1\right)-\omega \left(s\right))}^{\alpha -1}||{\omega }^{\prime }\left(s\right)||f\left(s\right)|ds\right.\hfill \\ & & +\left.{\int }_{t_1}^{t_2}{e}^{\left[{\int }_0^{\psi \left(s\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}{\omega }^{\prime }\left(s\right)\left|f\left(s\right)\right|ds\right)\hfill \\ & & +\left|e^{-\left[{\int }_0^{\psi \left(t_2\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}-e^{-\left[{\int }_0^{\psi \left(t_1\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\right|\hfill \\ & & \times \left|K_{\rho ,\psi }^{\alpha ,0,0}\left(e^{\left[{\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f(·)\right)\left(t_1\right)\right|\hfill \\ & & ={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^b\left|h_1\left(s\right)+h_2\left(s\right)\right|\left|f\left(s\right)\right|ds+\mathbf{J}(t_1,t_2),\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill \mathbf{J}(t_1,t_2)& :=& \left|e^{-\left[{\int }_0^{\psi \left(t_2\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}-e^{-\left[{\int }_0^{\psi \left(t_1\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\right|\hfill \\ & & \times \left|K_{\rho ,\psi }^{\alpha ,0,0}\left(e^{\left[{\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f(·)\right)\left(t_1\right)\right|,\hfill \end{array}$$

$$h_1\left(s\right):={\ell }_1(\rho ,s)\left({(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}-{(\omega \left(t_1\right)-\omega \left(s\right))}^{\alpha -1}\right){\omega }^{\prime }\left(s\right)e^{\left[{\int }_0^{\psi \left(s\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]},$$

for $s\in [a,t_1]$, and $h_1\left(s\right):=0\mathrm{for}s\in [t_1,b]$. Moreover,

$$h_2\left(s\right):={\ell }_1(\rho ,s)\left\{\begin{array}{cc}{\left(\omega \left(t_2\right)-\omega \left(s\right)\right)}^{\alpha -1}{\omega }^{\prime }\left(s\right)e^{{\displaystyle \left[{\int }_0^{\psi \left(s\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}\hfill & s\in [t_1,t_2],\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

We claim that $h_i\in L_{\tilde{\phi }}[a,b]$, $(i=1,2)$. Once this claim is established, in view of the generalized Hölder inequality in Orlicz spaces, we conclude that

$$\begin{array}{ccc}\hfill \left|K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t_2\right)-K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t_1\right)\right|& \le & {\displaystyle \frac{2\left[{∥h_1∥}_{\tilde{\phi }}+{∥h_2∥}_{\tilde{\phi }}\right]}{\Gamma \left(\alpha \right)}}{∥{\displaystyle \frac{f(·)}{{\ell }_1(\rho ,·)}}∥}_{\phi }+\mathbf{J}(t_1,t_2).\hfill \end{array}$$

(10)

It remains to prove our claim by showing that $h_i\in L_{\tilde{\phi }}[a,b],i=1,2$. For this, fix $k>0$ and define

$$▵:=e^{{\displaystyle \left[{\int }_0^{∥\psi ∥}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}},∥{\psi }^{\prime }∥:=\underset{t\in [a,b]}{max}\left|{\psi }^{\prime }\left(t\right)\right|.$$

An appropriate substitution, using the properties of $\tilde{\phi }$, i.e., $\tilde{\phi }\left(\lambda u\right)\le \lambda \tilde{\phi }\left(u\right),\lambda \in (0,1]$ and $\tilde{\phi }(u-v)\le \tilde{\phi }\left(u\right)-\tilde{\phi }\left(v\right),v\le u$, lead to the following estimation:

$$\begin{array}{ccc}& & {\int }_a^b\tilde{\phi }\left({\displaystyle \frac{|h_1\left(s\right)|{\ell }_1(\rho ,s)}{▵∥{\psi }^{\prime }∥k}}\right)ds\hfill \\ & =& {\int }_a^{t_1}\tilde{\phi }\left(e^{\left[{\int }_0^{\psi \left(s\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\displaystyle \frac{\left|{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}-{(\omega \left(t_1\right)-\omega \left(s\right))}^{\alpha -1}\right|}{▵∥{\psi }^{\prime }∥k}}{\omega }^{\prime }\left(s\right){\ell }_1(\rho ,s)\right)ds\hfill \\ & \le & {\int }_a^{t_1}\tilde{\phi }\left({\displaystyle \frac{\left|{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}-{(\omega \left(t_1\right)-\omega \left(s\right))}^{\alpha -1}\right|}{k}}\right){\displaystyle \frac{{\omega }^{\prime }\left(s\right){\ell }_1(\rho ,s)}{∥{\psi }^{\prime }∥}}ds\hfill \\ & \le & {\displaystyle \frac{∥{\ell }_1(\rho ,·)∥}{∥|{\psi }^{\prime }|∥}}{\int }_a^{t_1}\left[\tilde{\phi }\left({\displaystyle \frac{{(\omega \left(t_1\right)-\omega \left(s\right))}^{\alpha -1}}{k}}\right)-\tilde{\phi }\left({\displaystyle \frac{{(\omega \left(t_2\right)-\omega \left(s\right))}^{\alpha -1}}{k}}\right)\right]{\omega }^{\prime }\left(s\right)ds\hfill \\ & \le & {\displaystyle \frac{∥{\ell }_1(\rho ,·)∥k^{\frac{1}{\alpha -1}}}{∥|{\psi }^{\prime }|∥}}\left[{\int }_0^{k^{{\displaystyle \frac{1}{1-\alpha }}}(\omega \left(t_1\right)-\omega \left(a\right))}\tilde{\phi }\left(s^{\alpha -1}\right)ds-{\int }_{k^{{\displaystyle \frac{1}{1-\alpha }}}(\omega \left(t_2\right)-\omega \left(t_1\right))}^{k^{{\displaystyle \frac{1}{1-\alpha }}}(\omega \left(t_2\right)-\omega \left(a\right))}\tilde{\phi }\left(s^{\alpha -1}\right)ds\right]\hfill \\ & =& {\displaystyle \frac{∥{\ell }_1(\rho ,·)∥k^{\frac{1}{\alpha -1}}}{∥{\psi }^{\prime }∥}}\left[{\int }_0^{k^{{\displaystyle \frac{1}{1-\alpha }}}(\omega \left(t_1\right)-\omega \left(a\right))}\tilde{\phi }\left(s^{\alpha -1}\right)ds-{\int }_0^{k^{{\displaystyle \frac{1}{1-\alpha }}}(\omega \left(t_2\right)-\omega \left(a\right))}\tilde{\phi }\left(s^{\alpha -1}\right)ds\right.\hfill \\ & & +\left.{\int }_0^{k^{{\displaystyle \frac{1}{1-\alpha }}}(\omega \left(t_2\right)-\omega \left(t_1\right))}\tilde{\phi }\left(s^{\alpha -1}\right)ds\right]\le {\displaystyle \frac{∥{\ell }_1(\rho ,·)∥k^{\frac{1}{\alpha -1}}}{∥{\psi }^{\prime }∥}}{\int }_0^{k^{{\displaystyle \frac{1}{1-\alpha }}}(\omega \left(t_2\right)-\omega \left(t_1\right))}\tilde{\phi }\left(s^{\alpha -1}\right)ds.\hfill \end{array}$$

From this, we can conclude, by the definition of the norm in Orlicz spaces, that $h_1\in L_{\tilde{\phi }}[a,b]$ with ${∥h_1∥}_{\tilde{\phi }}\le ▵∥{\psi }^{\prime }∥K$, with $K:={\tilde{\Phi }}_{\alpha }\left(\right|\omega \left(t_2\right)-\omega \left(t_1\right)\left|\right)$, where ${\tilde{\Phi }}_{\alpha }(·)$ is defined as in (4). Arguing now similarly as above, we can show that

$$h_2\in L_{\tilde{\phi }}[a,b],\mathrm{and}{∥h_2∥}_{\tilde{\phi }}\le ▵∥{\psi }^{\prime }∥{\tilde{\Phi }}_{\alpha }\left(|\omega \left(t_2\right)-\omega \left(t_1\right)|\right).$$

Therefore, Equation (10) reads as follows:

$$\left|K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t_2\right)-K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t_1\right)\right|\le {\displaystyle \frac{4▵∥{\psi }^{\prime }∥{\tilde{\Phi }}_{\alpha }\left(|\omega \left(t_2\right)-\omega \left(t_1\right)|\right)}{\Gamma \left(\alpha \right)}}{∥{\displaystyle \frac{f(·)}{{\ell }_1(\rho ,·)}}∥}_{\phi }+\mathbf{J}(t_1,t_2).$$

(11)

According to our definition,

$$\begin{array}{ccc}& & e^{{\displaystyle \left[-{\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{K}_{\rho ,\psi }^{\alpha ,0,0}\left(e^{{\displaystyle \left[{\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}f(·)\right)\left(a\right)\hfill \\ & & =K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(a\right)=0,\hfill \end{array}$$

which follows that

$$\mathbf{J}(a,t)=0,\left|K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t\right)\right|\le {\displaystyle \frac{4▵∥{\psi }^{\prime }∥{\tilde{\Phi }}_{\alpha }\left(∥\omega ∥\right)}{\Gamma \left(\alpha \right)}}{∥{\displaystyle \frac{f(·)}{{\ell }_1(\rho ,·)}}∥}_{\phi }.$$

Hence,

$$\begin{array}{ccc}& & \left|K_{\rho ,\psi }^{\alpha ,0,0}\left(e^{{\displaystyle \left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}f\left(t\right)\right)\right|\hfill \\ & \le & {\displaystyle \frac{4e^{{\displaystyle \left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}▵∥{\psi }^{\prime }∥{\tilde{\Phi }}_{\alpha }\left(∥\omega ∥\right)}{\Gamma \left(\alpha \right)}}{∥{\displaystyle \frac{f(·)}{{\ell }_1(\rho ,·)}}∥}_{\phi }.\hfill \end{array}$$

Consequently,

$$\begin{array}{ccc}\hfill \mathbf{J}(t_1,t_2)& \le & \left|e^{{\displaystyle -\left[{\int }_0^{\psi \left(t_2\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}-e^{{\displaystyle -\left[{\int }_0^{\psi \left(t_1\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}\right|\hfill \\ & & \times \mathbf{K}{e}^{{\displaystyle \left[{\int }_0^{∥\psi ∥}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}},\hfill \end{array}$$

where the constant K is defined by

$$\mathbf{K}:={\displaystyle \frac{4▵∥{\psi }^{\prime }∥{\tilde{\Phi }}_{\alpha }\left(∥\omega ∥\right)}{\Gamma \left(\alpha \right)}}{∥{\displaystyle \frac{f(·)}{{\ell }_1(\rho ,·)}}∥}_{\phi }.$$

All in all, we arrive at

$$\begin{array}{ccc}\hfill & & \left|K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t_2\right)-K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t_1\right)\right|\le {\displaystyle \frac{4▵∥{\psi }^{\prime }∥{\tilde{\Phi }}_{\alpha }\left(|\omega \left(t_2\right)-\omega \left(t_1\right)|\right)}{\Gamma \left(\alpha \right)}}{∥{\displaystyle \frac{f(·)}{{\ell }_1(\rho ,·)}}∥}_{\phi }\hfill \\ & & +\left|e^{-\left[{\int }_0^{\psi \left(t_2\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}-e^{-\left[{\int }_0^{\psi \left(t_1\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\right|\hfill \\ & & \times \mathbf{K}{e}^{{\displaystyle \left[{\int }_0^{∥\psi ∥}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}.\hfill \end{array}$$

Therefore, $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }:L_{\phi }^{\ell }[a,b]\to C[a,b]$. Also,

$$∥K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f∥\le \left[1+\mathbf{K}\right]e^{{\displaystyle \left[{\int }_0^{∥\psi ∥}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}.$$

For this reason, the case is straightforward and the theorem is proven.  □

Lemma 4.

Let $\alpha \in (0,1)$, $\beta \in [0,1)$, $\mu \in {\mathbb{R}}^{+}$, and a positive increasing function $\psi \in C^1[a,b]$ such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0,$. Then, the operator $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }:C[a,b]\to C[a,b]$ is injective.

Proof. 

Let $f,g\in C[a,b]$ be such that $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t\right)=K_{\rho ,\psi }^{\alpha ,\ell ,\mu }g\left(t\right)$, for $t\in [a,b]$. Then, in view of the semi-group property, for almost all t, we have

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

Thus, due to the continuity of the function $f-g$, it follows that $f-g=0$ (so $f=g$) for every $t\in [a,b]$.  □

We can now present one of the main results regarding how the operator acts on generalized Hölder spaces. This result demonstrates the natural integral nature of the operator because, on this class of spaces, it increases the order of the function by the order $\alpha$ of this operator. Therefore, it is natural to consider the inverse differential operator of order $\alpha$ on these spaces. This operator will be shown in Theorem 3.

Theorem 2.

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

$${\mathbb{H}}^{\lambda ,0}[a,b]:=\left\{f\in {\mathbb{H}}^{\lambda }[a,b]:f\left(a\right)=0\right\}.$$

Then

$$K_{\rho ,\psi }^{\alpha ,\ell ,\mu }:{\mathbb{H}}^{\lambda ,0}[a,b]\to {\mathbb{H}}^{\alpha +\lambda ,0}[a,b].$$

Proof. 

Let $h>0,t,t+h\in [a,b]$ and $f\in {\mathbb{H}}^{\lambda ,0}[a,b]$. Define the following positive, increasing function:

$$\chi (·):={\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du={\int }_a^{(·)}\left[{\ell }_2\left(\rho ,u\right)+\mu {\ell }_1\left(\rho ,u\right)\right]{\omega }^{\prime }\left(u\right)du.$$

Hence, by the second mean value theorem for integrals, we obtain

$$\begin{array}{ccc}\hfill \chi (t+h)-\chi (t-s)& =& {\int }_{t-s}^{t+h}\left[{\ell }_2\left(\rho ,u\right)+\mu {\ell }_1\left(\rho ,u\right)\right]{\omega }^{\prime }\left(u\right)du=Z_1\left(\rho \right){\int }_{t-s}^{t+h}{\omega }^{\prime }\left(u\right)du\hfill \\ & =& Z_1\left(\rho \right)(\omega (t+h)-\omega (t-s)),\hfill \end{array}$$

where

$$Z_1\left(\rho \right):={\ell }_2\left(\rho ,{\varsigma }_1\right)+\mu {\ell }_1\left(\rho ,{\varsigma }_1\right),$$

for some ${\varsigma }_1\in [t-s,t+h]$, $s\in (-h,t-a)$. Also,

$$\begin{array}{ccc}\hfill \chi \left(t\right)-\chi (t-s)& =& {\int }_t^{t+h}\left[{\ell }_2\left(\rho ,u\right)+\mu {\ell }_1\left(\rho ,u\right)\right]{\omega }^{\prime }\left(u\right)du=Z_2\left(\rho \right){\int }_{t-s}^t{\omega }^{\prime }\left(u\right)du\hfill \\ & =& Z_2\left(\rho \right)(\omega \left(t\right)-\omega (t-s)),\hfill \end{array}$$

where

$$Z_2\left(\rho \right):={\ell }_2\left(\rho ,{\varsigma }_2\right)+\mu {\ell }_1\left(\rho ,{\varsigma }_2\right),$$

for some ${\varsigma }_2\in [t-s,t]$, $s\in (a,t-a)$. Now, trivial calculations show that

$$K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f(t+h)-K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t\right)=A+B+C,$$

where

$$\begin{array}{ccc}\hfill \left|A\right|& \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }{(t-a)}^{\lambda }}{\Gamma \left(\alpha \right)}}\left|{\int }_{-h}^{t-a}{\displaystyle \frac{e^{-\left(\chi (t+h)-\chi (t-s)\right)}}{{(\omega (t+h)-\omega (t-s))}^{1-\alpha }}}{\omega }^{\prime }(t-s)ds\right.\hfill \\ & & \left.-{\int }_0^{t-a}{\displaystyle \frac{e^{-\left(\chi \left(t\right)-\chi (t-s)\right)}}{{(\omega \left(t\right)-\omega (t-s))}^{1-\alpha }}}{\omega }^{\prime }(t-s)ds\right|,\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill \left|B\right|& \le & {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\left|{\int }_{-h}^0{\displaystyle \frac{e^{-\left(\chi (t+h)-\chi (t-s)\right)}}{{(\omega (t+h)-\omega (t-s))}^{1-\alpha }}}{\omega }^{\prime }(t-s)\left|f(t-s)-f\left(t\right)\right|ds\right|\hfill \\ & \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }}{\Gamma \left(\alpha \right)}}\left|{\int }_0^h{(\omega (t+h)-\omega (t+s))}^{\alpha -1}{\omega }^{\prime }(t+s)s^{\lambda }ds\right|\hfill \\ & \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }{h}^{\lambda }}{\Gamma \left(\alpha \right)}}\left|{\int }_0^h{(\omega (t+h)-\omega (t+s))}^{\alpha -1}{\omega }^{\prime }(t+s)ds\right|\hfill \\ & =& {\displaystyle \frac{{\left[f\right]}_{\lambda }{h}^{\lambda }}{\Gamma (1+\alpha )}}{(\omega (t+h)-\omega \left(t\right))}^{\alpha }\le {\displaystyle \frac{{\left[f\right]}_{\lambda }{∥{\omega }^{\prime }∥}^{\alpha }}{\Gamma (1+\alpha )}}{h}^{\lambda +\alpha },\hfill \end{array}$$

$$\left|C\right|\le {\displaystyle \frac{{\left[f\right]}_{\lambda }}{\Gamma \left(\alpha \right)}}\left|{\int }_0^{t-a}\left[{\displaystyle \frac{e^{-\left(\chi (t+h)-\chi (t-s)\right)}}{{(\omega (t+h)-\omega (t-s))}^{1-\alpha }}}-{\displaystyle \frac{e^{-\left(\chi \left(t\right)-\chi (t-s)\right)}}{{(\omega \left(t\right)-\omega (t-s))}^{1-\alpha }}}\right|{\omega }^{\prime }(t-s)s^{\lambda }ds\right).$$

After the substitutions $s\to \omega (t+h)-\omega (t-s)$ and $s\to \omega \left(t\right)-\omega (t-s)$, it follows that

$$\begin{array}{ccc}\hfill \left|A\right|& \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }{(t-a)}^{\lambda }}{\Gamma \left(\alpha \right)}}\left|{\int }_0^{\omega (t+h)}{e}^{-Z_1\left(\rho \right)s}{s}^{\alpha -1}ds-{\int }_0^{\omega \left(t\right)}{e}^{-Z_2\left(\rho \right)s}{s}^{\alpha -1}ds\right|\hfill \\ & \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }{(t-a)}^{\lambda }}{\Gamma \left(\alpha \right)}}\left[\left|{\int }_0^{\omega \left(t\right)}\left(e^{-Z_1\left(\rho \right)s}-e^{-Z_2\left(\rho \right)s}\right)s^{\alpha -1}ds+{\int }_{\omega \left(t\right)}^{\omega (t+h)}{s}^{\alpha -1}ds\right|\right]\hfill \\ & \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }{(t-a)}^{\lambda }}{\Gamma (1+\alpha )}}\left[{\left(\omega \left(t\right)\right)}^{\alpha }+{\left(\omega (t+h)\right)}^{\alpha }-{\left(\omega \left(t\right)\right)}^{\alpha }\right]\hfill \\ & =& {\displaystyle \frac{{\left[f\right]}_{\lambda }{(t-a)}^{\lambda }}{\Gamma (1+\alpha )}}\left[{(\omega (t+h)-w\left(a\right))}^{\alpha }\right]\hfill \\ & \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }{∥{\omega }^{\prime }∥}^{\alpha }{(t-a)}^{\lambda }}{\Gamma (1+\alpha )}}\left[{(t-a+h)}^{\alpha }\right].\hfill \end{array}$$

Thus, if $h\ge t-a$, then it follows that

$$\left|A\right|\le {\displaystyle \frac{2^{\alpha }{\left[f\right]}_{\lambda }{∥{\omega }^{\prime }∥}^{\alpha }{h}^{\lambda +\alpha }}{\Gamma (1+\alpha )}}.$$

Next, if $h<t-a$, then in view of (12), we once again apply the differences for $\chi (t+h)-\chi (t-s)$ and $\chi \left(t\right)-\chi (t-s)$. Let $v=\omega (t+h)-\omega (t-s)$ and $w=\omega \left(t\right)-\omega (t-s)$. Then,

$$\left|A\right|\le {\displaystyle \frac{{\left[f\right]}_{\lambda }{(t-a)}^{\lambda }}{\Gamma \left(\alpha \right)}}\left|{\int }_0^{\omega (t+h)}{\displaystyle \frac{e^{-Z_1\left(\rho \right)v}}{v^{1-\alpha }}}dv-{\int }_0^{\omega \left(t\right)}{\displaystyle \frac{e^{-Z_2\left(\rho \right)w}}{w^{1-\alpha }}}dw\right|,$$

as $\omega \left(a\right)=0$. We decompose the difference as

$$\begin{array}{ccc}\hfill {\int }_0^{\omega (t+h)}{\displaystyle \frac{e^{-Z_1\left(\rho \right)v}}{v^{1-\alpha }}}dv& -& {\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{e^{-Z_2\left(\rho \right)w}}{w^{1-\alpha }}}dw\hfill \\ & =& {\int }_0^{\omega \left(t\right)}\left({\displaystyle \frac{e^{-Z_1\left(\rho \right)v}}{v^{1-\alpha }}}-{\displaystyle \frac{e^{-Z_2\left(\rho \right)v}}{v^{1-\alpha }}}\right)dv+{\int }_{\omega \left(t\right)}^{\omega (t+h)}{\displaystyle \frac{e^{-Z_1\left(\rho \right)v}}{v^{1-\alpha }}}dv.\hfill \end{array}$$

Using the inequality $|e^{-a}-e^{-b}|\le |a-b|$ for $a,b\ge 0$,

$$\left|{\displaystyle \frac{e^{-Z_1\left(\rho \right)v}-e^{-Z_2\left(\rho \right)v}}{v^{1-\alpha }}}\right|\le |Z_1\left(\rho \right)-Z_2\left(\rho \right)|v^{\alpha }.$$

Since ${\ell }_1,{\ell }_2$ are bounded, $|Z_1\left(\rho \right)-Z_2\left(\rho \right)|\le 2M$. Thus,

$${\int }_0^{\psi \left(t\right)}|Z_1\left(\rho \right)-Z_2\left(\rho \right)|v^{\alpha }dv\le 2M{\displaystyle \frac{\psi {\left(t\right)}^{\alpha +1}}{\alpha +1}}.$$

To estimate the second term, we use an additional integral over $\left[\psi \right(t),\psi (t+h\left)\right]$:

$${\int }_{\omega \left(t\right)}^{\omega (t+h)}{\displaystyle \frac{e^{-Z_1\left(\rho \right)v}}{v^{1-\alpha }}}dv\le {\int }_{\omega \left(t\right)}^{\omega (t+h)}{v}^{\alpha -1}dv={\displaystyle \frac{{\left(\omega (t+h)\right)}^{\alpha }-{\left(\omega \left(t\right)\right)}^{\alpha }}{\alpha }}.$$

Since $\omega$ is Lipschitz, i.e., $\omega (t+h)-\omega \left(t\right)\le ∥{\omega }^{\prime }∥h$, and by the mean value theorem,

$${\left(\omega (t+h)\right)}^{\alpha }-{\left(\omega \left(t\right)\right)}^{\alpha }\le \alpha ∥{\omega }^{\prime }∥h\omega {\left(t\right)}^{\alpha -1}.$$

Substituting once again

$$\left|A\right|\le {\displaystyle \frac{{\left[f\right]}_{\lambda }{(t-a)}^{\lambda }}{\Gamma \left(\alpha \right)}}\left(2M{\displaystyle \frac{\omega {\left(t\right)}^{\alpha +1}}{\alpha +1}}+∥{\omega }^{\prime }∥h\psi {\left(t\right)}^{\alpha -1}\right).$$

For $h<t-a$, we have $\omega \left(t\right)\le ∥{\omega }^{\prime }∥(t-a)$. Therefore,

$$\left|A\right|\le {\displaystyle \frac{{\left[f\right]}_{\lambda }{(t-a)}^{\lambda }}{\Gamma \left(\alpha \right)}}\left(2M{\displaystyle \frac{{\left({\omega }_{max}^{\prime }(t-a)\right)}^{\alpha +1}}{\alpha +1}}+∥{\omega }^{\prime }∥h{\left(∥{\omega }^{\prime }∥(t-a)\right)}^{\alpha -1}\right).$$

Simplifying and noting that $h\le t-a$, we obtain $\left|A\right|\le W_1·h^{\alpha +\lambda }$, where

$$W_1={\displaystyle \frac{{\left[f\right]}_{\lambda }{∥{\omega }^{\prime }∥}^{\alpha }}{\Gamma \left(\alpha \right)}}\left({\displaystyle \frac{2M{(t-a)}^{\lambda +1}}{\alpha +1}}+{(t-a)}^{\lambda +\alpha -1}\right).$$

Finally, this type of estimation holds true for any h:

$$\left|A\right|\le W·h^{\alpha +\lambda },$$

For $\left|B\right|$, the expected estimate, i.e., a constant times $h^{\lambda +\alpha }$, has already been obtained.

It remains to estimate C. Obviously, since $\chi (t+h)-\chi (t+s)\ge 0$ for $s\in [0,h],t\in [a,b]$, we have, in view of the standard mean value theorem,

$$\begin{array}{ccc}\hfill \left|C\right|& \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }}{\Gamma \left(\alpha \right)}}\left[{\int }_0^{t-a}\left|{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_1\right)(s+h)}}{{\left({\omega }^{\prime }\left({\eta }_1\right)(h+s)\right)}^{1-\alpha }}}-{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)s}}{{\left({\omega }^{\prime }\left({\eta }_2\right)s\right)}^{1-\alpha }}}\right|∥{\omega }^{\prime }∥s^{\lambda }ds\right]\hfill \\ & \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }}{\Gamma \left(\alpha \right)}}\left[∥{\omega }^{\prime }∥h^{\lambda +\alpha }{\int }_0^{{\displaystyle \frac{t-a}{h}}}\left|{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_1\right)(1+s)h}}{{\left({\omega }^{\prime }\left({\eta }_1\right)(1+s)\right)}^{1-\alpha }}}-{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)hs}}{{\left({\omega }^{\prime }\left({\eta }_2\right)s\right)}^{1-\alpha }}}\right|s^{\lambda }ds\right]\hfill \\ & =& {\displaystyle \frac{{\left[f\right]}_{\lambda }}{\Gamma (1+\alpha )}}\left[∥{\omega }^{\prime }∥h^{\lambda +\alpha }{\int }_0^{{\displaystyle \frac{t-a}{h}}}\left|{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_1\right)(1+s)h}}{{\left({\omega }^{\prime }\left({\eta }_1\right)(1+s)\right)}^{1-\alpha }}}-{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)sh}}{{\left({\omega }^{\prime }\left({\eta }_1\right)(1+s)\right)}^{1-\alpha }}}\right.\right.\hfill \\ & & \left.\left.+{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)sh}}{{\left({\omega }^{\prime }\left({\eta }_1\right)(1+s)\right)}^{1-\alpha }}}-{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)hs}}{{\left({\omega }^{\prime }\left({\eta }_2\right)s\right)}^{1-\alpha }}}\right|s^{\lambda }ds\right]\hfill \\ & \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }∥{\omega }^{\prime }∥h^{\lambda +\alpha }}{\Gamma (1+\alpha )}}\left({\int }_0^{{\displaystyle \frac{t-a}{h}}}{\left({\omega }^{\prime }\left({\eta }_1\right)(1+s)\right)}^{\alpha -1}{s}^{\lambda }ds\right.\hfill \\ & & \left.+{\int }_0^{{\displaystyle \frac{t-a}{h}}}\left|{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)sh}}{{\left({\omega }^{\prime }\left({\eta }_1\right)(1+s)\right)}^{1-\alpha }}}-{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)hs}}{{\left({\omega }^{\prime }\left({\eta }_2\right)s\right)}^{1-\alpha }}}\right|s^{\lambda }ds\right),\hfill \end{array}$$

where ${\eta }_1,{\zeta }_1\in (t-s,t+h)$ and ${\eta }_2,{\zeta }_2\in (t-s,t)$. Hence, if $h\ge t-a,$ it follows that

$$\begin{array}{ccc}\hfill \left|C\right|& \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }∥{\omega }^{\prime }∥h^{\lambda +\alpha }}{\Gamma (1+\alpha )}}\left({\int }_0^1{\left({\omega }^{\prime }\left({\eta }_1\right)(1+s)\right)}^{\alpha -1}{s}^{\lambda }ds\right.\hfill \\ & & \left.+{\int }_0^1\left|{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)sh}}{{\left({\omega }^{\prime }\left({\eta }_1\right)(1+s)\right)}^{1-\alpha }}}-{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)hs}}{{\left({\omega }^{\prime }\left({\eta }_2\right)s\right)}^{1-\alpha }}}\right|s^{\lambda }ds\right)\hfill \\ & \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }∥{\omega }^{\prime }∥h^{\lambda +\alpha }}{\Gamma (1+\alpha )}}\left[{\displaystyle \frac{{\left(|{min}_{t\in [a,b]}{\omega }^{\prime }\left(t\right)|\right)}^{\alpha -1}{2}^{\lambda +\alpha }}{(\lambda +\alpha )}}\right].\hfill \end{array}$$

On the other hand, when $h<t-a,$ we have

$$\begin{array}{ccc}\hfill \left|C\right|& \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }∥{\omega }^{\prime }∥h^{\lambda +\alpha }}{\Gamma (1+\alpha )}}\left[{\int }_0^1\left|{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_1\right)(1+s)h}}{{\left({\omega }^{\prime }\left({\eta }_1\right)(1+s)\right)}^{1-\alpha }}}-{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)hs}}{{\left({\omega }^{\prime }\left({\eta }_2\right)s\right)}^{1-\alpha }}}\right|s^{\lambda }ds\right.\hfill \\ & & \left.+{\int }_1^{\infty }\left|{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_1\right)(1+s)h}}{{\left({\omega }^{\prime }\left({\eta }_1\right)(1+s)\right)}^{1-\alpha }}}-{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)hs}}{{\left({\omega }^{\prime }\left({\eta }_2\right)s\right)}^{1-\alpha }}}\right|s^{\lambda }ds\right]\hfill \\ & =& {\displaystyle \frac{{\left[f\right]}_{\lambda }∥{\omega }^{\prime }∥h^{\lambda +\alpha }}{\Gamma (1+\alpha )}}\left[{\displaystyle \frac{{\left(∥{\omega }^{\prime }∥\right)}^{\alpha -1}{2}^{\lambda +\alpha }}{(\lambda +\alpha )}}\right.\hfill \\ & & +{\int }_1^{\infty }\left|{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_1\right)(1+s)h}}{{\left({\omega }^{\prime }\left({\eta }_1\right)(1+s)\right)}^{1-\alpha }}}-{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)sh}}{{\left({\omega }^{\prime }\left({\eta }_2\right)(1+s)\right)}^{1-\alpha }}}\right.\hfill \\ & & \left.\left.+{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)sh}}{{\left({\omega }^{\prime }\left({\eta }_2\right)(1+s)\right)}^{1-\alpha }}}-{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)hs}}{{\left({\omega }^{\prime }\left({\eta }_2\right)s\right)}^{1-\alpha }}}\right|s^{\lambda }ds\right]\hfill \\ & \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }∥{\omega }^{\prime }∥h^{\lambda +\alpha }}{\Gamma (1+\alpha )}}\left({\displaystyle \frac{{\left(∥{\omega }^{\prime }∥\right)}^{\alpha -1}{2}^{\lambda +\alpha }}{(\lambda +\alpha )}}\right.\hfill \\ & & +{\int }_1^{\infty }\left|{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_1\right)(1+s)h}}{{\left({\omega }^{\prime }\left({\eta }_1\right)\right)}^{1-\alpha }}}-{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)sh}}{{\left({\omega }^{\prime }\left({\eta }_2\right)\right)}^{1-\alpha }}}\right|s^{\alpha +\lambda -1}ds\hfill \\ & & \left.+{\int }_1^{\infty }{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)sh}}{{\left({\omega }^{\prime }\left({\eta }_2\right)\right)}^{1-\alpha }}}\left|{(1+s)}^{\alpha -1}-s^{\alpha -1}\right|s^{\lambda }ds\right).\hfill \end{array}$$

Applying the mean value theorem to the function $t\to t^{\alpha -1}$ on the interval $[s,s+1]$ yields

$$\left|{(1+s)}^{\alpha -1}-s^{\alpha -1}\right|s^{\lambda }\le (1-\alpha )s^{\alpha +\lambda -2}.$$

Therefore,

$$\left|C\right|\le {\displaystyle \frac{{\left[f\right]}_{\lambda }∥{\omega }^{\prime }∥h^{\lambda +\alpha }}{\Gamma (1+\alpha )}}\left({\displaystyle \frac{{\left({min}_{t\in [a,b]}\left|{\omega }^{\prime }\left(t\right)\right|\right)}^{\alpha -1}{2}^{\lambda +\alpha }}{(\lambda +\alpha )}}+{\displaystyle \frac{\alpha -1}{{\left((\alpha +\lambda -1){\omega }^{\prime }\left({\eta }_2\right)\right)}^{1-\alpha }}}+\mathbf{J}\right),$$

where

$$\begin{array}{ccc}\hfill \mathbf{J}& \le & \left|e^{-{\chi }^{\prime }\left({\zeta }_1\right)h}{\int }_0^{\infty }{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_1\right)sh}{s}^{\alpha +\lambda -1}ds}{{\left({\omega }^{\prime }\left({\eta }_1\right)\right)}^{1-\alpha }}}-{\int }_0^{\infty }{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)sh}{s}^{\alpha +\lambda -1}ds}{{\left({\omega }^{\prime }\left({\eta }_2\right)\right)}^{1-\alpha }}}\right|\hfill \\ & =& \left|{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_1\right)h}}{{\left({\omega }^{\prime }\left({\eta }_1\right)\right)}^{1-\alpha }({\chi }^{\prime }{\left({\zeta }_1\right)}^{\lambda +\alpha }}}-{\displaystyle \frac{1}{{\left({\omega }^{\prime }\left({\eta }_2\right)\right)}^{1-\alpha }({\chi }^{\prime }{\left({\zeta }_2\right)}^{\lambda +\alpha }}}\right|{\displaystyle \frac{\Gamma (\lambda +\alpha )}{h^{\lambda +\alpha }}}.\hfill \end{array}$$

Consequently,

$$\begin{array}{ccc}& & \underset{h\to 0}{lim}{\displaystyle \frac{\frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)h}{\left({\omega }^{\prime }\left({\eta }_1\right)\right)}^{\alpha -1}}{({\chi }^{\prime }{\left({\zeta }_2\right)}^{\lambda +\alpha }}-\frac{{\left({\omega }^{\prime }\left({\eta }_2\right)\right)}^{\alpha -1}}{({\chi }^{\prime }{\left({\zeta }_2\right)}^{\lambda +\alpha }}}{h^{\lambda +\alpha }}}\hfill \\ & & =\underset{h\to 0}{lim}{\displaystyle \frac{e^{-{\chi }^{\prime }\left({\zeta }_2\right)h}{\left({\omega }^{\prime }\left({\eta }_1\right)\right)}^{\alpha -1}({\chi }^{\prime }{\left({\zeta }_2\right)}^{1-\lambda -\alpha }-0}{(\lambda +\alpha )}}{h}^{1-(\lambda +\alpha )}\to 0.\hfill \end{array}$$

□

3. Between Fractional and Non-Fractional Differential Operators

Given the operators that were previously examined, it is now worthwhile to consider a new class of generalized derivatives.

Definition 2.

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 with $\psi \left(a\right)=0$. The following definition of generalized derivatives is hereby proposed:

1. 

The generalized combination between fractional and non-fractional differential operator of order $n+\alpha ,\alpha \in (0,1),$$n\in \mathbb{N},$ with parameters $\rho \in (0,1)$ and $\mu \ge 0$ applied to the function $f\in L_1[a,b]$ is defined as

$${\mathfrak{D}}_{\psi ,\mu }^{n+\alpha ,\rho ,\ell }f={\left({\partial }_{\psi ,\mu }^{\rho ,\ell }\right)}^n{\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f,where{\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f={\partial }_{\psi ,\mu }^{\rho ,\ell }{K}_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }f.$$

(13)

2. 

The generalized combination between fractional and non-fractional Caputo differential operator of order $n+\alpha ,\alpha \in (0,1),$ $n\in \mathbb{N}$ with parameters $\rho \in (0,1)$ and $\mu \ge 0$ applied to the function $f\in AC[a,b]$ is defined as

$${}^C\mathfrak{D}_{\psi ,\mu }^{n+\alpha ,\rho ,\ell }f={\left({\partial }_{\psi ,\mu }^{\rho ,\ell }\right)}^n{}^C\mathfrak{D}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f,where{}^C\mathfrak{D}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f=K_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }{\partial }_{\psi ,\mu }^{\rho ,\ell }f.$$

(14)

If $\rho \to 1$, then the generalized combination between fractional and non-fractional Caputo differential operator is reduced to the classical Caputo fractional derivative of order $n+\alpha$.

3. 

The generalized combination between fractional and non-fractional Hilfer differential operator of order $n+\alpha ,\alpha \in (0,1),$ $n\in \mathbb{N},$ with parameters $\rho \in (0,1),\mu \ge 0$ and type $\beta \in [0,1]$ applied to the function $f\in L_1[a,b]$ is defined as

$${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{n+\alpha ,\beta ,\ell }f\left(t\right):={\left({\partial }_{\psi ,\mu }^{\rho ,\ell }\right)}^n{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\ell }f\left(t\right),where{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\ell }:=K_{\rho ,\psi }^{\beta (1-\alpha ),\ell ,\mu }{\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\ell ,\mu }.$$

Clearly, the generalized combination of the fractional and non-fractional Hilfer differential operators is used to interpolate between the generalized combinations of the fractional and non-fractional Riemann–Liouville and Caputo differential operators.

Through direct verification, it is easy to show that for any $t>a$ and any $0<\alpha <1$, we have

$$\begin{array}{ccc}\hfill & & {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\ell }\left\{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\left(\omega \left(t\right)\right)}^{\eta -1}\right\}\hfill \\ & & ={\displaystyle \frac{\Gamma \left(\eta \right)}{\Gamma (\eta -\alpha )}}{e}^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\left(\omega \left(t\right)\right)}^{\eta -\alpha -1},\hfill \end{array}$$

(15)

where $\eta >0$.

In fact, we have

$$\begin{array}{ccc}& & K_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\ell ,\mu }\left\{{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{1-\eta }}}\right\}\hfill \\ & & ={\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\Gamma \left(\eta \right)}{\Gamma (\eta +(1-\beta \left)\right(1-\alpha \left)\right)}}{\left(\omega \left(t\right)\right)}^{\eta +(1-\beta )(1-\alpha )-1}.\hfill \end{array}$$

Hence, in view of (5), it follows that

$$\begin{array}{ccc}& & {\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\ell ,\mu }\left\{{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{1-\eta }}}\right\}\hfill \\ & & ={\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{2-\eta -(1-\beta )(1-\alpha )}}}{\displaystyle \frac{\Gamma \left(\eta \right)[\eta -1+(1-\beta \left)\right(1-\alpha \left)\right]}{\Gamma (\eta +(1-\beta \left)\right(1-\alpha \left)\right)}}.\hfill \end{array}$$

Finally,

$$\begin{array}{ccc}& & {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\ell }\left\{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\left(\omega \left(t\right)\right)}^{\eta -1}\right\}\hfill \\ & & =K_{\rho ,\psi }^{\beta (1-\alpha ),\ell ,\mu }{\partial }_{\psi ,\mu }^{\rho ,\ell }{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\ell ,\mu }\left\{{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{1-\eta }}}\right\}\hfill \\ & & =K_{\rho ,\psi }^{\beta (1-\alpha ),\ell ,\mu }{\displaystyle \frac{e^{{\displaystyle \left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}}{{\left(\omega \left(t\right)\right)}^{2-\eta -(1-\beta )(1-\alpha )}}}{\displaystyle \frac{\Gamma \left(\eta \right)[\eta -1+(1-\beta \left)\right(1-\alpha \left)\right]}{\Gamma (\eta +(1-\beta \left)\right(1-\alpha \left)\right)}}\hfill \\ & & ={\displaystyle \frac{\Gamma \left(\eta \right)}{\Gamma (\eta -\alpha )}}{\left(\omega \left(t\right)\right)}^{\eta -\alpha -1}{e}^{{\displaystyle \left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}.\hfill \end{array}$$

In particular, when $\alpha =\eta -1$

$$\begin{array}{ccc}\hfill & & {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\ell }\left\{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\left(\omega \left(t\right)\right)}^{\alpha }\right\}\hfill \\ & & =\Gamma (1+\alpha )e^{\left[{\displaystyle -{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du}\right]}.\hfill \end{array}$$

(16)

Also, for $\eta =\alpha +\beta (1-\alpha ),$ it can be easily seen that (cf. (5) and (6))

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

(17)

Lemma 5.

For any $\alpha \in (0,1)$, $\mu \in {\mathbb{R}}^{+}$, and positive increasing function $\psi \in C^1[a,b]$ such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0,$ we have

$${\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }{K}_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t\right)=f\left(t\right),f\in L_1[a,b].$$

Proof. 

Let $f\in L_1[a,b].$ Then, by Lemma 1 and in view of Proposition 1, we obtain

$${\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }{K}_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t\right)={\partial }_{\psi ,\mu }^{\rho ,\ell }{K}_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }{K}_{\rho ,\psi }^{\alpha ,\ell ,\mu }f\left(t\right)={\partial }_{\psi ,\mu }^{\rho ,\ell }{K}_{\rho ,\psi }^{1,\ell ,\mu }f\left(t\right)={\partial }_{\psi ,\mu }^{\rho ,\ell }{\Im }_{\psi ,\mu }^{\rho ,\ell }f\left(t\right)=f\left(t\right).$$

□

Lemma 6.

For any $\alpha \in (0,1)$, $\mu \in {\mathbb{R}}^{+}$, and positive increasing function $\psi \in C^1[a,b]$ such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0,$ for any $f\in AC[a,b]$ we have

$${}^C\mathfrak{D}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right)={\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right)-{\displaystyle \frac{f\left(a\right)}{\Gamma (1-\alpha )}}\left(e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\left(\omega \left(t\right)\right)}^{-\alpha }\right).$$

Proof. 

Let $f\in AC[a,b]$. Then, for any $n\in \mathbb{N}$, we have

$$\begin{array}{ccc}\hfill {}^C\mathfrak{D}_{\psi ,\mu }^{n+\alpha ,\rho ,\ell }f\left(t\right)& =& {\left({\partial }_{\psi ,\mu }^{\rho ,\ell }\right)}^n{K}_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }{\partial }_{\psi ,\mu }^{\rho ,\ell }f\left(t\right)={\left({\partial }_{\psi ,\mu }^{\rho ,\ell }\right)}^n\left({\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }{K}_{\rho ,\psi }^{\alpha ,\ell ,\mu }\right)K_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }{\partial }_{\psi ,\mu }^{\rho ,\ell }f\left(t\right)\hfill \\ & =& \left[{\left({\partial }_{\psi ,\mu }^{\rho ,\ell }\right)}^n{\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }\right]\left(K_{\rho ,\psi }^{\alpha ,\ell ,\mu }{K}_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }\right){\partial }_{\psi ,\mu }^{\rho ,\ell }f\left(t\right)=\left[{\left({\partial }_{\psi ,\mu }^{\rho ,\ell }\right)}^n{\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }\right]K_{\rho ,\psi }^{1,\ell ,\mu }{\partial }_{\psi ,\mu }^{\rho ,\ell }f\left(t\right)\hfill \\ & =& {\mathfrak{D}}_{\psi ,\mu }^{n+\alpha ,\rho ,\ell }{\Im }_{\psi ,\mu }^{\rho ,\ell }{\partial }_{\psi ,\mu }^{\rho ,\ell }f\left(t\right)={\mathfrak{D}}_{\psi ,\mu }^{n+\alpha ,\rho ,\ell }\left(f\left(t\right)-e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f\left(a\right)\right).\hfill \end{array}$$

In particular, for $\alpha \in (0,1)$,

$$\begin{array}{ccc}\hfill {}^C\mathfrak{D}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right)& =& {\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }\left(f\left(t\right)-e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f\left(a\right)\right)\hfill \\ & =& {\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right)-f\left(a\right){\partial }_{\psi ,\mu }^{\rho ,\ell }{K}_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }\left\{e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\right\}\hfill \\ & =& {\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right)-{\displaystyle \frac{f\left(a\right)}{\Gamma (2-\alpha )}}{\partial }_{\psi ,\mu }^{\rho ,\ell }\left(e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\left(\omega \left(t\right)\right)}^{1-\alpha }\right)\hfill \\ & =& {\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right)-{\displaystyle \frac{f\left(a\right)}{\Gamma (1-\alpha )}}\left(e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\left(\omega \left(t\right)\right)}^{-\alpha }\right).\hfill \end{array}$$

(18)

□

Unfortunately, Definition 14 of the generalized combination between the fractional and non-fractional Caputo differential operators of order $\alpha \in (0,1)$ with parameters $\rho \in (0,1)$ and $\mu$ ($\mu \ge 0$) has the disadvantage of losing its meaning if f is not differentiable almost everywhere.

In fact, $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }$ does not map all of $C[a,b]$ into $AC[a,b]$ (see, e.g., [23,35]). Thus, the operator ${}^C\mathfrak{D}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f$ makes no sense for an arbitrary $f\in C[a,b]$. For this reason, we use the property of Lemma 6 to define the generalized combination between the fractional and non-fractional Caputo differential operators, i.e., we put

$${}^C\mathfrak{D}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right):={\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right)-{\displaystyle \frac{f\left(a\right)}{\Gamma (1-\alpha )}}\left(e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\left(\omega \left(t\right)\right)}^{-\alpha }\right).$$

(19)

According to Lemma 6, for absolutely continuous functions, Definition 18 coincides with the standard definition of the generalized combination between the fractional and non-fractional Caputo differential operators.

In the following lemma, we will show that, for any $\beta \in [0,1)$, the operator ${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\ell }$ coincides with ${\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }$ on the space $AC[a,b]$.

Corollary 1.

Let $\alpha \in (0,1)$, $\beta \in [0,1)$, $\mu \in {\mathbb{R}}^{+}$, and a positive increasing function $\psi \in C^1[a,b]$ such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Then, for any $f\in AC[a,b]$, we have

$${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\ell }f={\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f.$$

Proof. 

Let $f\in AC[a,b]$ and recall that, for any $n\in \mathbb{N}$ and $\beta \in [0,1)$, we have

$${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{n+\alpha ,\beta ,\ell }f={}^C\mathfrak{D}_{\psi ,\mu }^{n+1-\beta (1-\alpha ),\rho ,\ell }{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\ell ,\mu }f.$$

In view of (8), we know that

$$\begin{array}{ccc}\hfill {\partial }_{\psi ,\mu }^{\rho ,\ell }{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\ell ,\mu }f\left(t\right)& =& {\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{(1-\beta )(1-\alpha )-1}{e}^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\Gamma \left(\right(1-\beta \left)\right(1-\alpha \left)\right)}}\hfill \\ & & +K_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\ell ,\mu }{\partial }_{\psi ,\mu }^{\rho ,\ell }f\left(t\right).\hfill \end{array}$$

By Lemma 2 and Lemma 1, in view of (18), we obtain

$$\begin{array}{ccc}& & K_{\rho ,\psi }^{\beta (1-\alpha ),\ell ,\mu }{\partial }_{\psi ,\mu }^{\rho ,\ell }{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\ell ,\mu }f\left(t\right)\hfill \\ & =& {\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{-\alpha }{e}^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\Gamma (1-\alpha )}}+K_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }{\partial }_{\psi ,\mu }^{\rho ,\ell }f\left(t\right)\hfill \\ & =& {\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{-\alpha }{e}^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\Gamma (1-\alpha )}}{+}^C{\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right)\hfill \\ & =& {\displaystyle \frac{f\left(a\right){\left(\omega \left(t\right)\right)}^{-\alpha }{e}^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\Gamma (1-\alpha )}}+{\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right)\hfill \\ & & -{\displaystyle \frac{f\left(a\right)}{\Gamma (1-\alpha )}}\left(e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\left(\omega \left(t\right)\right)}^{-\alpha }\right)\hfill \\ & =& {\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right).\hfill \end{array}$$

Hence,

$${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{n+\alpha ,\beta ,\ell }f\left(t\right)={\mathfrak{D}}_{\psi ,\mu }^{n+\alpha ,\rho ,\ell }f\left(t\right){=}^H{\mathrm{D}}_{\psi ,\rho ,\mu }^{n+\alpha ,0,\ell }f\left(t\right),f\in AC[a,b].$$

□

We note that the generalized fractional operator defined by Definitions 2 and 1 generalizes several existing fractional integral operators. This new formalism allows us to treat several other classical models of fractional calculus as special cases, such as the Hadamard and Erdélyi–Kober fractional operators.

Fractional differential equations are significant because they appear in mathematical models of natural phenomena (see, for example, [5,6,7,8,9,44]). Many researchers have studied different aspects of the theory of terminal value problems, especially in [2,40,45,46,47,48].

We now present the main result regarding the operation of the generalized differential operator between Hölder spaces due to both applications and interesting generalizations of known mathematical results.

As we proved in Theorem 2, the image of a little Hölder space under a generalized integral operator is also a Hölder space. However, the order of the image is increased by the order $\alpha$ of the operator. So, to study if the operators can be inverted, we have good spaces to consider. We will show that the generalized differential operator acts in the opposite way on these spaces.

Theorem 3.

Let $0<\alpha <\lambda <1,\mu \ge 0$ and $\psi \in C^1[a,b]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0.$ Then, ${\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }$ maps the Hölder space ${\mathbb{H}}^{\lambda ,0}[a,b]$ onto ${\mathbb{H}}^{\lambda -\alpha ,0}[a,b]$, and

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

where we define $f\left(s\right):=0$ for $s<a.$

Proof. 

Let $f\in {\mathbb{H}}^{\lambda ,0}[a,b]$. Note that

$$f(·)e^{{\displaystyle \left[{\int }_0^{\psi (·)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}\in {\mathbb{H}}^{\lambda ,0}[a,b].$$

Define

$$\begin{array}{ccc}& & g\left(t\right):={\displaystyle \frac{\alpha }{\Gamma (1-\alpha )}}{\int }_{-\infty }^t{(\omega \left(t\right)-\omega \left(s\right))}^{-\alpha -1}\left(e^{\left[{\int }_0^{\psi \left(s\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f\left(t\right)\right.\hfill \\ & & \left.-e^{\left[{\int }_0^{\psi \left(s\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f\left(s\right)\right){\omega }^{\prime }\left(s\right)ds.\hfill \end{array}$$

The proof is divided into several points.

1.

Arguing similarly to the ([proof of Lemma 3] in [43]), it can be easily seen that

$${\mathfrak{D}}_{\psi ,0}^{\alpha ,\rho ,0}\left(e^{\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f\left(t\right)\right)=g\left(t\right).$$

2.

Now, we have

$$\begin{array}{ccc}\hfill {\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right)& =& {\partial }_{\psi ,\mu }^{\rho ,\ell }\left[e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{K}_{\rho ,\psi }^{1-\alpha ,0,0}\left(e^{\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f\left(t\right)\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{{\omega }^{\prime }\left(t\right)}}{e}^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\displaystyle \frac{d}{dt}}\left[K_{\rho ,\psi }^{1-\alpha ,0,0}\left(e^{\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f\left(t\right)\right)\right]\hfill \\ & =& e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\mathfrak{D}}_{\psi ,0}^{\alpha ,\rho ,0}\left(e^{\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f\left(t\right)\right)\hfill \\ & =& e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}g\left(t\right).\hfill \end{array}$$

Hence,

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

$$\begin{array}{ccc}& & {\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(a\right)={\displaystyle \frac{\alpha }{\Gamma (1-\alpha )}}{\int }_{-\infty }^a{e}^{-\left[{\int }_{\psi \left(s\right)}^0\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{(\omega \left(a\right)-\omega \left(s\right))}^{-\alpha -1}\left[f\left(a\right)-0\right]{\omega }^{\prime }\left(s\right)ds\hfill \\ & & =0.\hfill \end{array}$$

3.

Arguing similarly as in the proof of Theorem 4 [43], for all $h>0,t,t+h\in [a,b]$, there is a constant $L>0$ depending only on $\alpha$ and $\lambda$ such that

$$\begin{array}{ccc}\hfill \left|g(t+h)-g\left(t\right)\right|& \le & {\displaystyle \frac{\alpha L}{\Gamma (1-\alpha )}}{\left[e^{\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}f\left(t\right)\right]}_{\lambda }{h}^{\lambda -\alpha }\hfill \\ & \le & {\displaystyle \frac{\alpha L{e}^{\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\Gamma (1-\alpha )}}{\left[f\left(t\right)\right]}_{\lambda }{h}^{\lambda -\alpha }.\hfill \end{array}$$

$$\left|{\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f(t+h)-{\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right)\right|\le e^{{\displaystyle \left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{\displaystyle \frac{\alpha L}{\Gamma (1-\alpha )}}{\left[f\right]}_{\lambda }{h}^{\lambda -\alpha }.$$

□

Equipped with the aid of Lemma 4 and Theorem 3, we are able to demonstrate the following.

Corollary 2.

If the assumptions of Theorem 2 hold, then the operator

$$K_{\rho ,\psi }^{\alpha ,\ell ,\mu }:{\mathbb{H}}^{\lambda ,0}[a,b]\to {\mathbb{H}}^{\alpha +\lambda ,0}[a,b],$$

is bijective with a continuous inverse ${\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }$.

Proof. 

First, we note that, for any $f\in L_1[a,b]$, we have that, according to Lemma 5,

${\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }{K}_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }f=f$.

Next, define $g:={\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f,f\in {\mathbb{H}}^{\alpha +\lambda ,0}[a,b]$. Note that, by Theorem 3, $g\in {\mathbb{H}}^{\lambda ,0}[a,b].$ Since (cf. Remark 2) $K_{\rho ,\psi }^{2-\alpha ,\ell ,\mu }f\left(a\right)=0,$ it follows that

$$\begin{array}{ccc}& & K_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }{K}_{\rho ,\psi }^{\alpha ,\ell ,\mu }g\left(t\right)=K_{\rho ,\psi }^{1,\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }f\left(t\right)\hfill \\ & & =K_{\rho ,\psi }^{1,\ell ,\mu }\left[\left({\ell }_2(\rho ,t)+\mu \right)K_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }f\left(t\right)+{\displaystyle \frac{1}{{\omega }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}{K}_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }f\left(t\right)\right].\hfill \end{array}$$

Using integration by parts, we obtain the following:

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

Consequently, for all $t\in [a,b]$, we obtain

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

Therefore, since $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }$ is bijective (see Lemma 4), the left and right inverses of $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }$ are equal to each other and to ${\left(K_{\rho ,\psi }^{\alpha ,\ell ,\mu }\right)}^{-1}$:

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

Since ${\mathbb{H}}^{\alpha +\lambda ,0}[a,b],{\mathbb{H}}^{\alpha ,0}[a,b]$ are Banach spaces, the continuity follows from the properties of operators from ${\left(K_{\rho ,\psi }^{\alpha ,\ell ,\mu }\right)}^{-1}\equiv {\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }$ of $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }.$  □

4. Equivalence Results

Let us now discuss the equivalence problem between the previously defined combination of fractional and non-fractional Hilfer differential problems and the corresponding integral forms.

Consider the following initial value problem:

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

(20)

where $t\in [a,b]$, $\alpha >0$, $\beta \in [0,1]$, combined with an appropriate initial, terminal or boundary conditions. We need to determine the integral forms that correspond to the problem (20):

1.

If $\alpha \in (1,2]$, then (20) reads as follows:

$${\partial }_{\psi ,\mu }^{\rho ,\ell }{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha -1,\beta ,\ell }x\left(t\right)=f(t,x\left(t\right)).$$

Then, Proposition 1 along with (6) implies that

$${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha -1,\beta ,\ell }x\left(t\right)={\Im }_{\psi ,\mu }^{\rho ,\ell }f(t,x\left(t\right))+e^{{\displaystyle -\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{C}_0.$$

Keeping in mind (17) and arguing similarly as in [49], we can (only formally) show that

$$\begin{array}{ccc}\hfill x\left(t\right)& =& {\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(2-\alpha )}}}{C}_1+K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f(t,x\left(t\right))\hfill \\ & & +K_{\rho ,\psi }^{\alpha -1,\ell ,\mu }\left[e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{C}_0\right]\hfill \\ & =& {\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(2-\alpha )}}}{C}_1+K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f(t,x\left(t\right))\hfill \\ & & +e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\displaystyle \frac{C_0{\left(\omega \left(t\right)\right)}^{\alpha -1}}{\Gamma \left(\alpha \right)}},\hfill \end{array}$$

(21)

where $C_0$ and $C_0$ depend only on initial, terminal or boundary conditions.

2.

If $\alpha \in (0,1]$, then the integral form corresponding to (20) reads as follows:

$$x\left(t\right)={\displaystyle \frac{e^{\left[{\displaystyle -{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du}\right]}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(1-\alpha )}}}{C}_2+K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f(t,x\left(t\right)),$$

(22)

where $C_2$ depends only on initial, terminal or boundary conditions.

Now, let $f(·,x(·)))\in L_1[a,b]$ for any $x\in C[a,b].$ Assume that the integral Equations (21) and (22) admit a solution $x\in C[a,b]$. Then, it can be easily seen that

1.

For any $\beta \in [0,1]$ and $\alpha \in (1,2]$, any solution $x\in C[a,b]$ of (21) solves (20): Indeed, operating by $K_{\rho ,\psi }^{(1-\beta )(2-\alpha ),\ell ,\mu }$, ${\partial }_{\psi }^{\rho ,\ell }$ and $K_{\rho ,\psi }^{\beta (2-\alpha ),\ell ,\mu }$ in view of Lemma 1 and Lemma 2, yields the following, respectively, on both sides of (21):

$$\begin{array}{ccc}& & {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha -1,\beta ,\ell }x\left(t\right)=K_{\rho ,\psi }^{\beta (2-\alpha ),\ell ,\mu }{\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{(1-\beta )(2-\alpha ),\ell ,\mu }\left[{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(2-\alpha )}}}{C}_1\right.\hfill \\ & & +\left.K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f(t,x\left(t\right))+e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\displaystyle \frac{C_0{\left(\omega \left(t\right)\right)}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]\hfill \\ & & =K_{\rho ,\psi }^{\beta (2-\alpha ),\ell ,\mu }{\partial }_{\psi }^{\rho ,\ell }\left[e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\Gamma (1+(1-\beta )(\alpha -2))C_1\right.\hfill \\ & & +\left.K_{\rho ,\psi }^{1,\ell ,\mu }{K}_{\rho ,\psi }^{1-\beta (\alpha -2),\ell ,\mu }f(t,x\left(t\right))+e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\displaystyle \frac{C_0{\left(\omega \left(t\right)\right)}^{1-\beta (2-\alpha ))}}{\Gamma (2-\beta (2-\alpha \left)\right)}}\right].\hfill \end{array}$$

Therefore, recalling (5) and (6), it follows that

$$\begin{array}{ccc}\hfill {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha -1,\beta ,\ell }x\left(t\right)& =K_{\rho ,\psi }^{\beta (2-\alpha ),\ell ,\mu }{\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{(1-\beta )(2-\alpha ),\ell ,\mu }x\left(t\right)\hfill & \\ & =K_{\rho ,\psi }^{\beta (2-\alpha ),\ell ,\mu }\left[K_{\rho ,\psi }^{1-\beta (\alpha -2),\ell ,\mu }f(t,x\left(t\right))\right.\hfill \\ & +\left.e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{\displaystyle \frac{C_0{\left(\omega \left(t\right)\right)}^{-\beta (2-\alpha ))}}{\Gamma (2-\beta (2-\alpha \left)\right)}}\right]\hfill \\ & ={\Im }_{\psi ,\mu }^{\rho ,\ell }f(t,x\left(t\right))+e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{C}_0.\hfill \end{array}$$

Consequently, in view of (6), by Proposition 1, we get

$$\begin{array}{ccc}\hfill {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\ell }x\left(t\right)& =& {\partial }_{\psi ,\mu }^{\rho ,\ell }{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha -1,\beta ,\ell }x\left(t\right)\hfill \\ & =& {\partial }_{\psi ,\mu }^{\rho ,\ell }\left[{\Im }_{\psi ,\mu }^{\rho ,\ell }f(t,x\left(t\right))+e^{-\left[{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}{C}_0\right]=f(t,x\left(t\right)).\hfill \end{array}$$

2.

For any $\beta \in [0,1]$ and $\alpha \in (0,1]$, we will show that any solution $x\in C[a,b]$ of (22) satisfies (20): Since (cf. [23,35] $K_{\rho ,\psi }^{\alpha ,\ell ,\mu }$ does not map all of $C[a,b]$ into $AC[a,b]$, we are generally unable to proceed as in the previous case when $\alpha \in (1,2]$ by making use of Proposition 1 when $\beta \in (0,1]$. Therefore, we divide our discussion into two cases.

In what follows, assume that $f(·,x(·)))\in L_1[a,b]$ for any $x\in C[a,b].$

[Case I:] When $\beta =0$, it is easy to see that, for any $f\in L_1[a,b]$,

$$\begin{array}{ccc}\hfill {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,0,\ell }x\left(t\right)& =& {\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }x\left(t\right)\hfill \\ & & ={\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }\left[{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{1-\alpha }}}{C}_2+K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f(t,x\left(t\right))\right]\hfill \\ & & ={\partial }_{\psi }^{\rho ,\ell }\left[e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\Gamma \left(\alpha \right)C_2+K_{\rho ,\psi }^{1,\ell ,\mu }f(t,x\left(t\right))\right]\hfill \\ & & ={\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{1,\ell ,\mu }f(t,x\left(t\right))={\partial }_{\psi ,\mu }^{\rho ,\ell }{\Im }_{\psi ,\mu }^{\rho ,\ell }f(t,x\left(t\right)=f(t,x\left(t\right))).\hfill \end{array}$$

[Case II:] When $\beta \in (0,1),$ then for any $f\in {\mathbb{H}}^{\alpha +\lambda ,0}[a,b],\alpha +\lambda <1$, it is easy to see that

$$\begin{array}{ccc}& & {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,\beta ,\ell }x\left(t\right)\hfill \\ & & =K_{\rho ,\psi }^{\beta (1-\alpha ),\ell ,\mu }{\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\ell ,\mu }x\left(t\right)\hfill \\ & & =K_{\rho ,\psi }^{\beta (1-\alpha ),\ell ,\mu }{\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{(1-\beta )(1-\alpha ),\ell ,\mu }\left[{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(1-\alpha )}}}{C}_2+K_{\rho ,\psi }^{\alpha ,\ell ,\mu }f(t,x\left(t\right))\right]\hfill \\ & & =K_{\rho ,\psi }^{\beta (1-\alpha ),\ell ,\mu }{\partial }_{\psi }^{\rho ,\ell }\left[e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\Gamma (1+(1-\beta )(\alpha -1))C_2\right.\hfill \\ & & +\left.K_{\rho ,\psi }^{1-\beta (1-\alpha ),\ell ,\mu }f(t,x\left(t\right))\right]=K_{\rho ,\psi }^{\beta (1-\alpha ),\ell ,\mu }{\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{1-\beta (1-\alpha ),\ell ,\mu }f(t,x\left(t\right))\hfill \\ & & =K_{\rho ,\psi }^{\beta (1-\alpha ),\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{\beta (1-\alpha ),\rho ,\ell }f(t,x\left(t\right)).\hfill \end{array}$$

[Case III:] When $\beta =1$, and $f\in AC[a,b]$, it is not difficult to see, in view of (8), that

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

Hence, by Proposition 1, it follows that ${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\ell }x\left(t\right)=f(t,x\left(t\right))$, as required.

Remark 3.

We should note that our assumptions regarding [Case III:] are too restrictive. For this reason, we can relax these assumptions by proceeding in two non-standard ways, as follows:

• Our equivalence result in [Case III:] remains true if we replace in problem (20) $\alpha \in (0,1),\beta =1$ in some or all occurrences of the derivative ${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,0,\ell }\equiv {\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }$—by ${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\ell }{\equiv }^C{\mathfrak{D}}_{\psi ,\mu }^{\alpha ,\rho ,\ell }$ (in the sense of Definition (19)).
  We insert (19) into all occurrences of the ${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\ell }$ in problem (20) with $\alpha \in (0,1),\beta =1$, always using the corresponding initial, terminal or boundary conditions for the second term in (19). Thus, we can rewrite the given problem as an equivalent one in which only ${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,0,\ell }$ occurs. This new problem can also be written in the form (20) with $\alpha \in (0,1),\beta =1$ with a slightly modified function f (similarly to [Case I.]). This modified function evidently satisfies all the requirements of [Case I.], and thus, the equivalence follows from [Case I.].
• Let $\alpha \in (0,1),\beta =1$ and define the new operator $I_{\rho ,\psi }^{\alpha ,\ell ,\mu }:=K_{\rho ,\psi }^{1,\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{1-\alpha ,\rho ,\ell }.$ Apply the following by $I_{\rho ,\psi }^{\alpha ,\ell ,\mu }$ on both sides of (20):

  $$I_{\rho ,\psi }^{\alpha ,\ell ,\mu }{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\ell }x\left(t\right)=K_{\rho ,\psi }^{1,\ell ,\mu }\left({\mathfrak{D}}_{\psi ,\mu }^{1-\alpha ,\rho ,\ell }{K}_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }\right){\partial }_{\psi }^{\rho ,\ell }x\left(t\right)=I_{\rho ,\psi }^{\alpha ,\ell ,\mu }f(t,x\left(t\right)),t\in [a,b].$$
  Therefore, based on Lemma 5 and Proposition 1, we formally arrive at

  $$x\left(t\right)=C_0{e}^{\left[{\displaystyle -{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du}\right]}+K_{\rho ,\psi }^{1,\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{1-\alpha ,\rho ,\ell }f(t,x\left(t\right)),t\in [a,b].$$

  (23)
  Conversely, if $x\in C[a,b]$ solves (23) for some $f\in {\mathbb{H}}^{\alpha +\lambda ,0}[a,b],\alpha +\lambda <1$, then

  $${\partial }_{\psi }^{\rho ,\ell }x\left(t\right)={\mathfrak{D}}_{\psi ,\mu }^{1-\alpha ,\rho ,\ell }f(t,x\left(t\right)).$$
  Therefore, our assumption that $f\in {\mathbb{H}}^{\alpha +\lambda ,0}[a,b]$, where $\alpha +\lambda <1$ along with Corollary 2, results in

  $${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{\alpha ,1,\ell }x\left(t\right)=K_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }{\partial }_{\psi }^{\rho ,\ell }x\left(t\right)=\left(K_{\rho ,\psi }^{1-\alpha ,\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{1-\alpha ,\rho ,\ell }\right)f(t,x\left(t\right))=f(t,x\left(t\right)).$$

We summarize the above investigations as follows:

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

• For any $f\in L_1[a,b]$, the problem (20) and its corresponding integral form are equivalent whenever $\alpha \in (1,2]$.
• For any $f\in {\mathbb{H}}^{\alpha +\lambda ,0}[a,b]$, the problem (20) and its corresponding integral form are equivalent for any $\alpha \in (0,1),$ whenever $\alpha +\lambda <1$.

Motivated by the work established by many authors (see, e.g., [2,46,47,48,50,51,52]), we contribute to the literature by introducing and analyzing the following Langevin-type problem involving Hilfer fractional derivatives of two different orders:

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

(24)

where $t\in [a,b]$, $C\in \mathbb{R}$, $\beta \in [0,1]$, and ${\alpha }_1,{\alpha }_2\in (0,1)$, combined with the appropriate initial or boundary conditions. We will only consider the most interesting case when ${\alpha }_1,{\alpha }_2\in (0,1)$.

As claimed above, the abovementioned papers and many others are devoted to solving problems with such equations, wherein the proportional operators need to be introduced and justified. However, there is only one operator and one function $\psi$ used in these studies. Our paper is, in contrast, devoted to a broader study of a class of proportional operators, so we restrict our attention to the problem of equivalence of differential and integral problems. This is a basic tool for this study. We made sure that the studies are correct by using integral forms. Readers can choose which problem to study; this choice can be based on the boundary conditions. Choosing one specific application would not fully reflect the goals of this paper.

Let us (formally) convert the problem (24) into a corresponding integral form. Obviously, for any $\beta \in [0,1)$, we have

$${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,\beta ,\ell }x\left(t\right)+Cx\left(t\right)={\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(1-{\alpha }_1)}}}{C}_0+K_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right)),$$

where $\beta \in [0,1)$, ${\alpha }_1,{\alpha }_2\in (0,1)$. Hence,

$$\begin{array}{ccc}\hfill x\left(t\right)& =& {\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(1-{\alpha }_2)}}}{C}_1-C{K}_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }x\left(t\right)+K_{\rho ,\psi }^{{\alpha }_1+{\alpha }_2,\ell ,\mu }f(t,x\left(t\right))\hfill \\ & & +{\displaystyle \frac{\Gamma (\beta +(1-\beta ){\alpha }_1)}{\Gamma (\beta +(1-\beta ){\alpha }_1+{\alpha }_2)}}{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(1-{\alpha }_1)-{\alpha }_2}}}{C}_0,\hfill \end{array}$$

where $\beta \in [0,1)$, ${\alpha }_1,{\alpha }_2\in (0,1)$. Since we are seeking a Hölder continuous solution to (24), we set $C_1=0$. Thus, the integral form corresponding to (24) with $\beta \in [0,1)$ and ${\alpha }_1,{\alpha }_2\in (0,1)$ is

$$\begin{array}{ccc}\hfill x\left(t\right)& =& K_{\rho ,\psi }^{{\alpha }_1+{\alpha }_2,\ell ,\mu }f(t,x\left(t\right))-C{K}_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }x\left(t\right)\hfill \\ \hfill & +& {\displaystyle \frac{\Gamma (\beta +(1-\beta ){\alpha }_1)}{\Gamma (\beta +(1-\beta ){\alpha }_1+{\alpha }_2)}}\left\{\begin{array}{cc}{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(1-{\alpha }_1)-{\alpha }_2}}}{C}_0,\hfill & (1-\beta )(1-{\alpha }_1)\le {\alpha }_2,\hfill \\ 0,\hfill & (1-\beta )(1-{\alpha }_1)>{\alpha }_2.\hfill \end{array}\right.\hfill \end{array}$$

(25)

On the other hand, when $\beta =1$, we argue similarly as in (23) and arrive (formally) at

$${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\ell }x\left(t\right)+Cx\left(t\right)=C_0^{\prime }{e}^{\left[{\displaystyle -{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du}\right]}+I_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right)),$$

where $t\in [a,b]$, ${\alpha }_1,{\alpha }_2\in (0,1)$. Operating by $I_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }$ yields

$$\begin{array}{ccc}& & I_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\ell }x\left(t\right)+C{I}_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }x\left(t\right)\hfill \\ & & =I_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }\left(C_0^{\prime }{e}^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}+I_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right))\right)\hfill \\ & & ={\displaystyle \frac{C_0^{\prime }}{\Gamma \left({\alpha }_2\right)}}{K}_{\rho ,\psi }^{1,\ell ,\mu }\left[{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{1-{\alpha }_2}}}\right]+I_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }{I}_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right))\hfill \\ & & ={\displaystyle \frac{C_0^{\prime }}{\Gamma (1+{\alpha }_2)}}{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{-{\alpha }_2}}}+I_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }{I}_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right)).\hfill \end{array}$$

Using the semi-group property and Proposition 1, we have the following, in view of Corollary 2:

$$\begin{array}{ccc}\hfill I_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }{I}_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right))& =& K_{\rho ,\psi }^{1,\ell ,\mu }{\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }{K}_{\rho ,\psi }^{1,\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{1-{\alpha }_1,\rho ,\ell }f(t,x\left(t\right))\hfill \\ & =& K_{\rho ,\psi }^{1,\ell ,\mu }\left({\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{1,\ell ,\mu }\right)K_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{1-{\alpha }_1,\rho ,\ell }f(t,x\left(t\right))\hfill \\ & =& K_{\rho ,\psi }^{{\alpha }_1+{\alpha }_2,\ell ,\mu }\left(K_{\rho ,\psi }^{1-{\alpha }_1,\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{1-{\alpha }_1,\rho ,\ell }\right)f(t,x\left(t\right))\hfill \\ & =& K_{\rho ,\psi }^{{\alpha }_1+{\alpha }_2,\ell ,\mu }f(t,x\left(t\right)).\hfill \end{array}$$

Consequently, since

$$\begin{array}{ccc}\hfill I_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\ell }x\left(t\right)& =& K_{\rho ,\psi }^{1,\ell ,\mu }\left({\mathfrak{D}}_{\psi ,\mu }^{1-{\alpha }_2,\rho ,\ell }{K}_{\rho ,\psi }^{1-{\alpha }_2,\ell ,\mu }\right){\partial }_{\psi }^{\rho ,\ell }x\left(t\right)=K_{\rho ,\psi }^{1,\ell ,\mu }{\partial }_{\psi }^{\rho ,\ell }x\left(t\right)\hfill \\ & =& x\left(t\right)-C_1^{\prime }{e}^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]},\hfill \end{array}$$

the integral equation below still follows:

$$\begin{array}{ccc}\hfill x\left(t\right)& =& I_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }{I}_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right))\hfill \\ & & -C{I}_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }x\left(t\right)+\left(C_1^{\prime }+{\displaystyle \frac{C_0^{\prime }{\left(\omega \left(t\right)\right)}^{{\alpha }_2}}{\Gamma (1+{\alpha }_2)}}\right)e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]},\hfill \end{array}$$

(26)

where ${\alpha }_1,{\alpha }_2\in (0,1)$. Now, we will conclude our discussion using the following lemma.

Lemma 7.

If the assumptions of Theorem 2 hold, then for any $F\in {\mathcal{H}}_0^{\lambda +\alpha }[a,b]$ and sufficiently small ρ, the linear fractional integral equations

$$x\left(t\right)=F(t,x)-\nu K_{\rho ,\psi }^{\alpha ,\ell ,\mu }x\left(t\right),t\in [a,b],\nu \in \mathbb{R},F\in {\mathcal{H}}_0^{\lambda +\alpha }[a,b],\lambda +\alpha <1,$$

(27)

$$x\left(t\right)=F(t,x)-\nu I_{\rho ,\psi }^{\alpha ,\ell ,\mu }x\left(t\right),t\in [a,b],\nu \in \mathbb{R},F\in {\mathcal{H}}_0^{1+\lambda -\alpha }[a,b],\lambda <\alpha ,$$

(28)

admit Hölder continuous solutions $x\in {\mathcal{H}}_0^{\lambda +\alpha }[a,b]$ and ${\mathcal{H}}_0^{1+\lambda -\alpha }[a,b]$, respectively.

Proof. 

Consider the little Hölder space $\left({\mathcal{H}}_0^{\lambda +\alpha }[a,b],{\left[·\right]}_{\lambda +\alpha }\right)$, of functions satisfying $x\left(a\right)=0$, endowed with the norm

$${\left[u\right]}_{\lambda +\alpha }:=\underset{t,s\in [a,b],t\ne s}{sup}\left\{{\displaystyle \frac{|u\left(t\right)-u\left(s\right)|}{{|t-s|}^{\lambda +\alpha }}}\right\},u\in {\mathcal{H}}_0^{\lambda +\alpha }[a,b].$$

Define $T_1:{\mathcal{H}}_0^{\lambda +\alpha }[a,b]\to {\mathcal{H}}_0^{\lambda +\alpha }[a,b]$ by

$$T_{1}x\left(t\right)=F(t,x)-\nu K_{\rho ,\psi }^{\alpha ,\ell ,\mu }x\left(t\right),t\in [a,b],\nu \in \mathbb{R}.$$

Since ${\mathcal{H}}_0^{\lambda +\alpha }[a,b]\subset {\mathcal{H}}_0^{\lambda }[a,b]$, then Theorem 2 tells us that $T_1$ is well defined. Also, for every $x,y\in {\mathcal{H}}_0^{\lambda +\alpha }[a,b]$, there is a constant $M>0$ such that

$$\begin{array}{ccc}\hfill {\left[T_{1}x-T_{1}y\right]}_{\lambda +\alpha }& =& {\left[T\{x-y\}\right]}_{\lambda +\alpha }\le {\left[F(t,x)-F(t,y)\right]}_{\lambda +\alpha }+\left|\nu \right|{\left[K_{\rho ,\psi }^{\alpha ,\ell ,\mu }x-K_{\rho ,\psi }^{\alpha ,\ell ,\mu }y\right]}_{\lambda +\alpha }\hfill \\ & \le & M{\left[x-y\right]}_{\lambda +\alpha }.\hfill \end{array}$$

Therefore, since the pair $\left({\mathcal{H}}_0^{\lambda +\alpha }[a,b],{\left[·\right]}_{\lambda +\alpha }\right)$ forms a complete space, then by the Banach contraction principle, for sufficiently small $\nu$, the operator $T_1$ has a (unique) fixed point $x\in {\mathcal{H}}_0^{\lambda +\alpha }[a,b].$

Similarly, if we define $T_2:{\mathcal{H}}_0^{1+\lambda -\alpha }[a,b]\to {\mathcal{H}}_0^{1+\lambda -\alpha }[a,b]$ by

$$T_{2}x\left(t\right)=F(t,x)-\nu K_{\rho ,\psi }^{1,\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{1-\alpha ,\rho ,\ell }x\left(t\right),t\in [a,b],\nu \in \mathbb{R},$$

and then, by applying Theorem 3, ${\mathfrak{D}}_{\psi ,\mu }^{1-\alpha ,\rho ,\ell }\in {\mathbb{H}}^{\lambda ,0}[a,b]$ for any $x\in {\mathcal{H}}_0^{1+\lambda -\alpha }[a,b]$. Consequently, $K_{\rho ,\psi }^{1,\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{1-\alpha ,\rho ,\ell }\in C_0^1[a,b]\subset {\mathbb{H}}^{\lambda +\alpha ,0}[a,b]$, and so $T_2$ is well defined. Therefore, for a sufficiently small $\nu$, the operator $T_2$ admits a (unique) fixed point $x\in {\mathcal{H}}_0^{1+\lambda -\alpha }[a,b].$  □

Now, we can generalize Conclusion 1 by repeating the same proofs to show the equivalence between the Langevin-type problem of two different fractional orders (24) and the corresponding integral forms on Hölder spaces of some critical orders. Indeed, we have the following conclusion.

Conclusion 2. Let $\psi \in C^1[a,b]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Assume that $\lambda ,\rho \in (0,1)$, $\mu \ge 0$ and $\beta \in [0,1]$. If $f(·,x(·))\in {\mathbb{H}}^{\lambda +\beta (1-min\{{\alpha }_1,{\alpha }_2\}),0}[a,b]$ for any $x\in C[a,b]$, then the problem (24) and its corresponding integral form are equivalent for any ${\alpha }_1,{\alpha }_2\in (0,1)$ whenever $\lambda +\beta (1-min\{{\alpha }_1,{\alpha }_2\})<1$.

Proof. 

We will omit the details since they are nearly identical to those in the proofs of Conclusion 1, with a few necessary changes. First, note that under the assumption that $f\in {\mathbb{H}}^{\lambda +\beta (1-min\{{\alpha }_1,{\alpha }_2\}),0}[a,b]$, Theorem 2 implies that

$$K_{\rho ,\psi }^{{\alpha }_1+{\alpha }_2,\ell ,\mu }f\in {\mathbb{H}}^{\lambda +\beta (1-min\{{\alpha }_1,{\alpha }_2\}){\alpha }_1+{\alpha }_2,0}[a,b]\subset {\mathbb{H}}^{\lambda +\beta (1-min\{{\alpha }_1,{\alpha }_2\}),0}[a,b]\subset {\mathbb{H}}^{\lambda +\beta (1-{\alpha }_2\}),0}[a,b].$$

Therefore, by Lemma 7, the integral Equations (25) and (26) admit Hölder solutions of some critical orders less than one:

1.

Let $\beta \in [0,1)$ and assume that $x\in {\mathbb{H}}^{\lambda +\beta (1-{\alpha }_2\}),0}[a,b]$ solves (25). Then, in view of Corollary 2 and the semi-group property, it is easy to see that

$$\begin{array}{ccc}& & {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,\beta ,\ell }x\left(t\right){=}^H{\mathrm{D}}_{\psi ,\rho ,\mu }^{{\alpha }_2,\beta ,\ell }\left[K_{\rho ,\psi }^{{\alpha }_1+{\alpha }_2,\ell ,\mu }f(t,x\left(t\right))-C{K}_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }x\left(t\right)\right.\hfill \\ & +& \left.{\displaystyle \frac{\Gamma (\beta +(1-\beta ){\alpha }_1)}{\Gamma (\beta +(1-\beta ){\alpha }_1+{\alpha }_2)}}\left\{\begin{array}{cc}{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(1-{\alpha }_1)-{\alpha }_2}}}{C}_0,\hfill & (1-\beta )(1-{\alpha }_1)\le {\alpha }_2,\hfill \\ 0,\hfill & (1-\beta )(1-{\alpha }_1)>{\alpha }_2.\hfill \end{array}\right.\right].\hfill \end{array}$$

From (15), we know that

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

Therefore,

$${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,\beta ,\ell }{}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,\beta ,\ell }\left\{{\displaystyle \frac{e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}}{{\left(\omega \left(t\right)\right)}^{(1-\beta )(1-{\alpha }_1)-{\alpha }_2}}}\right\}=0.$$

Since $x\in {\mathbb{H}}^{\lambda +\beta (1-{\alpha }_2\}),0}[a,b]$, it follows from Corollary 2 that

$${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,\beta ,\ell }{K}_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }x=K_{\rho ,\psi }^{\beta (1-{\alpha }_2),\ell ,\mu }{\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{1-\beta (1-{\alpha }_2),\ell ,\mu }x=K_{\rho ,\psi }^{\beta (1-{\alpha }_2),\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{\beta (1-{\alpha }_2),\rho ,\ell }x=x.$$

Similarly,

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

Therefore,

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

As a result, we obtain the following differential problem:

$$\begin{array}{ccc}\hfill {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,\beta ,\ell }\left({}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,\beta ,\ell }+C\right)x\left(t\right)& =& {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,\beta ,\ell }{K}_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right))\hfill \\ & =& K_{\rho ,\psi }^{\beta (1-{\alpha }_1),\ell ,\mu }{\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{1-\beta (1-{\alpha }_1),\ell ,\mu }f(t,x\left(t\right))\hfill \\ & =& K_{\rho ,\psi }^{\beta (1-{\alpha }_1),\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{\beta (1-{\alpha }_1),\rho ,\ell }f(t,x\left(t\right))=f(t,x\left(t\right)),\hfill \end{array}$$

as required.

2.

Let $\beta =1$ and assume that $x\in {\mathbb{H}}^{\lambda +1-{\alpha }_2,0}[a,b]$ solves (26). Then, in view of the semi-group property, it is easy to see that

$${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\ell }{I}_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }x=K_{\rho ,\psi }^{1-{\alpha }_2,\ell ,\mu }\left({\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{1,\ell ,\mu }\right){\mathfrak{D}}_{\psi ,\mu }^{1-{\alpha }_2,\rho ,\ell }x=K_{\rho ,\psi }^{1-{\alpha }_2,\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{1-{\alpha }_2,\rho ,\ell }x=x,$$

$$\begin{array}{ccc}\hfill {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\ell }{I}_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }{I}_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right))& =& K_{\rho ,\psi }^{1-{\alpha }_2,\ell ,\mu }\left({\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{1,\ell ,\mu }\right){\mathfrak{D}}_{\psi ,\mu }^{1-{\alpha }_2,\rho ,\ell }{I}_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right))\hfill \\ & =& K_{\rho ,\psi }^{1-{\alpha }_2,\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{1-{\alpha }_2,\rho ,\ell }{I}_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right))=I_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right)).\hfill \end{array}$$

Also, it is easy to see that

$${}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\ell }\left(C_1^{\prime }+{\displaystyle \frac{C_0^{\prime }{\left(\omega \left(t\right)\right)}^{{\alpha }_2}}{\Gamma (1+{\alpha }_2)}}\right)e^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}=C_0^{\prime }{e}^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}.$$

Therefore,

$$\begin{array}{ccc}\hfill {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\ell }x\left(t\right)& =& {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\ell }{I}_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }{I}_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right))-C^H{\mathrm{D}}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\ell }{I}_{\rho ,\psi }^{{\alpha }_2,\ell ,\mu }x\left(t\right)\hfill \\ & & {+}^H{\mathrm{D}}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\ell }{\displaystyle \frac{C_0^{\prime }{\left(\omega \left(t\right)\right)}^{{\alpha }_2}}{\Gamma (1+{\alpha }_2)}}{e}^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}\hfill \\ & =& I_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right))-Cx\left(t\right)+C_0^{\prime }{e}^{\left[-{\int }_0^{\psi \left(t\right)}\left(\ell \left(\rho ,{\psi }^{-1}\left(u\right)\right)+\mu \right)du\right]}.\hfill \end{array}$$

Consequently, we arrive at the following differential problem:

$$\begin{array}{ccc}\hfill {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,1,\ell }\left({}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_2,1,\ell }+C\right)x\left(t\right)& =& {}^H\mathrm{D}_{\psi ,\rho ,\mu }^{{\alpha }_1,1,\ell }{I}_{\rho ,\psi }^{{\alpha }_1,\ell ,\mu }f(t,x\left(t\right))+0\hfill \\ & =& K_{\rho ,\psi }^{1-{\alpha }_1,\ell ,\mu }\left({\partial }_{\psi }^{\rho ,\ell }{K}_{\rho ,\psi }^{1,\ell ,\mu }\right){\mathfrak{D}}_{\psi ,\mu }^{1-{\alpha }_1,\rho ,\ell }f(t,x\left(t\right))\hfill \\ & =& K_{\rho ,\psi }^{1-{\alpha }_1,\ell ,\mu }{\mathfrak{D}}_{\psi ,\mu }^{1-{\alpha }_1,\rho ,\ell }f(t,x\left(t\right))=f(t,x\left(t\right)),\hfill \end{array}$$

as required.

□

5. Conclusions

This paper examines a family of operators that combines classical and fractional-order operators. Due to their construction, these operators are not typical fractional-order operators, despite the characteristics typically assumed for such operators. Nevertheless, these approaches enable us to study a broad class of operators in a unified manner. Specifically, we study the existence of inverse operators for differential and integral operators while preserving an important feature: changes in the regularity of the operator’s values with respect to the domain.

This is a critical problem when studying different classes of boundary problems for differential equations and transforming them into integral equations. This issue is often overlooked in existing classical problems and fields of such operators. In this paper, we examine generalized proportional operators on various function spaces, including absolutely continuous functions, as well as generalized Hölder and Orlicz spaces. These spaces are better suited to the equivalence of differential and integral forms.

This paper presents results on the equivalence of several classes of boundary problems, such as terminal problems, with appropriate integral forms in appropriately chosen function spaces. We emphasize this frequently overlooked aspect of the case in the paper.

An interesting topic for further study is the close relationship between the function spaces that ensure the equivalence of differential and integral problems and the parameters of these operators.