Regularity Results for Hybrid Proportional Operators on Hölder Spaces

Abstract

Recently, a new type of derivative has been introduced, known as Caputo proportional derivatives. These are motivated by the applications of such derivatives (which are a generalization of Caputo’s standard fractional derivative) and the need to incorporate such calculus into the research on operators. The investigation therefore focuses on the equivalence of differential and integral problems for proportional calculus problems. The operators are always studied in the appropriate function spaces. Furthermore, the investigation extends these results to encompass the more general notion of Hilfer hybrid derivatives. The primary aim of this study is to preserve the maximal regularity of solutions for this class of problems. To this end, we consider such operators not only in spaces of absolutely continuous functions, but also in particular in little Hölder spaces. It is widely acknowledged that these spaces offer a natural framework for the study of classical Riemann–Liouville integral operators as inverse operators with derivatives of fractional order. This paper presents a comprehensive study of this problem for proportional derivatives and demonstrates the application of the obtained results to Langevin-type boundary problems.

1. Introduction and Preliminaries

Proportional calculus is a developing discipline in applied mathematics (see, for example, [1,2,3,4] and the references therein for background on these topics). Its applications justify its adoption, as do other versions of calculus, especially with studies of the appropriate regularity of solutions to the problems it describes. Its applications include the fractional design of heating and cooling models and the study of incompressible and viscous fluid flows. In [5,6], the methodology was employed in the context of construction of heating and cooling models, with the objective of investigating unsteady and incompressible viscous flows of fluids. The paper [7] discusses, on the basis of proportional calculus, a Brinkman-type fluid model containing hybrid nanoparticles. Moreover, utilizing the hybrid Caputo-proportional operator, the authors of [3] explore the epidemiology of childhood diseases.

It is also important to note that this proportional calculus cannot be treated as a fractional calculus since it does not satisfy certain desiderata proposed for this calculus (see [8,9,10,11,12]). However, precisely because of the practical applications and, above all, the interesting properties of these operators, it is worthwhile studying them in the same context as the classical fractional operators. This facilitates the reader’s decision regarding the applicability of these operators in various models. It is important to note that, in the context of classical Cauchy problems, the utilization of these operators is not recommended [8].

However, it is worth emphasizing that although the differential operators under discussion share certain similarities with classical fractional-order operators, they do not qualify as fractional operators from a theoretical standpoint. As with any type of fractional calculus, there are some differences from classical calculus. Nevertheless, the investigation into the existence of inverse operators for such a defined derivative, and the function spaces in which such operators are well defined, is a matter of interest. It is imperative to note that our primary interest lies in the investigation of spaces in which the derived integral operators exhibit a similar improving character to that observed in classical integral operators. In this regard, it is possible to undertake a comparative analysis of these operators and fractional order operators.

The combination of the proportional operator and the standard Caputo fractional derivative has resulted in the proposal of a proportional Caputo operator. It is noteworthy that this particular operator, designated as fractional order $\alpha$, can be regarded as an interpolation between a function and its derivative. Consequently, one might anticipate a Hölder regularity of the operator’s values, similar to that observed for classical fractional order operators. However, to the best of our knowledge, no such study has been conducted. This gap in existing research is the subject of this study.

The focus will now be directed towards the problem under investigation, with the reader being referred to the literature for a comprehensive overview of the subject matter of proportional calculus. Building on the findings of this research, our study will include an introduction to equivalence problems with $\psi$-Hilfer generalized proportional derivatives (see Section 5).

We note that this case should not be regarded as a universal tool for research with fractional-order operators (cf. [8,11]). Nevertheless, given the above-mentioned practical applications and the intriguing possibility of treating such operators in a unified manner with other fractional-order operators, it is worthwhile to study them and thus demonstrate their properties, i.e., where their use is appropriate.

In the context of differential or integral calculus, a significant challenge arises in determining the maximum possible regularity of the solutions to the problems under study. In each situation, the existence of solutions is investigated, followed by an examination of their regularity. The latter is achieved by specifying the appropriate function space in which to find the solutions. For operators of fractional order, as established in Hardy and Littlewood’s paper [13], these are Hölder spaces (cf. also [14,15,16]). It is already known that the solutions of differential problems of fractional order are not only continuous or continuous in certain weighted spaces, but, first of all, they satisfy the condition of the Hölder of the order associated with the order of the derivative. The case of the space of absolutely continuous functions will also be considered.

It is important to note that no attempts have yet been made to determine the regularity of solutions to the proportional calculus. As we will prove in the paper, it is possible to construct inverse operators under certain conditions, namely, in little Hölder spaces.

Importantly, it is for such a derivative and integral, the definitions of which in a sense allow interpolation between the original function and its derivative or integral, that one should study.

An example of a practical application of the study of equations with proportional derivatives through their equivalent integral forms can be found in [3], where this method is applied to the study of modeling childhood disease epidemics. In such cases, Hölder spaces prove to be beneficial. It should be emphasized that correct results can only be obtained in Hölder spaces, otherwise the expected regularity of the solutions is lost. The development of this calculus is driven by practical motivations, and it is essential to establish a solid foundation that enables the treatment of such operators in a manner analogous to that of classical fractional order operators.

The results of this paper are complemented by results for the widely studied problem of equations of the Langevin type, for which we apply the results obtained for the right and left inverses of operators with proportional derivatives of the Caputo type (cf. [17]). This is a problem in which proportional derivatives naturally arise. Therefore, the present paper will examine the problem with the objective of identifying the maximum possible regularity of solutions. However, this is the first study of this type of calculus. This allows us to complete the results obtained so far and to eliminate errors in the previous results that arise from the bad choice of the spaces in which the operators operate. Our results will allow the proportional calculus to be treated in the broader context of operator studies with applications to fractional order calculus.

In the following pages, $C^1[a,b]$ will be considered as a Banach space of continuously differentiable functions on $[a,b],0\le a<b<\infty$. Also, the pair $\left({\mathcal{H}}^{\lambda }[a,b],{\parallel ·\parallel }_{\lambda }\right)$, $\lambda \in (0,1)$ denotes the Hölder space endowed with the norm

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

Also, if $f\left(a\right)=0$, we write $f\in {\mathcal{H}}_0^{\lambda }[a,b]$. However, the pair $\left({\mathcal{H}}_0^{\lambda }[a,b],{\left[·\right]}_{\lambda }\right)$ becomes a Banach space. We need the following well-known fact about the relations of spaces: for $0<{\lambda }_1<{\lambda }_2<1$, then

$$C^1[a,b]\subset {\mathcal{H}}^1[a,b]\subset {\mathcal{H}}^{{\lambda }_2}[a,b]\subset {\mathcal{H}}^{{\lambda }_1}[a,b]\subset C[a,b].$$

It is such inclusions of spaces, together with the properties of these embeddings, that will determine the significance and applications of the results obtained in this paper.

By convention, $\left(L_p[a,b],{∥·∥}_q\right)1\le p\le \infty$ will denote the standard Banach space measurable functions $f:[a,b]\to \mathbb{R}$, where ${\left|f\right|}^p$ is a Lebesgue integrable on $[a,b]$, and $L_{\infty }[a,b]$ denotes the Banach space of real-valued essentially bounded and measurable functions $f:[a,b]\to \mathbb{R}$. We consider that the pairs $p,q\in [1,\infty ]$ are conjugate exponents, namely, $p,q$ are connected by the relation $1/p+1/q=1$ for $1<p<\infty$ and with the convention that $1/\infty =0$.

Since we propose to include in the study of proportional derivatives also generalized operators with differentiation and integration with respect to another function, let us define the class of functions with respect to which we will perform such operations. 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 $\psi \left(a\right)=0$.

Definition 1 

(see [18,19,20]). Let $\alpha \in (0,1)$ and $n\in \mathbb{N}:=\{0,1,2,\cdots \}$. We accept the following definitions:

1.

(Riemann–Liouville fractional integral) We define the ψ-Riemann–Liouville fractional integral applied to a function $f\in L_1[a,b]$ of order $\alpha >0$ by

$${\mathbb{I}}_{\psi }^{\alpha }f\left(t\right):={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_a^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{\alpha -1}f\left(s\right){\psi }^{\prime }\left(s\right)ds,$$

(1)

where $(-\infty \le a<b\le \infty )$. For completeness, we also define ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(a\right):=0$.

2.

(Riemann–Liouville fractional derivative) We define the ψ-Riemann–Liouville fractional derivative of order $n+\alpha$, applied to a function $f\in L_1[a,b]$ by

$${\mathfrak{D}}_{\psi }^{n+\alpha }f\left(t\right)={\Delta }^n{\mathfrak{D}}_{\psi }^{\alpha }f\left(t\right),where{\mathfrak{D}}_{\psi }^{\alpha }:=\Delta {\mathbb{I}}_{\psi }^{1-\alpha },with\Delta :={\displaystyle \frac{1}{{\psi }^{\prime }}}{\displaystyle \frac{d}{dt}}.$$

(2)

For an appreciation of the role and importance of the $\psi$-function selection ability in practical applications, we recommend the paper [21].

A very important role in this paper will be played by the ’intermediate’ functions ${\mathsf{\ell }}_1, {\mathsf{\ell }}_2:[0,1]\times [a,b]\to [0,\infty )$, about which we will make the following assumptions:

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

(3)

In this connection, assume that

$${\mathsf{\ell }}_1(·,t),{\mathsf{\ell }}_2(·,·)\mathrm{are} \mathrm{continuous} \mathrm{and}\forall \alpha \in (0,1),{\left({\mathsf{\ell }}_1(\alpha ,·)\right)}^{-1},{\left({\mathsf{\ell }}_2(\alpha ,·)\right)}^{-1}\in L_q[a,b]$$

(4)

for some $q\ge 1$. Also, define

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

Correspondingly, let $\mathsf{\ell }(·,·)={\mathsf{\ell }}_2(·,·)/{\mathsf{\ell }}_1(·,·)\in L_1[a,b]$ and note that

$$∥\mathsf{\ell }∥:={\int }_a^b|\mathsf{\ell }\left(\alpha ,s\right)|ds\le \underset{t\in [a,b]}{\max }\left|{\mathsf{\ell }}_2(\alpha ,t)\right|{\left[{\int }_a^b{\left|{\displaystyle \frac{1}{{\mathsf{\ell }}_1(\alpha ,s)}}\right|}^{q}ds\right]}^{{\displaystyle \frac{1}{q}}}{(b-a)}^{{\displaystyle \frac{1}{p}}}=L_0{L}_1{(b-a)}^{{\displaystyle \frac{1}{p}}}.$$

(5)

A simple example of such functions might be ${\mathsf{\ell }}_1(\alpha ,t)=\alpha t^{1-\alpha }$, ${\mathsf{\ell }}_2(\alpha ,t)=(1-\alpha )t^{\alpha }$, $\alpha \in (0,1)$ and $t\in [0,1]$. In this case, (4) holds with $q=1$, more precisely, (4) holds for any $q\in \left[1,\min \{1/\alpha ,1/(1-\alpha \left)\right\}\right]$. Furthermore, ${\left({\mathsf{\ell }}_1(\alpha ,·)\right)}^{-1}\left(t\right)={\alpha }^{-1}·t^{\alpha -1}\in L^q[0,1]$ whenever $q>{\displaystyle \frac{1}{1-\alpha }}$ and ${\left({\mathsf{\ell }}_2(\alpha ,·)\right)}^{-1}\left(t\right)={\displaystyle \frac{1}{1-\alpha }}·t^{-\alpha }\in L^q[0,1]$ for for any $q<{\displaystyle \frac{1}{\alpha }}$. Meanwhile, (4) holds with $q=\infty$ if, for example, ${\mathsf{\ell }}_1(\alpha ,t)=\alpha e^{(1-\alpha )t}$ and ${\mathsf{\ell }}_2(\alpha ,t)=(1-\alpha )e^{\alpha t}$, $\alpha \in (0,1)$.

We are ready to recall the following generalizations of the standard differential operator (see, e.g., [1,2,4] and the references there for an overview of these topics). Importantly, this treatment of the topic provides not only a generalization of the classical proportional derivative, but also a fractional order calculus with derivatives depending on another function.

Define the following non-fractional differential operator of order $\alpha \in (0,1)$

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

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

This operator is related to the large and expanding theory of proportional derivatives. Since these operators combine features of fractional order operators, and the general class of differential operators, we can treat them like any other operator used in the research. In particular, we are interested in spaces where they are well-defined and invertible. Therefore, they are not necessarily “fractional” in any formal sense (does not have all the properties expected of fractional-order derivatives), so we have called them “non-fractional”. However, it has interesting properties as an operator, similar to fractional order operators, and will be investigated by us as expected for this class of operators.

Remark 1. 

According to (3), (4), we can “interpolate” between f and $\Delta f$. In fact, we have, formally,

$$\underset{\alpha \to 0}{lim}{\partial }_{p,\psi }^{\alpha ,\mathsf{\ell }}f=f,and\underset{\alpha \to 1}{lim}{\partial }_{\psi }^{\alpha ,\mathsf{\ell }}f=\Delta f.$$

Let us recall that the inverse of the operator ${\partial }_{\psi }^{\alpha ,\mathsf{\ell }}$ is given by

$${\displaystyle {\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right):={\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\frac{f\left(s\right){\psi }^{\prime }\left(s\right)ds}{{\mathsf{\ell }}_1(\alpha ,s)}.}$$

2. Basic Properties

In this context, according to (3), we can conclude (still only formally) that

$$\underset{\alpha \to 1}{lim}{\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f={\int }_a^{t}f\left(s\right){\psi }^{\prime }\left(s\right)ds.$$

Furthermore, if the assumptions (3) and (4) are satisfied, then for any $\alpha \in (0,1)$, and $t\in [a,b]$, we have

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

Therefore,

$${\partial }_{\psi }^{\alpha ,\mathsf{\ell }}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}\right\}=e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}\left[{\mathsf{\ell }}_2(\alpha ,t)-{\mathsf{\ell }}_2(\alpha ,t)\right]=0.$$

(6)

We will also have a look at the invertibility condition for the differential and integral operators under consideration. Composition in one direction is fairly straightforward. It can be proved even in the space of absolutely continuous functions.

Proposition 1 

(cf. [1]). Let $\alpha \in (0,1)$. If the assumptions (3) and (4) are satisfied, then for any $f\in L_p[a,b],p={\displaystyle \frac{1}{1-q}}$, we have

$${\partial }_{\psi }^{\alpha ,\mathsf{\ell }}{\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f=fa.e.,on[a,b].$$

Proof. 

The Hölder inequality implies that ${\displaystyle \frac{f(·)}{{\mathsf{\ell }}_1(\alpha ,·)}}f(·)·{\mathsf{\ell }}_1{(\alpha ,·)}^{-1}\in L_1[a,b]$, and then it is not so hard to see that

$${\partial }_{\psi }^{\alpha ,\mathsf{\ell }}{\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right)={\partial }_{\psi }^{\alpha ,\mathsf{\ell }}\left({\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{f\left(s\right){\psi }^{\prime }\left(s\right)ds}{{\mathsf{\ell }}_1(\alpha ,s)}}\right)=f\left(t\right).$$

□

However, there are different cases to consider for the inverse order of operations, depending on the function space we are operating on. As claimed above, in the case of absolutely continuous functions, the situation is relatively simple. We can prove the following:

Proposition 2. 

If the assumptions (3) and (4) are satisfied, then for any $f\in AC[a,b]$ and $\alpha \in (0,1)$, we have

$${\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}{\partial }_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right)=f\left(t\right)-e^{{\displaystyle -\left[{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}f\left(a\right),t\in [a,b].$$

Proof. 

Since $f\in AC[a,b]$, then f admits an integrable derivative defined a.e. on $[a,b]$. Hence,

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

Therefore, integration by parts provides

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

as required. □

And now, we present one of the most important results for the operators under consideration, concerning their (maximum possible) regularity. As proved by Hardy and Littlewood [13], for integral operators of fractional order, their images lie in certain Hölder spaces, and so it is this type of space that should be considered in general problems of searching for reciprocally inverse operators. This provides the basis for studying the equivalence of differential and integral problems, in which case the starting point for us will be proportional calculus. For the class of integral operators under consideration, we will now present a Hardy–Littlewood-type result showing the improving nature of integral operators.

Remark 2. 

This result in particular, as far as the full invertibility of the integration and differentiation operations is concerned, shows the importance of function spaces in which the functions satisfy the condition $f\left(a\right)=0$. Due to the different aims of the study of such operators, we will consider both spaces of the kind mentioned above (in our case, little Hölder spaces) and add general considerations. Of course, the zero value of the function at a is not so important—a fixed value is sufficient. By examining $g\left(x\right)=f\left(x\right)-f\left(a\right)$, we have functions that satisfy the initial condition with a zero value.

Theorem 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 $\alpha \in (0,1)$, $p\in \left(\max \left\{2,{\displaystyle \frac{1}{\alpha }}\right\},{\displaystyle \frac{2}{\alpha }}\right]$ and assume that ${\mathsf{\ell }}_1, {\mathsf{\ell }}_2$ satisfy the assumptions (3) and (4) with $q=p/(p-1)$. If there are $\lambda , {\zeta }_1<{\zeta }_2\in (0,1)$ such that

$$\alpha \in [{\zeta }_1,{\zeta }_2]\subset \left({\displaystyle \frac{1}{p}},{\displaystyle \frac{2}{p}}\right]and{\zeta }_2-{\zeta }_1\le \alpha -\lambda ,$$

then

$${\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda }[a,b]\to {\mathcal{H}}_0^{\lambda -\alpha +{\displaystyle \frac{2}{p}}+{\zeta }_2-{\zeta }_1}[a,b]\subset {\mathcal{H}}_0^{\lambda -\alpha +{\displaystyle \frac{2}{p}}}[a,b].$$

Proof. 

Let $f\in {\mathcal{H}}_0^{\lambda }[a,b]$, $h>0$ so that $a\le t<t+h\le b$ and note that $|f\left(t\right)-f\left(s\right)|\le {\left[f\right]}_{\lambda }{|t-s|}^{\lambda }$ for any $t,s\in [a,b].$ Therefore,

$$\begin{array}{ccc}& & {\displaystyle {\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f(t+h)-{\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right)={\int }_a^{t+h}{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\frac{f\left(s\right){\psi }^{\prime }\left(s\right)ds}{{\mathsf{\ell }}_1(\alpha ,s)}}\hfill \\ & & -{\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{f\left(s\right){\psi }^{\prime }\left(s\right)ds}{{\mathsf{\ell }}_1(\alpha ,s)}}\hfill \\ & =& {\displaystyle {\int }_{-h}^{t-a}{e}^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\frac{f(t-s){\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}}\hfill \\ & & -{\int }_0^{t-a}{e}^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{f(t-s){\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}}\hfill \end{array}$$

$$\begin{array}{ccc}& =& {\displaystyle {\int }_0^{t-a}\left\{e^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}-e^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\right\}}\hfill \\ & & ·{\displaystyle \frac{f(t-s){\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}}\hfill \\ & +& {\displaystyle {\int }_{-h}^0{e}^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\frac{f(t-s){\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}}\hfill \end{array}$$

$$\begin{array}{ccc}& =& {\displaystyle {\int }_0^{t-a}\left\{e^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}-e^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\right\}}\hfill \\ & & ·{\displaystyle \frac{(f(t-s)-f\left(t\right)){\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}}\hfill \\ & & {\displaystyle +{\int }_{-h}^0{e}^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\frac{(f(t-s)-f\left(t\right)){\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}}\hfill \\ & & {\displaystyle +{\int }_{-h}^{t-a}{e}^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\frac{(f\left(t\right)-f\left(a\right)){\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}}\hfill \\ & & {\displaystyle -{\int }_0^{t-a}{e}^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\frac{(f\left(t\right)-f\left(a\right)){\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}.}\hfill \end{array}$$

Hence,

$$\left|{\mathfrak{I}}_p^{\alpha }f(t+h)-{\mathfrak{I}}_p^{\alpha }f\left(t\right)\right|\le A+B+C,$$

where

$$\begin{array}{l}{\displaystyle A:=\left|\left\{{\int }_{-h}^{t-a}{e}^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}-{\int }_0^{t-a}{e}^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\right\}\right.}\\ \left.·{\displaystyle \frac{(f\left(t\right)-f\left(a\right)){\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}}\right|,\\ {\displaystyle B:={\int }_{-h}^0{e}^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\frac{|f(t-s)-f\left(t\right)|{\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)},}\\ {\displaystyle C:={\int }_0^{t-a}\left\{e^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}-e^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\right\}}\\ ·{\displaystyle \frac{|f(t-s)-f\left(t\right)|{\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}}.\end{array}$$

By noting that,

$$\begin{array}{ccc}\hfill A& \le & {\left[f\right]}_{\lambda }{(t-a)}^{\lambda }\left|{\int }_{-h}^{t-a}{\displaystyle \frac{e^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}{\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}}\right.\hfill \\ & & \left.-{\int }_0^{t-a}{\displaystyle \frac{e^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}{\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}}\right|.\hfill \end{array}$$

Using the substitutions

$$s\mapsto {\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\mathrm{and}s\mapsto {\int }_{\psi (t-s)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du,$$

respectively, and denoted by $a\left(\tau \right)={\displaystyle {\int }_0^{\psi \left(\tau \right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}$, we obtain

$$\begin{array}{ccc}\hfill A& \le & {\left[f\right]}_{\lambda }{(t-a)}^{\lambda }\left|{\displaystyle {\int }_0^{a(t+h)}\frac{e^{-s}ds}{{\mathsf{\ell }}_2(\alpha ,t-s)}-{\int }_0^{a\left(t\right)}\frac{e^{-s}ds}{{\mathsf{\ell }}_2(\alpha ,t-s)}}\right|\hfill \\ & =& {\left[f\right]}_{\lambda }{(t-a)}^{\lambda }\left|{\displaystyle {\int }_{a\left(t\right)}^{a(t+h)}\frac{e^{-s}ds}{{\mathsf{\ell }}_2(\alpha ,t-s)}}\right|\hfill \\ & \le & {\left[f\right]}_{\lambda }{(t-a)}^{\lambda }{\left({\int }_{a\left(t\right)}^{a(t+h)}ds\right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{t-a(t+h)}^{t-a\left(t\right)}{\left|{\displaystyle \frac{1}{{\mathsf{\ell }}_2(\alpha ,t-s)}}\right|}^{q}ds\right)}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

Since

$${\left({\int }_{t-a(t+h)}^{t-a\left(t\right)}{\left|{\displaystyle \frac{1}{{\mathsf{\ell }}_2(\alpha ,t-s)}}\right|}^{q}ds\right)}^{{\displaystyle \frac{1}{q}}}\le {\left({\int }_{{\displaystyle t-{\int }_0^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}^{{\displaystyle t-{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}{\left|{\displaystyle \frac{1}{{\mathsf{\ell }}_2(\alpha ,s)}}\right|}^{q}ds\right)}^{{\displaystyle \frac{1}{q}}}\le L_2,$$

it follows, by the aid of (5),

$$\begin{array}{ccc}\hfill A& \le & {\left[f\right]}_{\lambda }{(t-a)}^{\lambda }{L}_2{\left({\int }_{\psi \left(t\right)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le & {\left[f\right]}_{\lambda }{(t-a)}^{\lambda }{L}_2{\left({\int }_t^{t+h}\mathsf{\ell }\left(\alpha ,u)\right){\psi }^{\prime }\left(u\right)du\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le & {\left[f\right]}_{\lambda }{(t-a)}^{\lambda }{L}_2{\left(L_0{L}_1∥{\psi }^{\prime }∥\right)}^{{\displaystyle \frac{1}{p}}}{h}^{{\displaystyle \frac{2}{p}}}\hfill \\ & \le & {\left[f\right]}_{\lambda }{L}_2{\left(L_0{L}_1∥{\psi }^{\prime }∥\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & & ·\left\{\begin{array}{c}{\displaystyle h^{\lambda +\frac{2}{p}}=h^{\frac{2}{p}+\lambda -\alpha +{\zeta }_2-{\zeta }_1}{h}^{\alpha +{\zeta }_1-{\zeta }_2}\le h^{\frac{2}{p}+\lambda -\alpha +{\zeta }_2-{\zeta }_1}{(b-a)}^{\alpha +{\zeta }_1-{\zeta }_2},}\hfill \\ \mathrm{for}t-a\le h<b-a,\hfill \\ {\displaystyle {(t-a)}^{\lambda -\alpha +{\zeta }_2-{\zeta }_1}{(t-a)}^{\alpha +{\zeta }_1-{\zeta }_2}{h}^{\frac{2}{p}}\le h^{\frac{2}{p}+\lambda -\alpha +{\zeta }_2-{\zeta }_1}{(b-a)}^{\alpha +{\zeta }_1-{\zeta }_2},}\hfill \\ \mathrm{for}h\le t-a\le b-a.\hfill \end{array}\right.\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}\hfill B& \le & {\left[f\right]}_{\lambda }{\int }_{-h}^0{e}^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{{\psi }^{\prime }(t-s)ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}}\hfill \\ & =& {\displaystyle {\left[f\right]}_{\lambda }{\int }_0^{{\displaystyle {\int }_{\psi \left(t\right)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}\frac{s^{\lambda }{e}^{-s}ds}{{\mathsf{\ell }}_2(\alpha ,t-s)}}\hfill \\ & \le & {\left[f\right]}_{\lambda }{L}_2{\left({\displaystyle {\int }_0^{{\displaystyle {\int }_{\psi \left(t\right)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}{s}^{p\lambda }ds}\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }{L}_2}{\sqrt[p]{p\lambda +1}}}{\left({\displaystyle {\int }_{\psi \left(t\right)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}\right)}^{\lambda +{\displaystyle \frac{1}{p}}}\hfill \end{array}$$

$$\begin{array}{ccc}& \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }{L}_2{\left(L_0{L}_1∥{\psi }^{\prime }∥\right)}^{\lambda +\frac{1}{p}}}{\sqrt[p]{p\lambda +1}}}{h}^{\lambda +{\displaystyle \frac{2}{p}}}\hfill \\ & =& {\displaystyle \frac{{\left[f\right]}_{\lambda }{L}_2{\left(L_0{L}_1∥{\psi }^{\prime }∥\right)}^{\lambda +\frac{1}{p}}}{\sqrt[p]{p\lambda +1}}}{h}^{\alpha +{\zeta }_1-{\zeta }_2}{h}^{{\displaystyle \frac{2}{p}}+\lambda -\alpha +{\zeta }_2-{\zeta }_1}\hfill \\ & \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }{L}_2{\left(L_0{L}_1∥{\psi }^{\prime }∥\right)}^{\lambda +\frac{1}{p}}{(b-a)}^{\alpha +{\zeta }_1-{\zeta }_2}}{\sqrt[p]{p\lambda +1}}}{h}^{{\displaystyle \frac{2}{p}}+\lambda -\alpha +{\zeta }_2-{\zeta }_1}.\hfill \end{array}$$

It remains to estimate:

$${\displaystyle C\le {\left[f\right]}_{\lambda }{\int }_0^{t-a}\left|e^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}-e^{{\displaystyle \left[-{\int }_{\psi (t-s)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\right|\frac{s^{\lambda }ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}.}$$

By the integral mean value theorem, there exist ${\xi }_1\in (\psi (t-s),\psi (t+h))$ and ${\xi }_2\in (\psi (t-s),\psi \left(t\right))$ so that

$${\int }_{\psi (t-s)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du=\mathsf{\ell }(\alpha ,{\xi }_1)(\psi (t+h)-\psi (t-s)),$$

$${\int }_{\psi (t-s)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du=\mathsf{\ell }(\alpha ,{\xi }_2)(\psi \left(t\right)-\psi (t-s)).$$

Also, there exist ${\eta }_1\in (t-s,t+h)$ and ${\eta }_2\in t-s,t)$ such that

$$\psi (t+h)-\psi (t-s)={\psi }^{\prime }\left({\eta }_1\right)(h+s),\psi \left(t\right)-\psi (t-s)={\psi }^{\prime }\left({\eta }_2\right)s.$$

Therefore,

$$\begin{array}{ccc}& & {\displaystyle C\le {\left[f\right]}_{\lambda }{\int }_0^{t-a}\left|e^{{\displaystyle -\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)(h+s)}}-e^{{\displaystyle -\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)s}}\right|\frac{s^{\lambda }ds}{{\mathsf{\ell }}_1(\alpha ,t-s)}}\hfill \\ & \le & {\left[f\right]}_{\lambda }{\left({\int }_0^{t-a}{\displaystyle \frac{ds}{|{\mathsf{\ell }}_1{(\alpha ,t-s)|}^q}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & & ·{\left({\int }_0^{t-a}{\left|e^{{\displaystyle -\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)(h+s)}}-e^{{\displaystyle -\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)s}}\right|}^p{s}^{\lambda p}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le & {\left[f\right]}_{\lambda }{\left({\int }_a^t{\displaystyle \frac{ds}{|{\mathsf{\ell }}_1{(\alpha ,s)|}^q}}\right)}^{{\displaystyle \frac{1}{q}}}{\left({\int }_0^{{\displaystyle \frac{t-a}{h}}}{\left|e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)(s+1)h}-e^{-\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)sh}\right|}^p{s}^{\lambda p}{h}^{\lambda p+1}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le & {\left[f\right]}_{\lambda }{L}_1{h}^{\lambda +{\displaystyle \frac{1}{p}}}{\left({\int }_0^{{\displaystyle \frac{t-a}{h}}}{\left|e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)(s+1)h}-e^{-\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)sh}\right|}^p{s}^{\lambda p}ds\right)}^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

So if $t-a<h<b-a,$ we get

$$\begin{array}{ccc}\hfill C& \le & {\left[f\right]}_{\lambda }{L}_1{h}^{\lambda +{\displaystyle \frac{1}{p}}}{\left({\int }_0^{\infty }{\left|e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)(s+1)h}-e^{-\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)sh}\right|}^p{s}^{\lambda p}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le & {\left[f\right]}_{\lambda }{L}_1{h}^{{\displaystyle \frac{1}{p}}+\lambda -\alpha +{\zeta }_2-{\zeta }_1}{h}^{\alpha +{\zeta }_2-{\zeta }_1}{\left({\int }_0^1{\left|e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)(s+1)h}-e^{-\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)sh}\right|}^p{s}^{\lambda p}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =& {\left[f\right]}_{\lambda }{L}_1{h}^{{\displaystyle \frac{2}{p}}+\lambda -\alpha +{\zeta }_2-{\zeta }_1}\mathbf{I},\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill \mathbf{I}& :=& h^{\alpha +{\zeta }_1-{\zeta }_2-{\displaystyle \frac{1}{p}}}{\left({\int }_0^{\infty }{\left|e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)(s+1)h}-e^{-\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)sh}\right|}^p{s}^{\lambda p}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =& h^{\alpha +{\zeta }_1-{\zeta }_2-{\displaystyle \frac{1}{p}}}{\left({\int }_0^1{\left|e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h}-1\right|}^p{e}^{-p\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)sh}{s}^{\lambda p}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =& h^{\alpha +{\zeta }_1-{\zeta }_2-{\displaystyle \frac{1}{p}}}\left|e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h}-1\right|{\left({\int }_0^1{e}^{-p\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)sh}{s}^{\lambda p}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le & h^{\alpha +{\zeta }_1-{\zeta }_2-{\displaystyle \frac{1}{p}}}\left|e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h}-1\right|\hfill \\ & =& \left|{\displaystyle \frac{e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h}}{h^{-\alpha +{\zeta }_2-{\zeta }_1+\frac{1}{p}}}}-{\displaystyle \frac{1}{h^{-\alpha +{\zeta }_2-{\zeta }_1+\frac{1}{p}}}}\right|.\hfill \end{array}$$

Since ${\xi }_1\to {\xi }_2,{\eta }_1\to {\eta }_2$ (as $h\to 0$), taking into account that

$${\displaystyle \underset{h\to 0}{lim}\frac{e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h}-1}{h^{-\alpha +{\zeta }_2-{\zeta }_1+\frac{1}{p}}}=\underset{h\to 0}{lim}\frac{-e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h}\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h^{\alpha -\frac{1}{p}+{\zeta }_1-{\zeta }_2+1}}{\frac{1}{p}-\alpha +{\zeta }_2-{\zeta }_1}=0,}$$

we conclude that $\mathbf{I}<\infty$. Similarly, if $t-a>h,$ we obtain

$$\begin{array}{ccc}\hfill C& \le & {\left[f\right]}_{\lambda }{L}_1{h}^{\lambda +{\displaystyle \frac{1}{p}}}{\left({\int }_0^{\infty }{\left|e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)(s+1)h}-e^{-\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)sh}\right|}^p{s}^{\lambda p}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le & {\left[f\right]}_{\lambda }{L}_1{h}^{\lambda -\alpha +{\displaystyle \frac{2}{p}}+{\zeta }_2-{\zeta }_1}{h}^{\alpha +{\zeta }_1-{\zeta }_2-{\displaystyle \frac{1}{p}}}\hfill \\ & & ·{\left({\int }_0^{\infty }{\left|e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)(s+1)h}-e^{-\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)sh}\right|}^p{s}^{\lambda p}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =& {\left[f\right]}_{\lambda }{L}_1{h}^{\lambda -\alpha +{\displaystyle \frac{2}{p}}+{\zeta }_2-{\zeta }_1}\mathbf{J},\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill \mathbf{J}& :=& h^{\alpha +{\zeta }_1-{\zeta }_2-{\displaystyle \frac{1}{p}}}{\left({\int }_0^{\infty }{\left|e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)(s+1)h}-e^{-\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)sh}\right|}^p{s}^{\lambda p}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =& h^{\alpha +{\zeta }_1-{\zeta }_2-{\displaystyle \frac{1}{p}}}{\left({\int }_0^{\infty }{\left|e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h}-1\right|}^p{e}^{-p\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)sh}{s}^{\lambda p}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =& h^{\alpha +{\zeta }_1-{\zeta }_2-{\displaystyle \frac{1}{p}}}\left|e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h}-1\right|{\left({\int }_0^{\infty }{e}^{-p\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)sh}{s}^{\lambda p}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =& h^{\alpha +{\zeta }_1-{\zeta }_2-{\displaystyle \frac{1}{p}}}\left|{\displaystyle \frac{e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h}-1}{{\left(\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h\right)}^{\lambda +\frac{1}{p}}}}\right|{\left({\int }_0^{\infty }{e}^{-ps}{s}^{\lambda p}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =& {\displaystyle \frac{{\mathrm{\Gamma }}^{\frac{1}{p}}(1+p\lambda )}{p^{\lambda +\frac{1}{p}}}}\left|{\displaystyle \frac{e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h}-1}{{\left(\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h\right)}^{\lambda +\frac{2}{p}-\alpha +{\zeta }_2-{\zeta }_1}}}\right|\hfill \\ & =& {\displaystyle \frac{{\mathrm{\Gamma }}^{\frac{1}{p}}(1+p\lambda )}{p^{\lambda +\frac{1}{p}}}}\left|{\displaystyle \frac{e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h}}{{\left(\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)\right)}^{\lambda +\frac{1}{p}}{h}^{\lambda +\frac{2}{p}-\alpha +{\zeta }_2-{\zeta }_1}}}\right.\hfill \\ & & \left.-{\displaystyle \frac{1}{{\left(\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)\right)}^{\lambda +\frac{1}{p}}{h}^{\lambda +\frac{2}{p}-\alpha +{\zeta }_2-{\zeta }_1}}}\right|.\hfill \end{array}$$

Since ${\xi }_1\to {\xi }_2,{\eta }_1\to {\eta }_2$ (as $h\to 0$), taking into account that

$$\begin{array}{ccc}& & \underset{h\to 0}{lim}{\displaystyle \frac{e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h}{\left(\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)\right)}^{-\lambda -\frac{1}{p}}-{\left(\mathsf{\ell }(\alpha ,{\xi }_2){\psi }^{\prime }\left({\eta }_2\right)\right)}^{-\lambda -\frac{1}{p}}}{h^{\lambda +\frac{2}{p}-\alpha +{\zeta }_2-{\zeta }_1}}}\hfill \\ & & =\underset{h\to 0}{lim}{\displaystyle \frac{-e^{-\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)h}{h}^{\alpha -\lambda -\frac{2}{p}+{\zeta }_1-{\zeta }_2+1}}{(\lambda +\frac{2}{p}-\alpha +{\zeta }_2-{\zeta }_1){\left(\mathsf{\ell }(\alpha ,{\xi }_1){\psi }^{\prime }\left({\eta }_1\right)\right)}^{\lambda +\frac{1}{p}-1}}}=0,\hfill \end{array}$$

we conclude that $\mathbf{J}<\infty$. Hence ${\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f(t+h)-{\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right)\le K\left(\alpha \right){\left[f\right]}_{\lambda }{h}^{\lambda -\alpha +{\displaystyle \frac{2}{p}}+{\zeta }_2-{\zeta }_1}$, for some constant $K\left(\alpha \right)>0$. Namely,

$${\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda }[a,b]\to {\mathcal{H}}_0^{\lambda -\alpha +{\displaystyle \frac{2}{p}}+{\zeta }_2-{\zeta }_1}[a,b],{\left[{\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f\right]}_{\lambda -\alpha +{\displaystyle \frac{2}{p}}+{\zeta }_2-{\zeta }_1}\le K\left(\alpha \right){\left[f\right]}_{\lambda }.$$

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3. Hybrid Caputo-Proportional Derivatives

And now we will look at derivatives that combine the advantages of Caputo derivatives and proportional calculus (bearing in mind some objections to treating such derivatives as fractional order derivatives).

Let us restrict our attention in this section to the case of absolutely continuous functions. The following definition of the proportional Caputo operator (which is a hybrid operator combining the definitions of the proportional derivative and the Caputo fractional derivative) is derived from Baleanu et al. [2].

Definition 2 

((hybrid Caputo-proportional derivative) [1,2,3]). The hybrid Caputo-proportional derivative of order $n+\alpha ,\alpha \in (0,1)$, $n\in \mathbb{N},$ applied to the function $f\in AC[a,b]$ is defined as

$${\displaystyle \frac{d_{p,\psi }^{n+\alpha ,\mathsf{\ell }}}{d{t}^{n+\alpha }}}f\left(t\right)={\left({\partial }_{\psi }^{\alpha ,\mathsf{\ell }}\right)}^n{\displaystyle \frac{d_{p,\psi }^{\alpha ,\mathsf{\ell }}}{d{t}^{\alpha }}}f\left(t\right),where{\displaystyle \frac{d_{p,\psi }^{\alpha ,\mathsf{\ell }}}{d{t}^{\alpha }}}:={\mathbb{I}}_{\psi }^{1-\alpha }{\partial }_{\psi }^{\alpha ,\mathsf{\ell }}.$$

As we mentioned in the Introduction, an example of using the hybrid Caputo-proportional (for modeling childhood disease epidemics) can be found in [3].

By definition, we have

$${\displaystyle \frac{d_{p,\psi }^{n+\alpha ,\mathsf{\ell }}}{d{t}^{n+\alpha }}}f\left(t\right)={\left({\partial }_{\psi }^{\alpha ,\mathsf{\ell }}\right)}^n{\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_a^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha }\left[{\mathsf{\ell }}_2(\alpha ,t){\psi }^{\prime }\left(s\right)f\left(s\right)+{\mathsf{\ell }}_1(\alpha ,t)f^{\prime }\left(s\right)\right]ds.$$

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According to Remark 2 in [2], we can treat this definition as an interpolation between derivative and integral.

$$\underset{\alpha \to 0}{lim}{\displaystyle \frac{d_{p,t}^{\alpha ,0}}{d{t}^{\alpha }}}f\left(t\right)={\int }_0^{t}f\left(s\right)ds,\underset{\alpha \to 1}{lim}{\displaystyle \frac{d_{p,t}^{\alpha ,0}}{d{t}^{\alpha }}}f\left(t\right)=f^{\prime }\left(t\right).$$

We will therefore treat this derivative of $d_{p,\psi }^{\alpha ,\mathsf{\ell }}/d{t}^{\alpha }$ as an interpolation, in a sense, between the integral and the derivative of a function. In particular, when ${\mathsf{\ell }}_2$ and ${\mathsf{\ell }}_2$ are independent of $t\in [a,b]$, then ${\displaystyle \frac{d_{p,\psi }^{n+\alpha ,\mathsf{\ell }}}{d{t}^{n+\alpha }}}$ is known as a constant proportional-Caputo operator and is defined as

$${\displaystyle \frac{d_{p,\psi }^{n+\alpha ,\mathsf{\ell }}}{d{t}^{n+\alpha }}}f\left(t\right)={\mathsf{\ell }}_2\left(\alpha \right){\left({\partial }_{\psi }^{\alpha ,\mathsf{\ell }}\right)}^n{\mathbb{I}}_{\psi }^{1-\alpha }f+{\mathsf{\ell }}_1\left(\alpha \right){\left({\partial }_{\psi }^{\alpha ,\mathsf{\ell }}\right)}^n{\displaystyle \frac{d_{p,\psi }^{\alpha ,\mathsf{\ell }}}{d{t}^{\alpha }}}f.$$

In the following, we will construct an inverse operator that corresponds to the hybrid Caputo-proportional derivative.

Definition 3 

([1,2]). The integral inverse operator of order $\alpha \in (n-1,n),n\in \mathbb{N}$ is defined by

$${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right):={\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{{\mathfrak{D}}_{\psi }^{n-\alpha }f\left(s\right){\psi }^{\prime }\left(s\right)}{{\mathsf{\ell }}_1(\alpha ,s)}}ds={\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}{\mathfrak{D}}_{\psi }^{n-\alpha }f\left(t\right).$$

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Remark 3. 

Unfortunately, the integral operator ${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}$, for $\alpha \in (0,1)$ is not necessarily defined out of the space of absolutely continuous functions (and even in some Hölder spaces): According to Fact 3 in [22] (see also [15,23]), there exists a continuous (more precisely: Hölder continuous) f such that for any $\alpha \in (0,1)$, the function ${\mathbb{I}}_{\psi }^{\alpha }f$ is not absolutely continuous. Therefore, ${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f={\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}\Delta {\mathbb{I}}_{\psi }^{\alpha }f$ is “meaningless”.

So we cannot expect the existence of an inverse operator on every space. The results should depend on the domain of the operator, which we will fully justify in this paper.

As a consequence of Remark 3, outside the space of absolutely continuous functions $AC[a,b]$, the operator ${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}$, $\alpha \in (0,1)$ is not necessarily the right inverse of the operator $d_p^{\alpha }/d{t}^{\alpha }$, even in some Hölder spaces. This leads us to a lack of equivalence between differential and integral forms of the proportional Caputo-type problems outside $AC[a,b]$. Therefore, some results obtained (see, for example, [2,4]) are not correct.

Nevertheless, we can derive the following conclusion from Theorem 1:

Corollary 1. 

If the assumptions of Theorem 1 hold, then for any $p\in \left(\max \left\{2,{\displaystyle \frac{1}{\alpha }}\right\},{\displaystyle \frac{2}{\alpha }}\right]$, we have

$${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}:A{C}_0^p[a,b]\to {\mathcal{H}}_0^{{\displaystyle \frac{1}{p}}+{\zeta }_2-{\zeta }_1}[a,b]\subset {\mathcal{H}}_0^{{\displaystyle \frac{1}{p}}}[a,b],$$

where

$$A{C}_0^p[a,b]:=\left\{g\in AC[a,b]:g\left(a\right)=0,g^{\prime }\in L_p[a,b],\right\}.$$

Proof. 

Let $g\in A{C}_0^p[a,b]$. It is well known that a real-valued function is absolutely continuous; it can be obtained as an integral from its a.e. defined derivative, i.e, for the function g, we obtain

$$g\left(t\right)=g\left(a\right)+{\int }_a^t{g}^{\prime }\left(\theta \right)d\theta =g\left(a\right)+{\int }_a^t{\displaystyle \frac{g^{\prime }\left(\theta \right)}{{\psi }^{\prime }\left(\theta \right)}}{\psi }^{\prime }\left(\theta \right)d\theta ={\mathbb{I}}_{\psi }^1{\displaystyle \frac{g^{\prime }\left(t\right)}{{\psi }^{\prime }\left(t\right)}}.$$

Therefore,

$$\begin{array}{ccc}\hfill {\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}g\left(t\right)& =& {\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}\left({\displaystyle \frac{1}{{\psi }^{\prime }}}{\displaystyle \frac{d}{dt}}\left[{\mathbb{I}}_{\psi }^{\alpha }{\mathbb{I}}_{\psi }^1{\displaystyle \frac{g^{\prime }\left(t\right)}{{\psi }^{\prime }\left(t\right)}}\right]\right)={\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}\left({\displaystyle \frac{1}{{\psi }^{\prime }}}{\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\psi }^1{\mathbb{I}}_{\psi }^{\alpha }\left[{\displaystyle \frac{g^{\prime }\left(t\right)}{{\psi }^{\prime }\left(t\right)}}\right]\right)={\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}\mathfrak{W}\left(t\right)\hfill \\ & =& {\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{\mathfrak{W}\left(s\right){\psi }^{\prime }\left(s\right)}{{\mathsf{\ell }}_1(\alpha ,s)}}ds,\hfill \end{array}$$

where $\mathfrak{W}\left(t\right):={\mathbb{I}}_{\psi }^{\alpha }\left[{\displaystyle \frac{g^{\prime }\left(t\right)}{{\psi }^{\prime }\left(t\right)}}\right]\in L_p[a,b]$. Hence,

$$\left|{\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}g\left(t\right)\right|\le ∥{\psi }^{\prime }∥{\int }_a^t\left|{\displaystyle \frac{\mathfrak{W}\left(s\right)}{{\mathsf{\ell }}_1(\alpha ,s)}}\right|ds\to 0\mathrm{as}t\to a,\mathrm{namely},{\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}g\left(a\right)=0.$$

Since ${\displaystyle \frac{g^{\prime }(·)}{{\psi }^{\prime }(·)}}\in L_p[a,b]$, for any $p>1/\alpha$, it follows, in view of Remark 3 in [24] (see also Proposition 3.2(3) in [15]), that $\mathfrak{W}={\mathbb{I}}_{\psi }^{\alpha }\left[{\displaystyle \frac{g^{\prime }(·)}{{\psi }^{\prime }(·)}}\right]\in {\mathcal{H}}_0^{\alpha -{\displaystyle \frac{1}{p}}}[a,b]$. Now pick $p>\max \left\{2,{\displaystyle \frac{1}{\alpha }}\right\}$ and apply Theorem 1 with $\lambda =\alpha -{\displaystyle \frac{1}{p}}$; we conclude that

$${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}g={\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}\mathfrak{W}\in {\mathcal{H}}_0^{{\displaystyle \frac{1}{p}}+{\zeta }_2-{\zeta }_1}[a,b]\mathrm{as}\mathrm{claimed}.$$

□

Having the results of Theorem 1, we can prove the following corollary:

Corollary 2. 

If the assumptions of Theorem 1 hold, then

$${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda +1-\alpha }[a,b]\to {\mathcal{H}}_0^{\lambda +{\displaystyle \frac{2}{p}}-\alpha +{\zeta }_2-{\zeta }_1}[a,b]\subset {\mathcal{H}}_0^{\lambda +{\displaystyle \frac{2}{p}}-\alpha }[a,b].$$

There is also $k\ge 0$ such that ${\left[{\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f\right]}_{\lambda +{\displaystyle \frac{2}{p}}-\alpha +{\zeta }_2-{\zeta }_1}\le k\left(\alpha \right){\left[f\right]}_{\lambda +1-\alpha }$ for every $f\in {\mathcal{H}}_0^{\lambda +1-\alpha }[a,b].$

Proof. 

Let $f\in {\mathcal{H}}_0^{\lambda +1-\alpha }[a,b]$; then, by Lemma 3 in [24], we have ${\mathfrak{D}}_{\psi }^{1-\alpha }f\in {\mathcal{H}}_0^{\lambda }[a,b]$ whenever $\lambda \in (0,\alpha )$. Applying Theorem 1 yields

$${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}={\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}{\mathfrak{D}}_{\psi }^{1-\alpha }:{\mathcal{H}}_0^{\lambda +1-\alpha }[a,b]\to {\mathcal{H}}_0^{\lambda +{\displaystyle \frac{2}{p}}-\alpha +{\zeta }_2-{\zeta }_1}[a,b].$$

□

Corollary 3. 

If the assumptions of Theorem 1 hold, then for any $p\in \left(\max \left\{2,{\displaystyle \frac{1}{\alpha }}\right\},{\displaystyle \frac{2}{\alpha }}\right]$, we have,

$${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}:{\mathcal{H}}_0^{{\displaystyle \frac{1}{q}}}[a,b]\to {\mathcal{H}}_0^{{\displaystyle \frac{1}{p}}}[a,b].$$

Proof. 

Apply Corollary 2 with $\lambda =\alpha -{\displaystyle \frac{1}{p}}$; it follows that

$${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda +1-\alpha }[a,b]\equiv {\mathcal{H}}_0^{1-{\displaystyle \frac{1}{p}}}[a,b]\to {\mathcal{H}}_0^{\lambda +{\displaystyle \frac{2}{p}}-\alpha }[a,b]\equiv {\mathcal{H}}_0^{{\displaystyle \frac{1}{p}}}[a,b].$$

□

Example 1. 

Let $\alpha \in (0,1),\psi \left(t\right)=t,t\in [0,1]$ and define ${\mathsf{\ell }}_1(\alpha ,t):=\alpha t^{1-\alpha },{\mathsf{\ell }}_2(\alpha ,t):=(1-\alpha )t^{\alpha },$ (hence, $\mathsf{\ell }(\alpha ,t)={\displaystyle \frac{1-\alpha }{\alpha }}{t}^{2\alpha -1}$). Obviously, ${\mathsf{\ell }}_1, {\mathsf{\ell }}_2$ satisfy the assumptions (3) and (4) for any $q\in \left(1,\min \left\{2,{\displaystyle \frac{1}{\alpha }},{\displaystyle \frac{1}{1-\alpha }}\right\}\right)$. Now define $f\left(t\right):=e^{-t}{t}^{\lambda },\lambda \in (0,\alpha ),t\in [0,1]$ and note that $f\in {\mathcal{H}}_0^{\lambda }[0,1]$: Obviously, let $t,s\in [0,1]$ such that $0\le s<t$ and set $x=s/t\in [0,1)$. Then we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\left|f\right(t)-f(s\left)\right|}{{|t-s|}^{\lambda }}}& \le & e^{-t}{\displaystyle \frac{|t^{\lambda }-s^{\lambda }|}{{|t-s|}^{\lambda }}}+s^{\lambda }{\displaystyle \frac{|e^{-t}-e^{-s}|}{{|t-s|}^{\lambda }}}\le {\displaystyle \frac{|1-x^{\lambda }|}{{(1-x)}^{\lambda }}}+{\displaystyle \frac{∥e^{-t}∥|t-s|}{{|t-s|}^{\lambda }}}\hfill \\ & \le & {\displaystyle \frac{(1-x)}{(1-x)}}+{|t-s|}^{1-\lambda }\le 2.\hfill \end{array}$$

Set $\alpha ={\displaystyle \frac{1}{2}}$ and note that the assumptions of Theorem 1 are satisfied with $p=4,q={\displaystyle \frac{4}{3}}$. Consequently, we expect that ${\mathfrak{I}}_{p,t}^{{\displaystyle \frac{1}{2}},1}f\in {\mathcal{H}}_0^{\lambda }[0,1]$. Obviously, we have

$$\begin{array}{ccc}\hfill {\mathfrak{I}}_{p,t}^{{\displaystyle \frac{1}{2}},1}f\left(t\right)& =& {\displaystyle {\int }_0^t{e}^{{\displaystyle \left[-{\int }_s^{t}1du\right]}}\frac{2e^{-s}{s}^{\lambda }ds}{\sqrt{s}}=2{\int }_0^t{e}^{s-t}{e}^{-s}{s}^{\lambda -\frac{1}{2}}ds}\hfill \\ & =& {\displaystyle \frac{4e^{-t}}{2\lambda +1}}{t}^{\lambda +{\displaystyle \frac{1}{2}}}\in {\mathcal{H}}_0^{\lambda +{\displaystyle \frac{1}{2}}}[0,1]\subset {\mathcal{H}}_0^{\lambda }[0,1].\hfill \end{array}$$

Also, if we define $g\in {\mathcal{H}}_0^{\lambda +{\displaystyle \frac{1}{2}}}[0,1]$ by $g\left(t\right):=t^{\lambda +{\displaystyle \frac{1}{2}}},$ we expect (in view of Corollary 2) that ${\mathrm{I}}_{p,t}^{{\displaystyle \frac{1}{2}},\mathsf{\ell }}g\in {\mathcal{H}}_0^{\lambda +{\displaystyle \frac{1}{2}}}[0,1]$. Evidently, we have

$$\begin{array}{ccc}\hfill {\mathrm{I}}_{p,t}^{\alpha ,\mathsf{\ell }}g\left(t\right)& =& {\displaystyle {\mathfrak{I}}_{p,t}^{\frac{1}{2},\mathsf{\ell }}{\mathfrak{D}}_t^{\frac{1}{2}}g\left(t\right)={\int }_0^t{e}^{{\displaystyle \left[-{\int }_s^{t}1du\right]}}\frac{2{\mathfrak{D}}_s^{\frac{1}{2}}{s}^{\lambda +\frac{1}{2}}ds}{\sqrt{s}}=2\frac{\mathrm{\Gamma }(\lambda +\frac{3}{2})}{\mathrm{\Gamma }(\lambda +1)}{\int }_0^t{e}^{s-t}{s}^{\lambda -\frac{1}{2}}ds}\hfill \\ & =& 2{\displaystyle \frac{\mathrm{\Gamma }(\lambda +\frac{1}{2})}{\mathrm{\Gamma }(\lambda +1)}}\left[t^{\lambda +{\displaystyle \frac{1}{2}}}-{\int }_0^t{e}^{s-t}{s}^{\lambda +{\displaystyle \frac{1}{2}}}ds\right]\in {\mathcal{H}}_0^{\lambda +{\displaystyle \frac{1}{2}}}[0,1].\hfill \end{array}$$

The next counterexample shows that fact that the assumptions of Theorem 1 (and also Corollary 2) are really essential and that the results do not necessarily hold for arbitrary ${\mathsf{\ell }}_1,{\mathsf{\ell }}_2$ from $L_1[a,b]:$

Counterexample 1. 

Let $\alpha ={\displaystyle \frac{3}{4}},\lambda =\alpha -{\displaystyle \frac{1}{p}}$, where $p\in \left(2,{\displaystyle \frac{8}{3}}\right]\equiv \left(\max \left\{2,{\displaystyle \frac{1}{\alpha }}\right\},{\displaystyle \frac{2}{\alpha }}\right]$. Obviously, there is ${\zeta }_1<{\zeta }_2\in (0,1)$ such that

$$\alpha \in [{\zeta }_1,{\zeta }_2]\subset \left({\displaystyle \frac{1}{p}},{\displaystyle \frac{2}{p}}\right]and{\zeta }_2-{\zeta }_1\le \alpha -\lambda .$$

Then the assumptions of Corollary 2 hold whenever ${\mathsf{\ell }}_1,{\mathsf{\ell }}_2\in L_q[0,1]$ with $q\in \left(1,\min \left\{2,{\displaystyle \frac{1}{\alpha }},{\displaystyle \frac{1}{1-\alpha }}\right\}\right)$. Now define ${\mathsf{\ell }}_1(\alpha ,t):=\alpha t^{q(1-\alpha )},{\mathsf{\ell }}_2(\alpha ,t):=(1-\alpha )t^{\alpha },$ with $q=3\notin \left(1,{\displaystyle \frac{4}{3}}\right)\equiv \left(1,\min \left\{2,{\displaystyle \frac{1}{\alpha }},{\displaystyle \frac{1}{1-\alpha }}\right\}\right)$. Hence,

$${\mathsf{\ell }}_1\left({\displaystyle \frac{3}{4}},t\right)={\displaystyle \frac{3}{4}}{t}^{{\displaystyle \frac{3}{4}}},{\mathsf{\ell }}_2\left({\displaystyle \frac{3}{4}},t\right)={\displaystyle \frac{1}{4}}{t}^{{\displaystyle \frac{3}{4}}},\mathsf{\ell }\left({\displaystyle \frac{3}{4}},t\right)={\displaystyle \frac{1}{3}}.$$

Since ${\mathsf{\ell }}_1({\displaystyle \frac{3}{4}},·),{\mathsf{\ell }}_2({\displaystyle \frac{3}{4}},·)$ does not satisfy the assumptions (3) and (4), then we expect the existence of $f\in {\mathcal{H}}_0^{\lambda +1-\alpha }[0,1]\equiv {\mathcal{H}}_0^{\lambda +{\displaystyle \frac{1}{4}}}[0,1]$ such that ${\mathrm{I}}_{p,t}^{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{3}}}f\notin {\mathcal{H}}_0^{\lambda +{\displaystyle \frac{2}{p}}-{\displaystyle \frac{3}{4}}}[0,1]\equiv {\mathcal{H}}_0^{{\displaystyle \frac{1}{p}}}[0,1]$.

To see this, recall that, as claimed in [25], the translated Weierstrass function $f\left(t\right):=\omega \left(t\right)-\omega \left(0\right),t\in [0,1]$ (where ω denotes the well-known Weierstrass function) satisfies the Hölder condition of all orders less than one on $[0,1]$ with a bounded Riemann-Liouville fractional derivative of all orders less than one.

Since $M:=∥{\mathfrak{D}}_t^{{\displaystyle \frac{1}{4}}}f∥={\max }_{t\in [0,1]}|{\mathfrak{D}}_t^{{\displaystyle \frac{1}{4}}}f\left(t\right)|<\infty$, it follows in view of

$$\begin{array}{ccc}\hfill g\left(t\right)& :=& {\mathrm{I}}_{p,t}^{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{3}}}f\left(t\right)={\mathfrak{I}}_{p,t}^{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{3}}}{\mathfrak{D}}_t^{{\displaystyle \frac{1}{4}}}f\left(t\right)={\displaystyle \frac{4}{3}}{\int }_0^t{e}^{-{\displaystyle \frac{1}{3}}\left(t-s\right)}{\displaystyle \frac{{\mathfrak{D}}_s^{\frac{1}{4}}f\left(s\right)ds}{\sqrt[4]{s^3}}},\hfill \\ \hfill \left|g\right(t\left)\right|& \le & {\displaystyle \frac{4}{3}}M{\int }_0^t{\displaystyle \frac{ds}{\sqrt[4]{s^3}}}={\displaystyle \frac{16}{3}}M{t}^{{\displaystyle \frac{1}{4}}},t\in [0,1],\hfill \end{array}$$

that $g\left(0\right)=0,$. Consequently,

$${\displaystyle \frac{\left|g\right(t)-g(0\left)\right|}{{\left|t\right|}^{\lambda +\frac{2}{p}-\frac{3}{4}}}}\le {\displaystyle \frac{16}{3}}{M\left|t\right|}^{{\displaystyle \frac{1}{4}}-(\lambda +{\displaystyle \frac{2}{p}}-{\displaystyle \frac{3}{4}})}={\displaystyle \frac{15}{17}}M{\left|t\right|}^{{\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{p}}}\to \infty ,$$

for any $p\in \left(2,{\displaystyle \frac{8}{3}}\right]$.

Hence, $g\notin {\mathcal{H}}_0^{\lambda +{\displaystyle \frac{2}{p}}-{\displaystyle \frac{3}{4}}}[0,1]$, and so, the results we obtain show that Corollary 2 (also Theorem 1) has no analogue for $q=1$.

We will also be interested in the special case when $q=\infty$, e.g., the case of constant “proportional” when the functions ${\mathsf{\ell }}_1$ and ${\mathsf{\ell }}_2$ are constant with respect to t, depending only on an $\alpha$ (see, e.g., [2]): The analysis of the proof of the following theorem is the same as in the Proof of Theorem 1, with (small) necessary changes, so we omit the details.

Theorem 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 $\psi \left(a\right)=0$. Let $\alpha \in (0,1)$, and assume that ${\mathsf{\ell }}_1, {\mathsf{\ell }}_2$ satisfy the assumptions (3) and (4) with $q=\infty$. If there are $\lambda , {\zeta }_1<{\zeta }_2\in (0,1)$, such that

$$\alpha \in [{\zeta }_1,{\zeta }_2]and{\zeta }_2-{\zeta }_1\le \alpha -\lambda ,$$

then

$${\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda }[a,b]\to {\mathcal{H}}_0^{\lambda -\alpha +1+{\zeta }_2-{\zeta }_1}[a,b]\subset {\mathcal{H}}_0^{\lambda -\alpha +1}[a,b].$$

Hence,

$${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda +1-\alpha }[a,b]\to {\mathcal{H}}_0^{\lambda +1-\alpha +{\zeta }_2-{\zeta }_1}[a,b]\subset {\mathcal{H}}_0^{\lambda +1-\alpha }[a,b].$$

Proof. 

Arguing similarly as in the proof of Theorem 1, we arrive at

$$\left|{\mathfrak{I}}_p^{\alpha }f(t+h)-{\mathfrak{I}}_p^{\alpha }f\left(t\right)\right|\le A+B+C,f\in {\mathcal{H}}_0^{\lambda }[a,b],h>0\mathrm{so} \mathrm{that}a\le t<t+h\le b,$$

where

$$\begin{array}{ccc}\hfill A& \le & {\left[f\right]}_{\lambda }{L}_0{L}_1{L}_2∥{\psi }^{\prime }∥\hfill \\ \hfill ·& & \left\{\begin{array}{c}{\displaystyle h^{\lambda +1}=h^{1+\lambda -\alpha +{\zeta }_2-{\zeta }_1}{h}^{\alpha +{\zeta }_1-{\zeta }_2}\le h^{1+\lambda -\alpha +{\zeta }_2-{\zeta }_1}{(b-a)}^{\alpha +{\zeta }_1-{\zeta }_2},}\hfill \\ \mathrm{for}t-a\le h<b-a,\hfill \\ {\displaystyle {(t-a)}^{\lambda -\alpha +{\zeta }_2-{\zeta }_1}{(t-a)}^{\alpha +{\zeta }_1-{\zeta }_2}h\le h^{1+\lambda -\alpha +{\zeta }_2-{\zeta }_1}{(b-a)}^{\alpha +{\zeta }_1-{\zeta }_2},}\hfill \\ \mathrm{for}h\le t-a\le b-a.\hfill \end{array}\right.\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill B& \le & {\displaystyle \frac{{\left[f\right]}_{\lambda }{L}_2{\left(L_0{L}_1∥{\psi }^{\prime }∥\right)}^{\lambda +1}{(b-a)}^{\alpha +{\zeta }_1-{\zeta }_2}}{\lambda +1}}{h}^{1+\lambda -\alpha +{\zeta }_2-{\zeta }_1}.\hfill \end{array}$$

Similarly, there is a constant $M>0$ such that

$$C\le {\left[f\right]}_{\lambda }{L}_{1}M{h}^{\lambda -\alpha +1+{\zeta }_2-{\zeta }_1}.$$

Hence, ${\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f(t+h)-{\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right)\le K\left(\alpha \right){\left[f\right]}_{\lambda }{h}^{\lambda -\alpha +{\displaystyle \frac{2}{p}}+{\zeta }_2-{\zeta }_1}$, for some constant $K\left(\alpha \right)>0$. Namely,

$${\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda }[a,b]\to {\mathcal{H}}_0^{\lambda -\alpha +1+{\zeta }_2-{\zeta }_1}[a,b],{\left[{\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f\right]}_{\lambda -\alpha +1+{\zeta }_2-{\zeta }_1}\le K\left(\alpha \right){\left[f\right]}_{\lambda }.$$

Once again, we omit the detail since it is almost identical to that in the Proof of Theorem 1 and Corollary 2. □

Nevertheless, we can derive the following conclusion from Theorem 1:

Corollary 4. 

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 $\psi \left(a\right)=0$ and $\alpha \in (0,1)$. Assume that ${\mathsf{\ell }}_1, {\mathsf{\ell }}_2$ satisfy the assumptions (3) and (4) with $q=\infty$. Then, for any $p>1$ such that $\alpha \in \left({\displaystyle \frac{1}{p}},1-{\displaystyle \frac{1}{p}}\right]$, we have

$${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}:A{C}^p[a,b]\to {\mathcal{H}}_0^{\alpha }[a,b],$$

where

$$A{C}^p[a,b]:=\left\{g\in AC[a,b]:g^{\prime }\in L_p[a,b],\right\}.$$

Proof. 

Let $g\in A{C}^p[a,b]$. Arguing similarly as in the proof of Corollary 1, we conclude that

$$\begin{array}{ccc}\hfill {\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}g\left(t\right)& =& {\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}\left({\displaystyle \frac{1}{{\psi }^{\prime }}}{\displaystyle \frac{d}{dt}}\left[{\mathbb{I}}_{\psi }^{\alpha }g\left(a\right)+{\mathbb{I}}_{\psi }^{\alpha }{\mathbb{I}}_{\psi }^1{\displaystyle \frac{g^{\prime }\left(t\right)}{{\psi }^{\prime }\left(t\right)}}\right]\right)\hfill \\ & =& {\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}\left({\displaystyle \frac{g\left(a\right)}{\mathrm{\Gamma }\left(\alpha \right)}}{\left(\psi \left(t\right)\right)}^{\alpha -1}+{\displaystyle \frac{1}{{\psi }^{\prime }}}{\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\psi }^1{\mathbb{I}}_{\psi }^{\alpha }\left[{\displaystyle \frac{g^{\prime }\left(t\right)}{{\psi }^{\prime }\left(t\right)}}\right]\right)\hfill \\ & =& {\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{\mathfrak{W}\left(s\right){\psi }^{\prime }\left(s\right)}{{\mathsf{\ell }}_1(\alpha ,s)}}ds,\hfill \end{array}$$

where

$$\mathfrak{W}\left(t\right):={\displaystyle \frac{g\left(a\right)}{\mathrm{\Gamma }\left(\alpha \right)}}{\left(\psi \left(t\right)\right)}^{\alpha -1}+{\mathbb{I}}_{\psi }^{\alpha }\left[{\displaystyle \frac{g^{\prime }\left(t\right)}{{\psi }^{\prime }\left(t\right)}}\right]\in L_1[a,b].$$

Since $\mathfrak{W}(·)/{\mathsf{\ell }}_1(\alpha ,·)\in L_1[a,b]$, it follows

$$\left|{\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}g\left(t\right)\right|\le ∥{\psi }^{\prime }∥\left({\int }_a^t\left|{\displaystyle \frac{\mathfrak{W}\left(s\right)}{{\mathsf{\ell }}_1(\alpha ,s)}}\right|ds\right)\to 0\mathrm{as}t\to a, \mathrm{namely},{\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}g\left(a\right)=0.$$

On the other hand, we have ${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}g\left(t\right)={\displaystyle \frac{g\left(a\right)}{\mathrm{\Gamma }\left(\alpha \right)}}U\left(t\right)+V\left(t\right),$ where

$$\begin{array}{ccc}\hfill U\left(t\right)& :=& {\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{{\left(\psi \left(s\right)\right)}^{\alpha -1}{\psi }^{\prime }\left(s\right)}{{\mathsf{\ell }}_1(\alpha ,s)}}ds,\hfill \\ \hfill V\left(t\right)& :=& {\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{{\mathbb{I}}_{\psi }^{\alpha }\left[\frac{g^{\prime }\left(s\right)}{{\psi }^{\prime }\left(s\right)}\right]{\psi }^{\prime }\left(s\right)}{{\mathsf{\ell }}_1(\alpha ,s)}}ds.\hfill \end{array}$$

Since ${\displaystyle \frac{g^{\prime }(·)}{{\psi }^{\prime }(·)}}\in L_p[a,b]$, for any $p>1/\alpha$, it follows in view of Remark 3 in [24] (see also Proposition 3.2(3) in [15]), ${\mathbb{I}}_{\psi }^{\alpha }\left[{\displaystyle \frac{g^{\prime }(·)}{{\psi }^{\prime }(·)}}\right]\in {\mathcal{H}}_0^{\alpha -{\displaystyle \frac{1}{p}}}[a,b]$. Since $\alpha \in \left({\displaystyle \frac{1}{p}},1-{\displaystyle \frac{1}{p}}\right]$, then the assumptions of Theorem 2 are satisfied with $\lambda =\alpha -{\displaystyle \frac{1}{p}}$ and $[{\zeta }_1,{\zeta }_2]\subset \left(\alpha -{\displaystyle \frac{1}{2p}},\alpha +{\displaystyle \frac{1}{2p}}\right)$. Hence, we conclude that

$$V={\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}{\mathbb{I}}_{\psi }^{\alpha }\left[{\displaystyle \frac{g^{\prime }(·)}{{\psi }^{\prime }(·)}}\right]\in {\mathcal{H}}_0^{1-{\displaystyle \frac{1}{p}}}[a,b]\subseteq {\mathcal{H}}_0^{\alpha }[a,b].$$

Next, note that for any $h>0$ so that $a\le t<t+h\le b$, we have

$$\begin{array}{ccc}& & {\displaystyle U(t+h)-U\left(t\right)={\int }_a^{t+h}{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}\frac{{\left(\psi \left(s\right)\right)}^{\alpha -1}{\psi }^{\prime }\left(s\right)ds}{{\mathsf{\ell }}_1(\alpha ,s)}}\hfill \\ & -& {\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{{\left(\psi \left(s\right)\right)}^{\alpha -1}{\psi }^{\prime }\left(s\right)ds}{{\mathsf{\ell }}_1(\alpha ,s)}}\hfill \\ & =& {\int }_a^t\left[e^{{\displaystyle -{\int }_{\psi \left(s\right)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}-e^{{\displaystyle -{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}\right]{\displaystyle \frac{{\left(\psi \left(s\right)\right)}^{\alpha -1}{\psi }^{\prime }\left(s\right)ds}{{\mathsf{\ell }}_1(\alpha ,s)}}\hfill \\ & +& {\int }_t^{t+h}{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{{\left(\psi \left(s\right)\right)}^{\alpha -1}{\psi }^{\prime }\left(s\right)ds}{{\mathsf{\ell }}_1(\alpha ,s)}}\hfill \end{array}$$

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

Therefore,

$$\begin{array}{ccc}\hfill \left|U(t+h)-U\left(t\right)\right|& \le & \left|e^{{\displaystyle -{\int }_{\psi \left(t\right)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}-1\right|{\int }_a^t{\displaystyle \frac{{\left(\psi \left(s\right)\right)}^{\alpha -1}{\psi }^{\prime }\left(s\right)ds}{{\mathsf{\ell }}_1(\alpha ,s)}}\hfill \\ & & +{\int }_t^{t+h}{\displaystyle \frac{{\left(\psi \left(s\right)\right)}^{\alpha -1}{\psi }^{\prime }\left(s\right)ds}{{\mathsf{\ell }}_1(\alpha ,s)}}\hfill \\ & \le & {\displaystyle \frac{L_1}{\alpha }}\left[\left|e^{{\displaystyle -{\int }_{\psi \left(t\right)}^{\psi (t+h)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}-1\right|\left|{\left(\psi \left(s\right)\right)}^{\alpha }\right|\right.\hfill \\ & & \left.+\left|{\left(\psi (t+h)\right)}^{\alpha }-{\left(\psi \left(t\right)\right)}^{\alpha }\right|\right]\hfill \\ & \le & {\displaystyle \frac{L_1}{\alpha }}\left[\left|M_t\left(h\right)-M_t\left(0\right)\right|{∥\psi ∥}^{\alpha }+{\left|\left(\psi \right(t+h\left)\right)-\left(\psi \right(t\left)\right)\right|}^{\alpha }\right]\hfill \\ & \le & {\displaystyle \frac{L_1}{\alpha }}\left[\left|M_t\left(h\right)-M_t\left(0\right)\right|{∥\psi ∥}^{\alpha }+{\left|\underset{t\in [a,b]}{\max }{\psi }^{\prime }\left(t\right))h\right|}^{\alpha }\right],\hfill \end{array}$$

where $M_t(·):=e^{{\displaystyle -{\int }_{\psi \left(t\right)}^{\psi (t+·)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}.$

Applying the mean value theorem on $[0,h]$ for the function $M_t(·)$, we obtain

$$\left|U(t+h)-U\left(t\right)\right|\le {\displaystyle \frac{L_1}{\alpha }}\left[Mh{∥\psi ∥}^{\alpha }+h^{\alpha }{∥{\psi }^{\prime }∥}^{\alpha }\right]\le {\displaystyle \frac{L_1}{\alpha }}\left[M{h}^{\alpha }{(b-a)}^{1-\alpha }{∥\psi ∥}^{\alpha }+h^{\alpha }{∥{\psi }^{\prime }∥}^{\alpha }\right],$$

where $M:={\max }_{s\in [a,b]}\left|{\displaystyle \frac{d}{ds}}{M}_t\left(s\right)\right|.$

Hence, $U\in {\mathcal{H}}^{\alpha }[a,b]$. Consequently, given $V\in {\mathcal{H}}^{1-{\displaystyle \frac{1}{p}}}[a,b]$ and ${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}g\left(a\right)=0$, we conclude that ${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}g={\displaystyle \frac{g\left(a\right)}{\mathrm{\Gamma }\left(\alpha \right)}}U+V\in {\mathcal{H}}_0^{\alpha }[a,b]$, as claimed. □

Corollary 5. 

Let $\psi \in C^1[a,b]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Let $\alpha ,\beta \in (0,1)$ and assume that ${\mathsf{\ell }}_1, {\mathsf{\ell }}_2$ satisfy the assumptions (3) and (4) with $q=\infty$.

If there are $\lambda , {\zeta }_1<{\zeta }_2\in (0,1)$ such that

$$\alpha ,\beta \in [{\zeta }_1,{\zeta }_2]and{\zeta }_2-{\zeta }_1\le \min \{\alpha ,\beta \}-\lambda ,$$

then

$${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{\beta ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda +1-\beta }[a,b]\to {\mathcal{H}}_0^{\lambda +1-\alpha }[a,b].$$

There is also $k\ge 0$ such that ${\left[{\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{\beta ,\mathsf{\ell }}f\right]}_{\lambda +1-\alpha }\le k{\left[f\right]}_{\lambda +1-\beta }$ for every $f\in {\mathcal{H}}_0^{\lambda +1-\beta }[a,b]$.

Proof. 

Let $f\in {\mathcal{H}}_0^{\lambda +1-\beta }[a,b]$. Consider the following cases:

(1)

If $\beta \le \alpha$, then

$$1+\lambda +{\zeta }_2-{\zeta }_1-\beta \ge 1+\lambda +{\zeta }_2-{\zeta }_1-\alpha \ge 1+\lambda -\alpha$$

By Theorem 2, it follows that ${\mathrm{I}}_{\psi }^{\beta ,\mu }f\in {\mathcal{H}}_0^{1+\lambda +{\zeta }_2-{\zeta }_1-\beta }[a,b]\subseteq {\mathcal{H}}_0^{1+\lambda -\alpha }[a,b].$ Again by Theorem 2, it follows that ${\mathrm{I}}_{\psi }^{\alpha ,\mu }{\mathrm{I}}_{\psi }^{\beta ,\mu }f\in {\mathcal{H}}_0^{1+\lambda +{\zeta }_2-{\zeta }_1-\alpha }[a,b]\subset {\mathcal{H}}_0^{1+\lambda -\alpha }[a,b]$.

(2)

If $\alpha <\beta$, then by Theorem 2, it follows that ${\mathrm{I}}_{\psi }^{\beta ,\mu }f\in {\mathcal{H}}_0^{1+\lambda +{\zeta }_2-{\zeta }_1-\beta }[a,b].$ Since ${\zeta }_2-{\zeta }_1\ge \beta -\alpha$, then

$$1+\lambda +{\zeta }_2-{\zeta }_1-\beta \ge 1+\lambda -\alpha .$$

Therefore, ${\mathrm{I}}_{\psi }^{\beta ,\mu }f\in {\mathcal{H}}_0^{1+\lambda +{\zeta }_2-{\zeta }_1-\beta }[a,b]\subset {\mathcal{H}}_0^{1+\zeta -\alpha }[a,b]$. Also, by Theorem 2, we conclude that ${\mathrm{I}}_{\psi }^{\alpha ,\mu }{\mathrm{I}}_{\psi }^{\beta ,\mu }f\in {\mathcal{H}}_0^{1+\lambda -\alpha }[a,b]$.

□

The Banach fixed point theorem can be used to prove the following result about Hölder continuous solutions for linear integral equations with the generalized proportional integral operators. We will investigate Fredholm proportional integral equations of the second kind in Hölder spaces.

Lemma 1. 

Let the assumptions of Theorem 2 hold. Then, for any $F\in {\mathcal{H}}_0^{\lambda +1-\alpha }[a,b]$ and sufficiently small ρ, the linear fractional integral equation

$$x\left(t\right)=F\left(t\right)-\rho {\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}x\left(t\right),t\in [a,b],\rho \in \mathbb{R},$$

(9)

admits a Hölder continuous solution $x\in {\mathcal{H}}_0^{\lambda +1-\alpha }[a,b]$.

Proof. 

Note that to formulate the problem we are considering, we can also use the operators we have introduced. Namely, we have $[Id+\rho {\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}{\mathfrak{D}}_{\psi }^{1-\alpha }]x\left(t\right)=F\left(t\right)$ and then $[Id+\rho {\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}\Delta {\mathbb{I}}_{\psi }^{\alpha }]x\left(t\right)=F\left(t\right)$.

Consider the little Hölder space $({\mathcal{H}}_0^{\lambda +1-\alpha }[a,b],{\left[·\right]}_{\lambda +1-\alpha })$, endowed by the norm

$${\left[F\right]}_{\lambda +1-\alpha }:=\underset{t,s\in [a,b],t\ne s}{sup}\left\{{\displaystyle \frac{|F\left(t\right)-F\left(s\right)|}{{|t-s|}^{\lambda +1-\alpha }}}\right\},F\in {\mathcal{H}}_0^{\lambda +1-\alpha }[a,b].$$

Recall that in this case, the seminorm ${\left[·\right]}_{\lambda +1-\alpha }$ is, in fact, a norm on this space. Let $F\in {\mathcal{H}}_0^{\lambda +1-\alpha }[a,b]$ and define $T:{\mathcal{H}}_0^{\lambda +1-\alpha }[a,b]\to {\mathcal{H}}_0^{\lambda +1-\alpha }[a,b]$ by

$$Tx\left(t\right)=F\left(t\right)-\rho {\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}x\left(t\right),t\in [a,b],\rho \in \mathbb{R}.$$

Given Theorem 2, T becomes well defined and makes sense. Also, for every $x,y\in {\mathcal{H}}_0^{\lambda +1-\alpha }[a,b]$, there is a constant $M>0$ such that

$${\left[Tx-Ty\right]}_{\lambda +1-\alpha }={\left[T\{x-y\}\right]}_{\lambda +1-\alpha }\le \left|\rho \right|·{\left[{\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}(x-y)\right]}_{\lambda +1-\alpha }\le M\left|\rho \right|{\left[x-y\right]}_{\lambda +1-\alpha }.$$

Therefore, by the Banach contraction principle, for sufficiently small $\rho$, the operator T admits a (unique) fixed point $x\in {\mathcal{H}}_0^{\lambda +1-\alpha }[a,b].$ □

And now we will complete the consideration by studying another version of the question, closely related to certain forms of differential equations, which will allow it to be studied directly using the following lemma:

Lemma 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 $\psi \left(a\right)=0$ and $\alpha \in \left[{\displaystyle \frac{1}{2}},1\right)$. Assume that ${\mathsf{\ell }}_1, {\mathsf{\ell }}_2$ satisfy the assumptions (3) and (4) with $q=\infty$. If $p>1$ such that $\alpha \in \left({\displaystyle \frac{1}{p}},1-{\displaystyle \frac{1}{p}}\right]$, then for any $g\in A{C}^p[a,b]$, $f\in {\mathcal{H}}_0^{\alpha }[a,b]$ and sufficiently small ρ, the linear proportional integral equation

$$x\left(t\right)=f\left(t\right)+{\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}g\left(t\right)-\rho {\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}x\left(t\right),t\in [a,b],\rho \in \mathbb{R},$$

(10)

admits a Hölder continuous solution $x\in {\mathcal{H}}_0^{\alpha }[a,b]$.

Remark 4. 

Note that the assumptions of Lemma 1 are satisfied if we choose $\lambda \in (0,2\alpha -1],\alpha \ge {\displaystyle \frac{1}{2}}$. Obviously, since $\alpha \in [{\zeta }_1,{\zeta }_2]\subset \left({\displaystyle \frac{1}{p}},1-{\displaystyle \frac{1}{p}}\right]$ for some ${\zeta }_1,{\zeta }_2\in (0,1),{\zeta }_1<{\zeta }_2$, it follows that $1-\alpha \in \left({\displaystyle \frac{1}{p}},1-{\displaystyle \frac{1}{p}}\right]$ and that ${\zeta }_2-{\zeta }_1<{\displaystyle \frac{1}{p}}$. So $\alpha -\lambda \ge 1-\alpha >{\zeta }_2-{\zeta }_1.$ We also have $1+\lambda -\alpha \le \alpha$. Namely, for any $\alpha \in (0,1)$, there exists $p>{\displaystyle \frac{1}{\alpha }}$ for which the assumptions of Lemma 1 hold.

Proof. 

Let $\alpha \in \left[{\displaystyle \frac{1}{2}},1\right)$ and consider the Banach space $\left[{\mathcal{H}}_0^{\alpha }[a,b],{\left[·\right]}_{\alpha }\right].$

Define $F:=f+{\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}g$. By Corollary 1, we know that ${\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}g\in {\mathcal{H}}_0^{\alpha }[a,b]$. Hence, in view of Remark 4, $F\in {\mathcal{H}}_0^{\alpha }[a,b]\subseteq {\mathcal{H}}_0^{1+\lambda -\alpha }[a,b]$ (with $\lambda \in (0,2\alpha -1]$), and the result now follows by Lemma 1. □

The above results also hold for a certain class of Hölder spaces; as a consequence of Theorem 5 in [24], it is not hard to see that

Lemma 3. 

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 $\psi \left(a\right)=0$, $\alpha \in (0,1)$. If the assumptions (3) and (4) are satisfied, then for any $f\in {\mathcal{H}}_0^{\lambda +1-\alpha }[a,b],\lambda \in (0,\alpha )$, we have

$${\displaystyle \frac{d_{p,\psi }^{\alpha ,\mathsf{\ell }}}{d{t}^{\alpha }}}{\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right)=f\left(t\right).$$

Proof. 

Let $f\in {\mathcal{H}}_0^{\lambda +1-\alpha }[a,b]$. As a consequence of Lemma 3 in [24], we know that ${\mathfrak{D}}_{\psi }^{1-\alpha }f\in {\mathcal{H}}_0^{\lambda }[a,b]$. Hence, Proposition 1 together with Theorem 5 in [24] gives

$${\displaystyle \frac{d_{p,\psi }^{\alpha ,\mathsf{\ell }}}{d{t}^{\alpha }}}{\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right)={\mathbb{I}}_{\psi }^{1-\alpha }\left({\partial }_{\psi }^{\alpha ,\mathsf{\ell }}{\mathfrak{I}}_{p,\psi }^{\alpha ,\mathsf{\ell }}\right){\mathfrak{D}}_{\psi }^{1-\alpha }f\left(t\right)={\mathbb{I}}_{\psi }^{1-\alpha }{\mathfrak{D}}_{\psi }^{1-\alpha }f\left(t\right)=f\left(t\right).$$

□

We will now look at the left inverse of the differential operator and demonstrate the following

Lemma 4. 

For any $f\in AC[a,b]$ and $\alpha \in (0,1)$,

$$\begin{array}{ccc}\hfill {\mathrm{I}}_{\psi }^{\alpha ,\mathsf{\ell }}{\displaystyle \frac{d_{p,\psi }^{\alpha ,\mathsf{\ell }}}{d{t}^{\alpha }}}f\left(t\right)& =& {\mathfrak{I}}_{p,\psi }^{\alpha ,\mathsf{\ell }}\left({\mathfrak{D}}_{\psi }^{1-\alpha }{\mathbb{I}}_{\psi }^{1-\alpha }\right){\partial }_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right)={\mathfrak{I}}_{p,\psi }^{\alpha ,\mathsf{\ell }}\left(\delta {\mathbb{I}}_{\psi }^{\alpha }{\mathbb{I}}_{\psi }^{1-\alpha }\right){\partial }_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right)\hfill \\ & =& {\mathfrak{I}}_{p,\psi }^{\alpha ,\mathsf{\ell }}\left(\delta {\mathbb{I}}_{\psi }^{1,\mu }\right){\partial }_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right)={\mathfrak{I}}_{p,\psi }^{\alpha ,\mathsf{\ell }}{\partial }_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right)=f\left(t\right)-e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}f\left(a\right).\hfill \end{array}$$

4. Proportional Langevin’s Differential Problem

As an example of the application of our results, let us now consider a typical problem involving both classical fractional differential operators and proportional operators (compositions of operators of both types).

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 $\psi \left(a\right)=0$. In the following, we analyze the following proportional Caputo-type Langevin problem [26] with derivatives of two different orders

$${\displaystyle \frac{d_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}}{d{t}^{{\alpha }_1}}}\left({\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}+\rho \right)x\left(t\right)=f\left(t\right),t\in [a,b],\rho \in \mathbb{R},{\alpha }_1,{\alpha }_2\in (0,1),$$

(11)

combined with appropriate initial or boundary conditions. We note that the problem (11) has some physical motivations, and was proposed by Mainardi et al. [27,28] and arises from the study of generalized elastic models and protein dynamics (see, e.g., [29,30]). The problem (11) has aroused some interest and has been studied intensively in other papers, applications of which we mention for [16,23,31,32,33,34,35,36].

Let us (formally) convert (11) into a corresponding proportional integral form. According to Lemma 4, we obtain

$$\begin{array}{cc}\hfill & {\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}{\displaystyle \frac{d_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}}{d{t}^{{\alpha }_1}}}\left({\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}+\rho \right)x\left(t\right)={\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}f\left(t\right)\hfill \\ \hfill & \left({\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}+\rho \right)x\left(t\right)={\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}f\left(t\right)+e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\omega \left(a\right),\omega :={\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}x+\rho x\hfill \\ \hfill & {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}{\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}x\left(t\right)={\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}f\left(t\right)-\rho {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}x\left(t\right)+{\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}\left[e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\omega \left(a\right)\right].\hfill \end{array}$$

Thus

$$x\left(t\right)={\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}f\left(t\right)-\rho {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}x\left(t\right)+{\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}\left[e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\omega \left(a\right)\right]$$

(12)

$$+e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}{x}_0.$$

Put

$$y\left(t\right)=x\left(t\right)-x_0{e}^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}.$$

(13)

$$\begin{array}{ccc}\hfill y\left(t\right)& =& {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}f\left(t\right)-\rho {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}y\left(t\right)+{\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}\left[\omega \left(a\right)e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\right.\hfill \\ & & \left.-\rho x_0{e}^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}\right]\hfill \\ & =& {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}f\left(t\right)-\rho {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}y\left(t\right)+{\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}\rho x_0\left[e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\right.\hfill \\ & & \left.-e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}\right]\hfill \\ & +& (\omega \left(a\right)-\rho x_0){\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}\left[e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\right].\hfill \end{array}$$

Therefore,

$$y\left(t\right)=F\left(t\right)+g\left(t\right)-\rho {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}y\left(t\right),$$

(14)

$t\in [a,b]$, $\rho \in \mathbb{R}$, where

$$g\left(t\right):=(\omega \left(a\right)-\rho x_0){\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}u\left(t\right),\mathrm{where}u\left(t\right):=\left[e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\right],$$

$$F\left(t\right):={\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}f\left(t\right)+\rho x_0{\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}\left[e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}-e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}\right].$$

In our investigation, we assume that ${\mathsf{\ell }}_1, {\mathsf{\ell }}_2$ satisfy the assumptions (3) and (4) with $q=\infty$. In this case, we immediately obtain $u\in AC[a,b]$ and then

$$u^{\prime }\left(t\right)=-\mathsf{\ell }\left({\alpha }_2,t\right)e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}},t\in [a,b].$$

In view of (5) ${∥u^{\prime }∥}_{\infty }<\infty$, meaning that $u^{\prime }\in L_p[a,b]$, for all $p\ge 1.$

Hence, $u\in A{C}^p[a,b],\mathrm{for} \mathrm{all}p\ge 1$. Therefore, by Corollary 4, we know that

$$g=(\omega \left(a\right)-\rho x_0){\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}u\in {\mathcal{H}}_0^{{\alpha }_2}[a,b].$$

(15)

Similarly, we have

$$t\mapsto \left[e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}-e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}\right]\in A{C}_0^p[a,b],$$

(16)

for any $p\ge 1$. Arguing similarly to the proof of Corollary 1, we conclude that

$${\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}u\left(t\right)={\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{\mathfrak{W}\left(s\right){\psi }^{\prime }\left(s\right)}{{\mathsf{\ell }}_1({\alpha }_2,s)}}ds,$$

where $\mathfrak{W}\left(t\right):={\displaystyle \frac{{\left(\psi \left(t\right)\right)}^{{\alpha }_2-1}}{\mathrm{\Gamma }\left({\alpha }_2\right)}}+{\mathbb{I}}_{\psi }^{{\alpha }_2}\left({\displaystyle \frac{u^{\prime }\left(t\right)}{{\psi }^{\prime }\left(t\right)}}\right)$. Hence,

$$\begin{array}{ccc}& & {\partial }_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}u\left(t\right)=\left[{\mathsf{\ell }}_2({\alpha }_2,t)+{\mathsf{\ell }}_1({\alpha }_2,t)\Delta \right]\left({\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{\mathfrak{W}\left(s\right){\psi }^{\prime }\left(s\right)}{{\mathsf{\ell }}_1({\alpha }_2,s)}}ds\right)\hfill \\ & =& {\mathsf{\ell }}_2({\alpha }_2,t)\left({\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{\mathfrak{W}\left(s\right){\psi }^{\prime }\left(s\right)}{{\mathsf{\ell }}_1({\alpha }_2,s)}}ds\right)\hfill \\ & +& {\mathsf{\ell }}_1({\alpha }_2,t)\left({\displaystyle \frac{\mathfrak{W}\left(t\right)}{{\mathsf{\ell }}_1(\alpha ,t)}}-{\displaystyle \frac{{\mathsf{\ell }}_2({\alpha }_2,t)}{{\mathsf{\ell }}_1({\alpha }_2,t)}}{\int }_a^t{e}^{{\displaystyle \left[-{\int }_{\psi \left(s\right)}^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du\right]}}{\displaystyle \frac{\mathfrak{W}\left(s\right){\psi }^{\prime }\left(s\right)}{{\mathsf{\ell }}_1({\alpha }_2,s)}}ds\right)\hfill \\ & =& \mathfrak{W}\left(t\right).\hfill \end{array}$$

Therefore,

$${\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}{\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}u\left(t\right)={\mathbb{I}}_{\psi }^{1-{\alpha }_2}\left({\partial }_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}\right)u\left(t\right)={\mathbb{I}}_{\psi }^{1-{\alpha }_2}\mathfrak{W}\left(t\right)=1+{\mathbb{I}}_{\psi }^1\left({\displaystyle \frac{u^{\prime }\left(t\right)}{{\psi }^{\prime }\left(t\right)}}\right).$$

So, bearing in mind (6), it follows that

$${\partial }_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}\left({\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}{\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}\right)u\left(t\right)={\mathsf{\ell }}_2({\alpha }_1,t)+\left[{\mathsf{\ell }}_2({\alpha }_1,t)+{\mathsf{\ell }}_1({\alpha }_1,t)\Delta \right]\left({\int }_a^t{u}^{\prime }\left(s\right)ds\right)={\partial }_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}u\left(t\right)=0.$$

Hence,

$${\displaystyle \frac{d_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}}{d{t}^{{\alpha }_1}}}{\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}g\left(t\right)={\mathbb{I}}_{\psi }^{1-{\alpha }_1}{\partial }_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}\left({\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}{\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}\right)u\left(t\right)=0.$$

(17)

We are now ready to investigate the Langevin-type boundary value problems in Hölder spaces. This should be the starting point for any research in this area. Let us turn our attention to preserving the expected regularity of solutions for equations of fractional order, i.e., we are looking for solutions in Hölder spaces.

Theorem 3. 

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 ${\alpha }_1,{\alpha }_2\in (0,1)$, and assume that ${\mathsf{\ell }}_1, {\mathsf{\ell }}_2$ satisfy the assumptions (3) and (4) with $q=\infty$. If there are $\lambda , {\zeta }_1<{\zeta }_2\in (0,1)$ such that

$${\alpha }_1,{\alpha }_2\in [{\zeta }_1,{\zeta }_2]and{\zeta }_2-{\zeta }_1\le \min \{{\alpha }_1,{\alpha }_2\}-\lambda ,$$

then the problem (11) admits a Hölder continuous solution x for any $f\in {\mathcal{H}}_0^{\lambda +1-{\alpha }_1}[a,b]$, where

$$x\in \left\{\begin{array}{c}{\mathcal{H}}^{{\alpha }_2}[a,b],for all{\alpha }_1\in (0,1),{\alpha }_2\ge {\displaystyle \frac{1}{2}}whenever\omega \left(a\right)\ne \rho x_0,\hfill \\ {\displaystyle {\mathcal{H}}^{\lambda +1-{\alpha }_2}[a,b],for all{\alpha }_1,{\alpha }_2\in (0,1)whenever\omega \left(a\right)=\rho x_0.}\hfill \end{array}\right.$$

However, in all cases whenever ${\alpha }_2\ge {\displaystyle \frac{1}{2}}$, we conclude that $x\in {\mathcal{H}}^{{\alpha }_2}[a,b].$

• We prove the existence Hölder continuous solution to the integral form (14) (hence, the existence Hölder continuous solution to (12)): First, note that, for any ${\alpha }_1,{\alpha }_2\in (0,1)$ and $f\in {\mathcal{H}}_0^{\lambda +1-{\alpha }_1}[a,b]$, we have (in view of Corollary 5), ${\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}\in {\mathcal{H}}_0^{\lambda +1-{\alpha }_2}[a,b]$. Also by Corollary 1, in view of (16), it is easy to see that $F\in {\mathcal{H}}_0^{\lambda +1-{\alpha }_2}[a,b]$. Obviously, we apply Corollary 1 with some $p>2$ so that ${\displaystyle \frac{1}{p}}+{\zeta }_2-{\zeta }_1=\lambda +1-{\alpha }_2.$
  Now, we are looking for the closed form of the solutions for the integral Equation (14):
  • If $\omega \left(a\right)=\rho x_0$, then for any ${\alpha }_1,{\alpha }_2\in (0,1)$, we easily realize (in view of Lemma 1) that, for sufficiently small $\rho$, (14) admits a Hölder continuous solution $y\in {\mathcal{H}}_0^{\lambda +1-{\alpha }_2}[a,b]$. Hence, it follows (in view of (13)) that

    $$x=y+x_0{e}^{{\displaystyle -{\int }_0^{\psi (·)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}\in A{C}^{\infty }[a,b]\subset {\mathcal{H}}^{\lambda +1-{\alpha }_2}[a,b],$$

    solves the integral Equation (12).
  • If $\omega \left(a\right)\ne \rho x_0$, then for any ${\alpha }_1\in (0,1),{\alpha }_2\ge {\displaystyle \frac{1}{2}}$, we easily recognize (in view of Lemma 2) that, for sufficiently small $\rho$, (14) admits a Hölder continuous solution $y\in {\mathcal{H}}_0^{{\alpha }_2}[a,b]$. Since $u\in A{C}^{\infty }[a,b]$, it follows (in view of (13)) that

    $$x=y+x_{0}u=y+x_0{e}^{{\displaystyle -{\int }_0^{\psi (·)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}\in {\mathcal{H}}^{{\alpha }_2}[a,b],$$

    solves the integral Equation (12).
• We examine the inverse relationship from integral forms to the corresponding differential forms: Let x be the Hölder continuous solution of (12) be the corresponding to the solution y of (14). Obviously (in view of (6)),

  $$x\left(a\right)=y\left(a\right)+x_0=x_0,{\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}x={\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}y.$$
  Now let

  $$f\in \left\{\begin{array}{c}{\mathcal{H}}_0^{{\alpha }_2}[a,b],\mathrm{for} \mathrm{all}{\alpha }_1\in (0,1),{\alpha }_2\ge {\displaystyle \frac{1}{2}}\mathrm{whenever},\omega \left(a\right)\ne \rho x_0,\hfill \\ {\displaystyle {\mathcal{H}}_0^{\lambda +1-{\alpha }_2}[a,b],\mathrm{for} \mathrm{all}{\alpha }_1,{\alpha }_2\in (0,1)\mathrm{whenever},\omega \left(a\right)=\rho x_0.}\hfill \end{array}\right..$$
  • If ${\alpha }_1,{\alpha }_2\in (0,1)\mathrm{and}\omega \left(a\right)=\rho x_0$, it follows by making use of Lemma 3 (in view of (6) and $y\in {\mathcal{H}}_0^{\lambda +1-{\alpha }_2}[a,b]$)

    $$\begin{array}{ccc}& & {\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}y\left(t\right)={\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}f\left(t\right)-\rho y\left(t\right)+\rho x_0\left[e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\right.\hfill \\ & & \left.-e^{-{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}\right]\hfill \\ & & {\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}x\left(t\right)={\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}f\left(t\right)-\rho x\left(t\right)+\rho x_0\left[e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\right]\hfill \\ & & {\displaystyle \frac{d_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}}{d{t}^{{\alpha }_1}}}\left({\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}+\rho \right)x\left(t\right)=f\left(t\right)+{\mathbb{I}}_{\psi }^{1-{\alpha }_1}{\partial }_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\rho x_0\right\}\hfill \\ & & {\displaystyle \frac{d_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}}{d{t}^{{\alpha }_1}}}\left({\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}+\rho \right)x\left(t\right)=f\left(t\right).\hfill \end{array}$$
  • If ${\alpha }_1\in (0,1),{\alpha }_2\ge {\displaystyle \frac{1}{2}}\mathrm{and}\omega \left(a\right)\ne \rho x_0$, it follows by making use of Lemma 3 (in view of (6), (17) and $y\in {\mathcal{H}}_0^{{\alpha }_2}[a,b]$)

    $$\begin{array}{ccc}\hfill {\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}y\left(t\right)& =& {\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}f\left(t\right)-\rho y\left(t\right)+\rho x_0\left[e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\right.\hfill \\ & & \left.-e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}\right]+{\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}g\left(t\right)-{\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}x\left(t\right)\hfill \\ & =& {\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}f\left(t\right)-\rho x\left(t\right)+\rho x_0\left[e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\right]+{\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}g\left(t\right).\hfill \end{array}$$
    Thus,

    $$\begin{array}{ccc}\hfill {\displaystyle \frac{d_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}}{d{t}^{{\alpha }_1}}}\left({\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}+\rho \right)x\left(t\right)& =& f\left(t\right)+{\mathbb{I}}_{\psi }^{1-{\alpha }_1}{\partial }_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\rho x_0\right\}\hfill \\ & & +{\displaystyle \frac{d_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}}{d{t}^{{\alpha }_1}}}{\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}g\left(t\right),\hfill \\ \hfill {\displaystyle \frac{d_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}}{d{t}^{{\alpha }_1}}}\left({\displaystyle \frac{d_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}}{d{t}^{{\alpha }_2}}}+\rho \right)x\left(t\right)& =& f\left(t\right).\hfill \end{array}$$

Example 2. 

Let ${\mathsf{\ell }}_1(\alpha ,t)=\alpha e^t,{\mathsf{\ell }}_2(\alpha ,t)=(1-\alpha )e^t,\psi \left(t\right)=t, t\in [0,1]$. Hence, $\mathsf{\ell }(\alpha ,t)={\displaystyle \frac{1-\alpha }{\alpha }}$. Obviously, ${\mathsf{\ell }}_1, {\mathsf{\ell }}_2$ satisfy the assumptions (3) and (4) with $q=\infty$. Consider the following proportional Caputo-type Langevin problem

$${\displaystyle \frac{d_{p,\psi }^{\frac{1}{2},\mathsf{\ell }}}{d{t}^{\frac{1}{2}}}}{\displaystyle \frac{d_{p,\psi }^{\frac{1}{2},\mathsf{\ell }}}{d{t}^{\frac{1}{2}}}}x\left(t\right)=t^{{\displaystyle \frac{1}{2}}},t\in [0,1],$$

(18)

combined with an appropriate initial or boundary conditions. Put ${\zeta }_1={\displaystyle \frac{1}{4}}, {\zeta }_2={\displaystyle \frac{1}{2}}$, and pick $\lambda \in \left(0,{\displaystyle \frac{1}{4}}\right]$. Obviously

$${\alpha }_1={\displaystyle \frac{1}{2}}={\alpha }_2\in [{\zeta }_1,{\zeta }_2]and{\displaystyle \frac{1}{2}}-\lambda \ge {\zeta }_2-{\zeta }_1.$$

Since $f\in {\mathcal{H}}_0^{{\displaystyle \frac{1}{2}}}[0,1]$, the the assumptions of Theorem 3 hold, and so the formal integral equations corresponding to (18) read as

$$\begin{array}{ccc}\hfill x\left(t\right)& =& {\mathrm{I}}_{p,t}^{{\displaystyle \frac{1}{2}},1}{\mathrm{I}}_{p,t}^{{\displaystyle \frac{1}{2}},1}f\left(t\right)+{\mathrm{I}}_{p,t}^{{\displaystyle \frac{1}{2}},1}\left[e^{{\displaystyle -{\int }_0^{t}du}}\omega \left(0\right)\right]+e^{{\displaystyle -{\int }_0^{t}1du}}{x}_0\hfill \\ & =& {\mathrm{I}}_{p,t}^{{\displaystyle \frac{1}{2}},1}{\mathrm{I}}_{p,t}^{{\displaystyle \frac{1}{2}},1}f\left(t\right)+\omega \left(0\right){\mathrm{I}}_{p,t}^{{\displaystyle \frac{1}{2}},1}\left[e^{-t}\right]+x_0{e}^{-t}.\hfill \end{array}$$

(19)

We have

$${\mathrm{I}}_{p,t}^{{\displaystyle \frac{1}{2}},1}\left[e^{-t}\right]=2{\int }_0^t{e}^{s-t}{\displaystyle \frac{{\mathfrak{D}}_s^{\frac{1}{2}}{e}^{-s}ds}{e^s}}=2e^{-t}{\int }_0^t{\displaystyle \frac{{\mathbb{E}}_{1,\frac{1}{2}}(-s)ds}{\sqrt{s}}},$$

where $\mathbb{E}$ denotes the well-known Mittag–Leffler function. Also,

$$\begin{array}{ccc}& & {\mathrm{I}}_{p,t}^{{\displaystyle \frac{1}{2}},1}f\left(t\right)=2{\int }_0^t{e}^{s-t}{\displaystyle \frac{{\mathfrak{D}}_s^{\frac{1}{2}}{s}^{\frac{1}{2}}ds}{e^s}}=e^{-t}\mathrm{\Gamma }\left({\displaystyle \frac{1}{2}}\right)t\hfill \\ & & \Rightarrow {\mathrm{I}}_{p,t}^{{\displaystyle \frac{1}{2}},1}{\mathrm{I}}_{p,t}^{{\displaystyle \frac{1}{2}},1}f\left(t\right)=2\mathrm{\Gamma }\left({\displaystyle \frac{1}{2}}\right){\int }_0^t{e}^{s-t}{\displaystyle \frac{{\mathfrak{D}}_s^{\frac{1}{2}}s{e}^{-s}ds}{e^s}}=2e^{-t}\mathrm{\Gamma }\left({\displaystyle \frac{1}{2}}\right){\int }_0^t{\mathfrak{D}}_s^{{\displaystyle \frac{1}{2}}}s{e}^{-s}ds.\hfill \end{array}$$

Hence,

$$x\left(t\right)=2e^{-t}\mathrm{\Gamma }\left({\displaystyle \frac{1}{2}}\right){\int }_0^t{\mathfrak{D}}_s^{{\displaystyle \frac{1}{2}}}s{e}^{-s}ds+\omega \left(0\right)2e^{-t}{\int }_0^t{\displaystyle \frac{{\mathbb{E}}_{1,\frac{1}{2}}(-s)ds}{\sqrt{s}}}+x_0{e}^{-t}\in {\mathcal{H}}^{{\displaystyle \frac{1}{2}}}[0,1].$$

Since

$${\int }_t^{t+h}\left|{\mathfrak{D}}_s^{{\displaystyle \frac{1}{2}}}s{e}^{-s}\right|ds\le \underset{t\in [0,1]}{\max }\left|{\mathfrak{D}}_t^{{\displaystyle \frac{1}{2}}}t{e}^{-t}\right|h,{\int }_t^{t+h}\left|{\displaystyle \frac{{\mathbb{E}}_{1,\frac{1}{2}}(-s)ds}{\sqrt{s}}}\right|ds\le 2\underset{t\in [0,1]}{\max }\left|{\mathbb{E}}_{1,{\displaystyle \frac{1}{2}}}(-t)\right|h^{{\displaystyle \frac{1}{2}}},$$

it follows

$${\int }_0^t{\mathfrak{D}}_s^{{\displaystyle \frac{1}{2}}}s{e}^{-s}ds\in {\mathcal{H}}^{{\displaystyle \frac{3}{4}}}[0,1]\subset {\mathcal{H}}^{{\displaystyle \frac{1}{2}}}[0,1],{\int }_0^t{\displaystyle \frac{{\mathbb{E}}_{1,\frac{1}{2}}(-s)ds}{\sqrt{s}}}\in {\mathcal{H}}^{{\displaystyle \frac{1}{2}}}[0,1].$$

Therefore,

$$x\in \left\{\begin{array}{c}{\mathcal{H}}^{{\displaystyle \frac{1}{2}}}[0,1],whenever\omega \left(0\right)\ne 0,\hfill \\ {\displaystyle {\mathcal{H}}^{\frac{3}{4}}[0,1],whenever\omega \left(0\right)=0.}\hfill \end{array}\right.\Rightarrow in all casesx\in {\mathcal{H}}^{{\displaystyle \frac{1}{2}}}[0,1].$$

5. Hybrid $\mathit{\psi }$-Hilfer Proportional Derivatives

In order to complement the study of proportional operators, and at the same time to consider classical operators, we will present the problem of domains for generalized Hilfer derivatives.

Definition 4 

((hybrid $\psi$-Hilfer proportional fractional derivatives) [36,37]). The hybrid ψ-Hilfer proportional derivatives of order $n+\alpha ,\alpha \in (0,1)$, $n\in \mathbb{N}:=\{1,2,\cdots ,\}$ and type $\beta \in [0,1]$ applied to the function f is defined as

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

(20)

Define the anti-derivative of ${}^H\mathrm{D}_{p,\psi }^{n+\alpha ,\beta ,\mathsf{\ell }}$ by

$${\mathrm{I}}_{p,\psi }^{\alpha ,\beta ,\mathsf{\ell }}:={\mathfrak{D}}_{\psi }^{(1-\beta )(1-\alpha )}{\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}{\mathfrak{D}}_{\psi }^{\beta (1-\alpha )}.$$

(21)

Finally, summarizing the research carried out, we present a theorem discussing the characterization of the domain and its image of the studied operators in the class of Hölder spaces.

Theorem 4. 

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 $\alpha \in (0,1),\beta \in [0,1]$, and assume that ${\mathsf{\ell }}_1, {\mathsf{\ell }}_2$ satisfy the assumptions (3) and (4) with $q=\infty$. Assume that there are $\lambda , {\zeta }_1<{\zeta }_2\in (0,1)$ such that

$$\alpha \in [{\zeta }_1,{\zeta }_2]and{\zeta }_2-{\zeta }_1\le \left\{\begin{array}{c}\beta \alpha -\lambda ,\beta \in (0,1),\hfill \\ \alpha -\lambda ,\beta =0,1.\hfill \end{array}\right.$$

Then

$${\mathrm{I}}_{p,\psi }^{\alpha ,\beta ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda +\beta (1-\alpha )}[a,b]\to {\mathcal{H}}_0^{\lambda +\beta (1-\alpha )+{\zeta }_2-{\zeta }_1}[a,b].$$

Proof. 

The result holds when $\beta =1$ by Theorem 2. Consider now $\beta \in [0,1)$ and $f\in {\mathcal{H}}_0^{\lambda +\beta (1-\alpha )}[a,b].$ Since $\lambda <\alpha \beta \beta \in (0,1),$ (or $\lambda <\alpha \beta =0$), it follows $0<\lambda +\beta (1-\alpha )<1$ and so (cf. Lemma 3 in [24]) ${\mathfrak{D}}_{\psi }^{\beta (1-\alpha )}f\in {\mathcal{H}}_0^{\lambda }[a,b]$. Hence, by Theorem 2, ${\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}{\mathfrak{D}}_{\psi }^{\beta (1-\alpha )}f\in {\mathcal{H}}_0^{\lambda +1-\alpha +{\zeta }_2-{\zeta }_1}[a,b]$. Again, in Lemma 3 [24], we conclude ${\mathfrak{D}}_{\psi }^{(1-\beta )(1-\alpha )}{\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}{\mathfrak{D}}_{\psi }^{\beta (1-\alpha )}f\in {\mathcal{H}}_0^{\lambda +\beta (1-\alpha )+{\zeta }_2-{\zeta }_1}[a,b]$. □

Arguing similarly as in the proof of Corollary 5, the following can be proven:

Corollary 6. 

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 ${\alpha }_1,{\alpha }_2\in (0,1),\beta \in [0,1]$, and assume that ${\mathsf{\ell }}_1, {\mathsf{\ell }}_2$ satisfy the assumptions (3) and (4) with $q=\infty$. If there are $\lambda , {\zeta }_1<{\zeta }_2\in (0,1)$ such that

$${\alpha }_1,{\alpha }_2\in [{\zeta }_1,{\zeta }_2]and{\zeta }_2-{\zeta }_1\le \left\{\begin{array}{c}\beta \min \{{\alpha }_1,{\alpha }_2\}-\lambda ,\beta \in (0,1),\hfill \\ \min \{{\alpha }_1,{\alpha }_2\}-\lambda ,\beta =0,1,\hfill \end{array}\right.$$

then

$${\mathrm{I}}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{{\alpha }_1,\beta ,\mathsf{\ell }}:{\mathcal{H}}_0^{\lambda +\beta (1-{\alpha }_1)}[a,b]\to {\mathcal{H}}_0^{\lambda +\beta (1-{\alpha }_2)}[a,b].$$

(22)

Moreover, there is $k\ge 0$ such that ${\left[{\mathrm{I}}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{{\alpha }_1,\beta ,\mathsf{\ell }}f\right]}_{\lambda +\beta (1-{\alpha }_2)}\le {\left[f\right]}_{\lambda +\beta (1-{\alpha }_1)}$ for any $f\in {\mathcal{H}}_0^{\lambda +\beta (1-{\alpha }_1)}[a,b].$

Recall that, for any $f\in L_1[a,b],$ we have ${\mathfrak{D}}_{\psi }^{0}f={\displaystyle \frac{1}{{\psi }^{\prime }}}{\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\psi }^{0}f=f$. Also,

$${}^H\mathrm{D}_{p,\psi }^{\alpha ,0,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{\alpha ,0,\mathsf{\ell }}f\left(t\right)={\partial }_{\psi }^{\alpha ,\mathsf{\ell }}\left({\mathbb{I}}_{\psi }^{1-\alpha }{\mathfrak{D}}_{\psi }^{1-\alpha }\right){\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right)={\partial }_{\psi }^{\alpha ,\mathsf{\ell }}{\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}f\left(t\right)=f\left(t\right),f\in L_1[a,b].$$

(23)

Now we are in a position to state and prove the following interesting lemma, which shows that the integral operator is, in little Hölder spaces, the right inverse of the derivative operator under consideration.

Lemma 5. 

Let the assumptions of Theorem 4 be satisfied; then, for any $f\in {\mathcal{H}}_0^{\lambda +\beta (1-\alpha )}[a,b],$ we have

$${}^H\mathrm{D}_{p,\psi }^{n+\alpha ,\beta ,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{\alpha ,\beta ,\mathsf{\ell }}f\left(t\right)=f\left(t\right).$$

Proof. 

The case when $\beta =0$ follows by (23). Now, let $\beta \in (0,1]$ and $f\in {\mathcal{H}}_0^{\lambda +\beta (1-\alpha )}[a,b]$. As a consequence of Lemma 3 in [24], we know that ${\mathfrak{D}}_{\psi }^{\beta (1-\alpha )}f\in {\mathcal{H}}_0^{\lambda }[a,b]$. Therefore, Proposition 1 along with Theorem 5 in [24] give

$$\begin{array}{ccc}\hfill {}^H\mathrm{D}_{p,\psi }^{n+\alpha ,\beta ,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{\alpha ,\beta ,\mathsf{\ell }}f\left(t\right)& =& {\mathbb{I}}_{\psi }^{\beta (1-\alpha )}{\partial }_{\psi }^{\alpha ,\mathsf{\ell }}\left({\mathbb{I}}_{\psi }^{(1-\beta )(1-\alpha )}{\mathfrak{D}}_{\psi }^{(1-\beta )(1-\alpha )}\right){\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}{\mathfrak{D}}_{\psi }^{\beta (1-\alpha )}f\left(t\right)\hfill \\ & =& {\mathbb{I}}_{\psi }^{\beta (1-\alpha )}\left({\partial }_{\psi }^{\alpha ,\mathsf{\ell }}{\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}\right){\mathfrak{D}}_{\psi }^{\beta (1-\alpha )}f\left(t\right)={\mathbb{I}}_{\psi }^{1-\alpha }{\mathfrak{D}}_{\psi }^{1-\alpha }f\left(t\right)=f\left(t\right).\hfill \end{array}$$

□

The case of the left inverse can be proved for absolutely continuous functions. However, it requires an additional condition at the point a.

Lemma 6. 

For any $f\in AC[a,b]$ and $\alpha \in (0,1),\beta \in [0,1]$, we have

$$\begin{array}{ccc}& & {\mathrm{I}}_{p,\psi }^{\alpha ,\beta ,\mathsf{\ell }} {}^H\mathrm{D}_{p,\psi }^{\alpha ,\beta ,\mathsf{\ell }}f\left(t\right)={\mathfrak{D}}_{\psi }^{(1-\beta )(1-\alpha )}{\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}\left({\mathfrak{D}}_{\psi }^{\beta (1-\alpha )}{\mathbb{I}}_{\psi }^{\beta (1-\alpha )}\right){\partial }_{\psi }^{\alpha ,\mathsf{\ell }}{\mathbb{I}}_{\psi }^{(1-\beta )(1-\alpha )}f\left(t\right)\hfill \\ & =& {\mathfrak{D}}_{\psi }^{(1-\beta )(1-\alpha )}\left({\mathfrak{I}}_{\psi }^{\alpha ,\mathsf{\ell }}{\partial }_{\psi }^{\alpha ,\mathsf{\ell }}\right){\mathbb{I}}_{\psi }^{(1-\beta )(1-\alpha )}f\left(t\right)\hfill \\ & =& {\mathfrak{D}}_{\psi }^{(1-\beta )(1-\alpha )}\left[{\mathbb{I}}_{\psi }^{(1-\beta )(1-\alpha )}f\left(t\right)-e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}{\mathbb{I}}_{\psi }^{(1-\beta )(1-\alpha )}f\left(a\right)\right]\hfill \\ & =& f\left(t\right)-{\mathfrak{D}}_{\psi }^{(1-\beta )(1-\alpha )}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left(\alpha ,{\psi }^{-1}\left(u\right)\right)du}}{\mathbb{I}}_{\psi }^{(1-\beta )(1-\alpha )}f\left(a\right)\right\}.\hfill \end{array}$$

Now, to demonstrate the usefulness of the results obtained, we analyze the following proportional $\psi$-Hilfer Langevin’s problem

$${}^H\mathrm{D}_{p,\psi }^{{\alpha }_1,\beta ,\mathsf{\ell }}\left({}^H\mathrm{D}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}+\rho \right)x\left(t\right)=f\left(t\right),t\in [a,b],\rho \in \mathbb{R},{\alpha }_1,{\alpha }_2\in (0,1),\beta \in [0,1],$$

(24)

combined with an appropriate initial or boundary conditions.

The case when $\beta =1$ has been discussed above, and so we concentrate only when $\beta \in [0,1)$: Let us consider $\psi \in C^1[a,b]$ as a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$, and pick $\lambda ,{\zeta }_1,{\zeta }_2\in (0,1),{\zeta }_1<{\zeta }_2$ such that

$${\alpha }_1,{\alpha }_2\in [{\zeta }_1,{\zeta }_2]\mathrm{and}{\zeta }_2-{\zeta }_1\le \left\{\begin{array}{c}\beta \min \{{\alpha }_1,{\alpha }_2\}-\lambda ,\beta \in (0,1),\hfill \\ \\ min\{{\alpha }_1,{\alpha }_2\}-\lambda ,\beta =0,1.\hfill \end{array}\right..$$

Then, according to Lemma 6, we obtain (formally) that for any $f\in {\mathcal{H}}_0^{\lambda +\beta (1-{\alpha }_1)}[a,b]$, we have

$$\begin{array}{cc}\hfill & {\mathrm{I}}_{p,\psi }^{{\alpha }_1,\beta ,\mathsf{\ell }} {}^H\mathrm{D}_{p,\psi }^{{\alpha }_1,\beta ,\mathsf{\ell }}\left({}^H\mathrm{D}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}+\rho \right)x\left(t\right)={\mathrm{I}}_{p,\psi }^{{\alpha }_1,\beta ,\mathsf{\ell }}f\left(t\right)\hfill \\ \hfill & \left({}^H\mathrm{D}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}+\rho \right)x\left(t\right)={\mathrm{I}}_{p,\psi }^{{\alpha }_1,\beta ,\mathsf{\ell }}f\left(t\right)+{\mathfrak{D}}_{\psi }^{(1-\beta )(1-{\alpha }_1)}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}{C}_0\right\}\hfill \\ \hfill & {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }} {}^H\mathrm{D}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}x\left(t\right)={\mathrm{I}}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{{\alpha }_1,\beta ,\mathsf{\ell }}f\left(t\right)-\rho {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}x\left(t\right)\hfill \\ & +{\mathrm{I}}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}{\mathfrak{D}}_{\psi }^{(1-\beta )(1-{\alpha }_1)}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}{C}_0\right\}\hfill \end{array}$$

Thus, we easily recognize that the problem (24) can be expressed as:

$$\begin{array}{ccc}\hfill x\left(t\right)& =& {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{{\alpha }_1,\beta ,\mathsf{\ell }}f\left(t\right)-\rho {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}x\left(t\right)\hfill \\ & +& {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}{\mathfrak{D}}_{\psi }^{(1-\beta )(1-{\alpha }_1)}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}{C}_0\right\}\hfill \\ & & +{\mathfrak{D}}_{\psi }^{(1-\beta )(1-{\alpha }_2)}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}{C}_1\right\}.\hfill \end{array}$$

(25)

If we define

$$y\left(t\right):=x\left(t\right)-C_1{\mathfrak{D}}_{\psi }^{(1-\beta )(1-{\alpha }_2)}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}\right\},$$

for $t\in [a,b]$, we arrive at

$$y\left(t\right)={\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}{\mathrm{I}}_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}f\left(t\right)-\rho {\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}y\left(t\right)+F\left(t\right)+g\left(t\right),$$

(26)

$t\in [a,b]$, $\rho \in \mathbb{R}$, where

$$g\left(t\right):=(C_0-\rho C_1){\mathrm{I}}_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}u\left(t\right),$$

with

$$u\left(t\right):=\left[{\mathfrak{D}}_{\psi }^{(1-\beta )(1-{\alpha }_1)}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\right\}\right],$$

$$\begin{array}{ccc}\hfill F\left(t\right)& :=& \rho C_1{\mathrm{I}}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}\left[{\mathfrak{D}}_{\psi }^{(1-\beta )(1-{\alpha }_1)}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\right\}\right.\hfill \\ & & \left.-{\mathfrak{D}}_{\psi }^{(1-\beta )(1-{\alpha }_2)}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}\right\}\right].\hfill \end{array}$$

According to Lemma 1 together with Theorem 1, the above integral equation admits a Hölder continuous solution $x\in {\mathcal{H}}_0^{\gamma }[a,b]$ whenever $f\in {\mathcal{H}}_0^{\lambda +\beta (1-{\alpha }_1)}[a,b]$. Indeed, arguing now as in the proof of Theorem 3, it can be easily prove the following.

Theorem 5. 

Let $\psi \in C^1[a,b]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$ with $\psi \left(a\right)=0$. Let ${\alpha }_1,{\alpha }_2\in (0,1),\beta \in [0,1)$, and assume that ${\mathsf{\ell }}_1, {\mathsf{\ell }}_2$ satisfy the assumptions (3) and (4) with $q=\infty$. If there is $\lambda , {\zeta }_1<{\zeta }_2\in (0,1)$ such that

$${\alpha }_1,{\alpha }_2\in [{\zeta }_1,{\zeta }_2]and{\zeta }_2-{\zeta }_1\le \left\{\begin{array}{c}\beta \min \{{\alpha }_1,{\alpha }_2\}-\lambda ,\beta \in (0,1),\hfill \\ \min \{{\alpha }_1,{\alpha }_2\}-\lambda ,\beta =0.\hfill \end{array}\right.,$$

then for any $f\in {\mathcal{H}}_0^{\lambda +\beta (1-{\alpha }_1)}[a,b]$, the problem (24) admits a Hölder continuous solution x, where

$$x\in \left\{\begin{array}{c}{\mathcal{H}}^{{\alpha }_2}[a,b],for all{\alpha }_1\in (0,1),{\alpha }_2\ge {\displaystyle \frac{1}{2}}whenever{C}_0\ne \rho C_1,\hfill \\ {\displaystyle {\mathcal{H}}^{\lambda +\beta (1-{\alpha }_2)}[a,b],for all{\alpha }_1,{\alpha }_2\in (0,1)whenever{C}_0=\rho C_1}\hfill \end{array}\right..$$

However, in all cases whenever ${\alpha }_2\ge {\displaystyle \frac{1}{2}}$, we conclude that $x\in {\mathcal{H}}^{{\alpha }_2}[a,b]$.

Indeed, if $x\in {\mathcal{H}}^{{\alpha }_2}[a,b]$, it solves (25). Then,

$$\begin{array}{cc}\hfill & {}^H\mathrm{D}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}x\left(t\right)={\mathrm{I}}_{p,\psi }^{{\alpha }_1,\beta ,\mathsf{\ell }}f\left(t\right)-\rho x\left(t\right)\hfill \\ \hfill & + \rho C_1\left[{\mathfrak{D}}_{\psi }^{(1-\beta )(1-{\alpha }_1)}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}\right\}\right.\hfill \\ \hfill & \left.- {\mathfrak{D}}_{\psi }^{(1-\beta )(1-{\alpha }_2)}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}\right\}\right]\hfill \\ \hfill & +(C_0-\rho C_1){\mathbb{I}}_{\psi }^{\beta (1-{\alpha }_2)}{\partial }_{p,\psi }^{{\alpha }_2,\mathsf{\ell }}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_2,{\psi }^{-1}\left(u\right)\right)du}}{C}_1\right\}\hfill \\ \hfill & {}^H\mathrm{D}_{p,\psi }^{{\alpha }_1,\beta ,\mathsf{\ell }}\left({}^H\mathrm{D}_{p,\psi }^{{\alpha }_2,\beta ,\mathsf{\ell }}+\rho \right)x\left(t\right)=f\left(t\right)+{\mathbb{I}}_{\psi }^{\beta (1-{\alpha }_1)}{\partial }_{p,\psi }^{{\alpha }_1,\mathsf{\ell }}\left\{e^{{\displaystyle -{\int }_0^{\psi \left(t\right)}\mathsf{\ell }\left({\alpha }_1,{\psi }^{-1}\left(u\right)\right)du}}{C}_1\right\}+0.\hfill \end{array}$$

Thus

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

6. Conclusions

Hybrid proportional Caputo derivatives and their inverse integral counterparts are studied in this paper. This type of differential operator is, in a sense, an interpolation between a function and its derivative. Thus, one would expect a Hölder regularity of the values of this type of operator (as for fractional order operators), but no such study has been done. We make up for this lack. Moreover, as with any problem considered in differential or integral calculus, a very important question is to determine the maximum possible regularity of the solutions to the problems under study. In each situation, the existence of solutions is investigated and then their regularity. We prove that Hölder spaces (cf. [14,15,16]) are natural domains for differential and integral operators, allowing their invertibility and preserving the maximal regularity of functions.

Importantly, and quite unexpectedly, no such attempts have been made to determine the regularity of solutions to the proportional calculus. The correct results can only be obtained in Hölder spaces, otherwise we prove that the expected regularity of the solutions is lost.

The results of this paper are complemented by results for the widely studied problem of equations of the Langevin type, to which we apply the results obtained for the right and left inverses of operators with proportional derivatives of the Caputo type. This is the first study on this type of calculus. This allows us to complete the results obtained so far and to eliminate errors in the previous results that arise from the bad choice of the spaces in which the operators are acting.

Future research can focus on studying boundary problems for equations with such derivatives and on exploring the relationship between the interpolated derivatives and the spaces in which they are defined, which is an interesting problem from function spaces or in the vector-valued case (cf. [20]). The analogy can also be seen in this paper: the interpolated derivatives in the proportional calculus correspond to the scale of the interpolated (Hölder) spaces. An open question for future research is to demonstrate the formal connection and thus to obtain new research tools for the proportional calculus from interpolation theory.