Equivalence Between Fractional Differential Problems and Their Corresponding Integral Forms with the Pettis Integral

Abstract

The problem of equivalence between differential and integral problems is absolutely crucial when applying solution methods based on operators and their properties in function spaces. In this paper, we complement the solution of this important problem by considering the case of general derivatives and integrals of fractional order for vector functions for weak topology. Even if a Caputo differential fractional order problem has a right-hand side that is weakly continuous, the equivalence between the differential and integral forms may be affected. In this paper, we present a complete solution to this problem using fractional order Pettis integrals and suitably defined pseudo-derivatives, taking care to construct appropriate Hölder-type spaces on which the operators under study are mutually inverse. In this paper, we prove, in a number of cases, the equivalence of differential and integral problems in Hölder spaces and, by means of appropriate counter-examples, investigate cases where this property of the problems is absent.

1. Introduction

The problem of equivalence between differential and integral problems is fundamental to the use of operator methods to study solutions to such problems. A suitable integral operator allows us to prove the existence of fixed points and thus solutions to differential problems. The problem depends on how we define the solution or what we assume about the functions in the problem. It is not well studied for vector-valued functions with values in Banach spaces, and it is of particular interest to consider the case of weak topology (cf. [1,2]), which is different from that of real functions. The brief discussion of the case of strong derivatives, and why we study the weak topology here, is presented later, in Remark 1.

The use of methods based on the properties of integral operators to solve given differential problems always requires studies of the equivalence of the problems. As we show in this paper, especially for operators of fractional order, this is an absolutely fundamental step, without which most results about the existence or regularity of solutions will not hold. In practice, most papers on this subject start by presenting an integral problem equivalent to the initial differential problem, and this step is not always supported by correct reasoning, e.g., omitting the problem of the function space in which the problem is studied. Integral operators are much more convenient to study, and our results will provide a basis for their use in the study of fractional order differential problems. Our paper will verify and even extend such studies (for vector functions, among others).

The problem we are discussing is absolutely fundamental to the study of differential equations, including, of course, those of integer order. Surprisingly, the equivalence problem for vector-valued functions has so far been studied in the case of integrals and strong derivatives [1] of integer order. But, even in this base case we have a variety of different integrals and solution spaces. Basic results for weak solutions and pseudo-solutions are also collected in the same paper. In the cases considered there, however, the integral operator was only a Volterra operator. In the study of fractional order problems, we study weakly singular operators, so the problem has to be studied anew. We fill this gap in our paper.

The basic cases of the Riemann–Liouville integral and the Caputo derivative for real-valued functions are quite well studied [3,4]. However, many models based on other fractional order integrals (e.g., $\psi$-Hilfer integrals) remain to be studied. This problem is often overlooked or neglected by authors. And it requires both the identification or construction of corresponding differential and integral operators, and it also depends strongly on the function spaces considered on which we study these operators (and thus the solution spaces). Of course, the importance of checking the equivalence of differential and integral problems has been recognized many times. An excellent example is the article [5], where previous results on fractional order equations with impulse effects were reviewed and improved (with regulated solutions). In this paper, however, we will concentrate on problems for vector functions and weak derivatives. It is important to emphasize the role and usefulness of the study of conditions expressed in weak topology for practical issues. In this context, let us choose some examples. Applications of this topology include the study of random and stochastic equations [6,7], dynamical systems [8,9], or differential inclusions (cf. [10] for the case of fractional derivatives).

These studies are usually carried out on the most commonly defined fractional order integral so that all the previous ones are included in a single result. Thus, we will deal here with the most commonly used and very general notion of Stieltjes-type integrals, i.e., the $\psi$-Hilfer integral with respect to the $\psi$ function. The problem has recently been studied in [4] for real-valued functions. Note that the case of differential problems for vector functions with strong derivatives is similar to that for real functions and does not present significant mathematical difficulties for this theory. However, considering the interesting problems with weak differentiability of vector functions, we will concentrate on the still unsolved problem for integrable functions in the Pettis sense (and thus weak). We should note here that in the considered case we cannot expect the equivalence of weak-type problems in arbitrary space (cf. [11], where the lack of equivalence in a general case is proved). This is another reason to study the case of weak solutions and pseudo-solutions. Here, the foundations of the problem are given in [12,13] but only for some particular fractional operators. The case under consideration is fundamentally different from that of real functions since it is necessary to ensure that weak-type derivatives [1,14,15,16] are appropriate. For a description of this problem, see [17] or for fractional derivatives, see [13].

In this paper, we investigate solutions in the weak sense for fractional-order differential equations for a very general class of fractional operators and taking into account the natural requirement of Hölder-type continuity of solutions. As previously claimed, many fractional order derivatives and integrals exist. The aim of this paper is to obtain results for a wide range of them. This includes the definition of inverse differential and integral operators that go far beyond the classical cases of the Riemann–Liouville integral and the Caputo derivative. There is also the problem with vector-valued functions and weak topology. How is the derivative supposed to exist? This has been known since the study of integer order equations (see [1]) but is particularly relevant for fractional-order equations.

In the context of differential equations, the possibilities of using the Pettis integral were already noted in the papers [2,18]. These results were initially unappreciated, but after a systematic study of the relationship between weak-type solutions and Pettis integrals [1], research accelerated. This is a very different case from strong derivatives. Also of interest was the question of how to extend and unify ordinary differential equations with equations of fractional order. Over the years, more and more attention was paid to the theory of Pettis integrals, and there was a big step between the unification of fractional calculus and the theory of Pettis integrals (see [11] and references therein).

In particular, it was proved in [13] that even if the right-hand side of a pseudo/weak Caputo fractional order problem is weakly continuous, the equivalence between the differential and integral forms can be lost. It thus follows that if ${\mathfrak{D}}_{\omega ,\psi }^{\alpha ,\mu }:={\delta }_p{\mathbb{I}}_{\psi }^{1-\alpha ,\mu }$ (resp. ${\displaystyle \frac{d_{p,\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}}f:={\mathbb{I}}_{\psi }^{1-\alpha ,\mu }{\delta }_p)$ denotes the $\psi$-tempered Riemann–Liouville (resp. Caputo) fractional pseudo-derivative $\alpha \in (0,1)$, and with parameter $\mu \in {\mathbb{R}}^{+}$, then there exists $f\in C[I,E_{\omega }]$ such that

$${\displaystyle \frac{d_{p,\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}}{\mathbb{I}}_{\psi }^{\alpha ,\mu }f=f,\mathrm{or}{\displaystyle \frac{d_{\omega ,\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}}{\mathbb{I}}_{\psi }^{\alpha ,\mu }f=f\mathrm{is}\mathrm{meaningless}.$$

It is evident that this is also applicable in the case of tempered Hilfer fractional pseudo/weak derivatives.

Since the aim of this paper is to demonstrate the equivalence of differential and integral problems of fractional order with Pettis integrals, we should also pay attention to applications. This concerns the study of differential equations of fractional order with given initial or boundary conditions in the case of vector functions. Such studies have been carried out, among others, for the weakly singular Riemann–Liouville fractional Pettis integral, for pseudo-solutions of fractional BVP’s, and for the existence of random solutions for a class of partial random integral equations using Hadamard fractional integral, under the assumption of Pettis’s integrability. A further important step is the paper [12] and its consequences, i.e., [13].

In order not to prolong the introduction of this paper, we will place the discussion of some other results that are relevant to the results obtained here, and the comparison with previous results of this type, in Section 7.

Finally, it is worth mentioning here a paper in which the Pettis integral is combined with another operator of fractional order, namely the Hadamard operator. The paper [12] proves the existence of fractional point solutions of differential equations of the Caputo–Hadamard type of the Pettis integral. This is already a step towards the general integral operators of fractional order studied in this paper. With the introduction of the $\psi$-Hilfer fractional derivative, containing a wide class of particular cases from the choice of $\psi$, the parameters a, b and the limits $\beta \to 0$ or $\beta \to 1$. This same property is also valid for the $\psi$-Riemann–Liouville fractional integral. In the particular choice of the function $\psi \left(t\right)=lnt$, with $a>0$, we obtain the version of the Hadamard–Pettis fractional integral discussed, for example, in [12]. In order to maximize the generality of the results, this paper considers an equivalence problem for such a general class of fractional integrals and and corresponding pseudo-derivatives of fractional order.

It should be noted that the previous results concerned the special case of weak integrals of fractional order and a rather narrow class of E target spaces. Our results allow us to extend them considerably.

2. Preliminaries

Let E denote a real Banach space with a norm $∥·∥$ and $E^{*}$ its dual. Let $E_{\omega }$ denote the space E when it is endowed with the weak topology (i.e., generated by the continuous linear functionals on E). Moreover, let $C[I,E_{\omega }]$ denote the Banach space of continuous functions from I to E, with its weak topology. A function $x(·):I\to E$ is said to be weakly measurable on I if for every $y^{*}\in E^{*}$ the real function $t\to y^{*}x$ is Lebesgue measurable on I. Denote by I the compact interval $[a,b]\subset \mathbb{R}$.

We need the following definitions [19] in this paper:

Definition 1.

A weakly measurable function $x:I\to E$ is said to be Pettis integrable on I if

1.

x is Dunford integrable on I, that is, $\phi x\in L_1$ for every $\phi \in E^{*}$;

2.

For any measurable $A\subset I$, there exists an element in E denoted by ${\int }_{A}x\left(s\right)ds$ such that

$$\phi \left({\int }_{A}x\left(s\right)ds\right)={\int }_A\phi x\left(s\right)ds\mathrm{for}\mathrm{every}\phi \in E^{*}.$$

With $P[I,E]$, we denote the space of $E$—valued Pettis-integrable functions on the interval I. Immediately, it should be recalled that this space is normed but is not a complete space. Thus, we cannot reduce the case studied here to the results concerning Banach spaces (cf. [1]). Instead, this paper explores some subspaces of this function space. These subspaces consist of weakly Hölder continuous functions.

The problem of differentiability of weak integrals is more complicated than the strong differentiability. Let us recall, that there exists a strongly measurable and Pettis-integrable function $\theta (·):I\to E$ such that the indefinite Pettis integral

$$y\left(t\right)={\int }_0^t\theta \left(s\right)ds,t\in I$$

(1)

is not weakly differentiable on a set of positive Lebesgue measures or even nowhere weakly differentiable [17]. And this is one of the difficulties in achieving equivalence of differential and integral problems in weak topology.

The solution—although, as we shall see, not entirely sufficient—is to adopt a different, much more general definition of derivative [2].

Definition 2.

A function $x:I\to E$ is said to be pseudo-differentiable if there exists a function $y:I\to E$ such that, for every $\phi \in E^{*}$, the real-valued function $\phi \left(x\right(t\left)\right)$ is differentiable a.e., to the value $\phi \left(y\right(t\left)\right)$. In this case, the function y is called the pseudo-derivative of the function x and will be denoted by ${\mathfrak{D}}_p\left(x\right)$.

If the null set is independent on $\phi$, then x is said to be almost everywhere weakly differentiable on I and y (in this case) is called weak derivative of x (denoted by ${\mathfrak{D}}_{\omega }\left(x\right)$) and exists almost everywhere on I. In particular, when $E=\mathbb{R}$, it is clear that the pseudo- and almost everywhere weak derivatives coincide with the classical derivatives of real-valued functions.

If the space E has a total dual $E^{*}$. (i.e., the space $E^{*}$ contains a countable subset $\left(x_n^{*}\right)$ separating the points of E), then the pseudo-derivative, if any, is uniquely determined up to a set of measure zero. Note that the weak absolute continuity of Banach-valued functions does not necessarily imply strong or everywhere weak differentiability. This is another argument for the need to study the equivalence of fractional order derivatives problems beyond the superpositions of absolutely continuous functions.

Let ${\mathfrak{D}}_p$ denote the pseudo differential operator (resp. ${\mathfrak{D}}_{\omega }$ for the weak one). Similarly, for any $\psi \in C^1\left[I\right]$ being a positive increasing real-valued function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I$, we define some weak-type fractional derivative as follows

$${\delta }_{\omega }:=\mu +{\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\mathfrak{D}}_{\omega },{\delta }_p:=\mu +{\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\mathfrak{D}}_p.$$

Remember that the absolutely continuous real-valued function is a.e. differentiable. This is one of the few properties of the real-valued function that does not carry over to arbitrary Banach spaces: In fact, even in separable spaces, there is a weakly absolutely continuous function which has no pseudo-derivative [14]. We also note that a function can be pseudo-differentiable and at the same time neither weakly nor strongly differentiable. Let us first recall the relationship between the Pettis integral and the corresponding definition of differentiability.

As claimed above, with the notion of pseudo-derivative, we are able to make a step towards the equivalence of integral and differential (not yet fractional) problems:

Proposition 1

([16,19], Theorem VIII.3 in [14]). A function $f:I\to E$ is an indefinite Pettis integral if and only if f is weakly absolutely continuous having a pseudo-derivative ${\mathfrak{D}}_{p}f$ on I. In this case, ${\mathfrak{D}}_{p}f\in \mathrm{P}[I,E]$ and

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

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

Proposition 2

([20] and Theorem 5.1 in [21]). The indefinite integral of Pettis-integrable (resp. weakly continuous) function is wAC and it is pseudo- (resp. weakly) differentiable with respect to the right endpoint of the integration interval and its pseudo- (resp. weak) derivative equals the integrand at that point.

3. Fractional Integrals and Corresponding Function Spaces

We now turn to the case of derivatives and fractional integrals, and this idea of multiplication by a real and bounded function under the sign of the Pettis integral was already considered in the original paper [19]. We will now use this to define a very general fractional integral of a vector function.

What is new here? Compared to [19], here we have to consider operators on completely different function spaces (consisting of continuous functions), which will allow us to study the equivalence of differential (solutions are continuous or weakly continuous) and integral problems (i.e. the values of the integral operator should also be in such spaces).

Definition 3.

Let $\psi \in C^1\left[I\right]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I$. Let $f:I\to E$ be vector-valued, weakly measurable function. We define

1.

(ψ-tempered Riemann–Liouville fractional Pettis integral:) The tempered Riemann–Liouville ψ-fractional Pettis integral of f of order $\alpha >0$ and with parameter $\mu \ge 0$ is defined by

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

(2)

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

2.

(ψ-tempered Riemann–Liouville fractional derivative:) The ψ-tempered Riemann–Liouville fractional weak (resp. pseudo) derivative of order $n+\alpha \in (0,1)$, $n\in \mathbb{N}$, and with parameter $\mu \ge 0$ is defined as

$${\mathfrak{D}}_{\omega ,\psi }^{n+\alpha ,\mu }f\left(t\right):={\left({\delta }_{\omega }\right)}^n{\mathfrak{D}}_{\omega ,\psi }^{\alpha ,\mu }f,where{\mathfrak{D}}_{\omega ,\psi }^{\alpha ,\mu }f:={\delta }_{\omega }{\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f\left(t\right),$$

(3)

$$\left(resp.{\mathfrak{D}}_{p,\psi }^{n+\alpha ,\mu }f\left(t\right):={\left({\delta }_p\right)}^n{\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f,where{\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f:={\delta }_p{\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f\right),$$

(4)

3.

ψ-tempered Caputo fractional derivative: The ψ-tempered Caputo fractional weak (resp. pseudo-) derivative of order $n+\alpha \in (0,1)$, $n\in \mathbb{N}$, and with parameter $\mu \ge 0$ is defined as

$${\displaystyle \frac{d_{\omega ,\psi }^{n+\alpha ,\mu }}{d{t}^{n+\alpha }}}:={\left({\delta }_{\omega }\right)}^n{\displaystyle \frac{d_{\omega ,\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}}f,where{\displaystyle \frac{d_{\omega ,\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}}f:={\mathbb{I}}_{\psi }^{1-\alpha ,\mu }{\delta }_{\omega }f,$$

(5)

$$\left(resp.{\displaystyle \frac{d_{\omega ,\psi }^{n+\alpha ,\mu }}{d{t}^{n+\alpha }}}:={\left({\delta }_p\right)}^n{\displaystyle \frac{d_{p,\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}}f,where{\displaystyle \frac{d_{p,\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}}f:={\mathbb{I}}_{\psi }^{1-\alpha ,\mu }{\delta }_{p}f\right).$$

(6)

For any continuous $\psi :I\to \mathbb{R}$ with a positive continuous derivative ${\psi }^{\prime }$ on I (cf. Corollary 3.41 in [19]) ensures that ${\mathbb{I}}_{\psi }^{1,\mu }f$ exists for every $f\in \mathrm{P}[I,E]$ and by Proposition 2, it is easy to see that

$$\left\{\begin{array}{c}{\displaystyle {\delta }_{\omega }{\mathbb{I}}_{\psi }^{1,\mu }f=f,\mathrm{holds} \mathrm{for} \mathrm{any}f\in C[I,E_w],\left(D\omega I\right)}\hfill \\ {\displaystyle {\delta }_p{\mathbb{I}}_{\psi }^{1,\mu }f=f,\mathrm{holds} \mathrm{for} \mathrm{any}f\in \mathrm{P}[I,E].\left(DpI\right)}\hfill \end{array}\right.$$

Some remarks on the choice of integral and differential fractional operators. In recent years, there has been a tremendous growth both in the definition of fractional derivatives and in their applications. In short, the $\psi$-Hilfer operator discussed in this paper has some properties that are important for our problem. The generalized fractional calculus considered here unifies several existing ones, especially those considered in differential equations used to model some phenomena in physics. Although these conclusions are well known, we will briefly summarize them for the convenience of the reader. In particular, as special cases, certain choices of $\psi$ yield several classical models of fractional calculus (see [4,22]).

It also seems useful to compare studies of these operators in the case of norm topology. Originally they were studied in certain weight spaces of continuous or absolutely continuous functions [23], and only later the need to preserve the regularity of the values of integral operators by Hölder spaces was recognized (e.g., [4]). This includes several cases that have been studied separately.

In the case $\mu =0$ and $\psi \left(t\right)=t$, it is a classical fractional calculus with Riemann–Liouville and Caputo derivatives. There is a very large literature on this subject, so we will not go into the details.

TIn the case of $\mu \ne 0$ and $g\left(t\right)=t$, we obtain the tempered fractional calculus, which has been studied in recent years because of its applications, for example, in stochastic and dynamical systems. In this context of tempered calculus, fractional diffusion applies an exponential tempering factor to the particle jump density.

If $\mu =0$ and $\psi \left(t\right)=ln(1+t)$, with $\alpha >0$ we obtain the Hadamard calculus (the generalized version of the classical Hadamard calculus provided that $\mu \ne 0$). It is useful in the context of the study of the mean first-passage time as a function of $\psi \left(t\right)$ in order to highlight the difference with respect to the time-fractional model considered for the anomalous diffusion. The function g controls the anomalous diffusion at intermediate times.

Next, we consider the case $\psi \left(t\right)=t^{\rho }$ ($\rho >0$) and $\mu \in {\mathbb{R}}^{+}$. We obtain the Katugampola fractional integral operators (see also the case the Erdélyi–Kober fractional operators. When $\rho =2$ and $\alpha >{\displaystyle \frac{1}{2}}$, there are again some applications in the description of the anomalous subdiffusion [24,25].

In order not to focus the reader’s attention too much on this issue, let us simply note that the ability to select the $\psi$-function allows many real world phenomena to be modeled by its selection. Let us mention only $\psi \left(t\right)={\displaystyle \frac{t}{1+Bt}}[1+Aln(1+c{t}^{\gamma })]$ to describe subdiffusion in a medium with a time-evolving structure, or $\psi \left(t\right)=t+D_{\beta }{t}^{\beta /\alpha }$ in the study of the relation characteristic of superdiffusion processes.

However, finding and studying the case of weak topology leads to further complications.

Remark 1.

Instead, it is essential to keep in mind the differences between weak and strong topology, which we believe provides a strong basis for the separate research proposed in this paper. The case of the norm topology on E, as metrizable, is similar to the study for real functions with the replacement of the Lebesgue integral by the Bochner integral. If the functions under consideration are continuous in the sense of such a topology [26], the results obtained for real functions are similar to this case, the derivatives being in the strong sense [4]. In this paper, we will therefore concentrate on the much more complicated case of weak topology (as claimed in [27,28], for instance).

• As we have already mentioned, one reason for studying this case is the rather surprising non-equivalence of differential and integral problems in the general case is presented in [11] (which is why we study Hölder spaces), but there are others, such as the possibility of removing compactness conditions on functions appearing in differential equations [29] for those naturally satisfied under weak topology assumptions (cf. [2,18,30,31,32]).
• The fundamental differences between these cases will now be highlighted.

• Since the indefinite Pettis integral of a function $f\in \mathrm{P}[I,E]$ does not enjoy the strong property of being a.e. weakly differentiable (see [17]), then $\left(D\omega I\right)$ does not necessarily hold for arbitrary $f\in \mathrm{P}[I,E]$.
• The formula $\left(DpI\right)$ is not uniquely determined unless E has total dual $E^{*}$. Evidently, according to (e.g., [16], p. 2), it may happen that ${\delta }_p{\mathbb{I}}_{\psi }^{1,\mu }f=g,$ with g being weakly equivalent to f (but they need not be necessarily almost everywhere equal).

We are now going to look at what the inverse operators of the fractional integral operators we are studying are.

Let $\psi \in C^1\left[I\right]$ be a real-valued positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I$, $\alpha \in (0,1),\mu \ge 0$ and $f\in AC[I,E_{\omega }]$ be a vector-valued function with a pseudo-derivative in ${\mathrm{H}}^p\left(E\right),p>1/\alpha$. Then, by Proposition 1, we obtain the result for pseudo-derivatives:

$$\begin{array}{ccc}\hfill {\mathbb{I}}_{\psi }^{\alpha ,\mu }{\delta }_{p}f\left(t\right)& =& {\int }_a^t{e}^{-\mu \left(\psi \left(t\right)-\psi \left(s\right)\right)}\left[\mu f\left(s\right)+{\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\mathfrak{D}}_{p}f\left(s\right)\right]{\psi }^{\prime }\left(s\right)ds\hfill \\ & =& e^{-\mu \psi \left(t\right)}{\int }_a^t{\mathfrak{D}}_p\left[e^{\mu \psi \left(s\right)}f\left(s\right)\right]ds\hfill \\ & =& e^{-\mu \psi \left(t\right)}\left[e^{\mu \psi \left(t\right)}f\left(t\right)-e^{\mu \psi \left(a\right)}f\left(a\right)\right]=f\left(t\right)-f\left(a\right)e^{-\mu \psi \left(t\right)}.\hfill \end{array}$$

(7)

In particular, if ${\delta }_{\omega }f\in {\mathrm{H}}^p\left(E\right),p>1/\alpha$,

$${\mathbb{I}}_{\psi }^{\alpha ,\mu }{\delta }_{\omega }f\left(t\right)=f\left(t\right)-f\left(a\right)e^{-\mu \psi \left(t\right)}.$$

For the convenience of the reader, the following function spaces are provided:

Definition 4

( [19]). Let $\mathrm{P}[I,E]$ denote the space of E-valued Pettis-integrable functions on I. As its subspace, let us consider

$${\mathit{P}}^p\left(E\right):=\{\mathit{weakly} \mathit{measurable}f\in \mathrm{P}[I,E]\mathit{satisfying}\phi x\in L_p\left(I\right)\mathit{for} \mathit{every}\phi \in E^{*}\},$$

$${\tilde{\mathit{P}}}^p\left(E\right):=\{\mathit{strongly} \mathit{measurable}f\in \mathrm{P}[I,E]\mathit{satisfying}\phi x\in L_p\left(I\right)\mathit{for} \mathit{every}\phi \in E^{*}\}.$$

First, let us note some limitations on the consideration of fractional order integrals of vector functions. We are already aware of some results which indicate the need for in-depth studies of the equivalence problem between differential and integral equations. In our considerations, we need to go beyond the results of Pettis [19] (and [33] in the case of fractional operators for real-valued functions) and use the following results:

Proposition 3

([19]). A function $f\in {\mathit{P}}^p\left(E\right)$ if and only if $fg\in \mathrm{P}[I,E]$ for every $g\in L_q\left[I\right]$, with $1/p+1/q=1$.

Remark 2.

As known from our previous research (e.g., [4,11]), ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f,\alpha \in (0,1)$ exists (is convergent) if $f\in {\mathit{P}}^p\left(E\right),p>1/\alpha$. This is true for any $p\ge 1$ if E is a reflexive space. It should be added, however, that the Pettis integrability conditions have not yet been fully explored, even in the case of $\alpha =1$. A partial characterization and acting conditions are given in the following:

Proposition 4

(Lemma 2 in [12]). Let $\psi \in C^1\left[I\right]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I$. For any $\mu \ge 0$ and $\alpha ,\beta >0$, then

1.

For any $p>max\{1,1/\alpha \}$, we have ${\mathbb{I}}_{\psi }^{\alpha ,\mu }:{\mathit{P}}^p\left(E\right)\to C[I,E_{\omega }],$

2.

For any $p>max\{1,1/\alpha ,1/\beta \}$, we have ${\mathbb{I}}_{\psi }^{\alpha ,\mu }{\mathbb{I}}_{\psi }^{\beta ,\mu }\equiv {\mathbb{I}}_{\psi }^{\beta ,\mu }{\mathbb{I}}_{\psi }^{\alpha ,\mu }\equiv {\mathbb{I}}_{\psi }^{\alpha +\beta ,\mu }$ on ${\mathit{P}}^p\left(E\right).$

• If E is reflexive, this is also true for every $p\ge 1$.

The next step is to specify the appropriate function space as the domain of the operators. As in the case of real functions (cf. [4]), a variant of the Hölder condition plays an important role, as further demonstrated by Hardy and Littlewood in [33]. By reasoning as in [13], we know that there is $f\notin AC[I,E_{\omega }]$ such that ${\displaystyle \frac{d_{p,\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}}{\mathbb{I}}_{\psi }^{\alpha ,\mu }f$ is “meaningless" even for some Hölder continuous functions.

This points out that the equivalence between Caputo fractional differential and corresponding integral problems is no longer necessarily true outside the space of the absolutely continuous (or weakly absolutely continuous). In order to avoid such an equivalence problem, we proceed in a different way by showing the equivalence using the following definition

Definition 5.

Let $\psi \in C^1[a,b]$ be a real-valued positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in I$ and $f:I\to E$ is a weakly measurable vector-valued function. For any $\alpha \in (0,1),\mu \in {\mathbb{R}}^{+}$, we define a Caputo-type fractional derivative

$${\mathrm{I}}_{\psi }^{\alpha ,\mu }f:={\mathbb{I}}_{\psi }^{1,\mu }{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }f.$$

Let us introduce a subspace of the space of Pettis-integrable functions. To do it this, we need to recall Hölder spaces.

For $\lambda \in (0,1)$, the Hölder space ${\mathcal{H}}_0^{\lambda }[a,b]$ consists of all functions $f:[a,b]\to \mathbb{R}$ such that $f\left(a\right)=0$ and for $t,s\in [a,b]$, there exists a constant $M>0$ (independent on $t,s$) for which $|f\left(t\right)-f\left(s\right)|\le {M|t-s|}^{\lambda }$. Obviously, $f\in {\mathcal{H}}_0^{\lambda }[a,b]$ if the following seminorms

$${\left[f\right]}_{\lambda }:=\underset{t\ne s}{sup}\left\{{\displaystyle \frac{|f\left(t\right)-f\left(s\right)|}{{|t-s|}^{\gamma }}}\right\}$$

is finite. Define also its vector-valued counterpart:

$${\mathbf{H}}_0^{\lambda }[I,E_{\omega }]:=\{f\in C[I,E_{\omega }]:\mathrm{satisfying}\phi f\in {\mathcal{H}}_0^{\gamma }[a,b]\mathrm{for} \mathrm{every}\phi \in E^{*}\},$$

and

$${\mathbf{H}}_0^{\lambda }[I,E]:=\left\{f\in C[I,E]:f\left(a\right)=0\mathrm{and}{\left[\left[f\right]\right]}_{\lambda }<\infty \right\},$$

where ${\left[\left[f\right]\right]}_{\lambda }:={sup}_{t\ne s}{\displaystyle \frac{∥f\left(t\right)-f\left(s\right)∥}{{|t-s|}^{\lambda }}}$, and 0 denotes the zero element of E.

However, the equivalence problem is still not solved: recall that the weak absolute continuity of ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f$ is a necessary (but not sufficient) condition for the existence of a pseudo-derivative of ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f$. Moreover, ${\delta }_p{\mathbb{I}}_{\psi }^{\alpha ,\mu }f$ would exist on I, and then necessarily $f\in {\mathcal{H}}^p[I,E]$ with $p>{\displaystyle \frac{1}{\alpha }}.$

Obviously, ${\mathbf{H}}_0^{\lambda }[I,E]\subseteq {\mathbf{H}}_0^{\lambda }[I,E_{\omega }]$, and the equality holds when, e.g., $E=\mathbb{R}.$

Since the weak continuity of a function f implies strong measurability ([20], p. 73) and $\phi f\in L_{\infty }\left[I\right],\forall \phi \in E^{*}$, it obvious ${\mathbf{H}}_0^{\lambda }[I,E]\subseteq {\mathbf{H}}_0^{\lambda }[I,E_{\omega }]\subset {\mathbf{P}}^p\left(E\right)$. Also, for any $f\in C[I,E_{\omega }]$ we know that (cf. Remark 2) ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f$ exists and by definition of the Pettis integral, for any $\phi \in E^{*}$, we have

$$\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\right)={\mathbb{I}}_{\psi }^{\alpha ,\mu }\phi \left(f\right).$$

Spaces of the type we are considering (functions that satisfy the condition $f\left(a\right)=0$) are sometimes called little Hölder spaces. The Remark below shows why we are interested in these spaces.

Remark 3.

Let us remark that (see [34], Exercise 86(b), p. 353, [35], Lemma 3.1) for any $f\in C[I,E_{\omega }]$ we have

$$\begin{array}{ccc}\hfill \underset{t\to a}{lim}{\displaystyle \frac{1}{{\left(\psi \left(t\right)\right)}^{\alpha }}}\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}\left[e^{\mu \psi \left(t\right)}f\left(t\right)\right]\right)& =& \underset{t\to a}{lim}{\displaystyle \frac{1}{{\left(\psi \left(t\right)\right)}^{\alpha }}}{\mathbb{I}}_{\psi }^{\alpha ,0}\left[e^{\mu \psi \left(t\right)}\phi f\left(t\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{\Gamma (1+\alpha )}}\underset{t\to a}{lim}{e}^{\mu \psi \left(t\right)}\phi f\left(t\right),\alpha >0,\phi \in E^{*}.\hfill \end{array}$$

Therefore, for any $\phi \in E^{*}$,

$$\begin{array}{ccc}\hfill \underset{t\to a}{lim}\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)\right)& =& \underset{t\to a}{lim}{\mathbb{I}}_{\psi }^{\alpha ,\mu }\phi \left(f\left(t\right)\right)=\underset{t\to a}{lim}\left[e^{-\mu \psi \left(t\right)}{\displaystyle \frac{{\left(\psi \left(t\right)\right)}^{\alpha }}{{\left(\psi \left(t\right)\right)}^{\alpha }}}{\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}\phi f\left(t\right)\right)\right]\hfill \\ & =& \underset{t\to a}{lim}\left[e^{-\mu \psi \left(t\right)}{\left(\psi \left(t\right)\right)}^{\alpha }\right]{\displaystyle \frac{1}{\Gamma (1+\alpha )}}\underset{t\to a}{lim}{e}^{\mu \psi \left(t\right)}\phi f\left(t\right)=0=\phi \left(0\right).\hfill \end{array}$$

Hence, for any $f\in C[I,E_{\omega }]$ we have (in view of Proposition 4)

$$\underset{t\to a}{lim}{\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)={\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(a\right)=0.$$

(8)

Fractional integrability in the relevant sense of pseudo-derivatives will now be examined.

Lemma 1.

Let $g\in AC[a,b]$ and $f:[a,b]\to E$, weakly absolutely continuous with a pseudo-derivative ${\mathfrak{D}}_{p}f\in \mathrm{P}([a,b],E)$ be uniquely determined up to a negligible set. Then the Stieltjes integral ${\int }_a^{b}fdg$ exists and the following integration by the parts formula holds true:

$${\int }_a^{b}fdg={\int }_a^b{g}^{\prime }\left(s\right)f\left(s\right)ds={\left[g\left(s\right)f\left(s\right)\right]}_a^b-{\int }_a^{b}g\left(s\right){\mathfrak{D}}_{p}f\left(s\right)ds.$$

Proof. 

Since $f(·)g^{\prime }(·)$ (cf. Theorem 3.4 in [19]) and $g(·){\mathfrak{D}}_{p}f(·)\in \mathrm{P}([a,b],E),$ (cf. Corollary 3.41 in [19]), then by the definition of the Pettis integral on $[a,b]$, there exists an element in E denoted by ${\int }_a^{b}fdg$ such that for any $\phi \in E^{*}$

$$\phi \left({\int }_a^{b}fdg\right)={\int }_a^b{g}^{\prime }\left(s\right)\phi f\left(s\right)ds={\left[g\left(s\right)\phi f\left(s\right)\right]}_a^b-{\int }_a^{b}g\left(s\right){\left(\phi f\left(s\right)\right)}^{\prime }ds.$$

By the definition of the pseudo-derivative of f on $[a,b]$ we have for any $\phi \in E^{*},{\left(\phi f\left(s\right)\right)}^{\prime }=\phi \left({\mathfrak{D}}_{p}f\right)$. Consequently, since $g(·)\phi \left({\mathfrak{D}}_{p}f\right)(·)=\phi \left(g(·){\mathfrak{D}}_{p}f(·)\right)\in C[a,b]$ (namely, $g(·){\mathfrak{D}}_{p}f(·)\in \mathrm{P}([a,b],E)$), we obtain

$${\int }_a^{b}fdg={\int }_a^b{g}^{\prime }\left(s\right)f\left(s\right)ds={\left[g\left(s\right)f\left(s\right)\right]}_a^b-{\int }_a^{b}g\left(s\right){\mathfrak{D}}_{p}f\left(s\right)ds.$$

(9)

□

The next lemma gives a condition that ensures that the two integrals ${\mathrm{I}}_{\psi }^{\alpha ,\mu }$ and ${\mathbb{I}}_{\psi }^{\alpha ,\mu }$ coincide in the space of weakly absolutely continuous functions:

Lemma 2.

Let $\psi \in C^1\left[I\right]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I$, $\alpha \in (0,1),\mu \ge 0$ and $f\in AC[I,E_{\omega }]$, which is weakly absolutely continuous with a pseudo-derivative ${\mathit{P}}^p\left(E\right),p>1/\alpha$ uniquely determined up to a negligible set. Then, ${\mathrm{I}}_{\psi }^{\alpha ,\mu }f$ exists and

$${\mathrm{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)={\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right).$$

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

Proof. 

Let $f\in AC[I,E_{\omega }]$ with a pseudo-derivative in ${\mathit{P}}^p\left(E\right)$, $p>1/\alpha$. Then $f(·)e^{\mu \psi (·)}\in AC[I,E_{\omega }]$ having a pseudo-derivative in ${\mathbf{P}}^p\left(E\right)$, $p>1/\alpha$ as well. Since $C[I,E_{\omega }]\subset {\mathbf{P}}^{\infty }\left(E\right)$, it follows (cf. Remark 2) that ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f$ exists and by definition of Pettis integral, for any $\phi \in E^{*}$, we have

$$\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\right)={\mathbb{I}}_{\psi }^{\alpha ,\mu }\phi \left(f\right).$$

By Proposition 1, we have

$$f\left(t\right)e^{\mu \psi \left(t\right)}=f\left(a\right)+{\int }_a^t{\mathfrak{D}}_p\left(e^{\mu \psi \left(s\right)}f\left(s\right)\right)ds,$$

for $t\in I$. Therefore,

$${\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)={\mathbb{I}}_{\psi }^{\alpha ,0}f\left(a\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}\left({\int }_a^s{\mathfrak{D}}_p\left(e^{\mu \psi \left(\theta \right)}f\left(\theta \right)\right)d\theta \right){\psi }^{\prime }\left(s\right)ds.$$

Since the indefinite integral of ${\mathfrak{D}}_p\left(e^{\mu \psi (·)}f(·)\right)$ is weakly absolutely continuous and has a pseudo-derivative on I, it follows from (9) that

$${\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)={\mathbb{I}}_{\psi }^{\alpha ,0}f\left(a\right)+{\displaystyle \frac{1}{\Gamma (1+\alpha )}}{\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }{\mathfrak{D}}_p\left(e^{\mu \psi \left(s\right)}f\left(s\right)\right){\psi }^{\prime }\left(s\right)ds.$$

Keep in mind the following:

$${\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\mathfrak{D}}_p\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)=e^{\mu \psi \left(t\right)}{\delta }_{p}f\left(t\right)\in {\mathbf{P}}^p\left(E\right),$$

for $p>1/\alpha$, we obtain

$${\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)={\mathbb{I}}_{\psi }^{\alpha ,0}f\left(a\right)+{\displaystyle \frac{1}{\Gamma (1+\alpha )}}{\int }_a^t{e}^{\mu \psi \left(s\right)}{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }{\delta }_{p}f\left(t\right){\psi }^{\prime }\left(s\right)ds.$$

Hence,

$$\begin{array}{ccc}\hfill {\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)& =& e^{-\mu \psi \left(t\right)}{\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)\hfill \\ & =& e^{-\mu \psi \left(t\right)}{\displaystyle \frac{f\left(a\right){\left(\psi \left(t\right)\right)}^{\alpha }}{\Gamma (1+\alpha )}}+{\displaystyle \frac{1}{\Gamma (1+\alpha )}}{\int }_a^t{e}^{-\mu \left(\psi \right(t)-\psi (s\left)\right)}{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha }{\delta }_{p}f\left(s\right){\psi }^{\prime }\left(s\right)ds.\hfill \end{array}$$

Consequently, according to Proposition 4

$${\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)=e^{-\mu \psi \left(t\right)}{\displaystyle \frac{f\left(a\right){\left(\psi \left(t\right)\right)}^{\alpha }}{\Gamma (1+\alpha )}}+{\mathbb{I}}_{\psi }^{1,\mu }{\mathbb{I}}_{\psi }^{\alpha ,\mu }{\delta }_{p}f\left(t\right).$$

(10)

So ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f$ is weakly absolutely continuous and possesses a Pettis-integrable pseudo-derivative on I. By Proposition 1

$${\mathrm{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)={\mathbb{I}}_{\psi }^{1,\mu }{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }f\left(t\right)={\mathbb{I}}_{\psi }^{1,\mu }{\delta }_p\left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)\right)={\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)-{\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(a\right).$$

Obviously, ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(a\right)=0$, by virtue of Remark 3. Moreover, by Remark 2, this is also true for all $p\ge 1$ when E is reflexive. □

Remark 4.

In Lemma 2, the pseudo-derivative of ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f$ is, in fact, “an almost everywhere weak derivative” on I: To see this, we notice that $e^{\mu \psi (·)}{\delta }_{p}f(·)\in {\mathit{P}}^p\left(E\right),p>1/\alpha$. Hence, by Proposition 4, ${\mathbb{I}}_{\psi }^{\alpha ,0}{e}^{\mu \psi (·)}{\delta }_{p}f(·)\in C[I,E_{\omega }]$. By (10), we have

$${\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)=e^{\mu \psi \left(t\right)}{\displaystyle \frac{f\left(a\right)}{\Gamma \left(\alpha \right)}}{\int }_a^t{\left(\psi \left(s\right)\right)}^{\alpha -1}{\psi }^{\prime }\left(s\right)ds+e^{-\mu \psi \left(t\right)}{\mathbb{I}}_{\psi }^{1,0}{\mathbb{I}}_{\psi }^{\alpha ,0}{e}^{\mu \psi \left(t\right)}{\delta }_{p}f\left(t\right).$$

Recall [30] that the indefinite Pettis integral of a weakly continuous function is weakly absolutely continuous and possesses a weak derivative on I. This is in the view that the null set (where the derivative of $e^{\mu \psi (·)}\phi f\left(a\right){\left(\psi (·)\right)}^{\alpha }$ fails) is invariant for all $\phi \in E^{*}$, yields the almost everywhere weak differentiability of ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f$ on I.

And now one of the key results of the paper for vector functions, which is an analogue of the Hardy–Littlewood theorem [33] and the result of [4] for real-valued functions. Despite all the differences between integrals and topologies, fractional order operators preserve very well the Hölder orders of the functions on which they act.

Theorem 1.

Let $\psi \in C^1\left[I\right]$ be a real-valued positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I$, $\alpha \in (0,1)$, $p>1/\alpha$. Then, the vector-valued integral operator has values in ${\mathit{H}}^{\alpha -{\displaystyle \frac{1}{p}}}[I,E_{\omega }]$, i.e.,

$${\mathbb{I}}_{\psi }^{\alpha ,0}:{\mathit{P}}^p\left(E\right)\to {\mathit{H}}^{\alpha -{\displaystyle \frac{1}{p}}}[I,E_{\omega }],$$

for any $\phi \in E^{*}$, and moreover,

$${\left[\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}f\right)\right]}_{\alpha -{\displaystyle \frac{1}{p}}}\le {\displaystyle \frac{2{∥{\psi }^{\prime }∥}^{\alpha }}{\sqrt[q]{q(\alpha -1)+1}\Gamma \left(\alpha \right)}}{∥\phi f∥}_{L_p[a,b]},$$

where $1/p+1/q=1$.

Proof. 

Let $f\in {\mathit{P}}^p\left(E\right)$, $p>1/\alpha$; it follows that ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f$ exists (cf. Remark 2) and, by definition of the Pettis integral, for any $\phi \in E^{*}$, we have $\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}f\right)={\mathbb{I}}_{\psi }^{\alpha ,0}\phi \left(f\right)$. Now let $h>0,t,t+h\in I$. We have the following estimate

$$\begin{array}{ccc}& & \Gamma \left(\alpha \right)\left|\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}f(t+h)\right)-\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}f\left(t\right)\right)\right|=\Gamma \left(\alpha \right)\left|{\mathbb{I}}_{\psi }^{\alpha ,0}\phi f(t+h)-{\mathbb{I}}_{\psi }^{\alpha ,0}\phi f\left(t\right)\right|\hfill \\ & \le & \left|{\int }_t^{t+h}{\left(\psi (t+h)-\psi \left(s\right)\right)}^{\alpha -1}\phi f\left(s\right){\psi }^{\prime }\left(s\right)ds\right|+\hfill \\ & & +{\int }_a^t\left|{\left(\psi (t+h)-\psi \left(s\right)\right)}^{\alpha -1}-{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{\alpha -1}\right|\left|\phi f\left(s\right)\right|{\psi }^{\prime }\left(s\right)ds\hfill \\ & \le & {∥\phi f\left(t\right)∥}_{L_p[t,t+h]}{\left({\int }_t^{t+h}{\left(\psi (t+h)-\psi \left(s\right)\right)}^{q(\alpha -1)}{\left({\psi }^{\prime }\left(s\right)\right)}^{q}ds\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & & +{∥\phi f\left(t\right)∥}_{L_p[a,t]}{\left({\int }_a^t{\left|{\left(\psi (t+h)-\psi \left(s\right)\right)}^{\alpha -1}-{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{\alpha -1}\right|}^q{\left({\psi }^{\prime }\left(s\right)\right)}^{q}ds\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le & {\displaystyle \frac{{∥\phi f\left(t\right)∥}_{L_p[t,t+h]}}{\sqrt[q]{q(\alpha -1)+1}}}{∥{\psi }^{\prime }∥}^{{\displaystyle \frac{1}{p}}}{\left({\left(\psi (t+h)-\psi \left(t\right)\right)}^{q(\alpha -1)+1}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & & +{\displaystyle \frac{{∥\phi f\left(t\right)∥}_{L_p[a,t]}}{q(\alpha -1)+1}}{∥{\psi }^{\prime }∥}^{{\displaystyle \frac{1}{p}}}\hfill \\ & & ·{\left|{\left(\psi (t+h)-\psi \left(t\right)\right)}^{q(\alpha -1)+1}+\underset{\le 0}{\underbrace{{\left(\psi \left(t\right)\right)}^{q(\alpha -1)+1}-{\left(\psi (t+h)\right)}^{q(\alpha -1)+1}}}\right|}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le & {\displaystyle \frac{{∥\phi f\left(t\right)∥}_{L_p[t,t+h]}}{\sqrt[q]{q(\alpha -1)+1}}}{∥{\psi }^{\prime }∥}^{{\displaystyle \frac{1}{p}}}{\left({\left(h∥{\psi }^{\prime }∥\right)}^{q(\alpha -1)+1}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & & +{\displaystyle \frac{{∥\phi f\left(t\right)∥}_{L_p[a,t]}}{\sqrt[q]{q(\alpha -1)+1}}}{∥{\psi }^{\prime }∥}^{{\displaystyle \frac{1}{p}}}{\left[{\left(\psi (t+h)-\psi \left(t\right)\right)}^{q(\alpha -1)+1}\right]}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le & {\displaystyle \frac{2{∥{\psi }^{\prime }∥}^{\alpha }}{\sqrt[q]{q(\alpha -1)+1}}}{h}^{\alpha -{\displaystyle \frac{1}{p}}}max\left\{{∥\phi f\left(t\right)∥}_{L_p[t,t+h]},{∥\phi f\left(t\right)∥}_{L_p[a,t]}\right\}.\hfill \end{array}$$

That is,

$$\begin{array}{ccc}& & \left|\phi ({\mathbb{I}}_{\psi }^{\alpha ,0}f(t+h)-\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}f\right)\left(t\right)\right|\hfill \\ & & \le {\displaystyle \frac{2{∥{\psi }^{\prime }∥}^{\alpha }}{\sqrt[q]{q(\alpha -1)+1}\Gamma \left(\alpha \right)}}max\left\{{∥\phi f\left(t\right)∥}_{L_p[t,t+h]},{∥\phi f\left(t\right)∥}_{L_p[a,t]}\right\}{h}^{\alpha -{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

(11)

Hence, ${\mathbb{I}}_{\psi }^{\alpha ,0}$ maps ${\mathbf{P}}^p\left(E\right),p>1/\alpha$ into ${\mathbf{H}}^{\alpha -{\displaystyle \frac{1}{p}}}[I,E_{\omega }]$ and for any $\phi \in E^{*}$,

$${\left[\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\right)\right]}_{\alpha -{\displaystyle \frac{1}{p}}}\le {\displaystyle \frac{2{∥{\psi }^{\prime }∥}^{\alpha }}{\sqrt[q]{q(\alpha -1)+1}\Gamma \left(\alpha \right)}}{∥\phi f\left(t\right)∥}_{L_p[a,b]}.$$

□

We are now going to show an analogous result for $\mu >0$.

Since $e^{\mu \psi (·)}f(·){\mathbf{P}}^p\left(E\right)$ for every $f\in {\mathbf{P}}^p\left(E\right)$ for $p>1/\alpha$ and since

$${\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)=e^{-\mu \psi \left(t\right)}{\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right),$$

We can repeat our argument with slight modifications. Namely, if one argues as in the proof of Theorem 1, then for all $h>0,t,t+h\in I$ and $\phi \in E^{*}$, we have

$$\begin{array}{ccc}& & \left|\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}\left[e^{\mu \psi (t+h)}f(t+h)\right]\right)-\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}\left[e^{\mu \psi \left(t\right)}f\left(t\right)\right]\right)\right|\hfill \\ & \le & {\displaystyle \frac{2{∥{\psi }^{\prime }∥}^{\alpha }}{\sqrt[q]{q(\alpha -1)+1}\Gamma \left(\alpha \right)}}max\left\{{∥\phi e^{\mu \psi \left(t\right)}f\left(t\right)∥}_{L_p[t,t+h]},{∥\phi e^{\mu \psi \left(t\right)}f\left(t\right)∥}_{L_p[a,t]}\right\}{h}^{\alpha -{\displaystyle \frac{1}{p}}}\hfill \\ & \le & {\displaystyle \frac{2{∥{\psi }^{\prime }∥}^{\alpha }{e}^{\mu \psi \left(t\right)}}{\sqrt[q]{q(\alpha -1)+1}\Gamma \left(\alpha \right)}}max\left\{{∥\phi f\left(t\right)∥}_{L_p[t,t+h]},{∥\phi f\left(t\right)∥}_{L_p[a,t]}\right\}{h}^{\alpha -{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}\hfill \Gamma \left(\alpha \right)\left|\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}\left[e^{\mu \psi \left(t\right)}f\left(t\right)\right]\right)\right|& \le & {∥\phi f\left(t\right)∥}_{L_p[a,t]}{\left({\int }_a^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{q(\alpha -1)}{\left({\psi }^{\prime }\left(s\right)\right)}^{q}ds\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le & {\displaystyle \frac{{∥{\psi }^{\prime }∥}^{\alpha }{e}^{\mu \psi \left(t\right)}}{\sqrt[q]{q(\alpha -1)+1}}}{\left(\psi \left(t\right)\right)}^{\alpha -{\displaystyle \frac{1}{p}}}{∥\phi f\left(t\right)∥}_{L_p[a,t]}.\hfill \end{array}$$

Since ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)=e^{-\mu \psi \left(t\right)}{\mathbb{I}}_{\psi }^{\alpha ,\mu }\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right),$ we obtain

$$\begin{array}{ccc}\hfill {\mathbb{I}}_{\psi }^{\alpha ,\mu }f(t+h)-{\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)& =& e^{-\mu \psi (t+h)}{\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(\right(t+h\left)\right)}f(t+h)\right)-e^{-\mu \psi (t+h)}{\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)\hfill \\ & +& e^{-\mu \psi (t+h)}{\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)-e^{-\mu \psi \left(t\right)}{\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)\hfill \\ & =& e^{-\mu \psi (t+h)}\left[{\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(\right(t+h\left)\right)}f(t+h)\right)-{\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)\right]\hfill \\ & +& {\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)\left[e^{-\mu \psi (t+h)}-e^{-\mu \psi \left(t\right)}\right].\hfill \end{array}$$

In view of the mean value theorem,

$|exp\{-\mu \psi (t+h)\}-exp\{-\mu \psi \left(t\right)\}|\le \mu e^{-\mu \psi \left(t\right)}∥{\psi }^{\prime }∥h,$ and then

$$\begin{array}{ccc}& & \left|\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f(t+h)\right)-\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)\right)\right|\hfill \\ & \le & e^{-\mu \psi (t+h)}\left|\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}\left[e^{\mu \psi (t+h)}f(t+h)\right]\right)-\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}\left[e^{\mu \psi \left(t\right)}f\left(t\right)\right]\right)\right|\hfill \\ & +& \left|e^{-\mu \psi (t+h)}-e^{-\mu \psi \left(t\right)}\right|\left|\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}\left[e^{\mu \psi \left(t\right)}f\left(t\right)\right]\right)\right|\hfill \\ & \le & e^{-\mu \psi \left(t\right)}\left|\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}\left[e^{\mu \psi (t+h)}f(t+h)\right]\right)-\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}\left[e^{\mu \psi \left(t\right)}f\left(t\right)\right]\right)\right|\hfill \\ & +& \mu e^{-\mu \psi \left(t\right)}∥{\psi }^{\prime }∥h\left|\phi \left({\mathbb{I}}_{\psi }^{\alpha ,0}\left[e^{\mu \psi \left(t\right)}f\left(t\right)\right]\right)\right|\hfill \\ & \le & {\displaystyle \frac{2{∥{\psi }^{\prime }∥}^{\alpha }}{\sqrt[q]{q(\alpha -1)+1}\Gamma \left(\alpha \right)}}max\left\{{∥\phi f\left(t\right)∥}_{L_p[t,t+h]},{∥\phi f\left(t\right)∥}_{L_p[a,t]}\right\}{h}^{\alpha -{\displaystyle \frac{1}{p}}}\hfill \\ & +& \mu h{\displaystyle \frac{{∥{\psi }^{\prime }∥}^{\alpha +1}{∥\psi ∥}^{\alpha -\frac{1}{p}}}{\sqrt[q]{q(\alpha -1)+1}}}{∥\phi f\left(t\right)∥}_{L_p[a,t]}.\hfill \end{array}$$

Since $h=h^{\alpha -{\displaystyle \frac{1}{p}}}{h}^{1-\alpha +{\displaystyle \frac{1}{p}}}\le h^{\alpha -{\displaystyle \frac{1}{p}}}{(b-a)}^{1-\alpha +{\displaystyle \frac{1}{p}}},$ we obtain the following

$$\begin{array}{ccc}& & \left|\phi {\mathbb{I}}_{\psi }^{\alpha ,\mu }f(t+h)-\phi {\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)\right|\le {\displaystyle \frac{{∥{\psi }^{\prime }∥}^{\alpha }}{\sqrt[q]{q(\alpha -1)+1}\Gamma \left(\alpha \right)}}\left[2max\left\{{∥\phi f∥}_{L_p[t,t+h]},{∥\phi f∥}_{L_p[a,t]}\right\}\right.\hfill \\ & & +\left.\mu ∥{\psi }^{\prime }∥{(b-a)}^{1-\alpha +{\displaystyle \frac{1}{p}}}{∥\psi ∥}^{\alpha -{\displaystyle \frac{1}{p}}}{∥\phi f∥}_{L_p[a,t]}\right]h^{\alpha -{\displaystyle \frac{1}{p}}}\hfill \\ & & \le {\displaystyle \frac{{∥{\psi }^{\prime }∥}^{\alpha +1}}{\sqrt[q]{q(\alpha -1)+1}\Gamma \left(\alpha \right)}}\left[{\displaystyle \frac{2}{∥{\psi }^{\prime }∥}}+\mu {(b-a)}^{1-\alpha +{\displaystyle \frac{1}{p}}}{∥\psi ∥}^{\alpha -{\displaystyle \frac{1}{p}}}\right]{∥\phi f∥}_{L_p[a,b]}{h}^{\alpha -{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

(12)

Thus, in the vector-valued case we have also shows interesting results for $\mu \ne 0$:

Theorem 2.

Let $\psi \in C^1\left[I\right]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I$, $\alpha \in (0,1),p>1/\alpha$. Then for any $\mu \ge 0,$

$${\mathbb{I}}_{\psi }^{\alpha ,\mu }:{\mathit{P}}^p\left(E\right)\to {\mathit{H}}^{\alpha -{\displaystyle \frac{1}{p}}}[I,E_{\omega }],$$

for any $\phi \in E^{*}$,

$${\left[\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\right)\right]}_{\alpha -{\displaystyle \frac{1}{p}}}\le {\displaystyle \frac{{∥{\psi }^{\prime }∥}^{\alpha +1}}{\sqrt[q]{q(\alpha -1)+1}\Gamma \left(\alpha \right)}}\left[{\displaystyle \frac{2}{∥{\psi }^{\prime }∥}}+\mu {(b-a)}^{1-\alpha +{\displaystyle \frac{1}{p}}}{∥\psi ∥}^{\alpha -{\displaystyle \frac{1}{p}}}\right]{∥\phi f\left(t\right)∥}_{L_p[a,b]}.$$

It is worth noting that the theorem proved is a far-reaching extension of the classical Hardy–Littlewood (Theorem 4 in [33]) result. In the case $E=\mathbb{R}$ and $\psi \left(t\right)=t$, we have:

Corollary 1

( Theorem 4 in [33]). Suppose that $f\in L^p\left(I\right)$, where $p>1$, $0<\alpha <1/p$, $q={\displaystyle \frac{p}{1-p\alpha }}$. Then the Riemann–Liouville fractional integral operator has values in $L^q\left(I\right)$, $I^{\alpha }:L^p\left(I\right)\to L^q\left(I\right)$, and

$$\parallel I^{\alpha }{\left(f\right)\parallel }_q\le K·{\parallel f\parallel }_p,$$

where $K=K(p,\alpha )$ is a function of p and α only.

In the above result, the Riemann–Liouville integral operator $I^{\alpha }$ still acts between Lebesgue spaces. In the context of Hölder continuity of images of $L^p$-integrable functions, we also extend the following:

Corollary 2

(Proposition 3.2(3) in [3]). For $1/p<\alpha <1+1/p$ or $p=1$ and $1\le \alpha <2$, the fractional Riemann–Liouville integral operator $I^{\alpha }$ is bounded from $L^p$ into a Hölder space ${\mathit{H}}_0^{\alpha -1/p}$, and hence, for $f\in L^p$, $I^{\alpha }f$ is Hölder continuous with exponent $\alpha -1/p$.

It remains to examine the operator once we have restricted its domain to the appropriate Hölder space.

Proposition 5.

Let $\psi \in C^1\left[I\right]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I$, $\mu \ge 0,\alpha \in (0,1),p>{\displaystyle \frac{1}{\alpha }}$. Then

$${\mathbb{I}}_{\psi }^{\alpha ,\mu }:{\mathbf{H}}_0^{\lambda }[I,E_{\omega }]\to {\mathbf{H}}_0^{\alpha -{\displaystyle \frac{1}{p}}}[I,E_{\omega }],\mathit{holds} \mathit{for} \mathit{any}\lambda \in (0,1).$$

Also, there exists a constant $L\ge 0$ such that

$${\left[\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)\right)\right]}_{\alpha -{\displaystyle \frac{1}{p}}}\le L{\left[\phi f\left(t\right)\right]}_{\lambda }.$$

Proof. 

Let $f\in {\mathbf{H}}_0^{\lambda }[I,E_{\omega }]\subset {\mathbf{P}}_0^p\left(E\right):=\{f\in {\mathbf{P}}^p\left(E\right):f\left(a\right)=0\}$ for any $p\ge 1$ (in particular for any $p>{\displaystyle \frac{1}{\alpha }}$). It follows (cf. Remark 2) that ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f$ exists and by definition of the Pettis integral, and for any $\phi \in E^{*}$, we have $\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\right)={\mathbb{I}}_{\psi }^{\alpha ,\mu }\phi \left(f\right).$ Also, by (8), we have ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(a\right)=0$. Obviously, since $\alpha <1-\lambda ,p>1/\alpha$, then automatically $p>1/(1-\lambda )$ or $\lambda +{\displaystyle \frac{1}{p}}<1$. Since

$$\begin{array}{ccc}\hfill {∥\phi f∥}_{L_p[a,b]}^p& =& {\int }_a^b{\left|\phi f\left(s\right)\right|}^{p}ds={\int }_a^b{\left|\phi f\left(s\right)-\phi f\left(a\right)\right|}^{p}ds\le {\left[\phi f\left(s\right)\right]}_{\lambda }^p{\int }_a^b{\left(s-a\right)}^{p\lambda }ds\hfill \\ & =& {\displaystyle \frac{{\left[\phi f\right]}_{\lambda }^p}{p\lambda +1}}{\left(b-a\right)}^{p\lambda +1},\hfill \end{array}$$

it follows

$${∥\phi f∥}_{L_p[a,b]}\le {\displaystyle \frac{{\left[\phi f\left(s\right)\right]}_{\lambda }}{\sqrt[p]{p\lambda +1}}}{\left(b-a\right)}^{\lambda +{\displaystyle \frac{1}{p}}}.$$

Therefore, the result follows from Theorem 2. □

Finally, we proved a vector-valued extension for the case of generalized fractional integral operators acting between Hölder spaces:

Corollary 3

( Theorem 1 in [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 $\psi \left(a\right)=0$. For arbitrary $\mu >0$, $\alpha ,\delta ,\zeta \in (0,1)$ such that $\alpha -\delta \ge \zeta$, we have

$${\mathrm{I}}_{\psi }^{\alpha ,\mu }:{\mathcal{H}}_0^{\zeta -\alpha +1}[a,b]\to {\mathcal{H}}_0^{\zeta -\alpha +1+\delta }[a,b].$$

In particular, for any $\alpha ,\zeta \in (0,1)$ with $\alpha >\zeta ,$ the operator ${\mathrm{I}}_{\psi }^{\alpha ,\mu }$ maps the Hölder space ${\mathcal{H}}_0^{\zeta -\alpha +1}[a,b]$ into itself.

Theorem 3.

Let $\psi \in C^1\left[I\right]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I$, $\mu \ge 0,\alpha ,\lambda \in (0,1)$ such that $\alpha +\lambda <1$. Then

$${\mathbb{I}}_{\psi }^{\alpha ,\mu }:{\mathit{H}}_0^{\lambda }[I,E_{\omega }]\to {\mathit{H}}_0^{\lambda +\alpha }[I,E_{\omega }],\lambda +\alpha <1.$$

Also, there exists a constant $L\ge 0$ such that

$${\left[\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\right)\right]}_{\lambda +\alpha }\le L{\left[\phi f\right]}_{\lambda }.$$

As we have already noted, it is possible to consider the norm topology on E in a way that is similar to the case of the weak topology (but not vice versa). So, we have:

Corollary 4.

Let $\psi \in C^1\left[I\right]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I$, $\mu \ge 0,\alpha ,\lambda \in (0,1)$ such that $\alpha +\lambda <1$. Then,

$${\mathbb{I}}_{\psi }^{\alpha ,\mu }:{\mathit{H}}_0^{\lambda }[I,E]\to {\mathit{H}}_0^{\lambda +\alpha }[I,E],\lambda +\alpha <1.$$

Also, there exists $k\ge 0$ such that

$${\left[\left[{\mathbb{I}}_{\psi }^{\alpha ,\mu }f\right]\right]}_{\alpha +\lambda }\le K{\left[\left[f\right]\right]}_{\lambda }.$$

Proof. 

We will give a sketch of the proof showing how to obtain results for this case using our main theorem. Let $f\in {\mathbf{H}}_0^{\lambda }[I,E]$. Since $f\in {\mathbf{H}}_0^{\lambda }[I,E]\subseteq {\mathbf{H}}_0^{\lambda }[I,E_{\omega }],$ we obtain ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f\in {\mathbf{H}}_0^{\lambda +\alpha }[I,E_{\omega }]$. Also, any $\phi \in E^{*}$ we know that

$${\left[\phi f\right]}_{\lambda }=\underset{t\ne s}{sup}{\displaystyle \frac{\left|\phi \left(f\right(t)-f(s\left)\right)\right|}{{|t-s|}^{\lambda }}}\le {∥\phi ∥}_{E^{*}}\underset{t\ne s}{sup}{\displaystyle \frac{∥f\left(t\right)-f\left(s\right)∥}{{|t-s|}^{\lambda }}}={∥\phi ∥}_{E^{*}}{\left[\left[f\right]\right]}_{\lambda }.$$

Without loss of generality, assume that ${\mathbb{I}}_{\psi }^{\alpha ,\mu }f(t+h)-{\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)\ne 0$. Then, there exists (as an immediate consequence of the Hahn-Banach theorem) $\phi \in E^{*}$ with $∥\phi ∥=1$ such that (cf. Theorem 3)

$$\begin{array}{ccc}\hfill ∥{\mathbb{I}}_{\psi }^{\alpha ,\mu }f(t+h)-{\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)∥& =& \left|\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f(t+h)-{\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)\right)\right|\hfill \\ & =& \left|\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\right)(t+h)-\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\right)\left(t\right)\right|.\hfill \end{array}$$

Note that the function $\phi \in E^{*}$ is chosen with fixed t and h, i.e., to obtain a universal estimate for the Hölder constant we must estimate these expressions independently of these quantities. However, by definition, we have estimated

$$\left|\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\right)(t+h)-\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }f\right)\left(t\right)\right|\le K{\left[\phi f\right]}_{\lambda }{h}^{\alpha +\lambda }\le K{∥\phi ∥}_{E^{*}}{\left[\left[f\right]\right]}_{\lambda }\le K{h}^{\alpha +\lambda }{\left[\left[f\right]\right]}_{\lambda }.$$

That explains how this came about. □

As a result, we obtain the following

Corollary 5.

Let $\psi \in C^1\left[I\right]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I$, $\mu \ge 0$ and $\delta ,\lambda \in (0,1),\delta >\lambda$. Then, we have

$${\mathbb{I}}_{\psi }^{1,\mu }f\in {\mathit{H}}_0^{1+\lambda -\delta }[I,E_{\omega }], \mathit{holds} \mathit{for} \mathit{any}f\in {\mathit{H}}_0^{\lambda }[I,E_{\omega }]⟺{\mathbb{I}}_{\psi }^{1,\mu }:{\mathit{H}}_0^{\lambda }[I,E_{\omega }]\to {\mathit{H}}_0^{1+\lambda -\delta }[I,E_{\omega }].$$

(13)

In particular,

$${\mathbb{I}}_{\psi }^{1,\mu }:{\mathit{H}}_0^{\lambda }[I,E_{\omega }]\to {\mathit{H}}_0^{\lambda }[I,E_{\omega }].$$

Proof. 

Choose $\eta \in (\delta -\lambda ,1-\lambda )$. Since $\lambda +\eta <1,$ it follows from Theorem 3,

$${\mathbb{I}}_{\psi }^{\eta ,\mu }:{\mathbf{H}}_0^{\lambda }[I,E_{\omega }]\to {\mathbf{H}}_0^{\lambda +\eta }[I,E_{\omega }].$$

Since ${\mathbf{H}}_0^{\lambda +\eta }[I,E_{\omega }]\subset {\mathbf{H}}_0^{\lambda +\eta -\delta }[I,E_{\omega }],\lambda +\eta -\delta +1-\eta =1+\lambda -\delta <1,$ we obtain

$${\mathbb{I}}_{\psi }^{1-\eta ,\mu }:{\mathbf{H}}_0^{\lambda +\eta }[I,E_{\omega }]\subset {\mathbf{H}}_0^{\lambda +\eta -\delta }[I,E_{\omega }]\to {\mathbf{H}}_0^{1+\lambda -\delta }[I,E_{\omega }].$$

Therefore,

$${\mathbb{I}}_{\psi }^{1,\mu }={\mathbb{I}}_{\psi }^{1-\eta ,\mu }{\mathbb{I}}_{\psi }^{\eta ,\mu }:{\mathbf{H}}_0^{\lambda }[I,E_{\omega }]\to {\mathbf{H}}_0^{1+\lambda -\delta }[I,E_{\omega }].$$

□

4. Fractional Pseudo-Differentiability

Now, recall that the Riemann–Liouville fractional pseudo-derivative ${\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f$, of the function $f:I\to E$ exists if there is a function $g:I\to E$ and a null set $N\left(\phi \right)\subset I$ (depending only on $\phi \in E^{*}$) such that

$$\phi \left({\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f\right)\left(t\right)=\delta \phi \left({\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f\right)\left(t\right)=\delta {\mathbb{I}}_{\psi }^{1-\alpha ,\mu }\phi \left(f\right)\left(t\right)=\phi \left(g\left(t\right)\right),$$

for every $t\in I\setminus N\left(\phi \right)$. In this case, $g={\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f$. It is important to emphasize the differences between a derivative and a pseudo-derivative, which exclude the automatic application ’by analogy’ of results from a strong topology to a weak topology. As far as we know, the first systematic study of pseudo-derivatives was performed by Solomon [14]. We know that the simplest case, where the pseudo-derivative is uniquely determined up to a set of measure zero, is to assume that E is total dual (Theorem VI.8 in [14]). However, this is not a necessary condition, so we have applied the assumption of singularity of the determination of the pseudo-derivative almost everywhere immediately.

Let us start with the sufficiency condition for the pseudo-differential of the fractional-order vector integral, which is of course a necessary step in the study of the equivalence of differential and integral problems. Note that the situation is different from that of the real-valued Lebesgue integral.

Lemma 3.

Let $\psi \in C^2\left[I\right]$ be a real-valued positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I$ and $0<\alpha <\lambda <1$. For any $f\in {\mathit{H}}_0^{\lambda }[I,E_{\omega }]$, ${\mathbb{I}}_{\psi }^{1-\alpha ,0}f$ exists and is pseudo-differentiable on I with

$${\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\mathfrak{D}}_p{\mathbb{I}}_{\psi }^{1-\alpha ,0}f\left(t\right)={\mathfrak{D}}_{p,\psi }^{\alpha ,0}f\left(t\right)=g\left(t\right),$$

where $g\left(t\right):={\displaystyle \frac{\alpha }{\Gamma (1-\alpha )}}{\int }_{-\infty }^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}[f\left(t\right)-f\left(s\right)]{\psi }^{\prime }\left(s\right)ds$ and where we define $f\left(s\right):=0$ for $s<a$.

Proof. 

Let $f\in {\mathbf{H}}_0^{\lambda }[I,E_{\omega }]$. In view of $f\left(s\right):=0,s\le a$, we have

$$g\left(a\right)={\mathfrak{D}}_{p,\psi }^{\alpha ,0}f\left(a\right)={\displaystyle \frac{\alpha }{\Gamma (1-\alpha )}}{\int }_{-\infty }^a{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}[f\left(a\right)-0]{\psi }^{\prime }\left(s\right)ds=0.$$

Note that since $f\left(a\right)=0$, then for every $\phi \in E^{*},\phi f\left(a\right)=0$ and we have

$$|\phi f\left(t\right)-\phi f\left(s\right)|\le {\left[\phi f\right]}_{\lambda }{|t-s|}^{\lambda },$$

for $t,s\in (-\infty ,b]$. Let $ϵ\in (0,b-a]$, $t\in I$. Define

$$g_{ϵ}\left(t\right):={\int }_a^{t-ϵ}{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha }f\left(s\right){\psi }^{\prime }\left(s\right)ds={\int }_{a+ϵ}^t{\left(\psi \left(t\right)-\psi (s-ϵ)\right)}^{-\alpha }f(s-ϵ){\psi }^{\prime }(s-ϵ)ds.$$

Observe that for any $t\in I$, the function ${\left(\psi \left(t\right)-\psi (·)\right)}^{-\alpha }{\psi }^{\prime }(·)\in L_q[a,t-ϵ]$ for any $q\in [1,1/\alpha ).$ Since for any $\phi \in E^{*}$ and any $p>1/(1-\alpha )$, we have $\phi f\in L_p[a,t-ϵ]$, it follows from Proposition 3, that $g_{ϵ}$ exists on $[a,t-ϵ]$. Similarly, by Remark 2, ${\mathbb{I}}_{\psi }^{1-\alpha ,0}f$ exists too. Also, for any $\phi \in E^{*},$ we have

$$\phi g_{ϵ}\left(t\right)={\int }_{a+ϵ}^t{\left(\psi \left(t\right)-\psi (s-ϵ)\right)}^{-\alpha }{\psi }^{\prime }(s-ϵ)\phi f(s-ϵ)ds.$$

For any $\phi \in E^{*},$ we have $\phi g_{ϵ}\in C^1[a+ϵ,b]$. So for any $t\in [a+ϵ,b]$,

$$\begin{array}{ccc}\hfill {\left(\phi g_{ϵ}\right)}^{\prime }\left(t\right)& =& {\left(\psi \left(t\right)-\psi (t-ϵ)\right)}^{-\alpha }{\psi }^{\prime }(t-ϵ)\phi f(t-ϵ)\hfill \\ & & -\alpha {\int }_{a+ϵ}^t{\left(\psi \left(t\right)-\psi (s-ϵ)\right)}^{-\alpha -1}{\psi }^{\prime }(s-ϵ){\psi }^{\prime }\left(t\right)\phi f(s-ϵ)ds.\hfill \end{array}$$

(14)

Note that for $t\le ϵ+a$, we have $f\left(t\right)=0$, and so, by definition $g_{ϵ}\left(t\right)=0$, for $t<ϵ+a$, ${\left(\phi g_{ϵ}\right)}^{\prime }\left(t\right)=0$. Furthermore, the left-hand derivative vanishes at $t=a+ϵ$. According to (14), the right-hand derivative also vanishes at $t=a+ϵ$. It follows from (14) that ${\left(\phi g_{ϵ}\right)}^{\prime }$ is continuously differentiable on I with ${\left(\phi g_{ϵ}\right)}^{\prime }\left(t\right)=0$ for $t\le a+ϵ$. With the above convention, (14) is also valid for $t\in [a,a+ϵ]$. Consequently, for any $t\in I$,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d\phi g_{ϵ}}{dt}}\left(t\right)& =& {\left(\psi \left(t\right)-\psi (t-ϵ)\right)}^{-\alpha }{\psi }^{\prime }(t-ϵ)\phi f(t-ϵ)\hfill \\ & & -\alpha {\int }_{-\infty }^{t-ϵ}{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}{\psi }^{\prime }\left(s\right){\psi }^{\prime }(t-ϵ)\phi f\left(t\right)ds\hfill \\ & & -\alpha {\int }_a^{t-ϵ}{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}{\psi }^{\prime }\left(s\right){\psi }^{\prime }\left(t\right)\phi f\left(s\right)ds\hfill \\ & & +\alpha {\int }_{-\infty }^{t-ϵ}{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}{\psi }^{\prime }\left(s\right){\psi }^{\prime }(t-ϵ)\phi f\left(t\right)ds\hfill \\ & =& {\left(\psi \left(t\right)-\psi (t-ϵ)\right)}^{-\alpha }{\psi }^{\prime }(t-ϵ)[\phi f(t-ϵ)-\phi f\left(t\right)]\hfill \\ & & +\alpha \left[{\int }_{-\infty }^{t-ϵ}{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}{\psi }^{\prime }\left(s\right)\left({\psi }^{\prime }(t-ϵ)\phi f\left(t\right)-{\psi }^{\prime }\left(t\right)\phi f\left(s\right)\right)ds\right]\hfill \\ & =& {\left(\psi \left(t\right)-\psi (t-ϵ)\right)}^{-\alpha }{\psi }^{\prime }(t-ϵ)[\phi f(t-ϵ)-\phi f\left(t\right)]\hfill \\ & & +\alpha \phi f\left(t\right)\left({\psi }^{\prime }(t-ϵ)-{\psi }^{\prime }\left(t\right)\right){\int }_{-\infty }^{t-ϵ}{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}{\psi }^{\prime }\left(s\right)ds\hfill \\ & & +\alpha {\psi }^{\prime }\left(t\right)\left[{\int }_{-\infty }^{t-ϵ}{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}{\psi }^{\prime }\left(s\right)\left(\phi f\left(t\right)-\phi f\left(s\right)\right)ds\right]\hfill \\ & =& {\left(\psi \left(t\right)-\psi (t-ϵ)\right)}^{-\alpha }{\psi }^{\prime }(t-ϵ)[\phi f(t-ϵ)-\phi f\left(t\right)]\hfill \\ & & +\alpha \phi f\left(t\right)\left({\psi }^{\prime }(t-ϵ)-{\psi }^{\prime }\left(t\right)\right){\int }_{-\infty }^{t-ϵ}{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}{\psi }^{\prime }\left(s\right)ds\hfill \\ & & -\alpha {\psi }^{\prime }\left(t\right)\left[{\int }_{t-ϵ}^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}{\psi }^{\prime }\left(s\right)\left(\phi f\left(t\right)-\phi f\left(s\right)\right)ds\right]\hfill \\ & & +{\psi }^{\prime }\left(t\right)\Gamma (1-\alpha )\phi g\left(t\right).\hfill \end{array}$$

The first summand converges as $ϵ\to 0$ because of

$${\left(\psi \left(t\right)-\psi (t-ϵ)\right)}^{-\alpha }{\psi }^{\prime }(t-ϵ)[\phi f(t-ϵ)-\phi f\left(t\right)]\le {\left[\phi f\right]}_{\lambda }(min|{\psi }^{\prime }\left(t\right){\left|\right)}^{-\alpha }∥{\psi }^{\prime }∥{ϵ}^{\lambda -\alpha }.$$

Also, since $\psi \in C^2[a,b],$ the second summand converges as $ϵ\to 0$ because of

$$\begin{array}{ccc}\hfill {\int }_{-\infty }^{t-ϵ}{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}{\psi }^{\prime }\left(s\right)ds& =& {\int }_{ϵ}^{\infty }{\left(\psi \left(t\right)-\psi (t-s)\right)}^{-\alpha -1}{\psi }^{\prime }(t-s)ds\hfill \\ & & \le ∥{\psi }^{\prime }∥(min|{\psi }^{\prime }{\left|\right)}^{-\alpha -1}{\int }_{ϵ}^{\infty }{s}^{-\alpha -1}ds,\hfill \end{array}$$

$$\begin{array}{ccc}& & \left|\phi f\left(t\right)\left({\psi }^{\prime }(t-ϵ)-{\psi }^{\prime }\left(t\right)\right){\int }_{-\infty }^{t-ϵ}{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}{\psi }^{\prime }\left(s\right)ds\right|\hfill \\ & \le & \left|{\psi }^{\prime }(t-ϵ)-{\psi }^{\prime }\left(t\right)\right|{\displaystyle \frac{∥\phi f∥∥{\psi }^{\prime }∥(min|{\psi }^{\prime }{\left|\right)}^{-\alpha -1}}{\alpha }}{ϵ}^{-\alpha }\hfill \\ & \le & {\displaystyle \frac{∥\phi f∥∥{\psi }^{\prime }∥(min|{\psi }^{\prime }{\left|\right)}^{-\alpha -1}}{\alpha }}\left[{\displaystyle \frac{{\psi }^{\prime }(t-ϵ)-{\psi }^{\prime }\left(t\right)}{ϵ}}\right]{ϵ}^{1-\alpha }.\hfill \end{array}$$

Moreover, we have

$$\begin{array}{ccc}& & \left|{\int }_{t-ϵ}^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}{\psi }^{\prime }\left(s\right)\left(\phi f\left(t\right)-\phi f\left(s\right)\right)ds\right|\hfill \\ & \le & {\displaystyle \frac{{\left[\phi f\right]}_{\lambda }∥{\psi }^{\prime }∥}{(min|{\psi }^{\prime }\left(t\right){\left|\right)}^{1+\alpha }}}\left|{\int }_{t-ϵ}^t{\left(t-s\right)}^{-\alpha -1}{\left(t-s\right)}^{\lambda }ds\right|\hfill \\ & \le & {\displaystyle \frac{{\left[\phi f\right]}_{\lambda }∥{\psi }^{\prime }∥}{(\lambda -\alpha )(min|{\psi }^{\prime }\left(t\right){\left|\right)}^{1+\alpha }}}{ϵ}^{\lambda -\alpha }.\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d\phi g_{ϵ}\left(t\right)}{dt}}\to \Gamma (1-\alpha )\phi g\left(t\right),\mathrm{as}ϵ\to 0,\mathrm{for} \mathrm{all}\phi \in E^{*}\hfill \\ & & ⟺{\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d{g}_{ϵ}}{dt}}\to \Gamma (1-\alpha )\phi g.\hfill \end{array}$$

Moreover,

$$\begin{array}{ccc}\hfill ∥{\mathbb{I}}_{\psi }^{1-\alpha }f\left(t\right)-{\displaystyle \frac{g_{ϵ}\left(t\right)}{\Gamma (1-\alpha )}}∥& \le & {\displaystyle \frac{∥f\left(t\right)∥}{\Gamma (1-\alpha )}}\left|{\int }_{t-ϵ}^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha }{\psi }^{\prime }\left(s\right)ds\right|\hfill \\ & \le & {\displaystyle \frac{∥\left|f\right|∥∥{\psi }^{\prime }∥}{\Gamma (2-\alpha )(min|{\psi }^{\prime }\left(t\right){\left|\right)}^{\alpha }}}{ϵ}^{1-\alpha }\to 0\hfill \end{array}$$

as $ϵ\to 0$. Hence,

$$g_{ϵ}\left(t\right)\to \Gamma (1-\alpha ){\mathbb{I}}_{\psi }^{1-\alpha }f\left(t\right),\mathrm{as}ϵ\to 0,\mathrm{strongly}.$$

Therefore, ${\mathbb{I}}_{\psi }^{1-\alpha }f$ is weakly continuous, pesudo-differentiable on I and for all $\phi \in E^{*},$

$${\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\phi \left({\mathbb{I}}_{\psi }^{1-\alpha }f\right)\left(t\right)={\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\psi }^{1-\alpha }\phi \left(f\left(t\right)\right)=\phi \left(g\left(t\right)\right)⟹{\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\psi }^{1-\alpha }f\left(t\right)=g\left(t\right).$$

□

We can now prove a key result for the acting conditions of fractional pseudo-differential operators on Hölder spaces.

Theorem 4.

Let $\psi \in C^2\left[I\right]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I$ and $0<\alpha <\lambda <1$. Then, for any $f\in {\mathit{H}}_0^{\lambda }[I,E_{\omega }]$, $f\left(s\right):=0$ for $s<a$, we have ${\mathfrak{D}}_{p,\psi }^{\alpha }f\in {\mathit{H}}_0^{\lambda -\alpha }[I,E_{\omega }]$. That is

$${\mathfrak{D}}_{p,\psi }^{\alpha ,0}:{\mathit{H}}_0^{\lambda }[I,E_{\omega }]\to {\mathit{H}}_0^{\lambda -\alpha }[I,E_{\omega }].$$

Proof. 

Let $f\in {\mathbf{H}}_0^{\lambda }[I,E_{\omega }]$. Note that

$${\mathfrak{D}}_{p,\psi }^{\alpha ,0}f={\displaystyle \frac{\alpha }{\Gamma (1-\alpha )}}G\left(t\right),$$

where $G\left(t\right):={\int }_{-\infty }^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}[f\left(t\right)-f\left(s\right)]{\psi }^{\prime }\left(s\right)ds$. For any $\phi \in E^{*},$ we have

$$\begin{array}{ccc}& & \left|\phi G(t+h)-\phi G\left(t\right)\right||=\left|{\int }_{-\infty }^{t+h}{\left(\psi (t+h)-\psi \left(s\right)\right)}^{-\alpha -1}\left[\phi \right(f(t+h)-\phi \left(f\left(s\right)\right]{\psi }^{\prime }\left(s\right)ds\right.\hfill \\ & & \left.-{\int }_{-\infty }^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}[\phi f\left(t\right)-\phi f\left(s\right)]{\psi }^{\prime }\left(s\right)ds\right|\hfill \\ & =& \left|{\int }_0^{\infty }{\left(\psi (t+h)-\psi (t+h-s)\right)}^{-\alpha -1}[\phi f(t+h)-\phi f(t+h-s)]{\psi }^{\prime }(t+h-s)ds\right.\hfill \\ & & \left.-{\int }_h^{\infty }{\left(\psi \left(t\right)-\psi (t+h-s)\right)}^{-\alpha -1}[\phi f\left(t\right)-\phi f(t+h-s)]{\psi }^{\prime }(t+h-s)ds\right|\hfill \\ & \le & \left|{\int }_0^h{\left(\psi (t+h)-\psi (t+h-s)\right)}^{-\alpha -1}[\phi f(t+h)-\phi f(t+h-s)]{\psi }^{\prime }(t+h-s)ds\right|\hfill \\ & & +\left|{\int }_h^{\infty }{\left(\psi (t+h)-\psi (t+h-s)\right)}^{-\alpha -1}[\phi f(t+h)-\phi f(t+h-s)]{\psi }^{\prime }(t+h-s)ds\right.\hfill \\ & & \left.-{\int }_h^{\infty }{\left(\psi \left(t\right)-\psi (t+h-s)\right)}^{-\alpha -1}[\phi f\left(t\right)-\phi f(t+h-s)]{\psi }^{\prime }(t+h-s)ds\right|\hfill \\ & \le & \left|{\int }_0^h{\left(\psi (t+h)-\psi (t+h-s)\right)}^{-\alpha -1}[\phi f(t+h)-\phi f(t+h-s)]{\psi }^{\prime }(t+h-s)ds\right|\hfill \\ & & +\left|{\int }_h^{\infty }{\left(\psi (t+h)-\psi (t+h-s)\right)}^{-\alpha -1}[\phi f(t+h)-\phi f\left(t\right)]{\psi }^{\prime }(t+h-s)ds\right.\hfill \\ & & -{\int }_h^{\infty }\left[{\left(\psi \left(t\right)-\psi (t+h-s)\right)}^{-\alpha -1}-{\left(\psi (t+h)-\psi (t+h-s)\right)}^{-\alpha -1}\right]\hfill \\ & & \left.·[\phi f\left(t\right)-\phi f(t+h-s)]{\psi }^{\prime }(t+h-s)ds\right|.\hfill \end{array}$$

We denote the components of the above sum by $A,B$, and C respectively. This means that we have

$$\left|A\right|\le ∥{\psi }^{\prime }∥(min|{\psi }^{\prime }{\left|\right)}^{-\alpha -1}{\left[\phi f\right]}_{\lambda }{\int }_0^h{s}^{\lambda -\alpha -1}ds={\displaystyle \frac{∥{\psi }^{\prime }∥(min|{\psi }^{\prime }{\left|\right)}^{-\alpha -1}{\left[\phi f\right]}_{\lambda }}{\lambda -\alpha }}{h}^{\lambda -\alpha },$$

$$\begin{array}{ccc}\hfill \left|B\right|& \le & ∥{\psi }^{\prime }∥(min|{\psi }^{\prime }{\left|\right)}^{-\alpha -1}{\left[\phi f\right]}_{\lambda }{h}^{\lambda }{\int }_h^{\infty }{s}^{-\alpha -1}ds={\displaystyle \frac{∥{\psi }^{\prime }∥(min|{\psi }^{\prime }{\left|\right)}^{-\alpha -1}{\left[\phi f\right]}_{\lambda }{h}^{\lambda }}{\alpha }}{h}^{-\alpha }\hfill \\ & =& {\displaystyle \frac{∥{\psi }^{\prime }∥(min|{\psi }^{\prime }{\left|\right)}^{-\alpha -1}{\left[\phi f\right]}_{\lambda }}{\alpha }}{h}^{\lambda -\alpha },\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \left|C\right|& \le & {\left[\phi f\right]}_{\lambda }{\int }_h^{\infty }\left|{\left(\psi \left(t\right)-\psi (t+h-s)\right)}^{-\alpha -1}-{\left(\psi (t+h)-\psi (t+h-s)\right)}^{-\alpha -1}\right|\hfill \\ & & ·{(s-h)}^{\lambda }{\psi }^{\prime }(t+h-s)ds.\hfill \end{array}$$

In the last integral, the substitution $s\mapsto sh$ gives the following estimates

$$\left|C\right|\le {\left[f\right]}_{\lambda }{h}^{\lambda +1}∥{\psi }^{\prime }∥W_1^{\infty },$$

where

$$W_1^t:={\int }_1^t\left|{\left(\psi \left(t\right)-\psi (t+h(1-s\left)\right)\right)}^{-\alpha -1}-{\left(\psi (t+h)-\psi (t+h(1-s\left)\right)\right)}^{-\alpha -1}\right|{(s-1)}^{\lambda }ds,$$

and $W_1^{\infty }={lim}_{t\to \infty }{W}_1^t$. Clearly, since $\psi \left(t\right)-\psi (t+h(1-s\left)\right)\le \psi (t+h)-\psi (t+h(1-s\left)\right)$, $s\in [1,2)$, we obtain

$$\begin{array}{ccc}\hfill W_1^2& \le & {\int }_1^2{\left(\psi \left(t\right)-\psi (t+h(1-s\left)\right)\right)}^{-\alpha -1}{(s-1)}^{\lambda }ds\hfill \\ & \le & (min|{\psi }^{\prime }{\left|\right)}^{-\alpha -1}{\int }_1^2{h}^{-\alpha -1}{(s-1)}^{\lambda -1-\alpha }ds={\displaystyle \frac{(min|{\psi }^{\prime }{\left|h\right)}^{-\alpha -1}}{\lambda -\alpha }}.\hfill \end{array}$$

Also, for $s\ge 2$, define a function $\omega :J\to [0,\infty )$, where J is the interval $\left[\psi \right(t)-\psi (t-h(s-1)),\psi (t+h)-\psi (t-h(s-1)\left)\right]$ by

$$\omega \left(\theta \right):={\theta }^{-\alpha -1},\theta \in [\psi \left(t\right)-\psi (t-h(s-1)),\psi (t+h)-\psi (t-h(s-1))].$$

Then we have

$${\omega }^{\prime }\left(\theta \right)=-(1+\alpha ){\theta }^{-\alpha -2}\ge -(1+\alpha ){\left(\psi \left(t\right)-\psi (t-h(s-1\left)\right)\right)}^{-\alpha -2}.$$

Therefore,

$$\begin{array}{ccc}& & {\left(\psi \left(t\right)-\psi (t+h(1-s\left)\right)\right)}^{-\alpha -1}-{\left(\psi (t+h)-\psi (t+h(1-s\left)\right)\right)}^{-\alpha -1}\hfill \\ & =& \omega \left(\psi \right(t)-\psi (t-h(s-1))-\omega (\psi (t+h)-\psi (t-h(s-1\left)\right))\hfill \\ & \le & (1+\alpha ){\left(\psi \left(t\right)-\psi (t-h(s-1\left)\right)\right)}^{-\alpha -2}\hfill \\ & \le & (1+\alpha )(min|{\psi }^{\prime }{\left|\right)}^{-\alpha -2}{\left(h(s-1)\right)}^{-\alpha -2}[\psi (t+h)-\psi \left(t\right)]\hfill \\ & \le & (1+\alpha )∥{\psi }^{\prime }∥(min|{\psi }^{\prime }{\left|\right)}^{-\alpha -2}{\left(h(s-1)\right)}^{-\alpha -2}h.\hfill \end{array}$$

Consequently,

$$W_2^{\infty }\le (1+\alpha )(min|{\psi }^{\prime }{\left|h\right)}^{-\alpha -2}∥{\psi }^{\prime }∥{\int }_2^{\infty }{(s-1)}^{\lambda -\alpha -2}ds={\displaystyle \frac{(1+\alpha )∥{\psi }^{\prime }∥}{1-\lambda +\alpha }}(min|{\psi }^{\prime }{\left|\right)}^{-\alpha -2}{h}^{-\alpha -1}.$$

Then,

$$\begin{array}{ccc}\hfill \left|C\right|& \le & {\left[\phi f\right]}_{\lambda }{h}^{\lambda +1}∥{\psi }^{\prime }∥\left[W_1^2+W_2^{\infty }\right]\hfill \\ & \le & {\left[\phi f\right]}_{\lambda }∥{\psi }^{\prime }∥\left[{\displaystyle \frac{(min|{\psi }^{\prime }{\left|\right)}^{-\alpha -1}}{\lambda -\alpha }}+{\displaystyle \frac{(1+\alpha )∥{\psi }^{\prime }∥}{1-\lambda +\alpha }}(min|{\psi }^{\prime }{\left|\right)}^{-\alpha -2}\right]h^{\lambda -\alpha }.\hfill \end{array}$$

It follows for all $h>0,t,t+h\in I$ and $\phi \in E^{*},$

$$|\phi G(t+h)-\phi G\left(t\right)|\le \left|A\right|+\left|B\right|+\left|C\right|\le L{\left[\phi f\right]}_{\lambda }{h}^{\lambda -\alpha },$$

where L depends only on $\lambda ,\alpha$. Hence, for any $f\in {\mathbf{H}}_0^{\lambda }[I,E_{\omega }],$ we have $g={\mathfrak{D}}_{p,\psi }^{\alpha ,0}f\in {\mathbf{H}}_0^{\lambda -\alpha }[I,E_{\omega }]$. □

And now the general result:

Theorem 5.

Let $\psi \in C^2\left[I\right]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in I,\mu \ge 0$ and $0<\alpha <\lambda <1$. Then, for any $f\in {\mathit{H}}_0^{\lambda }[I,E_{\omega }]$, $f\left(s\right):=0$ for $s<a$, we have ${\mathfrak{D}}_{p,\psi }^{\alpha }f\in {\mathit{H}}_0^{\lambda -\alpha }[I,E_{\omega }]$. That is

$${\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }:{\mathit{H}}_0^{\lambda }[I,E_{\omega }]\to {\mathit{H}}_0^{\lambda -\alpha }[I,E_{\omega }],$$

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

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Proof. 

Since

$$\begin{array}{ccc}\hfill {\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f\left(t\right)& =& {\delta }_p\left[e^{-\mu \psi \left(t\right)}{\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)\right]=e^{-\mu \psi \left(t\right)}{\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\left[{\mathbb{I}}_{\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)\right]\hfill \\ & =& e^{-\mu \psi \left(t\right)}{\mathfrak{D}}_{p,\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right),\hfill \end{array}$$

by defining

$$g_{ϵ}^{\mu }\left(t\right):={\int }_a^{t-ϵ}{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha }\left(e^{\mu \psi \left(s\right)}f\left(s\right)\right){\psi }^{\prime }\left(s\right)ds,$$

$$g_{\mu }\left(t\right):={\displaystyle \frac{\alpha }{\Gamma (1-\alpha )}}{\int }_{-\infty }^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}\left[e^{\mu \psi \left(s\right)}f\left(t\right)-e^{\mu \psi \left(s\right)}f\left(s\right)\right]{\psi }^{\prime }\left(s\right)ds$$

we can observe that

$${\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d\phi g_{ϵ}^{\mu }\left(t\right)}{dt}}\to \Gamma (1-\alpha )\phi g_{\mu }\left(t\right)asϵ\to 0\mathrm{for} \mathrm{all}\phi \in E^{*}.$$

Moreover,

$$\begin{array}{ccc}\hfill ∥{\mathbb{I}}_{\psi }^{1-\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)-\Gamma (1-\alpha )g_{ϵ}^{\mu }\left(t\right)∥& \le & ∥\left|f{e}^{\mu \psi (·)}\right|∥\left|{\int }_{t-ϵ}^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha }{\psi }^{\prime }\left(s\right)ds\right|\hfill \\ & \le & {\displaystyle \frac{∥\left|f{e}^{\mu \psi (·)}\right|∥∥{\psi }^{\prime }∥}{\alpha (min|{\psi }^{\prime }\left(t\right){\left|\right)}^{\alpha }}}{ϵ}^{1-\alpha }\to 0asϵ\to 0,\hfill \end{array}$$

$$g_{ϵ}^{\mu }\left(t\right)\to \Gamma (1-\alpha ){\mathbb{I}}_{\psi }^{1-\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)asϵ\to 0.$$

Hence,

$${\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\psi }^{1-\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)=g_{\mu }\left(t\right),$$

and then,

$$\begin{array}{ccc}\hfill {\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f\left(t\right)& =& {\delta }_p\left[e^{-\mu \psi \left(t\right)}{\mathbb{I}}_{\psi }^{1-\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)\right]=e^{-\mu \psi \left(t\right)}{\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\psi }^{1-\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)\hfill \\ & =& e^{-\mu \psi \left(t\right)}{g}_{\mu }\left(t\right)\hfill \\ & =& {\displaystyle \frac{\alpha }{\Gamma (1-\alpha )}}{\int }_{-\infty }^t{e}^{-\mu \left(\psi \right(t)-\psi (s\left)\right)}{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}[f\left(t\right)-f\left(s\right)]{\psi }^{\prime }\left(s\right)ds.\hfill \end{array}$$

Since $f\left(s\right):=0,s\le a,$ it follows that

$${\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f\left(a\right)={\displaystyle \frac{\alpha }{\Gamma (1-\alpha )}}{\int }_{-\infty }^a{e}^{-\mu \left(\psi \right(t)-\psi (s\left)\right)}{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}[f\left(a\right)-0]{\psi }^{\prime }\left(s\right)ds=0.$$

Finally,

$${\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f\left(t\right)=e^{-\mu \psi \left(t\right)}{\mathfrak{D}}_{p,\psi }^{\alpha ,0}\left(e^{\mu \psi \left(t\right)}f\left(t\right)\right)={\displaystyle \frac{\alpha e^{-\mu \psi \left(t\right)}}{\Gamma (1-\alpha )}}G\left(t\right),$$

where

$$G\left(t\right)={\int }_{-\infty }^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{-\alpha -1}\left[f\left(t\right)e^{\mu \psi \left(t\right)}-f\left(s\right)e^{\mu \psi \left(s\right)}\right]{\psi }^{\prime }\left(s\right)ds,$$

Arguing as in the proof of Theorem 4, then for all $h>0,t,t+h\in I$ and $\phi \in E^{*},$

$$|\phi e^{-\mu \psi (t+h)}G(t+h)-\phi e^{-\mu \psi \left(t\right)}G\left(t\right)|\le L{\left[\phi e^{\mu \psi \left(t\right)}f\left(t\right)\right]}_{\lambda }{h}^{\lambda -\alpha }=e^{\mu \psi \left(t\right)}L{\left[\phi f\left(t\right)\right]}_{\lambda }{h}^{\lambda -\alpha },$$

where L depends only on $\lambda ,\alpha$. Namely,

$$\begin{array}{ccc}& & |\phi {\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f(t+h)-\phi {\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f\left(t\right)|\le {\displaystyle \frac{\alpha e^{\mu \psi \left(t\right)}}{\Gamma (1-\alpha )}}L{\left[\phi f\left(t\right)\right]}_{\lambda }{h}^{\lambda -\alpha },\hfill \\ & & {\left[\phi {\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f\right]}_{\lambda -\alpha }\le {\displaystyle \frac{\alpha e^{\mu \psi \left(t\right)}}{\Gamma (1-\alpha )}}L{\left[\phi f\right]}_{\lambda }.\hfill \end{array}$$

(16)

Hence, ${\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f\in {\mathbf{H}}_0^{\lambda -\alpha }[I,E_{\omega }]$, for any $f\in {\mathbf{H}}_0^{\lambda }[I,E_{\omega }].$ □

5. Equivalence Result

In view of the differences between the topologies, the following facts should be taken into account:

1.

Since the pseudo-derivative need not have to be uniquely determined, and since two pseudo-derivatives need not to be equal almost everywhere, the map ${\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }:{\mathbf{H}}_0^{\lambda }[I,E_{\omega }]\to {\mathbf{H}}_0^{\lambda -\alpha }[I,E_{\omega }]$ cannot be bijective in general. But, the case of ${\mathfrak{D}}_{\omega ,\psi }^{\alpha ,\mu }$ is much more suitable for studying equivalence problems (as will be seen below).

2.

Let $\psi \in C^2\left[I\right]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in I,\mu \ge 0$ and $\alpha ,\lambda ,\delta \in (0,1)$ such that $\alpha -\delta >\lambda$. Since $0<\lambda -\alpha +1<1$, by Theorem 5, we obtain ${\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }:{\mathbf{H}}_0^{\lambda -\alpha +1}[I,E_{\omega }]\to {\mathbf{H}}_0^{\lambda }[I,E_{\omega }]$. Also, by Corollary 5, we obtain

$${\mathrm{I}}_{\psi }^{\alpha ,\mu }={\mathbb{I}}_{\psi }^{1,\mu }{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }:{\mathbf{H}}_0^{\lambda -\alpha +1}[I,E_{\omega }]\to {\mathbf{H}}_0^{1+\lambda -\alpha +\delta }[I,E_{\omega }].$$

(17)

3.

Let $\alpha ,\lambda \in (0,1)$ such that $\alpha +\lambda <1$. We will show that ${\mathbb{I}}_{\psi }^{\alpha ,\mu }{\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f=f$ for any $f\in C[I,E_{\omega }]$: Firstly, we will show that ${\Im }_{p,\psi }^{\alpha ,\mu }$ is injective: Let $f_1,f_2\in {\mathbf{H}}_0^{\lambda }[I,E_{\omega }]$, such that ${\Im }_{p,\psi }^{\alpha ,\mu }{f}_1={\Im }_{p,\psi }^{\alpha ,\mu }{f}_1$ on I and $u:=f_1-f_2$. Then

$$0=\phi \left(0\right)={\mathbb{I}}_{\psi }^{1-\alpha ,\mu }{\mathbb{I}}_{\psi }^{\alpha ,\mu }\phi u\left(t\right)={\mathbb{I}}_{\psi }^{1,\mu }\phi u\left(t\right),forallt\in I.$$

Hence, $\phi u\left(t\right)=0$ ($u\left(t\right)=0$) (or $f_1\left(t\right)=f_2\left(t\right)$) for all $t\in I$.

Now, we can prove in which case the integral and differential operators are mutually inverse.

Theorem 6.

Let $\psi \in C^2\left[I\right]$ be a positive increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in I,\mu \ge 0$ and $0<\alpha <\lambda <1$. Then, for any $f\in {\mathit{H}}_0^{\lambda }[I,E_{\omega }]$, we have

$${\mathbb{I}}_{\psi }^{\alpha ,\mu }{\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f=f.$$

Proof. 

Let $f\in {\mathbf{H}}_0^{\lambda }[I,E_{\omega }]$ and define $g:={\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f={\delta }_p{\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f.$ By Theorem 5 $g\in {\mathbf{H}}_0^{\lambda -\alpha }[I,E_{\omega }].$ It means that ${\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f\in C[I,E_{\omega }]$ which is pseudo-differentiable on I with ${\delta }_p{\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f\in \mathrm{P}[I,E]$. Given

$$e^{\mu \psi \left(t\right)}{\delta }_p\left({\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f\left(t\right)\right)={\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\mathfrak{D}}_p\left(e^{\mu \psi \left(t\right)}{\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f\left(t\right)\right),$$

it follows from Proposition 1

$$\begin{array}{ccc}\hfill {\mathbb{I}}_{\psi }^{1-\alpha ,\mu }{\mathbb{I}}_{\psi }^{\alpha ,\mu }g\left(t\right)& =& {\mathbb{I}}_{\psi }^{1,\mu }g\left(t\right)={\mathbb{I}}_{\psi }^{1,\mu }\left({\delta }_p{\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f\right)\left(t\right)={\mathbb{I}}_{\psi }^{1,\mu }\left[{\displaystyle \frac{e^{-\mu \psi \left(t\right)}}{{\psi }^{\prime }\left(t\right)}}{\mathfrak{D}}_p\left(e^{\mu \psi \left(t\right)}{\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f\left(t\right)\right)\right]\hfill \\ & =& e^{-\mu \psi \left(t\right)}{\int }_a^t{\mathfrak{D}}_p\left(e^{\mu \psi \left(s\right)}{\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f\left(s\right)\right)ds={\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f\left(t\right).\hfill \end{array}$$

Since ${\mathbb{I}}_{\psi }^{1-\alpha ,\mu }$ is injective and ${\mathbb{I}}_{\psi }^{\alpha ,\mu }g\in {\mathbf{H}}_0^{\lambda +\alpha }[I,E_{\omega }]$, we conclude that ${\mathbb{I}}_{\psi }^{\alpha ,\mu }{\mathfrak{D}}_{p,\psi }^{\alpha ,\mu }f={\mathbb{I}}_{\psi }^{\alpha ,\mu }g=f$ on I.  □

6. Application

Next, we will prove the following for all $\alpha \in (0,1)$ without imposing an absolute continuity condition on f.

Let $\alpha <\lambda \in (0,1)$ and $f\in {\mathbf{H}}_0^{\lambda -\alpha +1}[I,E_{\omega }]$. Since ${\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }f\in C[I,E_{\omega }]\subset {\mathbf{P}}^p\left(E\right)$ for any $p>1$, Proposition 2 can be combined with Proposition 4 to ensure (in view of Theorem 6) that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d_{p,\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}}{\mathrm{I}}_{\psi }^{\alpha ,\mu }f& =& {\mathbb{I}}_{\psi }^{1-\alpha ,\mu }{\delta }_p{\mathbb{I}}_{\psi }^{1,\mu }\left({\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }f\right)={\mathbb{I}}_{\psi }^{1-\alpha ,\mu }\left({\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }f\right)=f.\hfill \end{array}$$

(18)

Also, for any $f\in AC[[a,b],E_{\omega }]$ we have

$$\begin{array}{ccc}\hfill {\mathrm{I}}_{\psi }^{\alpha ,\mu }{\displaystyle \frac{d_{p,\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}}f\left(t\right)& =& {\mathbb{I}}_{\psi }^{1,\mu }{\delta }_p{\mathbb{I}}_{\psi }^{\alpha ,\mu }\left({\mathbb{I}}_{\psi }^{1,\mu }{\delta }_{p}f\left(t\right)\right)={\mathbb{I}}_{\psi }^{1,\mu }\left({\delta }_p{\mathbb{I}}_{\psi }^{1,\mu }\right){\mathbb{I}}_{\psi }^{\alpha ,\mu }{\delta }_{p}f\left(t\right)\hfill \\ & =& {\mathbb{I}}_{\psi }^{1,\mu }{\mathbb{I}}_{\psi }^{\alpha ,\mu }{\delta }_{p}f\left(t\right))={\mathbb{I}}_{\psi }^{\alpha ,\mu }{\mathbb{I}}_{\psi }^{1,\mu }{\delta }_{p}f\left(t\right)\hfill \\ & =& {\mathbb{I}}_{\psi }^{\alpha ,\mu }\left[f\left(t\right)-f\left(a\right)e^{-\mu \psi \left(t\right)}\right]={\mathbb{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)-{\displaystyle \frac{f\left(a\right){\left(\psi \left(t\right)\right)}^{\alpha }{e}^{-\mu \psi \left(t\right)}}{\Gamma (1+\alpha )}}.\hfill \end{array}$$

(19)

Let $c_0$ be the separable space of all real-valued sequences that converge to 0. The topological dual $c_0^{*}$ is known to be isometrically isomorphic to ${\ell }_1$ (the space of all summable real-valued sequences). Define $\mathcal{W}:[0,1]\to c_0$ by

$$\mathcal{W}\left(t\right):=\left\{{\left({\displaystyle \frac{1}{b}}\right)}^n\left(e^{i{b}^{\gamma n}t}-1\right)\right\},b>1\mathrm{and}\gamma \in [{\displaystyle \frac{3}{2}},2].$$

(20)

It is immediate that $\mathcal{W}$ acts from $[0,1]$ into $c_0$ and is weakly measurable (even strongly measurable by the Pettis measurability theorem [19]). We recall that there exists a unique $\left\{{\lambda }_n\right\}\in {\ell }_1$ corresponding to each $\phi \in c_0^{*}$ such that for any $h>0$ such that $0\le t<t+h\le 1$ we have

$$\left|\phi \mathcal{W}(t+h)-\phi \mathcal{W}\left(t\right)\right|\le \sum _{n=0}\left|{\lambda }_n{\left({\displaystyle \frac{1}{b}}\right)}^n\right|\left|e^{i{b}^{n\gamma }(t+h)}-e^{i{b}^{n\gamma }\left(t\right)}\right|.$$

(21)

Of course, the estimate

$$\sum _{n=0}|{\lambda }_n|\left|{\displaystyle \frac{e^{i{b}^{n\gamma }(t+h)}-e^{i{b}^{n\gamma }\left(t\right)}}{b^n}}\right|\le h\sum _{n=0}2\left|{\lambda }_n\right|<\infty ,$$

yields (in view of $\mathcal{W}\left(0\right)=0$) that $\mathcal{W}\in {\mathcal{H}}_0^{\lambda }\left[[0,1],E_{\omega }\right],E\equiv c_0$ for any $\lambda \in (0,1)$. According to [13], $\mathcal{W}$ has the Riemann–Liouville fractional pseudo-derivative of any order $\alpha \in (0,{\displaystyle \frac{1}{2}}]$, while the pseudo-derivative does not exist on $[0,1]$. Furthermore, Theorem 6 gives ${\mathbb{I}}_t^{\alpha ,0}{\mathfrak{D}}_{p,t}^{\alpha ,0}\mathcal{W}=\mathcal{W}$.

Example 1.

Define $f:[0,1]\to c_0$ by $f:={\mathfrak{D}}_{p,t}^{{\displaystyle \frac{1}{2}},0}\mathcal{W}\in {\mathcal{H}}_0^{\lambda -\alpha +1}\left[[0,1],E_{\omega }\right]$ (with $\lambda <\alpha ={\displaystyle \frac{1}{2}}$). Consider the following

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

(22)

combined with appropriate initial or boundary conditions. Of course, (22) and the corresponding “standard" integral form

$$x\left(t\right)=C_0+C_1{t}^{{\displaystyle \frac{1}{2}}}+{\int }_0^{t}f\left(s\right)ds,t\in [0,1],$$

(23)

are not (in general) equivalent outside the space of absolutely continuous functions. Even the (restrictive) assumption that $f\in {\mathcal{H}}_0^{\lambda -\alpha +1}\left[[0,1],E_{\omega }\right],\lambda <\alpha ={\displaystyle \frac{1}{2}}$ is not sufficient for the equivalence between (22) and (23): Obviously,

$${\displaystyle \frac{d_{p,t}^{\frac{1}{2},0}}{d{t}^{\frac{1}{2}}}}x\left(t\right)=C_1+{\mathbb{I}}_t^{{\displaystyle \frac{1}{2}},0}f\left(t\right)=C_1+{\mathbb{I}}_t^{{\displaystyle \frac{1}{2}},0}{\mathfrak{D}}_{p,t}^{{\displaystyle \frac{1}{2}},0}\mathcal{W}\left(t\right)=\mathcal{W}\left(t\right).$$

Since ${\delta }_p\mathcal{W}$ does not exist, x given by (23) is not a solution of (22).

On the other hand, in view of (19), we can convert (22) into the following integral form:

$$x\left(t\right)=C_0+C_1{t}^{{\displaystyle \frac{1}{2}}}+{\mathrm{I}}_t^{{\displaystyle \frac{1}{2}},0}{\mathrm{I}}_t^{{\displaystyle \frac{1}{2}},0}f\left(t\right),t\in [0,1].$$

(24)

In view of (18), the equivalence between (22) and (27) holds and we simply arrive at (22).

Remark 5.

Recall that Theorem 6 (hence (18) is true for any Banach space E even if E has no total dual. In such a case, any two pseudo-derivatives of a function $f:I\to E$ are weakly equivalent on I. Suppose ${\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f$ exists and $f_1,f_2$ are two pseudo-derivatives of ${\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f$; then, for every $\phi \in E^{*}$ there exists a null set $\mathfrak{N}\left(\phi \right)$ such that

$${\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(\psi \left(t\right)-\psi \left(s\right)\right)}\phi f_1\left(s\right){\psi }^{\prime }\left(s\right)={\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(\psi \left(t\right)-\psi \left(s\right)\right)}\phi f_2\left(s\right){\psi }^{\prime }\left(s\right),$$

for $s\in [a,t]\setminus \mathfrak{N}\left(\phi \right)$. Hence,

$$\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }{f}_1\left(t\right)\right)={\mathbb{I}}_{\psi }^{\alpha ,\mu }\phi \left(f_1\left(t\right)\right)={\mathbb{I}}_{\psi }^{\alpha ,\mu }\phi \left(f_2\left(t\right)\right)=\phi \left({\mathbb{I}}_{\psi }^{\alpha ,\mu }{f}_2\left(t\right)\right).$$

Consequently, the results of Theorem 6 and (18) are independent of the choice of a pseudo-derivative of ${\mathbb{I}}_{\psi }^{1-\alpha ,\mu }f$.

In the following, we construct an example that shows this: The equivalence between the Caputo-type differential problems and the corresponding “standard” integral form may be lost for the functions with values in the Banach spaces without total dual. However (given (18), the equivalence holds in arbitrary spaces.

The following example is also a warning against too quickly attempting to take equivalence results for real functions as true for results for vector-valued functions.

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

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

(25)

We know that [19], Example 9.1, $f\in \mathrm{P}\left[[0,1],\mathbb{B}\right]$ with $\phi f\equiv 0,$ for each $\phi \in {\mathbb{B}}^{*}[0,1]$. So, it is quite obvious that $f\in {\mathcal{H}}_0^{\lambda -\alpha +1}\left[[0,1],{\mathbb{B}}_{\omega }\right]$ for any $\lambda <\alpha ={\displaystyle \frac{1}{2}}.$

Example 2.

Consider the problem (22) with f defined by (25). The corresponding “standard" integral form is

$$x\left(t\right)=C_0+C_1{t}^{{\displaystyle \frac{1}{2}}}+{\int }_0^{t}f\left(s\right)ds,$$

(26)

for $t\in [0,1]$. According to Example 9.1 in [19], the function $t\to {\int }_0^{t}f\left(s\right)ds$ has two pseudo-derivatives 0 and f on $[0,1]$, that differ on the set of positive measure $\mathfrak{J}$. Therefore, either

$${\delta }_{p}x\left(t\right)=2C_1{t}^{-{\displaystyle \frac{1}{2}}}$$

or

$${\delta }_{p}x\left(t\right)=2C_1{t}^{-{\displaystyle \frac{1}{2}}}+f\left(t\right).$$

Consequently, we conclude that on J we have ${\displaystyle \frac{d_{p,t}^{\frac{1}{2},0}}{d{t}^{\frac{1}{2}}}}{\displaystyle \frac{d_{p,t}^{\frac{1}{2},0}}{d{t}^{\frac{1}{2}}}}x\ne f$.

On the other hand, given (19), we can transform (22) into the following integral form:

$$x\left(t\right)=C_0+C_1{t}^{{\displaystyle \frac{1}{2}}}+\left\{\begin{array}{c}{\mathrm{I}}_t^{{\displaystyle \frac{1}{2}},0}{\mathrm{I}}_t^{{\displaystyle \frac{1}{2}},0}{\chi }_t,t\in \mathfrak{J},\hfill \\ 0,t\notin \mathfrak{J}.\hfill \end{array}\right.$$

(27)

Since $f\in {\mathcal{H}}_0^{\lambda -\alpha +1}\left[[0,1],E_{\omega }\right]$ for arbitrary $\lambda <{\displaystyle \frac{1}{2}},$ it follows, in view of (18) (bearing in mind Remark 5), the equivalence between (22) and (26) holds, and we simply arrive at (22). Namely, for any $t\in [0,1]$, we have

$${\displaystyle \frac{d_{p,t}^{\frac{1}{2},0}}{d{t}^{\frac{1}{2}}}}{\displaystyle \frac{d_{p,t}^{\frac{1}{2},0}}{d{t}^{\frac{1}{2}}}}x\left(t\right)=f\left(t\right).$$

Remark 6.

Recall that due to Theorem 2 in [13], for any infinite dimensional Banach space E that fails cotype, there is a function $h\in {\tilde{\mathit{P}}}^1\left(E\right)$ and a weakly absolutely continuous, nowhere weakly differentiable function g defined on $[0,1]$ such that

$${\mathfrak{D}}_{\omega ,t}^{\alpha ,0}g={\mathbb{I}}_t^{1-\alpha ,0}h,\alpha \in (0,{\displaystyle \frac{1}{2}}].$$

(28)

We also know that ${\mathbb{I}}_t^{1,0}h=g$, by reasoning as in proof of Theorem 1 in [13].

Consider now the problem (22) with $f={\mathfrak{D}}_{\omega ,t}^{{\displaystyle \frac{1}{2}},0}g\in {\mathcal{H}}_0^{\lambda -\alpha +1}\left[[0,1],E_{\omega }\right]$, with $\lambda <\alpha ={\displaystyle \frac{1}{2}}$. Recall that

$$x\left(t\right)=C_0+C_1{t}^{{\displaystyle \frac{1}{2}}}+{\int }_0^t{\mathfrak{D}}_{\omega ,s}^{{\displaystyle \frac{1}{2}},0}g\left(s\right)ds=C_0+C_1{t}^{{\displaystyle \frac{1}{2}}}+{\int }_0^t{\mathbb{I}}_s^{{\displaystyle \frac{1}{2}},0}h\left(s\right)ds.$$

(29)

Obviously,

$${\displaystyle \frac{d_{\omega ,t}^{\frac{1}{2},0}}{d{t}^{\frac{1}{2}}}}x\left(t\right)=C_1+{\mathbb{I}}_t^{1,0}h\left(t\right)=C_1+g\left(t\right).$$

This yields ${\displaystyle \frac{d_{\omega ,t}^{\frac{1}{2},0}}{d{t}^{\frac{1}{2}}}}x$ is weakly absolutely continuous and nowhere weakly differentiable on $[0,1]$. So, the problem (22) with Caputo-type fractional weak derivatives would make no sense.

In many papers, authors will not consider the Hölder continuity. They will look for solutions of differential or integral equations only in spaces of continuous functions. In such a case, the following conclusion is useful:

Corollary 6.

The operator ${\mathrm{I}}_{\psi }^{\alpha ,\mu }:{\mathbf{H}}_0^{\lambda -\alpha +1}[I,E_{\omega }]\to C[I,E_{\omega }]$, for $\alpha ,\lambda \in (0,1)$, $\alpha >\lambda$ is weakly compact.

Proof. 

Let $\mathcal{Q}$ be bounded subset of ${\mathbf{H}}_0^{\lambda -\alpha +1}[I,E_{\omega }]$. Let $\left(f_n\right)$ be a sequence in $\mathcal{Q}$ convergent weakly to f in $\mathcal{Q}$. From Theorem 5, it follows that ${\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }{f}_n\in C[I,E_{\omega }]$.

Since $\left(f_n\right)$ is weakly convergent in $\mathcal{Q}\subset C[I,E_{\omega }]$, it is bounded, weakly equicontinuous, and weakly pointwisely convergent. Again, by applying Theorem 5, we obtain the same property for $\left({\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }{f}_n\right)$ (with the estimation of of the modulus of the weak continuity by ${\displaystyle \frac{\alpha e^{\mu \psi \left(t\right)}}{\Gamma (1-\alpha )}}L{\left[\phi f_n\right]}_{\lambda }$), we conclude that the latter is also weakly convergent in $C[I,E_{\omega }]$. Indeed, for each $\phi \in E^{*}$ and $t\in I$ we obtain

$$\begin{array}{ccc}\hfill \left|\phi \left({\mathrm{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)-{\mathrm{I}}_{\psi }^{\alpha ,\mu }{f}_n\left(t\right)\right)\right|& =& \left|\phi ({\mathbb{I}}_{\psi }^{1,\mu }{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }f\left(t\right)-{\mathbb{I}}_{\psi }^{1,\mu }{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }{f}_n\left(t\right))\right|\hfill \\ & \le & {\int }_a^t{e}^{-\mu \left(\psi \right(t)-\psi (s\left)\right)}\left|\phi \left({\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }f\left(s\right)-{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }{f}_n\left(s\right)\right)\right|{\psi }^{\prime }\left(s\right)ds.\hfill \end{array}$$

Moreover, for any $t,\tau \in I,\tau <t$ and $\phi \in E^{*}$, we have

$$\begin{array}{ccc}\hfill \phi \left({\mathrm{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)-{\mathrm{I}}_{\psi }^{\alpha ,\mu }f\left(\tau \right)\right)& =& {\mathbb{I}}_{\psi }^{1,\mu }{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }\phi f\left(t\right)-{\mathbb{I}}_{\psi }^{1,\mu }{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }\phi f\left(\tau \right)\hfill \\ & =& {\int }_a^t{e}^{-\mu \left(\psi \right(t)-\psi (s\left)\right)}{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }\phi f\left(s\right){\psi }^{\prime }\left(s\right)ds\hfill \\ & & -{\int }_a^{\tau }{e}^{-\mu \left(\psi \right(\tau )-\psi (s\left)\right)}{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }\phi f\left(s\right){\psi }^{\prime }\left(s\right)ds\hfill \\ & =& {\int }_{\tau }^t{e}^{-\mu \psi \left(t\right)}{e}^{\mu \left(\psi \right(s)}{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }\phi f\left(s\right){\psi }^{\prime }\left(s\right)ds\hfill \\ & & +{\int }_a^{\tau }\left[e^{-\mu \psi \left(t\right)}-e^{-\mu \psi \left(\tau \right))}\right]e^{\mu \left(\psi \right(s)}{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }\phi f\left(s\right){\psi }^{\prime }\left(s\right)ds.\hfill \end{array}$$

Hence, in view of (16)

$$\begin{array}{ccc}\hfill \left|{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }\phi f\left(s\right)\right|& =& \left|{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }\phi f\left(s\right)-{\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }\phi f\left(a\right)\right|\le {\left[\phi {\mathfrak{D}}_{p,\psi }^{1-\alpha ,\mu }f\left(s\right)\right]}_{\lambda }\hfill \\ & & \le {\displaystyle \frac{(1-\alpha )e^{\mu \psi \left(s\right)}}{\Gamma \left(\alpha \right)}}L{\left[\phi f\left(s\right)\right]}_{\lambda -\alpha +1}{|s-a|}^{\lambda },\hfill \end{array}$$

it follows that

$$\begin{array}{ccc}& & \left|\phi \left({\mathrm{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)-{\mathrm{I}}_{\psi }^{\alpha ,\mu }f\left(\tau \right)\right)\right|\le {\displaystyle \frac{(1-\alpha )}{\Gamma \left(\alpha \right)}}L{\left[\phi f\left(s\right)\right]}_{\lambda -\alpha +1}{\int }_{\tau }^t{e}^{-\mu \psi \left(t\right)}{(s-a)}^{\lambda }{\psi }^{\prime }\left(s\right)ds\hfill \\ & & +{\displaystyle \frac{(1-\alpha )}{\Gamma \left(\alpha \right)}}L{\left[\phi f\left(s\right)\right]}_{\lambda -\alpha +1}{\int }_a^{\tau }\left|e^{-\mu \psi \left(t\right)}-e^{-\mu \psi \left(\tau \right))}\right|{(s-a)}^{\lambda }{\psi }^{\prime }\left(s\right)ds\hfill \\ & \le & {\displaystyle \frac{(1-\alpha )∥{\psi }^{\prime }∥}{(\lambda +1)\Gamma \left(\alpha \right)}}L{\left[\phi f\left(s\right)\right]}_{\lambda -\alpha +1}\left[\left|{(t-a)}^{\lambda +1}-{(\tau -a)}^{\lambda +1}\right|\right.\hfill \\ & & \left.+{(b-a)}^{\lambda +1}\left|e^{-\mu \psi \left(t\right)}-e^{-\mu \psi \left(\tau \right))}\right|\right].\hfill \end{array}$$

Hence, the weak sequential equicontinuity of the image of $\mathcal{Q}$ under ${\mathrm{I}}_{\psi }^{\alpha ,\mu }$ follows. Also, in view of ${\mathrm{I}}_{\psi }^{\alpha ,\mu }f\left(a\right)=0$, there exists $\phi \in E^{*},∥\phi ∥=1$ such that

$${∥{\mathrm{I}}_{\psi }^{\alpha ,\mu }f∥}_{\infty }=\underset{t\in [a,b]}{sup}∥{\mathrm{I}}_{\psi }^{\alpha ,\mu }f\left(t\right)∥\le {\displaystyle \frac{(1-\alpha )∥{\psi }^{\prime }∥}{(\lambda +1)\Gamma \left(\alpha \right)}}L{\left[{∥f∥}_{\infty }\right]}_{\lambda -\alpha +1}2{(b-a)}^{\lambda +1},$$

the uniform boundedness also follows.

Thus, ${\mathrm{I}}_{\psi }^{\alpha ,\mu }$ is weakly sequentially continuous and the set ${\mathrm{I}}_{\psi }^{\alpha ,\mu }\left(\mathcal{Q}\right)$ is weakly sequentially compact in $C[I,E_{\omega }]$. By the Eberlein–Šmulan thorem (e.g. [36]), it is also weakly compact in $C[I,E_{\omega }]$. Therefore, ${\mathrm{I}}_{\psi }^{\alpha ,\mu }$ maps bounded sets in ${\mathbf{H}}_0^{\lambda -\alpha +1}[I,E_{\omega }]$ into weakly compact sets in $C[I,E_{\omega }]$. Thus, ${\mathrm{I}}_{\psi }^{\alpha ,\mu }$ is weakly compact. □

7. Comparison with Previous Results: Final Comments

The problem of the equivalence of differential and integral problems studied in this paper has a long history. Our results directly extend those of fractional order operators and are applied to the study of fractional order differential equations. For generalized fractional order integral operators, i.e., differential problems with derivatives with respect to another function (with some special choice of such functions), this applies to the case of real-valued functions (e.g., [3,4,35]) for the case of vector-valued functions with strong topology (e.g., [26]) and the basis for vector-valued functions with derivatives considered in the weak sense or pseudo-derivatives in [11,37]. We will now compare the more relevant of these (and other) papers with the results obtained.

As we have already mentioned, it is now possible to extend the known results, among others mentioned in the Introduction, to the case of pseudo-solutions, i.e., for $\psi$-Hilfer integrals of fractional order in the integral form of the considered problem. However, this can be performed directly by repeating the proofs of other papers, inserting only our results on equivalence problems, and would not show the relevance of our theorems.

In order to focus the readers’ attention on the main objectives and results of this paper, we will present only some extensions of the already known results, choosing those most closely related to fractional order operators and derivatives in weak topology. We will not focus on correcting errors in previous papers (as in the paper of [5]) but will point out the possibility of considering the issues under study with much more general derivatives and integrals of fractional type, which, in addition to obtaining generalizations of previous results, allows them to be unified. The derivatives and integrals considered, as we have already mentioned, include many previously studied operators.

We will limit ourselves mainly to a few of the papers cited in the Introduction and to the most advanced ones.

First, let us recall the result from [38] about the existence of ‘weak-mild’ solutions for the problem

$${}^{c}D^{\alpha }x\left(t\right)=f(t,x\left(t\right)),\mathrm{for}\mathrm{each}t\in I=[0,T]$$

with boundary-value condition $x\left(0\right)+\mu {\int }_0^{T}x\left(s\right)ds=x\left(T\right)$ in reflexive spaces.

Using the approach from our paper, first of all, the solution definitions should be improved to refer to pseudo-derivatives (which is not there), cf. Proposition 1. According to the results of our paper, to improve the assumptions that guarantee that the solution operator is well defined (cf. Theorem 1). Furthermore, the problem can be considered with the fractional derivatives and the integrals as in Definition 3. Of course, we also obtain greater regularity of solutions (not necessarily in spaces of absolutely continuous functions). This paper can be corrected and extended significantly.

In the next step, the research was extended to the existence of pseudo-solutions and weak solutions for nonlocal boundary problems using Riemann–Liouville operators. This required a complete study of the Pettis integral of fractional order and the corresponding spaces of integral functions in the Pettis sense.

This research showed that the close relationship of the spaces associated with fractional-order operators allows for weaker assumptions that guarantee a good definition of the operators. It was also an indication that such results could and should be extended to a larger class of fractional-order operators and their greater regularity obtained. Further important steps in this direction can be found in the papers [4,11,12,13].

The situation is similar in the paper [37], where, among others, the existence of solutions to the fractional differential problem given by

$$D_p^{\alpha }y\left(t\right)=f(t,y\left(t\right)),y\left(0\right)=y_0,$$

where $D_p^{\alpha }(·)$ is a fractional pseudo-derivative of a weakly absolutely continuous and pseudo-differentiable function $y(·):T\to E$ is established. The results go beyond reflexive spaces (cf. [39]) and are carefully proved. These proofs can be carried out without major changes for the case proposed in this paper, i.e., with the inclusion of much more general fractional order operators (cf. again Definition 3). This allows them not to be proved separately for, e.g., the Hadamard operator. It is worth mentioning also a paper in which the Pettis integral is combined with the Hadamard operator. In the paper [12], the existence of solutions of differential equations of the Caputo-Hadamard type of the Pettis integral is proved. This was a step towards the general integral operators of fractional order studied in this paper (cf. [40]). Similarly, the multi-term fractional differential equation studied there can be generalized using our treatment of the topic. Again, we can consider Hölder spaces and solutions more regularly.

In order not to overstretch the concluding remarks, we will only point out that the direction discussed in the paper to study the regularity of solutions of fractional order equations in Hölder-type spaces is currently being pursued for the case of real functions (and thus without problems related to weak topology). Let us mention a recent paper [4].

8. Conclusions

The problem of equivalence between differential and integral problems is absolutely crucial when applying solution methods based on operators and their properties in function spaces. In this paper, we complement the solution of this important problem by considering the case of general derivatives and integrals of fractional order for vector functions for weak topology.

This paper proves that it is worth considering methods for studying fractional differential equations with weak-type derivatives but also that one should carefully choose the spaces for such studies. It turns out that Hölder-type spaces are natural cases, and we have studied them in this paper, but there remains a very important open problem for further research: which spaces allow us to obtain maximal regularity of solutions to differential problems and whether they can be found commonly for any choice of the function $\psi$.