On the Generalization of Tempered-Hilfer Fractional Calculus in the Space of Pettis-Integrable Functions

Abstract

We propose here a general framework covering a wide range of fractional operators for vector-valued functions. We indicate to what extent the case in which assumptions are expressed in terms of weak topology is symmetric to the case of norm topology. However, taking advantage of the differences between these cases, we emphasize the possibly less-restrictive growth conditions. In fact, we present a definition and a serious study of generalized Hilfer fractional derivatives. We propose a new version of calculus for generalized Hilfer fractional derivatives for vector-valued functions, which generalizes previously studied cases, including those for real functions. Note that generalized Hilfer fractional differential operators in terms of weak topology are studied here for the first time, so our results are new. Finally, as an application example, we study some n-point boundary value problems with just-introduced general fractional derivatives and with boundary integral conditions expressed in terms of fractional integrals of the same kind, extending all known cases of studies in weak topology.

1. Introduction

The usefulness in mathematical models of fractional derivatives probably does not require detailed discussion and we will not focus on it. However, notice the multitude of different definitions of fractional-order operators, resulting in duplication of work on various derivatives and fractional integrals. In the work, we will focus on this issue and propose the unification of the topic within one general definition. In fact, we will focus on the case of vector functions and problems where assumptions are expressed in the language of weak topology. This is an interesting and unexplored case in which care must be taken—especially the problem of the equivalence of differential problems and their integral forms. This problem is often wrongly overlooked by many authors, and here we will show what assumptions are necessary to obtain differential integral forms. Note that the integrals are taken in the Pettis sense and that we emphasize the appropriate function spaces related to our operators and problems.

The paper contains a detailed introduction to the theory of fractional differential and integral operators, including as special cases practically all those previously considered and being an extension of this theory. In particular, the results obtained here extend to those of [1] and allow inclusion in the calculus for fractional-order-tempered derivatives ($\mu \ne 0$). For more on the advantages of tempered fractional calculus over typical fractional calculus, i.e., for $\mu =0$, see [2,3,4] or [5]. Practical applications of such a calculus, together with an introduction to computational methods, can be found, for example, in [6].

In particular, we emphasize the case of vector functions, which is more complicated than real-valued functions, and we must also control the properties of the space in which the functions under consideration have values. Since our approach to generalized tempered-Hilfer derivatives allows us to unify the many types of fractional integrals and derivatives studied separately, we try to list all of them and include a full set of references.

Motivated, among other things, by [2,7,8,9,10,11,12], we examine the existence of solutions to the following n-point fractional-order boundary value problem:

$$\left\{\begin{array}{c}{}^H{\tilde{\mathbb{D}}}_{a,g,p}^{\alpha ,\beta ,\mu }x\left(t\right)=\lambda f(t,x\left(t\right)),t\in [a,b],\alpha \in (1,2),\beta \in [0,1],\mu \in {\mathbb{R}}^{+},\lambda \in \mathbb{R}\hfill \\ {\displaystyle x\left(a\right)=0,x\left(b\right)=\sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{{\delta }_i,\mu }x\left({\xi }_i\right),{\delta }_i>0,{\xi }_i\in [a,b].}\hfill \end{array}\right.$$

(1)

Here, ${}^H{\tilde{\mathbb{D}}}_{a,g,p}^{\alpha ,\beta ,\mu }$ denotes the so-called g-Hilfer fractional derivative [13,14,15], which will be defined later. Let us remark that

• If we take $g\left(t\right)=t,t\in (0,1),\alpha \in (1,2)$ and $\beta \to 1$, then Equation (1) turns into a boundary-value problem with the concept of the Caputo fractional derivative.
• If we take $g\left(t\right)=t,t\in (0,1),\alpha \in (1,2)$ and $\beta \to 0$, then Equation (1) turns into a boundary-value problem with the concept of the Riemann-Liouville fractional derivative.
• If we take $g\left(t\right)=ln(1+t),t\in (0,1),\alpha \in (1,2)$ and $\beta \to 1$, then Equation (1) turns into boundary-value problem with the concept of the Caputo–Hadamard fractional derivative.
• If we take $g\left(t\right)=ln(1+t),t\in (0,1),\alpha \in (1,2)$ and $\beta \to 0$, then Equation (1) turns into a boundary-value problem with the concept of the Hadamard fractional derivative.

Note that in light of the above Remarks 1 and 2, the Hilfer fractional derivative, given as $g\left(t\right)=t$ and with respect to $\beta$, forms a kind of interpolation between Caputo and Riemann-Liouville derivatives. The same property is true for $g\left(t\right)=ln(1+t)$ for Hadamard and Caputo–Hadamard derivatives (Remarks 3 and 4). This suggests that the appropriate choice of g can serve us to study boundary value problems with any fractional order derivative studied so far, and has further potentially interesting extensions. For more information on the advantages of Hilfer-type derivatives, see [16], for instance. Our paper will extend this to the case of vector functions and derivatives of weak types. It is worth noting that the use of g-type fractional calculus is not limited to boundary value problems, although we will present applications of just this area. For the sake of completeness, we should necessarily mention that the fractional tempered calculus is currently also being extended towards its discrete version ([17], cf. also [3] for more details). Further research should be devoted to such a topic for vector functions, but this is definitely beyond the intended purpose of this article, and we refer those interested to the literature.

What about the choice of g? Let us mention, for instance, a recent result about the sub-diffusion process [18] with appropriately chosen g, i.e., with $g\left(t\right)=\frac{t}{1+Bt}[1+Aln(1+c{t}^{\gamma })]$, $g\left(t\right)=t+\frac{t[Aln(1+C{t}^{\gamma })-Bt]}{1+Bt}$ or $g\left(t\right)=t+D_{\beta }{t}^{\beta /\alpha }$ (where $D_{\beta }$ is an ultraslow diffusion coefficient). This is also a very natural area of applications for g-type fractional calculus, and thus for the calculus proposed in this paper.

Apart from this, the boundary value problem (1) for different values of $\mu ,\beta$ and g includes the study of implicit fractional boundary-value problems involving the fractional differential operators: Hilfer, Hadamard, Katugampola, Chen, Jumarie, Prabhakar, Erdélyi-Kober, Riesz, Feller, Weyl, Cassar, and many other fractional differential operators listed in [8,14,15,19]. For instance, when $\mu =0$, the outcomes acquired in the present paper incorporate the results of, e.g., [9,11,12]. In this paper, we introduce this calculus for vector-valued functions in all its generality, highlighting, of course, the changes required to consider $\mu \ne 0$.

2. Preliminaries

When studying fractional order equations, we have to talk about functions in many interesting function spaces. Let us collect all the auxiliary facts about these spaces and about the operators acting between them, as well as all the necessary definitions, in keeping the paper self-explanatory.

In order to make it easier for readers who have so far learned the problems of real functions to use the paper, we will now present the necessary concepts and theorems for important function spaces and the weak topology.

2.1. Function Spaces

Let us first recall the concepts related to Orlicz spaces, which play an important role in this paper. At the origin of fractional calculus and fractional operators in function spaces, it turned out that the natural domain and counter-domain for such operators included Orlicz spaces [20]. Moreover, the study of operators in spaces of weakly (i.e., Pettis) integrable functions does not change this situation and these spaces are useful also in this context. Recall that Pettis integrability is also closely related to certain weak integrability conditions in Orlicz spaces.

We should provide some facts about such spaces, since in many works the growth constraints on the nonlinear component f of the problem under study are expressed in terms of Lebesgue spaces. In this work, despite the consideration of vector functions, we will use weaker assumptions in terms of Orlicz spaces, which are more general even in the case of real-valued functions.

We say that a function $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a Young function if $\psi$ is increasing, convex, and continuous with $\psi \left(0\right)=0$ and ${lim}_{u\to \infty }\psi \left(u\right)=\infty$. For any Young function $\psi$, the function $\tilde{\psi }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ defined by $\tilde{\psi }\left(u\right)={sup}_{v\ge 0}\{vu-\psi \left(v\right)\}$ is called the Young complementary function for $\psi$. Furthermore, $\tilde{\psi }$ is also a Young-type function as well. The Orlicz space $L_{\psi }=L_{\psi }([a,b],\mathbb{R})$ consists of all (classes of) measurable functions $x:[a,b]\to \mathbb{R}$ for which

$${∥x∥}_{\psi }:=inf\{k>0:{\int }_a^b\psi \left(\frac{\left|x\right(s\left)\right|}{k}\right)ds<1\},$$

(2)

is finite (see, e.g., [21]).

It is clear, that the particular choice $\psi \left(u\right)={\psi }_p\left(u\right):=\frac{1}{p}{\left|u\right|}^p,p\in [1,\infty )$ leads to the classical case of Lebesgue space $L_p=L_p([a,b],\mathbb{R}),p\in [1,\infty )$. In this case, it can be easily seen that ${\tilde{\psi }}_p={\psi }_{\tilde{p}}$ with $\frac{1}{p}+\frac{1}{\tilde{p}}=1$ for $p>1$.

In this connection it is worth recalling that for any Young function $\psi$ we have $\psi (u-v)\le \psi \left(u\right)-\psi \left(v\right)$ and $\psi \left(\rho u\right)\le \rho \psi \left(u\right)$ hold for any $u,v\in \mathbb{R}$ and $\rho \in [0,1]$. Furthermore, for a non-trivial Young function $\psi$, $L_{\infty }\subset L_{\psi }.$ For further properties of Young functions and Orlicz spaces generated by such functions, we refer the reader to [21,22].

In the subsequent pages, E will be considered as a Banach space with a certain fixed norm $∥·∥$ and with its dual space $E^{*}$. Furthermore, $E_w$ denotes the space E when endowed with its weak topology $\sigma (E,E^{*})$. By $C[I,E]$, we denote the Banach space of (strongly) continuous functions $x:I\to E$ endowed by the classical norm ${∥x∥}_0={sup}_{t\in I}∥x\left(t\right)∥$. Let $C[I,E_w]$ denote the Banach space of all weakly continuous functions $x:I\to E$ with its weak topology (i.e., generated by continuous linear functionals on E). We recall that a function $x:I\to E$ is weakly absolutely continuous wAC on I if, for every $\phi \in E^{*}$, a real valued function $\phi x$ is absolutely continuous on I.

By $\mathbb{P}[I,E]$, we denote the space of E-valued Pettis integrable functions defined on I, which is in general a normed space, but not a complete space. For a particular case $E=\mathbb{R}$, the space $\mathbb{P}[I,\mathbb{R}]=L_1[I,\mathbb{R}]$. We need to introduce more meaningful function spaces. For convenience, we recall the following.

Definition 1

([23,24]). For any Young function ψ, we define a class of functions ${\mathcal{H}}^{\psi }\left(E\right)$ as

$${\mathcal{H}}^{\psi }\left(E\right):=\{x:I\to E:x weakly measurable and satisfying\phi x\in L_{\psi }\left(I\right)for every\phi \in E^{*}\}.$$

As a subspace of it, let us consider

$${\tilde{\mathcal{H}}}^{\psi }\left(E\right):=\left\{x:I\to E:xstrongly measurable and satisfying\phi x\in L_{\psi }\left(I\right)forevery\phi \in E^{*}\right\}.$$

Moreover, the class ${\mathcal{H}}_0^{\psi }\left(E\right)$ (resp. ${\tilde{\mathcal{H}}}_0^{\psi }\left(E\right)$) is defined to be the subspace of ${\mathcal{H}}^{\psi }\left(E\right)$ (resp. ${\tilde{\mathcal{H}}}^{\psi }\left(E\right)$) composed of Pettis integrable functions on I, that is

$${\mathcal{H}}_0^{\psi }\left(E\right):=\{x\in {\mathcal{H}}^{\psi }\left(E\right):x\in \mathbb{P}[I,E]\},{\tilde{\mathcal{H}}}_0^{\psi }\left(E\right):=\{x\in {\tilde{\mathcal{H}}}^{\psi }\left(E\right):x\in \mathbb{P}[I,E]\}.$$

In particular, the well-known class ${\mathcal{H}}_0^p\left(E\right)$ denotes the class ${\mathcal{H}}_0^{\psi }\left(E\right)$ for the particular choice $\psi \equiv \frac{{|·|}^p}{p}$.

Obviously ${\tilde{\mathcal{H}}}_0^{\psi }\left(E\right)\subseteq {\mathcal{H}}_0^{\psi }\left(E\right)\subseteq {\mathcal{H}}^{\psi }\left(E\right)$ and ${\tilde{\mathcal{H}}}_0^{\psi }\left(E\right)\equiv {\mathcal{H}}_0^{\psi }\left(E\right)$ holds true whenever E is separable (cf. [24], Corollary 1.11). Some special facts about these spaces are known (cf. [24,25,26]):

Proposition 1.

(1) If E is reflexive, then ${\mathcal{H}}^1\left(E\right)\equiv {\mathcal{H}}_0^1\left(E\right)$.

(2) For any Young function ψ with ${lim}_{u\to \infty }\psi \left(u\right)/u\to \infty ,$ ${\tilde{\mathcal{H}}}^{\psi }\left(E\right)\subseteq {\mathcal{H}}_0^{\psi }\left(E\right)$. In particular, ${\tilde{\mathcal{H}}}^p\left(E\right)\subseteq {\mathcal{H}}_0^p\left(E\right)$ holds true for any $p>1$. If additionally E is weakly complete or even more generally, contains no isomorphic copy of $c_0$, it is also true for any Young function ψ. That is, ${\tilde{\mathcal{H}}}^1\left(E\right)\subseteq {\mathcal{H}}_0^1\left(E\right)$ whenever E satisfies this additional condition.

Clearly, since the weak continuity of a function implies strong measurability (cf. [27], p. 73), in light of Proposition 1, it implies that

Corollary 1.

For any non-trivial Young function ψ, the space $C[I,E_w]$ is a proper subset of ${\tilde{\mathcal{H}}}_0^{\psi }\left(E\right)$.

Let us emphasize that the connection between Pettis integrals and Orlicz spaces is much deeper than that presented in [26] (see also [28]). In what follows, we will integrate vector-valued functions with respect to some real-valued ones. For this reason, we recall the results that complement some of those from [22,24], on the pointwise products of Pettis integrable functions and real-valued functions.

Proposition 2

([29], Proposition 5). If $x\in {\mathcal{H}}_0^{\psi }\left(E\right)$, then $x(·)y(·)\in \mathbb{P}[I,E]$ for every $y\in L_{\tilde{\psi }}$.

2.2. Differential and Integral Operators

Let us recall the necessary definitions and facts about weak-type derivatives in Banach spaces. Let us collect all of them that are applicable to the problems described in this paper.

Definition 2

([24,25]). The vector-valued function $x:I\to E$ is said to be pseudo-differentiable on I if

• For every $\phi \in E^{*}$, the real-valued functions $\phi x$ are differentiable almost everywhere on I,
• There exists a function $y:I\to E$ such that for every $\phi \in E^{*}$ there exists a null set $N\left(\phi \right)\subset I$ with

  $${\left(\phi x\left(t\right)\right)}^{\prime }=\phi y\left(t\right),foreveryt\in I\setminus N\left(\phi \right).$$

In such a case, y is called a pseudo-derivative of x. If the null sets are not dependent on φ, then x is then said to be almost everywhere weakly differentiable on I and in this case y is called almost everywhere weak derivative of x. It 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.

Throughout this paper, we let ${\mathfrak{D}}_p$ denotes the pseudo-differential operator (resp.${\mathfrak{D}}_{\mathrm{\omega }}$ for the weak one). This is definitely a more general case than ordinary derivatives, and we need to introduce readers to the topic.

Remark 1.

It should be noted that, unlike the real-valued functions space, the weak absolute continuity of a function $x:I\to E$ gives no guarantee for the existence of ${\mathfrak{D}}_{p}x$ even if E is separable (see, e.g., [30]).

We remark also that, the indefinite Pettis integral of Pettis integrable function does not enjoys the stronger property of being a.e. weakly differentiable.

The following lemma will be used for our study of pseudo-derivatives of weakly absolutely continuous [wAC] functions (cf. [24] Section 8, [31] Theorem 5.1 or [27]):

Lemma 1.

(1) The (indefinite) integral of any Pettis integrable (resp. weakly continuous) function is wAC and is pseudo- (resp. weakly) differentiable with respect to the right endpoint of the integration interval, and its pseudo- (resp. weak) derivative is equal to the integrand at that point.

(2) A function $x:I\to E$ is an indefinite Pettis integral if and only if x is wAC and has a pseudo-derivative ${\mathfrak{D}}_{p}x$ on I. In this case, ${\mathfrak{D}}_{p}x\in \mathbb{P}[I,E]$ and 

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

Before proceeding to the next section, note that it is natural to assume that the space E has a total dual, i.e., there exists a countable determining set. If E is separable, then both E and $E^{*}$ have a total dual, so the spaces as $BV\left(I\right)$ or $L_{\infty }\left(I\right)$ have this property. In these relations, all considered pseudo-derivatives of functions from I to E are uniquely determined up to a set of measure zero. In-depth results on this problem can be found in ([32], Corollary 3.4, Theorem 3.6).

In the tempered context of fractional calculus, we should remember that for any continuous $g:I\to \mathbb{R}$ having a positive continuous derivative $g^{\prime }$ on I, the result ([24], Corollary 3.41) ensures that $exp\{\mu g(·)\}x(·)g^{\prime }(·)\in \mathbb{P}[I,E]$ (resp. $exp\{\mu g(·)\}x(·)g^{\prime }(·)\in C[I,E_w]$) is true for every $x\in \mathbb{P}[I,E]$ (resp. $x\in C[I,E_w]$). From which, based on the definition of the weak (resp. pesudo) derivative, in light of Lemma 1, it can easily be seen that

$$\left\{\begin{array}{c}{\displaystyle \left(\frac{1}{g^{\prime }\left(t\right)}{\mathfrak{D}}_{\mathrm{\omega }}+\mu \right){\int }_a^t{e}^{-\mu \left(g\right(t)-g(s\left)\right)}x\left(s\right)g^{\prime }\left(s\right)ds=x,\mathrm{holds} \mathrm{for} \mathrm{any}x\in C[I,E_w],(✠)}\hfill \\ {\displaystyle \left(\frac{1}{g^{\prime }\left(t\right)}{\mathfrak{D}}_p+\mu \right){\int }_a^t{e}^{-\mu \left(g\right(t)-g(s\left)\right)}x\left(s\right)g^{\prime }\left(s\right)ds=x,\mathrm{holds} \mathrm{for} \mathrm{any}x\in \mathbb{P}[I,E].(◊)}\hfill \end{array}\right.$$

Remark 2.

Let us note that

• Since the (indefinite) Pettis integral of a function $x\in \mathbb{P}[I,E]$ does not have the property of being a.e. weakly differentiable (see [33]), the formula $(✠)$ need not occur for any $x\in \mathbb{P}[I,E]$.
• The next formula $(◊)$ is not uniquely determined, except when E has a total dual $E^{*}$. Of course, according to ([30], p. 2 and [34]), it can happen that

  $$\left(\frac{1}{g^{\prime }\left(t\right)}{\mathfrak{D}}_{\mathrm{\omega }}+\mu \right){\int }_a^t{e}^{-\mu \left(g\right(t)-g(s\left)\right)}x\left(s\right)g^{\prime }\left(s\right)ds=y,$$

  whereby y is weakly equivalent to x (but they do not have to be equal a.e.).

2.3. Other Useful Tools

Since our considerations involve vector functions, certain compactness conditions will be necessary for the existence of solutions, as well as the continuity of operators appropriate to the topology under study. We will briefly recall the necessary facts.

Let us recall some important notion of operator continuity. A mapping $T:X\to Y$, where X and Y are Banach spaces are called weakly–weakly sequentially continuous (or: $ww$-sequentially continuous) if and only if it maps weakly convergent sequences $\left(x_n\right)$ to $x\in E$ into sequences ($T\left(x_n\right)$) weakly convergent to $T\left(x\right)$ in Y.

Definition 3

([35]). Let ${\mathcal{M}}_E$ be the family of all nonempty bounded subsets of E. Let $B_1$ denote the closed unit ball of E centered at 0. The measure of weak noncompactness of De Blasi

$$\mathit{\mu }:{\mathcal{M}}_E\to [0,\infty )$$

is defined by

$$\mathit{\mu }\left(X\right):=inf\{ϵ>0:there exists a weakly compact subsetWaofE:X\subset ϵB_1+W\}.$$

For the properties of $\mathit{\mu }$ let us refer to [35]. The Ambrosetti-type lemma will be useful in the paper.

Lemma 2.

Let V be bounded and strongly equicontinuous subset of $C[I,E]$. Then,

• $t\mapsto \mathit{\mu }\left(V\left(t\right)\right)\in C[I,{\mathbb{R}}^{+}],whereV\left(t\right):=\{v\left(t\right):v\in V,t\in I\}$,
• ${\displaystyle {\mathit{\mu }}_C\left(V\right)=\underset{t\in I}{sup}\mathit{\mu }\left(V\left(t\right)\right)=\mathit{\mu }\left(V\left(I\right)\right)}$,

where ${\mathit{\mu }}_C$ denotes the De Blasi measure of weak noncompactness in the space $C[I,E]$.

For our purposes, one of the basic tools will be the following Mönch-type fixed point theorem:

Theorem 1

([36]). Let $\mathcal{Q}$ be a nonempty, closed, convex and equicontinuous subset of a metrizable locally convex vector space $C[I,E_w]$ such that $0\in \mathcal{Q}$. Suppose that $T:\mathcal{Q}\to \mathcal{Q}$ is weakly–weakly sequentially continuous mapping. If the implication

$$\begin{array}{c}\hfill \overline{V}=\overline{conv}(\{0\}\cup T\left(V\right))\Rightarrow Vis relatively weakly compact,\end{array}$$

(3)

and holds true for any subset V of $\mathcal{Q}$, then the operator T has a fixed point in the set $\mathcal{Q}$.

3. Generalized Fractional Integral Operators

There are various modification and generalizations of classical fractional operators, which are widely used both in theory and in applications. The below definition allows us to unify the different fractional integrals and, consequently, to solve some initial and/or boundary value problems with different types of fractional integrals and derivatives in a unified way.

Definition 4.

Let $g\in C^1[a,b]$ be a positive increasing function such that $g^{\prime }\left(t\right)\ne 0$, for all $t\in [a,b].$ The generalized fractional (or briefly g-fractional) integral of a function $x:[a,b]\to E$ of order $\alpha >0$ and parameter $\mu \in {\mathbb{R}}^{+}$ is that defined by

$${\Im }_{a,g}^{\alpha ,\mu }x\left(t\right):=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_a^t{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}x\left(s\right)g^{\prime }\left(s\right)ds,(-\infty \le a<b\le \infty ).$$

(4)

For completeness, we define ${\Im }_{a,g}^{a,\mu }x\left(a\right):=0$. In the previous definition, the sign “∫” denotes the Pettis integral. This of course includes the case of the Lebesgue integral when $E=\mathbb{R}$.

Remark 3.

If the g-fractional integral ${\Im }_{a,g}^{\alpha ,\mu }x$ exists for some $x\in \mathbb{P}[I,E]$, it follows accordingly to the definition of Pettis integral for any $t\in I$ there exists an element of E denoted by ${\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)$ such that

$$\begin{array}{ccc}\hfill \phi \left({\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)\right)& =& {\int }_a^t\phi \left(\frac{e^{-\mu \left(g\right(t)-g(s\left)\right)}x\left(s\right)g^{\prime }\left(s\right)}{{(g\left(t\right)-g\left(s\right))}^{1-\alpha }\mathrm{\Gamma }\left(\alpha \right)}\right)ds\hfill \\ & =& {\int }_a^t\frac{e^{-\mu \left(g\right(t)-g(s\left)\right)}\phi \left(x\left(s\right)\right)g^{\prime }\left(s\right)ds}{{(g\left(t\right)-g\left(s\right))}^{1-\alpha }\mathrm{\Gamma }\left(\alpha \right)}={\Im }_{a,g}^{\alpha ,\mu }\phi \left(x\left(t\right)\right),\hfill \end{array}$$

(5)

holds true for every $\phi \in E^{*}$.

We include a proof of the following well-known facts since we did not find a suitable reference for it:

Fact 1.

For any $\alpha >0$, $\mu \in {\mathbb{R}}^{+}$ and positive increasing function $g\in C^1[a,b]$, having $g^{\prime }\left(t\right)\ne 0,$ we have ${\Im }_{a,g}^{\alpha ,\mu }$ maps $L_{\infty }[a,b]$ into $C[a,b]$. Moreover, if g is a Lipschitz-continuous function, then ${\Im }_{a,g}^{\alpha ,\mu }x$ is also Hölder continuous.

Proof. 

The case when $\alpha \ge 1$ is trivial. We follow the idea from ([1], Theorem 2, Remark 3). We consider the case when $\alpha \in (0,1)$: Obviously, ${\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)=e^{-\mu g\left(t\right)}{\Im }_{a,g}^{\alpha ,0}\left(e^{\mu g\left(t\right)}x\left(t\right)\right)$. Put $x\in L_{\infty }[a,b]$ and note that for each $a<\tau <t\le b$, we have

$$\begin{array}{ccc}\hfill & & \left|{\Im }_{a,g}^{\alpha ,0}{e}^{\mu g\left(t\right)}x\left(t\right)-{\Im }_{a,g}^{\alpha ,0}{e}^{\mu g\left(\tau \right)}x\left(\tau \right)\right|\hfill \\ & \le & \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\left({\int }_a^{\tau }\left|\frac{e^{\mu g\left(s\right)}}{{(g\left(t\right)-g\left(s\right))}^{1-\alpha }}-\frac{e^{\mu g\left(s\right)}}{{(g\left(\tau \right)-g\left(s\right))}^{1-\alpha }}\right|\left|x\left(s\right)\right|g^{\prime }\left(s\right)ds\right.\hfill \\ \hfill & +& \left.{\int }_{\tau }^t\frac{e^{\mu g\left(s\right)}}{{(g\left(t\right)-g\left(s\right))}^{1-\alpha }}\left|x\left(s\right)\right|g^{\prime }\left(s\right)ds\right)\hfill \\ & \le & \frac{ess\; sup\left(\left|x(·)\right|e^{\mu g(·)}\right)}{\mathrm{\Gamma }\left(\alpha \right)}\left({\int }_a^{\tau }\left|{(g\left(t\right)-g\left(s\right))}^{\alpha -1}-{(g\left(\tau \right)-g\left(s\right))}^{\alpha -1}\right|g^{\prime }\left(s\right)ds\right.\hfill \\ & +& \left.{\int }_{\tau }^t{(g\left(t\right)-g\left(s\right))}^{\alpha -1}{g}^{\prime }\left(s\right)ds\right)\hfill \\ & \le & \frac{2{∥x∥}_{\infty }{e}^{\mu ∥g∥}}{\mathrm{\Gamma }(1+\alpha )}{(g\left(t\right)-g\left(\tau \right))}^{\alpha }.\hfill \end{array}$$

(6)

From which, in view of our definition that ${\Im }_{a,g}^{\alpha ,\mu }x\left(a\right):=0$, it follows ${\Im }_{a,g}^{\alpha ,\mu }x(·)\in C[a,b]$, as required. □

Fact 2

(semigroup property). For any $\alpha ,\beta >0$, $\mu \in {\mathbb{R}}^{+}$ and a positive increasing function $g\in C^1[a,b]$, having $g^{\prime }\left(t\right)\ne 0,$ we obtain that

$${\Im }_{a,g}^{\alpha ,\mu }{\Im }_{a,g}^{\beta ,\mu }x={\Im }_{a,g}^{\beta ,\mu }{\Im }_{a,g}^{\alpha ,\mu }x={\Im }_{a,g}^{\alpha +\beta ,\mu }x$$

is true for any $x\in L_1[a,b]$.

Proof. 

It is known that the semigroup property in the case when $\mu =0$ holds (see, e.g., [37,38]). Thus, since ${\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)=e^{-\mu g\left(t\right)}{\Im }_{a,g}^{\alpha ,0}\left(e^{\mu g\left(t\right)}x\left(t\right)\right)$ we obtain

$$\begin{array}{ccc}\hfill {\Im }_{a,g}^{\alpha ,\mu }{\Im }_{a,g}^{\beta ,\mu }x\left(t\right)& =& e^{-\mu g\left(t\right)}{\Im }_{a,g}^{\alpha ,0}\left(e^{\mu g\left(t\right)}{\Im }_{a,g}^{\beta ,\mu }x\left(t\right)\right)=e^{-\mu g\left(t\right)}{\Im }_{a,g}^{\alpha ,0}\left(e^{\mu g\left(t\right)}{e}^{-\mu g\left(t\right)}{\Im }_{a,g}^{\beta ,0}{e}^{\mu g\left(t\right)}x\left(t\right)\right)\hfill \\ & =& e^{-\mu g\left(t\right)}{\Im }_{a,g}^{\alpha ,0}\left({\Im }_{a,g}^{\beta ,0}{e}^{\mu g\left(t\right)}x\left(t\right)\right)=e^{-\mu g\left(t\right)}{\Im }_{a,g}^{\alpha +\beta ,0}\left(e^{\mu g\left(t\right)}x\left(t\right)\right)={\Im }_{a,g}^{\alpha +\beta ,\mu }x\left(t\right).\hfill \end{array}$$

Similarly, we can show that ${\Im }_{a,g}^{\alpha ,\mu }{\Im }_{a,g}^{\beta ,\mu }x={\Im }_{a,g}^{\beta ,\mu }{\Im }_{a,g}^{\alpha ,\mu }x.$ □

Remark 4.

We note that the generalized fractional operator defined by our Definition 4 allows us to generalize several existing fractional integral operators (and even in the context of the norm topology, i.e., with the Bochner integral instead of the Pettis integral). Of course, we should note that this new formalism allows us to consider as special cases several other models of fractional calculus, such as the Erdélyi-Kober and the Hadamard fractional operators:

(1) 

${\Im }_{0,ln(1+t)}^{\alpha ,\mu }$, $t\in [0,1]$, where $\alpha >0,\mu \in {\mathbb{R}}^{+}$ is a generalized version of the classical Hadamard model of fractional calculus. Obviously, in the particular choice $\mu =0$, we cover the standard version of the Hadamard fractional integral, discussed, among others, by Cichoń and Salem in [29,39,40] (cf. also [22,41,42,43,44,45]), for the existence of solutions to the fractional Cauchy problem.

(2) 

${\Im }_{0,t}^{\alpha ,0}$, $t\in [0,1]$ is the classical fractional calculus of the Riemann-Liouville -type.

(3) 

${\Im }_{a,t}^{\alpha ,\mu },t\in [a,b]$, we obtain the tempered fractional calculus, as in [4,46], which has been studied in recent years because of its applications in dynamical and stochastic systems.

(4) 

${\displaystyle {\Im }_{a,t^{\rho }}^{\alpha ,0},t\in [a,b]}$, with $a>0,\alpha >0,\mu \in {\mathbb{R}}^{+}$ we obtain the Katugampola fractional integral calculus (see [47]) (and fractional integral operators with respect to $t^{\rho }$ as defined by Erdélyi-Kober).

(5) 

${\rho }^{-\alpha }{\Im }_{a,t}^{\alpha ,\frac{1-\rho }{\rho }},t\in [a,b]\rho \in (0,1]$ is the generalized proportional fractional calculus (cf. [37,38]).

The above particular choice of functions g, μ and a shows that our model with general analytic kernels can be extended to even more classical models of fractional calculus: not only the Riemann–Liouville model and related formulas with different kernels, but also the Hadamard and Katugampola models and their generalizations. However, from the particular choice of g, it is possible to obtain connections with papers so far discussed in the literature. In addition, several papers show the advantages that the parameter μ provides in the discussion of solutions of fractional differential equations.

In this regard, using the substitution $u=\frac{g\left(s\right)-g\left(a\right)}{g\left(t\right)-g\left(a\right)}$, it can be verified that

$${\Im }_{a,g}^{\alpha ,\mu }\left\{e^{-\mu g\left(t\right)}{\left(g\left(t\right)-g\left(a\right)\right)}^{\beta -1}\right\}=\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }(\alpha +\beta )}{e}^{-\mu g\left(t\right)}{\left(g\left(t\right)-g\left(a\right)\right)}^{\alpha +\beta -1},\alpha ,\beta >0,t>a.$$

(7)

Remark 5.

The above formula also suggests an approach through Stieltjes-type integrals:

$${\int }_a^{t}x\left(s\right)d_{s}h(t,s)$$

with appropriately chosen $h(t,s)$. The Pettis–Stieltjes integral was first introduced in the context of stochastic differential equations ([48]), but it is also used for fractional differential equations ([49]). In the current research, another extension is usually used, namely the Kurzweil–Stieltjes integral. Clearly, the set of assumptions on h is related to expected properties of derivatives, and the approach we propose seems ready for applications. It should be noted that in fractional calculus, if we try to cover existing integrals, the h functions in the above formula are quite complicated to study—in the approach proposed here, it is based on a lower incomplete Gamma function. In our opinion, fractional problems should be investigated directly, as in this paper.

However, if we expect discontinuous solutions for some fractional differential problems, such as impulsive differential equations, an interesting approach might be to consider the discrete part for $h(t,s)$ (including jumps at some points). For instance, we might consider

$$h(t,s)=e^{-\mu g\left(t\right)}{\left(g\left(t\right)-g\left(a\right)\right)}^{\alpha +\beta -1}+\sum _{i=1}^k(g\left(t_k\right)-g\left(t_k^{-}\right)){\chi }_{[t_k,b]},$$

for some $t_1,\cdots ,t_k\in [a,b]$. Due to discontinuity of h in such a case we must replace the Pettis (or: Bochner) integral by Stieltjes-type integrals. However, this topic exceeds the scope of this paper and will be investigated in a forthcoming paper.

Example 1.

Let $\alpha >0$ and let J be a subset of $I:=[a,b]$ of positive measure. Consider the Banach space $E=B\left[I\right]$ of bounded real-valued functions defined on I. Let us define a weakly measurable function $x:I\to B\left[I\right]$ by

$$x\left(t\right):=\left\{\begin{array}{c}{\displaystyle {\chi }_{\{t\}}(·),t\in J,}\hfill \\ {\displaystyle \theta ,t\notin J.}\hfill \end{array}\right.$$

Since every $\phi \in B^{*}\left[I\right]$ is of the form $x\to {\int }_{I}x\left(t\right)d\zeta \left(t\right)$ for some countable additive measure $\zeta$, we conclude that $\phi x\left(t\right)=0$ for every $\phi \in B^{*}\left[I\right]$. Also, for every measurable $\mathrm{\Sigma }\subset I$, we obtain

$${\int }_{\mathrm{\Sigma }}\phi \left(x\left(s\right)\right)ds={\int }_{\mathrm{\Sigma }}\left({\int }_J{\chi }_{\{s\}}d\zeta \right)ds=\phi \left(\theta \right).$$

From which, based on the definition of the Pettis integral, we conclude that $x\in P[I,B[I\left]\right]$. Furthermore, for any measurable set $\mathrm{\Sigma }\subset I$, we have the following:

$$\begin{array}{ccc}& & {\int }_{\mathrm{\Sigma }}\phi \left({(g\left(t\right)-g\left(s\right))}^{\alpha -1}{e}^{-\mu \left(g\right(t)-g(s\left)\right)}x\left(s\right)\right)g^{\prime }\left(s\right)ds\hfill \\ & =& {\int }_{\mathrm{\Sigma }}{(g\left(t\right)-g\left(s\right))}^{\alpha -1}{e}^{-\mu \left(g\right(t)-g(s\left)\right)}\phi \left(x\left(s\right)\right)g^{\prime }\left(s\right)ds=0=\phi \left(\theta \right).\hfill \end{array}$$

This means that, according to the definition of the Pettis integral, ${\Im }_{a,g}^{\alpha ,\mu }x$ exists on the interval I and ${\Im }_{a,g}^{\alpha ,\mu }x=\theta$.

Remark 6.

Since $s\to {(g\left(t\right)-g\left(s\right))}^{\alpha -1}{e}^{-\mu \left(g\right(t)-g(s\left)\right)}{g}^{\prime }\left(s\right)\in L_{\infty }[a,t]$ for a.e. $t\in [a,b]$ and any $\alpha \ge 1,\mu >0$, it follows in view of ([24], Corollary 3.41), ${\Im }_{a,g}^{\alpha ,\mu }x$ exists for any $x\in {\mathcal{H}}_0^1\left(E\right)$.

Lemma 3.

If $x\left(·\right),y\left(·\right)\in \mathbb{P}[I,E]$ are weakly equivalent on the interval $I:=[a,b]$ so that ${\Im }_{a,g}^{\alpha ,\mu }x$ and ${\Im }_{a,g}^{\alpha ,\mu }y$ exist, then ${\Im }_{a,g}^{\alpha ,\mu }x={\Im }_{a,g}^{\alpha ,\mu }y$ on I.

Proof. 

If $x\left(·\right),y\left(·\right)\in \mathbb{P}[I,E]$ are weakly equivalent on I, then for every $\phi \in E^{*}$ there exists a null set $N\left(\phi \right)$ such that for every $t\in I$ we have that $\phi x\left(s\right)=\phi y\left(s\right)$ for $s\in \left[a,t\right]\setminus N\left(\phi \right)$. If follows that

$$g^{\prime }\left(s\right)\frac{{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}}{\mathrm{\Gamma }\left(\alpha \right)}\phi x\left(s\right)=g^{\prime }\left(s\right)\frac{{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}}{\mathrm{\Gamma }\left(\alpha \right)}\phi y\left(s\right).$$

for $s\in \left[a,t\right]\setminus N\left(\phi \right)$ and $t\in I$, and so

$$\begin{array}{ccc}& & {\int }_a^t{g}^{\prime }\left(s\right)\frac{{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}}{\mathrm{\Gamma }\left(\alpha \right)}\phi x\left(s\right)ds\hfill \\ & =& {\int }_a^t{g}^{\prime }\left(s\right)\frac{{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}}{\mathrm{\Gamma }\left(\alpha \right)}\phi y\left(s\right)ds,\hfill \end{array}$$

for $t\in I$. Therefore, by (5) it follows $\phi \left({\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)\right)=\phi \left({\Im }_{a,g}^{\alpha ,\mu }y\left(t\right)\right)$ for every $\phi \in E^{*}$ and every $t\in I$ and thus ${\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)={\Im }_{a,g}^{\alpha ,\mu }y\left(t\right)$ on I. □

It is a long-known fact that the g-fractional integral operator ${\Im }_{a,g}^{\alpha ,\mu }$ enjoys the smooth property of being continuous map from $L_p[a,b]$ into $C[a,b]$ for some $p>max\{1,1/\alpha \}$ (see, e.g., [50]). However, the following result tells us that the g-fractional integral operator has a similar smoothing property from Orlicz spaces into the space into $C[a,b]$. Indeed, the following theorem provides a useful characterization of the g-fractional integral and is complementary to similar results in ([44], Lemma 1) and ([29], Theorem 2) (see also [1,50]).

Theorem 2.

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

$$\begin{array}{c}\hfill {\int }_0^t\tilde{\psi }\left(s^{\alpha -1}\right)ds<\infty ,t>0,\end{array}$$

(8)

the operator ${\Im }_{a,g}^{\alpha ,\mu }$ maps the space ${\mathcal{H}}_0^{\psi }\left(E\right)$ into $C[I,E_w]$. Moreover, for any $x\in {\mathcal{H}}_0^{\psi }\left(E\right)$ there is $\phi \in E^{*},$ with $∥\phi ∥=1$ such that

$$∥{\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)∥\le \frac{4e^{\mu (∥g∥-g\left(a\right))}∥g^{\prime }∥{\tilde{\mathrm{\Psi }}}_{\alpha }\left(∥g∥\right)}{\mathrm{\Gamma }\left(\alpha \right)}{∥\phi \left(x\right)∥}_{\psi }.$$

In particular, ${\Im }_{a,g}^{\alpha ,\mu }:C[I,E_w]\to C[I,E_w]$.

In order to simplify the proof of Theorem 2, we have divided it into several steps, giving some facts.

Firstly, in the same spirit as in the proof of ([29], Proposition 2), it can be easily proved that

Lemma 4.

For any $\alpha \in (0,1]$, the function ${\tilde{\mathrm{\Psi }}}_{\alpha }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ defined by

$${\tilde{\mathrm{\Psi }}}_{\alpha }\left(\sigma \right):=inf\left\{k>0:\frac{k^{\frac{1}{\alpha -1}}}{∥g^{\prime }∥}{\int }_0^{\sigma k^{\frac{1}{1-\alpha }}}\tilde{\psi }\left(s^{\alpha -1}\right)ds\le 1\right\},\sigma \ge 0.$$

(9)

is Hölder-continuous function, i.e., ${\tilde{\mathrm{\Psi }}}_{\alpha }$ is well defined, increasing and continuous with ${\tilde{\mathrm{\Psi }}}_{\alpha }\left(0\right)=0$.

Our next step is to prove the following:

Lemma 5.

Fix $\alpha \in (0,1]$ and let $\mu \in {\mathbb{R}}^{+}$. For any Young function ψ with its Young complement $\tilde{\psi }$ satisfying (8), the integral ${\Im }_{a,g}^{\alpha ,\mu }x$ exists (is convergent) for any $x\in {\mathcal{H}}_0^{\psi }\left(E\right)$. Moreover, it remains true for every $x\in {\tilde{\mathcal{H}}}^{\psi }\left(E\right)$ provided the function ψ satisfies the additional property that ${lim}_{u\to \infty }\psi \left(u\right)/u\to \infty$.

Proof. 

Let us remark that ${\displaystyle {\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)=e^{-\mu g\left(t\right)}{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t\right)}x\left(t\right)\right],t\in [a,b]}$. Define $y:[a,b]\to {\mathbb{R}}^{+}$ by

$${\displaystyle y\left(s\right):=\left\{\begin{array}{cc}{(g\left(t\right)-g\left(s\right))}^{\alpha -1}{e}^{\mu g\left(s\right)}{g}^{\prime }\left(s\right),\hfill & s\in [a,t],t>a\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.}$$

Let us note that for any $t\in [a,b]$, the function ${\displaystyle f_t\left(\eta \right)}$ defined as $\eta -\frac{1}{∥g^{\prime }∥}{\int }_0^{\eta \left(g\right(t)-g(a\left)\right)}$ $\tilde{\psi }\left(s^{\alpha -1}\right)ds$, for some sufficiently big $\eta >0$ has a positive derivative ($\tilde{\psi }\left(\rho \right)\to 0$ as $\rho \to 0$). Consequently, for any $t\in [a,b]$, there exists a sufficiently large number $\eta >0$ such that $f_t\left(\eta \right)>0$, and hence for any $t\in [a,b]$ the following set,

$$\left\{k>0:\frac{1}{∥g^{\prime }∥}{\int }_0^{k^{\frac{1}{1-\alpha }}(g\left(t\right)-g\left(a\right))}\tilde{\psi }\left(s^{\alpha -1}\right)ds\le k^{\frac{1}{1-\alpha }}\right\},$$

(10)

is non-empty. This is in accordance with $\tilde{\psi }\left(\lambda u\right)\le \lambda \tilde{\psi }\left(u\right),\lambda \in (0,1]$ and with the following observations:

$$\begin{array}{ccc}\hfill {\int }_a^b\tilde{\psi }\left(\frac{\left|y\right(s\left)\right|}{e^{\mu ∥g∥}∥g^{\prime }∥k}\right)ds& =& {\int }_a^t\tilde{\psi }\left(e^{\mu g\left(s\right)}\frac{{(g\left(t\right)-g\left(s\right))}^{\alpha -1}{g}^{\prime }\left(s\right)}{e^{\mu ∥g∥}∥g^{\prime }∥k}\right)ds\hfill \\ & \le & {\int }_a^t\tilde{\psi }\left(\frac{{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}}{k}\right)\frac{g^{\prime }\left(s\right)ds}{∥g^{\prime }∥}\hfill \\ & \le & \frac{k^{\frac{1}{\alpha -1}}}{∥g^{\prime }∥}{\int }_0^{k^{\frac{1}{1-\alpha }}(g\left(t\right)-g\left(a\right))}\tilde{\psi }\left(s^{\alpha -1}\right)ds,\hfill \end{array}$$

occurs for any $k>0$. From which, in the light of (10), it follows that $y\in L_{\tilde{\psi }}\left([a,b]\right)$. Thus, by virtue of Proposition 2, ${\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t\right)}x\left(t\right)\right]$ (hence ${\Im }_{a,g}^{\alpha ,\mu }x$) exists for every $x\in {\tilde{\mathcal{H}}}^{\psi }\left(E\right))$. Moreover, the second assertion of our lemma follows directly from Proposition 1 (Part 2). □

We can now provide a proof of Theorem 2.

Proof of Theorem 2.

Let $a\le t_1\le t_2\le b$ and $x\in {\mathcal{H}}_0^{\psi }\left(E\right)$. According to Lemma 5, we ensure that ${\Im }_{a,g}^{\alpha ,0}{e}^{\mu g(·)}x(·)$ makes sense. Also, Remark 3 allows for the formulation of the following inequalities:

$$\begin{array}{ccc}& & \left|\phi \left({\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(t_2\right)-{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(t_1\right)\right)\right|\hfill \\ & =& \left|{\Im }_{a,g}^{\alpha ,0}\phi \left(\left[e^{\mu g(·)}x(·)\right]\right)\left(t_2\right)-{\Im }_{a,g}^{\alpha ,0}\phi \left(\left[e^{\mu g(·)}x(·)\right]\right)\left(t_1\right)\right|\hfill \\ & \le & \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\left({\int }_a^{t_1}{e}^{\mu g\left(s\right)}\left|{(g\left(t_2\right)-g\left(s\right))}^{\alpha -1}-{(g\left(t_1\right)-g\left(s\right))}^{\alpha -1}\right||g^{\prime }\left(s\right)\left|\right|\phi \left(x\left(s\right)\right)|ds\right.\hfill \\ & +& \left.{\int }_{t_1}^{t_2}{e}^{\mu g\left(s\right)}{(g\left(t_2\right)-g\left(s\right))}^{\alpha -1}{g}^{\prime }\left(s\right)\left|\phi \left(x\left(s\right)\right)\right|ds\right)\hfill \\ & =& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_a^b\left|h_1\left(s\right)+h_2\left(s\right)\right|\left|\phi \left(x\left(s\right)\right)\right|ds,\hfill \end{array}$$

where

$$h_1\left(s\right):=\left\{\begin{array}{cc}\left({(g\left(t_2\right)-g\left(s\right))}^{\alpha -1}-{(g\left(t_1\right)-g\left(s\right))}^{\alpha -1}\right)g^{\prime }\left(s\right)e^{\mu g\left(s\right)},\hfill & s\in [a,t_1],\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

and

$$h_2\left(s\right):=\left\{\begin{array}{cc}\left(g\left(t_2\right)-g\left(s\right){)}^{\alpha -1}\right)g^{\prime }\left(s\right)e^{\mu g\left(s\right)}\hfill & s\in [t_1,t_2],\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

We claim that $h_i\in L_{\tilde{\psi }}\left([a,b]\right)$, $(i=1,2)$. Having established our claim, in the light of the Hölder inequality in Orlicz spaces, we can conclude that

$$\begin{array}{ccc}\hfill \left|\phi \left({\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(t_2\right)-{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(t_1\right)\right)\right|& \le & \frac{2\left[{∥h_1∥}_{\tilde{\psi }}+{∥h_2∥}_{\tilde{\psi }}\right]}{\mathrm{\Gamma }\left(\alpha \right)}{∥\phi \left(x\right)∥}_{\psi }.\hfill \end{array}$$

(11)

It is left to prove our claim by demonstrating that $h_i\in L_{\tilde{\psi }}\left([a,b]\right),i=1,2$. To prove this, let us fix $k>0$. A suitable substitution, using the properties $\tilde{\psi }\left(\lambda u\right)\le \lambda \tilde{\psi }\left(u\right),\lambda \in (0,1]$ and $\tilde{\psi }(u-v)\le \tilde{\psi }\left(u\right)-\tilde{\psi }\left(v\right),v\le u$, lead to the following estimation:

$$\begin{array}{ccc}& & {\int }_a^b\tilde{\psi }\left(\frac{|h_1\left(s\right)|}{e^{\mu ∥g∥}∥g^{\prime }∥k}\right)ds\hfill \\ & =& {\int }_a^{t_1}\tilde{\psi }\left(e^{\mu g\left(s\right))}\frac{\left|{(g\left(t_2\right)-g\left(s\right))}^{\alpha -1}-{(g\left(t_1\right)-g\left(s\right))}^{\alpha -1}\right|}{e^{\mu ∥g∥}∥g^{\prime }∥k}{g}^{\prime }\left(s\right)\right)ds\hfill \\ \hfill & \le & {\int }_a^{t_1}\tilde{\psi }\left(\frac{\left|{(g\left(t_2\right)-g\left(s\right))}^{\alpha -1}-{(g\left(t_1\right)-g\left(s\right))}^{\alpha -1}\right|}{k}\right)\frac{g^{\prime }\left(s\right)}{∥g^{\prime }∥}ds\hfill \\ & \le & \frac{1}{∥g^{\prime }∥}{\int }_a^{t_1}\left[\tilde{\psi }\left(\frac{{(g\left(t_1\right)-g\left(s\right))}^{\alpha -1}}{k}\right)-\tilde{\psi }\left(\frac{{(g\left(t_2\right)-g\left(s\right))}^{\alpha -1}}{k}]\right)\right]g^{\prime }\left(s\right)ds\hfill \\ & \le & \frac{k^{\frac{1}{\alpha -1}}}{∥g^{\prime }∥}\left[{\int }_0^{k^{\frac{1}{1-\alpha }}(g\left(t_1\right)-g\left(a\right))}\tilde{\psi }\left(s^{\alpha -1}\right)ds-{\int }_{k^{\frac{1}{1-\alpha }}(g\left(t_2\right)-g\left(t_1\right))}^{k^{\frac{1}{1-\alpha }}(g\left(t_2\right)-g\left(a\right))}\tilde{\psi }\left(s^{\alpha -1}\right)ds\right]\hfill \\ & =& \frac{k^{\frac{1}{\alpha -1}}}{∥g^{\prime }∥}\left[{\int }_0^{k^{\frac{1}{1-\alpha }}(g\left(t_1\right)-g\left(a\right))}\tilde{\psi }\left(s^{\alpha -1}\right)ds-{\int }_0^{k^{\frac{1}{1-\alpha }}(g\left(t_2\right)-g\left(a\right))}\tilde{\psi }\left(s^{\alpha -1}\right)ds\right.\hfill \\ & +& \left.{\int }_0^{k^{\frac{1}{1-\alpha }}(g\left(t_2\right)-g\left(t_1\right))}\tilde{\psi }\left(s^{\alpha -1}\right)ds\right]\le \frac{k^{\frac{1}{\alpha -1}}}{∥g^{\prime }∥}{\int }_0^{k^{\frac{1}{1-\alpha }}(g\left(t_2\right)-g\left(t_1\right))}\tilde{\psi }\left(s^{\alpha -1}\right)ds.\hfill \end{array}$$

From which, in view of (10) together with the definition of the norm in the Orlicz space, we can deduce that $h_1\in L_{\tilde{\psi }}\left([a,b]\right)$ with the estimation ${∥h_1∥}_{\tilde{\psi }}\le e^{\mu ∥g∥}∥g^{\prime }∥K$, where

$$K:=inf\left\{k>0:\frac{1}{∥g^{\prime }∥}{\int }_0^{k^{\frac{1}{1-\alpha }}(g\left(t_2\right)-g\left(t_1\right))}\tilde{\psi }\left(s^{\alpha -1}\right)ds\le k^{\frac{1}{1-\alpha }}\right\}={\tilde{\mathrm{\Psi }}}_{\alpha }\left(\right|g\left(t_2\right)-g\left(t_1\right)\left|\right)$$

Arguing similarly as above, we can show that

$$h_2\in L_{\tilde{\psi }}([a,b],\mathrm{and}{∥h_2∥}_{\tilde{\psi }}\le e^{\mu ∥g∥}∥g^{\prime }∥{\tilde{\mathrm{\Psi }}}_{\alpha }\left(|g\left(t_2\right)-g\left(t_1\right)|\right).$$

Thus, for any $\phi \in E^{*}$, Equation (11) takes the form

$$\left|\phi \left({\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(t_2\right)-{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(t_1\right)\right)\right|\le \frac{4e^{\mu ∥g∥}∥g^{\prime }∥{\tilde{\mathrm{\Psi }}}_{\alpha }\left(|g\left(t_2\right)-g\left(t_1\right)|\right)}{\mathrm{\Gamma }\left(\alpha \right)}{∥\phi \left(x\right)∥}_{\psi }.$$

(12)

One can combine this with the Hahn–Banach theorem to ensure that

$$∥{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(t_2\right)-{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(t_1\right)∥\le \frac{4e^{\mu ∥g∥}∥g^{\prime }∥{\tilde{\mathrm{\Psi }}}_{\alpha }\left(|g\left(t_2\right)-g\left(t_1\right)|\right)}{\mathrm{\Gamma }\left(\alpha \right)}{∥\phi \left(x\right)∥}_{\psi },$$

holds true for some $\phi \in E^{*}$ with $∥\phi ∥=1$. Hence, in view of

$${\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)=e^{-\mu g\left(t\right)}{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t\right)}x\left(t\right)\right]$$

and our definition ${\Im }_{a,g}^{\alpha ,0}x\left(a\right):=0$, we observe that

$$\begin{array}{ccc}\hfill ∥{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(t\right)∥& =& ∥{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(t\right)-{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(a\right)∥\hfill \\ & \le & \frac{4e^{\mu ∥g∥}∥g^{\prime }∥{\tilde{\mathrm{\Psi }}}_{\alpha }\left(∥g∥\right)}{\mathrm{\Gamma }\left(\alpha \right)}{∥\phi \left(x\right)∥}_{\psi },\hfill \end{array}$$

Because of

$$\begin{array}{ccc}\hfill {\Im }_{a,g}^{\alpha ,\mu }x\left(t_2\right)& -& {\Im }_{a,g}^{\alpha ,\mu }x\left(t_1\right)\hfill \\ & =& e^{-\mu g\left(t_2\right)}\left({\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t_2\right)}x\left(t_2\right)\right]-{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t_1\right)}x\left(t_1\right)\right]\right)\hfill \\ & +& {\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t_1\right)}x\left(t_1\right)\right]\left(e^{-\mu g\left(t_2\right)}-e^{-\mu g\left(t_1\right)}\right),\hfill \end{array}$$

(13)

it follows

$$\begin{array}{ccc}\hfill ∥{\Im }_{a,g}^{\alpha ,\mu }x\left(t_2\right)-{\Im }_{a,g}^{\alpha ,\mu }x\left(t_1\right)∥& =& e^{-\mu g\left(a\right)}∥{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(t_2\right)-{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(t_1\right)∥\hfill \\ & +& ∥{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}x(·)\right]\left(t_1\right)∥\left|e^{-\mu g\left(t_2\right)}-e^{-\mu g\left(t_1\right)}\right|,\hfill \end{array}$$

so ${\Im }_{a,g}^{\alpha ,\mu }:{\mathcal{H}}_0^{\psi }\left(E\right)\to C[I,E_w]$. Moreover,

$$∥{\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)∥=∥{\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)-{\Im }_{a,g}^{\alpha ,\mu }x\left(a\right)∥\le \frac{4e^{\mu (∥g∥-g\left(a\right))}∥g^{\prime }∥{\tilde{\mathrm{\Psi }}}_{\alpha }\left(∥g∥\right)}{\mathrm{\Gamma }\left(\alpha \right)}{∥\phi \left(x\right)∥}_{\psi }.$$

(14)

In this connection, a particular case follows from Corollary 1 and the theorem is then proved. □

Remark 7.

We should remark that, if ${\Im }_{a,g}^{\alpha ,\mu }x$ not exists for some $x\in {\mathcal{H}}_0^{\psi }\left(E\right)$, then it cannot exists if we “enlarge” the space E into F. To see this, we argue by contradiction assuming that ${\Im }_{a,g}^{\alpha ,\mu }x$ (when we consider x as a function from ${\mathcal{H}}_0^{\psi }\left(F\right)$) exists. In this case, for the particular choice for the functional $\phi \in F^{*}$ having ${\phi |}_E=\theta$ we conclude, in view of (5) and $x\left(I\right)\subseteq E$, that $\phi \left({\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)\right)={\Im }_{a,g}^{\alpha ,\mu }\phi \left(x\left(t\right)\right)=0$. From which ${\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)\in E$. It would lead to a contradiction.

Example 2.

Let $\alpha \in (0,1)$ and $\psi \left(u\right)={\psi }_p\left(u\right):=\frac{1}{p}{\left|u\right|}^p,p\in (1,\infty )$. In such a case, we have ${\tilde{\psi }}_p={\psi }_{\tilde{p}}$ with $\frac{1}{p}+\frac{1}{\tilde{p}}=1$. It is easy to see that (8) is true if and only if $p>\frac{1}{\alpha }$. Hence, we conclude that ${\Im }_{a,g}^{\alpha ,\mu }$ maps the Bochner space $L_p[[a,b],E],p>\frac{1}{\alpha }$ into the space $C[[a,b],E_w]$. For example, in view of the above observation, ${\Im }_{a,g}^{\alpha ,\mu }:L_2[[a,b],\mathbb{R}]\to C[[a,b],\mathbb{R}]$ for any $\alpha \in (0.5,1)$ and $\mu \in {\mathbb{R}}^{+}$.

Remark 8.

Theorem 2 can be combined with ([29], Example 1) to ensure the existence of a Young function ψ (for example $\psi \left(u\right):=e^{\left|u\right|}-\left|u\right|-1$), for which the operator ${\Im }_{a,g}^{\alpha ,\mu }$ maps ${\mathcal{H}}_0^{\psi }\left(E\right)$ into the space $C[[a,b],E_w]$ “for all” $\alpha \in (0,1],\mu \in {\mathbb{R}}^{+}$. According to Example 2, this interesting phenomenon has no counterpart in the case of the Lebesgue spaces $L_p[[a,b],\mathbb{R}]$.

Lemma 6.

Let $\alpha \in (0,1)$. If $x\in {\mathcal{H}}_0^{\psi }\left(E\right)$, where ψ is a Young function with its complement $\tilde{\psi }$ satisfying

$$\begin{array}{c}\hfill {\int }_0^t\tilde{\psi }\left(s^{\alpha -1}\right)ds<\infty ,t>0,\end{array}$$

(15)

then

$${\Im }_{a,g}^{1+\alpha ,\mu }x\left(t\right)={\Im }_{a,g}^{1,\mu }{\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)={\Im }_{a,g}^{\alpha ,\mu }{\Im }_{a,g}^{1,\mu }x\left(t\right),onI.$$

(16)

is true for every $x\in {\mathcal{H}}_0^{\psi }\left(E\right)$. In particular, the property (16) is true for every function $x\in C[I,E_w]$.

Proof. 

Let $x\in {\mathcal{H}}_0^{\psi }\left(E\right)$. Lemma 5 may be combined with Remark 6 in order to assure the existence of ${\Im }_{a,g}^{1+\alpha ,\mu }x$ and ${\Im }_{a,g}^{\alpha ,\mu }x$. Also by Theorem 2, we know that ${\Im }_{a,g}^{\alpha ,\mu }x\in C[I,E_w]$. From which we conclude the existence of ${\Im }_{a,g}^{1,\mu }{\Im }_{a,g}^{\alpha ,\mu }x$. Now, by the semigroup property proved in Fact 2, we have

$$\phi \left({\Im }_{a,g}^{1+\alpha ,\mu }x\right)={\Im }_{a,g}^{1+\alpha ,\mu }\phi \left(x\right)={\Im }_{a,g}^{1,\mu }{\Im }_{a,g}^{\alpha ,\mu }\phi \left(x\right)={\Im }_{a,g}^{1,\mu }\phi \left({\Im }_{a,g}^{\alpha ,\mu }x\right)=\phi \left({\Im }_{a,g}^{1,\mu }{\Im }_{a,g}^{\alpha ,\mu }x\right),$$

for any $\phi \in E^{*}$. It means that

$$\phi \left({\Im }_{a,g}^{1+\alpha ,\mu }x-{\Im }_{a,g}^{1,\mu }{\Im }_{a,g}^{\alpha ,\mu }x\right)=0,\mathrm{for} \mathrm{every}\phi \in E^{*}.$$

Hence, ${\Im }_{a,g}^{1+\alpha ,\mu }x={\Im }_{a,g}^{1,\mu }{\Im }_{a,g}^{\alpha ,\mu }x.$ Similarly, it is not hard to show that ${\Im }_{a,g}^{\alpha ,\mu }{\Im }_{a,g}^{1,\mu }x={\Im }_{a,g}^{1+\alpha ,\mu }x.$ □

Consequently, in view of the semigroup property proved in Fact 2, an analogous reasoning as in ([40], Lemma 5) (see also [29], Lemma 2) gives us the following:

Lemma 7.

Let $\alpha ,\beta \in (0,1)$ be fixed. If $x\in {\mathcal{H}}_0^{\psi }\left(E\right)$, where ψ is a Young function with its complementary function $\tilde{\psi }$, which satisfies the condition

$$\begin{array}{c}\hfill {\int }_0^t\tilde{\psi }\left(s^{-\nu }\right)ds<\infty ,t>0,where\nu :=max\{1-\alpha ,1-\beta \},\end{array}$$

(17)

then

$${\Im }_{a,g}^{\beta ,\mu }{\Im }_{a,g}^{\alpha ,\mu }x={\Im }_{a,g}^{\alpha +\beta ,\mu }x={\Im }_{a,g}^{\alpha ,\mu }{\Im }_{a,g}^{\beta ,\mu }xonI.$$

(18)

In particular, the property (18) holds true for every $x\in C[I,E_w].$

4. Generalized $\mathit{g}-$Hilfer Fractional Differential Operators

From now, the definitions of the $g-$fractional derivatives of x becomes a natural requirement. So, we include also the definitions of the generalized fractional derivatives. The most typical of these is as follows.

From this point onwards, definitions of g-fractional derivatives of x become a natural requirement. We therefore also include definitions of generalized fractional derivatives. The most typical of these is the following:

Definition 5.

The generalized $g-$Riemann-Liouville fractional-pseudo- (resp. weak) derivative of order $\alpha \in (m,m+1)$, $m\in \mathbb{N}:=\{0,1,2,\cdots \},$ with parameter $\mu \in {\mathbb{R}}^{+}$ applied to the function $x\in \mathbb{P}[I,E]$ is defined as

$${\mathfrak{D}}_{a,g,p}^{\alpha ,\mu }x:={\delta }_p^{m+1}{\Im }_{a,g}^{m+1-\alpha ,\mu }x,\left(\mathit{resp}.{\mathfrak{D}}_{a,g,\mathrm{\omega }}^{\alpha ,\mu }x:={\delta }_{\mathrm{\omega }}^{m+1}{\Im }_{a,g}^{m+1-\alpha ,\mu }x\right),t\in I.$$

(19)

Here, ${\delta }_p$ and ${\delta }_{\mathrm{\omega }}$ are defined as follows:

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

Let us characterize this operator as acting on Orlicz spaces:

Lemma 8.

Let $\alpha ,\beta \in (0,1)$. Assume that the space ${\mathcal{H}}_0^{\psi }\left(E\right)$ is generated by the Young function ψ with its complementary Young function $\tilde{\psi }$ satisfying

$$\begin{array}{c}\hfill {\int }_0^t\tilde{\psi }\left(s^{-\nu }\right)ds<\infty ,t\in [a,b],\nu =1-\alpha +\beta .\end{array}$$

(20)

Then, we have ${\mathfrak{D}}_{a,g,p}^{0,\mu }x=x$ and

$${\mathfrak{D}}_{a,g,p}^{\beta ,\mu }{\Im }_{a,g}^{\alpha ,\mu }x=\left\{\begin{array}{c}{\displaystyle {\mathfrak{D}}_{a,g,p}^{\beta -\alpha ,\mu }x,a.e.,\beta \ge \alpha }\hfill \\ {\displaystyle {\Im }_{a,g}^{\alpha -\beta ,\mu },\beta <\alpha .}\hfill \end{array}\right.$$

(21)

Specifically, when $\alpha =\beta$, the formula (21) means that the operator ${\mathfrak{D}}_{a,g,p}^{\alpha ,\mu }{\Im }_{a,g}^{\alpha ,\mu }$ is defined on the space ${\mathcal{H}}_0^{\psi }\left(E\right)$ and that ${\mathfrak{D}}_{a,g,p}^{\alpha ,\mu }$ is the left-inverse of ${\Im }_{a,g}^{\alpha ,\mu }$.

Proof. 

At the beginning, let us recall (cf. [29], Proposition 1) that for any Young function $\tilde{\psi }$ we have

$${\int }_0^t\tilde{\psi }\left(s^{-{\nu }_1}\right)ds<\infty ,\mathrm{with}{\nu }_1>0,⟹{\int }_0^t\tilde{\psi }\left(s^{-{\nu }_2}\right)ds<\infty ,\mathrm{for}\mathrm{any}{\nu }_2\in (-\infty ,{\nu }_1).$$

(22)

Now, let $x\in {\mathcal{H}}_0^{\psi }\left(E\right)$. The first claim, i.e., ${\delta }_p{\Im }_{a,g}^{1,\mu }x$, follows from $(◊)$. Also, in view of Theorem 2, ${\Im }_{a,g}^{\alpha ,\mu }x$ exists and is a weakly continuous function defined on I. By noting that $max\{1-\alpha ,\beta \}\le \nu$, it follows by Lemmas 6 and 7 in view of (22) and (20) that

$${\mathfrak{D}}_{a,g,p}^{\beta ,\mu }{\Im }_{a,g}^{\alpha ,\mu }x={\delta }_p{\Im }_{a,g}^{1-\beta ,\mu }{\Im }_{a,g}^{\alpha ,\mu }x={\delta }_p{\Im }_{a,g}^{1-(\beta -\alpha ),\mu }x=\left\{\begin{array}{c}{\displaystyle {\mathfrak{D}}_{a,g,p}^{\beta -\alpha ,\mu }x,a.e.,\beta >\alpha }\hfill \\ {\displaystyle {\delta }_p{\Im }_{a,g}^{1,\mu }{\Im }_{a,g}^{\alpha -\beta ,\mu }={\Im }_{a,g}^{\alpha -\beta ,\mu }x,\beta \le \alpha .}\hfill \end{array}\right.$$

When $\alpha =\beta$, the result follows from the first claim. □

Besides the $g-$Riemann-Liouville fractional derivatives the g-Caputo fractional derivatives are also of special interest:

Definition 6.

The g-Caputo fractional pseudo (resp. weak) derivative of order $\alpha \in (m,m+1)$, $m\in \mathbb{N},$ with parameter $\mu \in {\mathbb{R}}^{+}$ applied to the function $x\in \mathbb{P}[I,E]$ is defined as

$$\frac{d_{a,g,p}^{\alpha ,\mu }}{d{t}^{\alpha }}x:={\Im }_{a,g}^{m+1-\alpha ,\mu }{\delta }_p^{m+1}x,\left(\mathit{resp}.\frac{d_{a,g,\mathrm{\omega }}^{\alpha ,\mu }}{d{t}^{\alpha }}x:={\Im }_{a,g}^{m+1-\alpha ,\mu }{\delta }_{\mathrm{\omega }}^{m+1}x\right).$$

(23)

Alongside the fractional derivative in the Riemann–Liouville sense and the Caputo sense, we introduce the so-called $g-$ Hilfer fractional derivative [13,14,15], which unifies both derivatives.

Definition 7.

The g-Hilfer fractional pseudo- (resp. weak) derivative of order $\alpha \in (m,m+1)$, $m\in \mathbb{N},$ with parameter $\mu \in {\mathbb{R}}^{+}$ and type $\beta \in [0,1]$ applied to the function $x\in \mathbb{P}[I,E]$ is defined as

$$\begin{array}{ccc}\hfill {}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }x& :=& {\Im }_{a,g}^{\beta (m+1-\alpha ),\mu }{\delta }_p^{m+1}{\Im }_{a,g}^{(1-\beta )(m+1-\alpha ),\mu }x,\hfill \\ & & \left(\mathit{resp}.{}^H{\mathbb{D}}_{a,g,\mathrm{\omega }}^{\alpha ,\beta ,\mu }x:={\Im }_{a,g}^{\beta (m+1-\alpha ),\mu }{\delta }_{\mathrm{\omega }}^{m+1}{\Im }_{a,g}^{(1-\beta )(m+1-\alpha ),\mu }x\right).\hfill \end{array}$$

(24)

Recall that the g-Hilfer fractional derivative interpolates the Riemann–Liouville fractional derivative and the Caputo fractional derivative. Indeed, the g-Hilfer fractional derivative of type $\beta \to 0$ (resp. $\beta \to 1$) is in fact the g-Riemann–Liouville (resp. Caputo) fractional derivative.

Remark 9.

Let us remark that, ${\rho }^{\alpha }{}^H{\mathbb{D}}_{a,t^{\rho },p}^{\alpha ,0,0}$ is the Katugampola fractional derivative, ${}^H{\mathbb{D}}_{0,ln(1+t),p}^{\alpha ,0,0}$ is the Hadamard fractional derivative and ${\rho }^{\alpha }{}^H{\mathbb{D}}_{a,t^{\rho },p}^{\alpha ,1,0}$ is the Caputo–Katugampola fractional derivative, ${}^H{\mathbb{D}}_{a,lnt,p}^{\alpha ,1,0}$ is the Caputo–Hadamard fractional derivative. From the above particular choice of the function g and a, we obtain other possible variations of fractional pseudo- and weak derivatives in the Pettis space.

Through direct verification using (7), there is no difficulty in demonstrating that

$${}_{}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }\left\{e^{-\mu g\left(t\right)}{\left(g\left(t\right)-g\left(a\right)\right)}^{\eta -1}\right\}=\frac{\mathrm{\Gamma }\left(\eta \right)}{\mathrm{\Gamma }(\eta -\alpha )}{e}^{-\mu g\left(t\right)}{\left(g\left(t\right)-g\left(a\right)\right)}^{\eta -\alpha -1},t>a,\eta >0.$$

In particular,

$${}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }\left\{e^{-\mu g\left(t\right)}{\left(g\left(t\right)-g\left(a\right)\right)}^{\alpha }\right\}=\frac{\mathrm{\Gamma }(1+\alpha )}{\mathrm{\Gamma }(1+\alpha -\alpha )}{e}^{-\mu g\left(t\right)}{\left(g\left(t\right)-g\left(a\right)\right)}^0=\mathrm{\Gamma }(1+\alpha )e^{-\mu g\left(t\right)}.$$

Also,

$${}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }\left\{e^{-\mu g\left(t\right)}{\left(g\left(t\right)-g\left(a\right)\right)}^{\gamma -1}\right\}=0,\mathrm{with}\gamma =\alpha +\beta (m+1-\alpha ).$$

Remark 10.

Evidently, unless the space E has total dual $E^{*}$ (cf. [34]), the g-Hilfer pseudo-derivative ${}^H{\mathbb{D}}_{a,g,p}^{\alpha ,0,\mu }x$ of the Pettis-integrable function $x\in \mathbb{P}[I,E]$ (if it exists) need not be uniquely determined.

Also, in virtue of Lemma 3, in view of the fact that the pseudo-derivatives of the pseudo-differentiable function are weakly equivalent, there is no difficulty to conclude that ${}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }x$ with $\beta \in (0,1]$ (if exists), is necessarily uniquely determined, even if E does not have a total dual.

The relationship of the g-Hilfer fractional derivative to the g-Riemann–Liouville fractional derivative is well-known and easy to see:

$${}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }x={\Im }_{a,g}^{\gamma -\alpha ,\mu }{\mathfrak{D}}_{a,g,p}^{\gamma ,\mu }x,\left({}^H{\mathbb{D}}_{a,g,\mathrm{\omega }}^{\alpha ,\mu }x={\Im }_{a,g}^{\gamma -\alpha ,\mu }{\mathfrak{D}}_{a,g,\mathrm{\omega }}^{\gamma ,\mu }\right)x,\mathrm{where}\gamma =\alpha +\beta (m+1-\alpha ).$$

(25)

Also, the connection of the g-Hilfer fractional derivative with the g-Caputo fractional derivative is known:

$${}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }x=\frac{d_{a,g,p}^{\gamma ,\mu }}{d{t}^{\alpha }}{\Im }_{a,g}^{\gamma -\alpha ,\mu }x,\left({}^H{\mathbb{D}}_{a,g,\mathrm{\omega }}^{\alpha ,\beta ,\mu }x=\frac{d_{a,g,\mathrm{\omega }}^{\gamma ,\mu }}{d{t}^{\alpha }}{\Im }_{a,g}^{\gamma -\alpha ,\mu }x\right),$$

(26)

where $\gamma =m+1-\beta (m+1-\alpha )>\alpha$. With appropriate assumptions imposed on the function $x\in \mathbb{P}[I,E]$, using the semigroup property of ${\Im }_{a,g}^{\alpha ,\beta ,\mu }$ (cf. Lemma 7), we obtain

$${\Im }_{a,g}^{\alpha ,\mu }{}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }x={\Im }_{a,g}^{\gamma ,\mu }{\mathfrak{D}}_{a,g,p}^{\gamma ,\mu }x,\mathrm{with}\gamma =\alpha +\beta (m+1-\alpha ).$$

(27)

An analogous result for the converse composition is the following:

$${}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }{\Im }_{a,g}^{\alpha ,\mu }x=\frac{d_{a,g,p}^{\gamma ,\mu }}{d{t}^{\alpha }}{\Im }_{a,g}^{\gamma ,\mu }x,\mathrm{with}\gamma =m+1-\beta (m+1-\alpha ).$$

(28)

Remark 11.

Note that one of the important topics in fractional calculus is to establish the equivalence or implication between linear (or homogeneous) fractional differential equations and the corresponding integral equations. In this context, we may point out that even in Hölder spaces, but outside the space of absolutely continuous functions, the operator ${}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }$ does not enjoy (in general) the behavior of being the left inverse operator of ${}^I{m}^{{\alpha ,\mu }_{a,g}}$. In other words, outside the space of absolutely continuous functions, the equivalence of the fractional generalized integral equations and the corresponding fractional differential g-Hilfer problem fails even in Hölder spaces. Actually, in the following we will show that for real-valued Hölder continuous functions the inverse implication from the fractional integral equations to the corresponding g-Hilfer differential form is no longer necessarily true.

To see this, let us now consider a particular fractional differential operator ${}^H{\mathbb{D}}_{0,t,p}^{\alpha ,1,0}={\Im }_{0,t}^{m+1-\alpha ,0}{\delta }_p^{m+1}$ for some $\alpha \in (m,m+1)$ where $t\in [0,1]$, $E=\mathbb{R}$. Let y be Hölder continuous (but nevertheless nowhere differentiable on $[0,1]$) function of some critical order $\lambda <1$. According to ([51], Theorem 13.13), we know that there is $\gamma :=\alpha -m\in (0,1)$ depending only on λ and a Hölder continuous function $x\notin AC[0,1]$ such that ${\Im }_{0,t}^{\gamma ,0}x=y$. From which we can deduce that ${\delta }_p^m{\Im }_{0,t}^{\alpha ,0}x={\delta }_p^m{\Im }_{0,t}^{m+\gamma ,0}x={\Im }_{0,t}^{\gamma ,0}x=y$. This gives reason to believe that ${}^H{\mathbb{D}}_{0,t,p}^{\alpha ,1,0}{\Im }_{0,t}^{\alpha ,0}x={\Im }_{0,t}^{m+1-\alpha ,0}{\delta }_{p}y$ is “meaningless”. This implies that even on Hölder spaces (but outside of the space of absolutely continuous functions), the operator ${}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }$ is not (in general) the left inverse of ${\Im }_{a,g}^{\alpha ,\mu }$ as required. Consequently, we conclude that the results in, e.g., [2,7,9,10,11,12] are false.

Nevertheless, the following example illustrates that on the space $C[I,E_w]$, but still outside of the space of weakly absolutely continuous functions, it is no more necessarily true that ${}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }$ is a left inverse operator for ${\Im }_{a,g}^{\alpha ,\mu }$, for any $\alpha >0$ and $\beta \in (0,1]$.

Counterexample 1.

Let $\alpha \in (m,m+1],m\in \mathbb{N}$, $\beta \in (0,1]$ and $\gamma =m+1-\beta (m+1-\alpha )\in (m,m+1)$. Define $x:[a,b]\to L_1[a,b]$ by

$$x\left(t\right):=\frac{e^{-\mu g\left(t\right)}}{\mathrm{\Gamma }(m+1-\alpha )}{(g\left(t\right)-g(·))}^{m-\gamma }{\chi }_{[a,t]}(·),t\in [a,b].$$

Let $\phi \in L_1^{*}$ and $\varphi \in L_{\infty }\cong L_1^{*}$ be the corresponding to φ and note that $m-\gamma \ne -1$, so

$$\begin{array}{ccc}\hfill \phi \left(x\right(t\left)\right)& =& {\int }_a^b\varphi \left(s\right)\frac{e^{-\mu g\left(t\right)}}{\mathrm{\Gamma }(m+1-\gamma )}{(g\left(t\right)-g\left(s\right))}^{m-\gamma }{\chi }_{[a,t]}\left(s\right)ds\hfill \\ & =& {\int }_a^t\varphi \left(s\right)\frac{e^{-\mu \left(g\right(t)-g(s\left)\right)}}{\mathrm{\Gamma }(m+1-\gamma )}{(g\left(t\right)-g\left(s\right))}^{m-\gamma }\frac{g^{\prime }\left(s\right)e^{-\mu g\left(s\right)}}{g^{\prime }\left(s\right)}ds\hfill \\ & =& {\Im }_{a,g}^{m+1-\gamma ,\mu }\left(\frac{\varphi \left(t\right)}{g^{\prime }\left(t\right)}{e}^{-\mu g\left(t\right)}\right)\in C[[a,b],\mathbb{R}],\left(byFact1\right).\hfill \end{array}$$

From which, we conclude that x is weakly continuous (but not wAC) on $[a,b]$. Consequently, in view of Theorem 2, we know that ${\Im }_{a,g}^{\gamma -m,\mu }x$ exists on $[a,b]$. In this context, we can show that

$${\Im }_{a,g}^{\gamma -m,\mu }x\left(t\right)(·)=e^{-\mu g\left(t\right)}{\chi }_{[a,t]}(·),holdsforany\gamma \in (m,m+1).$$

(29)

We do this by permitting $\varphi \in L_{\infty }$ to correspond to $\phi \in L_1^{*}$ and performing necessary calculations using Fact 2 as follows:

$$\begin{array}{ccc}& & {\int }_a^t\frac{\phi \left(e^{-\mu \left(g\right(t)-g(s\left)\right)}{[g\left(t\right)-g\left(s\right)]}^{\gamma -m-1}{g}^{\prime }\left(s\right)x\left(s\right)\right)}{\mathrm{\Gamma }(\gamma -m)}ds\hfill \\ & =& {\Im }_{a,g}^{\gamma -m,\mu }\phi \left(x\left(t\right)\right)={\Im }_{a,g}^{\gamma -m,\mu }\left[{\Im }_{a,g}^{m+1-\gamma ,\mu }\left(\frac{\varphi \left(t\right)}{g^{\prime }\left(t\right)}{e}^{-\mu g\left(t\right)}\right)\right]\hfill \\ & =& {\Im }_{a,g}^{1,\mu }\left(\frac{\varphi \left(t\right)}{g^{\prime }\left(t\right)}{e}^{-\mu g\left(t\right)}\right)={\int }_a^t{e}^{-\mu g\left(t\right)}\varphi \left(s\right)ds\hfill \\ & =& {\int }_a^b{e}^{-\mu g\left(t\right)}\varphi \left(s\right){\chi }_{[a,t]}\left(s\right)ds=\phi \left(e^{-\mu g\left(t\right)}{\chi }_{[a,t]}\right).\hfill \end{array}$$

That is, for any $\phi \in L_1^{*}$, we have

$${\int }_a^t\phi \left(\frac{e^{-\mu \left(g\right(t)-g(s\left)\right)}{[g\left(t\right)-g\left(s\right)]}^{\gamma -m-1}{g}^{\prime }\left(s\right)x\left(s\right)}{\mathrm{\Gamma }(\gamma -m)}\right)ds=\phi \left(e^{-\mu g\left(t\right)}{\chi }_{[a,t]}\right),$$

as needed for (29). Accordingly, in view of the weak continuity of x, Theorem 2 may be combined with Lemma 7 in order to assure that

$${\Im }_{a,g}^{\gamma ,\mu }x\left(t\right)={\Im }_{a,g}^{m,\mu }{\Im }_{a,g}^{\gamma -m,\mu }x\left(t\right)={\Im }_{a,g}^{m,\mu }{e}^{-\mu g\left(t\right)}{\chi }_{[a,t]}.$$

From (◊), it follows that ${\delta }_p^m{\Im }_{a,g}^{\gamma ,\mu }x=e^{-\mu g\left(t\right)}{\chi }_{[a,t]}$.

Next, we claim that the function $f\left(t\right):=e^{-\mu g\left(t\right)}{\chi }_{[a,t]}(·),t\in [a,b]$ is wAC but with no pseudo- (and therefore no weak) derivatives on $[a,b]$. Having established our claim, we can conclude that the derivative ${\delta }_p^{m+1}{\Im }_{a,g}^{\gamma ,\mu }x={\delta }_{p}f\left(t\right)$ is “meaningless”. More specifically,

$${}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }{\Im }_{a,g}^{\alpha ,\mu }x=\frac{d_p^{\gamma ,g}}{d{t}^{\gamma }}{\Im }_{a,g}^{\gamma ,\mu }x\ne x.\beta \ne 0,$$

according to our requirements.

It remains to demonstrate our claim. But the weak absolute continuity of f on $[a,b]$ is obvious because

$$\phi \left(e^{-\mu g\left(t\right)}{\chi }_{[a,t]}\right)={\int }_a^b{e}^{-\mu g\left(t\right)}{\chi }_{[a,t]}\left(s\right)\varphi \left(s\right)ds=e^{-\mu g\left(t\right)}{\int }_a^t\varphi \left(s\right)ds,t\in [a,b],$$

is satisfied for any $\phi \in L_1^{*}[a,b]$. Now, we proceed by contradiction in order to show that

$${\delta }_p\left(e^{-\mu g\left(t\right)}{\chi }_{[a,t]}\right)=\left(\frac{1}{g^{\prime }\left(t\right)}{\mathfrak{D}}_p+\mu \right)e^{-\mu g\left(t\right)}{\chi }_{[a,t]}=\frac{1}{g^{\prime }\left(t\right)}{\mathfrak{D}}_p{e}^{-\mu g\left(t\right)}{\chi }_{[a,t]}+\mu e^{-\mu g\left(t\right)}{\chi }_{[a,t]},$$

does not exist on $[a,b]$. To see this, it is suffices to show that there does not exist a function $y:[a,b]\to L_1[a,b]$ with the property that

$${\left[\phi f\right]}^{\prime }\left(t\right)=\phi y\left(t\right),for almost everyt\in [a,b],wheref\left(t\right):=e^{-\mu g\left(t\right)}{\chi }_{[a,t]},$$

(30)

holds for every $\phi \in L_1^{*}[a,b]$. For this goal, we construct a set of elements from $L_1^{*}[a,b]$ verifying (30) to obtain a contradiction. Pick a sequence of measurable functions $\{{\varphi }_n\}\subset L_{\infty },n\in \mathbb{N}$ converges in measure to the null function on $[a,b]$, but satisfies ${\varphi }_n\neg \to 0$ (for instance, pick $\{{\varphi }_n\}$ to be the well-known typewriter sequence which indicate functions of intervals of decreasing length, marching across $[a,b]$ over and over again). If y satisfying (30) would exist, there would in particular exist a null set $\mathfrak{N}\subseteq [a,b]$ such that

$${\left[{\phi }_{n}f\right]}^{\prime }\left(t\right)={\phi }_{n}y\left(t\right),forallt\in [a,b]/\mathfrak{N},n\in \mathbb{N}.$$

(31)

To lead (31) to a contradiction, we use that by the definitions of ${\phi }_n$ and f we have

$${\phi }_{n}f\left(t\right)={\int }_a^b{\varphi }_n\left(s\right)e^{-\mu g\left(s\right)}{\chi }_{[a,t]}\left(s\right)ds={\int }_a^t{\varphi }_n\left(s\right)e^{-\mu g\left(s\right)}ds.$$

From which it follows by direct differentiation that ${\left[{\phi }_{n}f\right]}^{\prime }\left(t\right)={\varphi }_n\left(t\right)e^{-\mu g\left(t\right)}$ holds a.e. in $[a,b].$ Thus, in view of (31) we conclude the existence of null set $\mathfrak{N}\subseteq [a,b]$ such that

$${\int }_a^b{\varphi }_n\left(s\right)y\left(s\right)ds={\varphi }_n\left(t\right)e^{-\mu g\left(t\right)},t\in [a,b]/\mathfrak{N},n\in \mathbb{N}.$$

(32)

Since the right-hand side of (32) does not converge to 0 as $n\to \infty$ for every fixed $t\in [a,b]$, while the left-hand side does by the Vital’s convergence theorem, we obtain the desired contradiction, which is what we wished to show.

In order to avoid such an equivalence problem with the g-Hilfer boundary value problem of fractional order $\alpha >1$ and the corresponding integral form, we will modify (slightly) our definition of the g-Hilfer fractional differential operator to a more appropriate form.

Definition 8.

The modified g-Hilfer fractional pseudo (resp. weak) derivative “in brief  MHFPD  (resp.  MHFWD )” of order $m+\alpha$, $m\in \mathbb{N},\alpha \in (0,1)$ with parameter $\mu \in {\mathbb{R}}^{+}$ and type $\beta \in [0,1]$ applied to the Pettis integrable function $x\in P[I,E]$ is defined as

$$\begin{array}{ccc}\hfill {}^H{\tilde{\mathbb{D}}}_{a,g,p}^{m+\alpha ,\beta ,\mu }x& :=& {\delta }_p^m{}^H{\mathbb{D}}_{a,g,p}^{\alpha ,\beta ,\mu }x={\delta }_p^m\left({\Im }_a^{\beta (1-\alpha ),g}{\delta }_p{\Im }_{a,g}^{(1-\beta )(1-\alpha ),\mu }\right)x={\delta }_p^m\left({\Im }_{a,g}^{\gamma -\alpha ,\mu }{\mathfrak{D}}_{a,g,p}^{\gamma ,\mu }\right)x,\hfill \\ \hfill {}^H{\tilde{\mathbb{D}}}_{a,g,\mathrm{\omega }}^{m+\alpha ,\beta ,\mu }x& :=& {\delta }_p^m{}^H{\mathbb{D}}_{a,g,\mathrm{\omega }}^{\alpha ,\beta ,\mu }x={\delta }_p^m\left({\Im }_{a,g}^{\beta (1-\alpha ),\mu }{\delta }_{\mathrm{\omega }}{\Im }_{a,g}^{(1-\beta )(1-\alpha ),\mu }\right)x={\delta }_{\mathrm{\omega }}^m\left({\Im }_{a,g}^{\gamma -\alpha ,\mu }{\mathfrak{D}}_{a,g,\mathrm{\omega }}^{\gamma ,\mu }\right)x,\hfill \end{array}$$

where $\gamma =\alpha +\beta (1-\alpha ).$

Obviously, unless the space E has its total dual $E^{*}$ (see [34]), the g-Hilfer fractional pseudo-derivative (in the sense of the above Definition of x need not be uniquely determined.

The upcoming result will be a cornerstone of the solution of the boundary value problem (1). However, before embarking into the next theorem, in what follows we assume (without loss of generality) that $g\left(a\right)=0$.

Theorem 3.

Let $\alpha \in (1,2)$, $\beta \in [0,1]$, $\mu \in [0,\infty )$, ${\delta }_i>0$ and ${\xi }_i\in [a,b]$. Assume that ψ is a Young function such that its complementary Young function $\tilde{\psi }$ satisfies

$${\int }_0^t\tilde{\psi }\left(s^{-\nu }\right)ds<\infty ,t>0,\nu :=\underset{i}{max}\{2-{\delta }_i-\alpha ,\alpha -\gamma ,1+\gamma -\alpha \},$$

(33)

where $\gamma =\alpha -1+\beta (2-\alpha )$. Assume that $f:I\times E\to E$, (where E has total dual) is a function such that $f(·,x(·))\in {\mathcal{H}}_0^{\psi }\left(E\right)$ for any $x\in C[[a,b],E_w]$. If ${\eta }_i\in \mathbb{R},(i=1,2,\cdots ,n-2)$ are constants such that

$$\mathrm{\Lambda }:=\frac{e^{-\mu ∥g∥}{∥g∥}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}-\sum _{i=1}^{n-2}\frac{{\eta }_i{e}^{-\mu g\left({\xi }_i\right)}{\left(g\left({\xi }_i\right)\right)}^{\alpha +{\delta }_i-1}}{\mathrm{\Gamma }(\alpha +{\delta }_i)}\ne 0,$$

then $x\in C[I,E_w]$ solves the problem (1) if and only if x satisfies the following integral equation:

$$x\left(t\right)=\frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{c}_0+\lambda {\Im }_{a,g}^{\alpha ,\mu }f(t,x\left(t\right)),$$

(34)

where

$$c_0=\frac{\lambda }{\mathrm{\Lambda }}\left[\sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{\alpha +{\delta }_i,\mu }f({\xi }_i,x\left({\xi }_i\right))-{\Im }_{a,g}^{\alpha ,\mu }f(b,x\left(b\right))\right].$$

Proof. 

Let $x\in C[I,E_w]$ satisfy the problem (1). Then, we formally have

$${\delta }_p{}^H{\mathbb{D}}_{a,g,p}^{\alpha -1,\beta ,\mu }x\left(t\right)=\lambda f(t,x\left(t\right))\Rightarrow {}^H{\mathbb{D}}_{a,g,p}^{\alpha -1,\beta ,\mu }x\left(t\right)=\lambda {\Im }_{a,g}^{1,\mu }f(t,x\left(t\right)+c_0{e}^{-\mu g\left(t\right)}$$

with some quantity $c_0$. Arguing similarly as in a classical case $E=\mathbb{R}$ (cf. [52]), we can show that (still formally)

$$\begin{array}{ccc}\hfill x\left(t\right)& =& \frac{{\left(g\left(t\right)\right)}^{(2-\alpha )(\beta -1)}}{\mathrm{\Gamma }\left(\right(2-\alpha \left)\right(\beta -1)+1)}{c}_1+{\Im }_{a,g}^{\alpha -1,\mu }\left[\lambda {\Im }_{a,g}^{1,\mu }f(t,x\left(t\right))+c_0{e}^{-\mu g\left(t\right)}\right]\hfill \\ & =& \frac{{\left(g\left(t\right)\right)}^{(2-\alpha )(\beta -1)}}{\mathrm{\Gamma }\left(\right(2-\alpha \left)\right(\beta -1)+1)}{c}_1+\lambda {\Im }_{a,g}^{\alpha ,\mu }f(t,x\left(t\right)+c_0\frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\hfill \end{array}$$

(35)

with some (presently unknown) quantities $c_0,c_1$. The boundary condition $x\left(a\right)=0$ results in $c_1=0$. Thus,

$$x\left(t\right)=c_0\frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\lambda {\Im }_{a,g}^{\alpha ,\mu }f(t,x\left(t\right)).$$

(36)

Now, we solve (36) for $c_0$ by $x\left(b\right)={\sum }_{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{{\delta }_i,\mu }x\left({\xi }_i\right)$, it follows that

$$\begin{array}{ccc}\hfill c_0\frac{e^{-\mu ∥g∥}{\left(∥g∥\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}& +& \lambda {\Im }_{a,g}^{\alpha ,\mu }f(b,x\left(b\right))\hfill \\ & =& \sum _{i=1}^{n-2}{\eta }_i\left[c_0\frac{e^{-\mu g\left({\xi }_i\right)}{\left(g\left({\xi }_i\right)\right)}^{\alpha +{\delta }_i-1}}{\mathrm{\Gamma }(\alpha +{\delta }_i)}+\lambda {\Im }_{a,g}^{\alpha +{\delta }_i,\mu }f({\xi }_i,x\left({\xi }_i\right))\right]\hfill \\ & =& c_0\sum _{i=1}^{n-2}\frac{{\eta }_i{e}^{-\mu g\left({\xi }_i\right)}{\left(g\left({\xi }_i\right)\right)}^{\alpha +{\delta }_i-1}}{\mathrm{\Gamma }(\alpha +{\delta }_i)}+\lambda \sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{\alpha +{\delta }_i,\mu }f({\xi }_i,x\left({\xi }_i\right)).\hfill \end{array}$$

Because ${\eta }_i\in \mathbb{R},(i=1,2,\cdots ,n-2)$ are the constants such that $\mathrm{\Lambda }\ne 0$, it follows that

$$c_0=\frac{\lambda }{\mathrm{\Lambda }}\left[\sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{\alpha +{\delta }_i,\mu }f({\xi }_i,x\left({\xi }_i\right))-{\Im }_{a,g}^{\alpha ,\mu }f(b,x\left(b\right))\right].$$

In this connection, let us recall that the integral form (34) of the problem (1) makes sense: This is a direct consequence of Remark 6.

On the other hand, in view of the above considerations, the sufficiency condition is obvious. To see this, let us first note that $2-\alpha ,<\nu ,$ which, in view of (33) and (22), Theorem 2, guarantees the existence and continuity of the operator ${\Im }_{a,g}^{\alpha -1,\mu }f(·,x(·))$. Consequently, it follows from Lemma 6 that

$$\begin{array}{ccc}\hfill x\left(t\right)& =& c_0\frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\lambda {\Im }_{a,g}^{1,\mu }{\Im }_{a,g}^{\alpha -1,\mu }f(t,x\left(t\right))\hfill \\ & =& c_0\frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\lambda {\int }_a^t{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right){\Im }_{a,g}^{\alpha -1,\mu }f(s,x\left(s\right))ds.\hfill \end{array}$$

Thus, $x\left(a\right)=0$.

Operating now by ${}^H{\mathbb{D}}_{a,g,p}^{\alpha -1,\beta ,\mu }$ on both sides of (34), we obtain

$$\begin{array}{ccc}\hfill {}^H{\mathbb{D}}_{a,g,p}^{\alpha -1,\beta ,\mu }x\left(t\right)& =& {}^H{\mathbb{D}}_{a,g,p}^{\alpha -1,\beta ,\mu }\frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{c}_0+{\lambda }^H{\mathbb{D}}_{a,g,p}^{\alpha -1,\beta ,\mu }{\Im }_{a,g}^{\alpha ,\mu }f(t,x\left(t\right))\hfill \\ & =& c_0{e}^{-\mu g\left(t\right)}+\lambda {\Im }_{a,g}^{1+\gamma -\alpha ,\mu }{\mathfrak{D}}_{a,g,p}^{\gamma ,\mu }{\Im }_{a,g}^{\alpha ,\mu }f(t,x\left(t\right)).\hfill \end{array}$$

Therefore, putting in mind (33) and making use of Lemma 8 result in

$${}^H{\mathbb{D}}_{a,g,p}^{\alpha -1,\beta ,\mu }x\left(t\right)=c_0{e}^{-\mu g\left(t\right)}+\lambda {\Im }_{a,g}^{1+\gamma -\alpha ,\mu }{\Im }_{a,g}^{\alpha -\gamma ,\mu }f(t,x\left(t\right)=c_0{e}^{-\mu g\left(t\right)}+\lambda {\Im }_{a,g}^{1,\mu }f(t,x\left(t\right)).$$

From which we conclude in view of $(◊)$ that

$${}^H{\tilde{\mathbb{D}}}_{a,g,p}^{\alpha ,\beta ,\mu }x\left(t\right)={\delta }_p\left(c_0{e}^{-\mu g\left(t\right)}+\lambda {\Im }_{a,g}^{1,\mu }f(t,x\left(t\right)\right)=\lambda f(t,x\left(t\right)),t\in [a,b].$$

Accordingly, it can easily be seen that $x\left(b\right)={\sum }_{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{{\delta }_i,\mu }x\left({\xi }_i\right),$ as required. Evidently, the continuity of ${\Im }_{a,g}^{1,\mu }f(·,x(·))$ yields in view of Lemmas 6 and 7 that

$$\begin{array}{ccc}& & \sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{{\delta }_i,\mu }x\left({\xi }_i\right)=\sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{{\delta }_i,\mu }\left[c_0\frac{e^{-\mu g\left({\xi }_i\right)}{\left(g\left({\xi }_i\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\lambda {\Im }_{a,g}^{\alpha ,\mu }f({\xi }_i,x\left({\xi }_i\right))\right]\hfill \\ & =& c_0\sum _{i=1}^{n-2}{\eta }_i\frac{e^{-\mu g\left({\xi }_i\right)}{\left(g\left({\xi }_i\right)\right)}^{{\delta }_i+\alpha -1}}{\mathrm{\Gamma }({\delta }_i+\alpha )}+\lambda \sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{{\delta }_i,\mu }{\Im }_{a,g}^{\alpha -1,\mu }{\Im }_{a,g}^{1,\mu }f({\xi }_i,x\left({\xi }_i\right))\hfill \\ & =& c_0\sum _{i=1}^{n-2}{\eta }_i\frac{e^{-\mu g\left({\xi }_i\right)}{\left(g\left({\xi }_i\right)\right)}^{{\delta }_i+\alpha -1}}{\mathrm{\Gamma }({\delta }_i+\alpha )}+\lambda \sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{{\delta }_i+\alpha -1,\mu }{\Im }_{a,g}^{1,\mu }f({\xi }_i,x\left({\xi }_i\right))\hfill \\ & =& c_0\sum _{i=1}^{n-2}{\eta }_i\frac{e^{-\mu g\left({\xi }_i\right)}{\left(g\left({\xi }_i\right)\right)}^{{\delta }_i+\alpha -1}}{\mathrm{\Gamma }({\delta }_i+\alpha )}+\lambda \sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{{\delta }_i+\alpha ,\mu }f({\xi }_i,x\left({\xi }_i\right))\hfill \\ & =& c_0\left[\frac{e^{-\mu ∥g∥}{∥g∥}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}-\mathrm{\Lambda }\right]+\lambda \sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{{\delta }_i+\alpha ,\mu }f({\xi }_i,x\left({\xi }_i\right))\hfill \\ & =& c_0\frac{e^{-\mu ∥g∥}{∥g∥}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}-\lambda \left[\sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{\alpha +{\delta }_i,\mu }f({\xi }_i,x\left({\xi }_i\right))-{\Im }_{a,g}^{\alpha ,\mu }f(b,x\left(b\right))\right]\hfill \\ & +& \lambda \sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{{\delta }_i+\alpha ,\mu }f({\xi }_i,x\left({\xi }_i\right))=c_0\frac{e^{-\mu ∥g∥}{∥g∥}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\lambda {\Im }_{a,g}^{\alpha ,\mu }f(b,x\left(b\right))=x\left(b\right).\hfill \end{array}$$

□

Remark 12.

Due to the fact that the (indefinite) Pettis integral of a function f need not have the property of being a.e. weakly differentiable (see [33] for an example of Pettis integrable function whose indefinite Pettis integral is even nowhere weakly differentiable), it is immediately apparent that the result obtained in Theorem 3 has no counterpart if we replace ${\displaystyle {}^H{\tilde{\mathbb{D}}}_{a,g,p}^{\alpha ,\beta ,\mu }}$ by the space ${\displaystyle {}^H{\tilde{\mathbb{D}}}_{a,g,\mathrm{\omega }}^{\alpha ,\beta ,\mu }}$.

The following counterexample demonstrates that our assumption that E has a total dual is crucial in Theorem 3 and cannot be ignored even if x is weakly absolutely continuous on I. Out of the context of such spaces, we should instead assume that the considered derivatives should be of (Jordan) bounded variation (cf. [32]).

Counterexample 2.

Let $\alpha \in (1,2],$ and assume that $B\left[I\right]$ be the Banach space of bounded real-valued functions on I. Note that the choice of this space in the counterexample is not accidental, since $B\left[I\right]$ does not have a total dual. Define the weakly measurable function $f(·,x(·\left)\right):[a,b]\times B\left[I\right]\to B\left[I\right]$ by

$$f(t,x\left(t\right)):=\left\{\begin{array}{c}{\displaystyle {\chi }_{\{t\}}(·),t\in J,J\subset [a,b]bis a set of positive measure,}\hfill \\ {\displaystyle \theta ,t\notin J,}\hfill \end{array}\right.$$

According to Example 1, we know that $f\in {\mathcal{H}}_0^{\psi }\left(E\right),$ for any Young function ψ having ${\Im }_{a,g}^{\delta ,\mu }f(t,x\left(t\right))=\theta$ for any $\delta >0$. From which it follows that the integral equation

$$x\left(t\right)=\frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{c}_0+\lambda {\Im }_{a,g}^{\alpha ,\mu }f(t,x\left(t\right))=\theta .$$

(37)

has a unique weakly continuous solution $x\left(t\right)=\theta ,$ for any $t\in [a,b].$ Accordingly, $x\left(t\right)=\theta ,$ is a weakly continuous solution to the problem

$$\left\{\begin{array}{c}{}^H{\tilde{\mathbb{D}}}_{a,g,p}^{\alpha ,\beta ,\mu }x\left(t\right)=\theta ,t\in [a,b],\beta =0,\mu \in {\mathbb{R}}^{+},\hfill \\ {\displaystyle x\left(a\right)=0,x\left(b\right)=\sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{{\delta }_i,\mu }x\left({\xi }_i\right),{\delta }_i>0,{\xi }_i\in [a,b],}\hfill \end{array}\right.$$

(38)

which differ from the problem (1) on a set of positive measure.

5. Existence of Solutions of Differential Equations with Fractional Pseudo-Derivatives

For the sake of completeness of the article, it remains to prove a result about the existence of solutions to the nonlinear n-point boundary value problem (1). First, we need to define the notion of an expected solution under our (very mild) assumptions. In our discussion, we emphasized that the equivalence of differential and integral problems is a more subtle problem. We need a new notion:

Definition 9.

A weakly continuous function $x\left(·\right):I\to E$ is said to be a solution of the problem (1) if

1.

$x\left(·\right)$ has g-MHFPDof order α, $\alpha \in (1,2]$ with parameter $\mu \in {\mathbb{R}}^{+}$ and type $\beta \in [0,1]$ such that

$${}^H{\tilde{\mathbb{D}}}_{a,g,p}^{\alpha ,\beta ,\mu }x\left(t\right)=\lambda f(t,x\left(t\right)),t\in [a,b],\lambda \in \mathbb{R},$$

2.

${\displaystyle x\left(a\right)=0,x\left(b\right)=\sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{{\delta }_i,\mu }x\left({\xi }_i\right),{\delta }_i>0,{\xi }_i\in [a,b].}$

Recall that a function $f\left(·\right):E\to E$ is said to be sequentially continuous from $E_w$ into $E_w$ (or: weakly–weakly sequentially continuous) if for every weakly convergent sequence ${\left\{{\theta }_n\right\}}_{n\ge 1}\subset E$ the sequence ${\left\{f\left({\theta }_n\right)\right\}}_{n=1}$ is weakly convergent in E.

We need to prove an important property of generalized g-fractional integrals. It is a version of the Goebel–Rzymowski lemma with Pettis integrals and the De Blasi measures of weak non-compactness. It will be an important tool in our considerations, but it is interesting in itself and could be useful in many similar problems.

Lemma 9.

Let ψ be a Young function with its complementary Young function $\tilde{\psi }$ satisfying

$$\begin{array}{c}\hfill {\int }_0^t\tilde{\psi }\left(s^{\alpha -1}\right)ds<\infty ,t>0.\end{array}$$

Moreover, assume that g is a positive increasing function on I, having positive continuous derivative. Next, let μ be the De Blasi measure of weak non-compactness on E.

For any $\alpha >0$, $t\in I$ and any bounded strongly equicontinuous set $V\subset C[I,E_w]$, put $\mathit{\mu }\left({\Im }_{a,g}^{\alpha ,\mathit{\mu }}V\left(t\right)\right):=\mathit{\mu }\left(\{{\Im }_{a,g}^{\alpha ,\mathit{\mu }}v\left(t\right):v\in V\}\right)$. Then,

$$\mathit{\mu }\left({\Im }_{a,g}^{\alpha ,\mathit{\mu }}V\left(t\right)\right)\le {\Im }_{a,g}^{\alpha ,\mu }\mathit{\mu }\left(V\left(t\right)\right)\le \frac{4e^{\mu ∥g∥}∥g^{\prime }∥{\tilde{\mathrm{\Psi }}}_{\alpha }\left(∥g∥\right)}{\mathrm{\Gamma }\left(\alpha \right)}·{\mu }_C\left(V\right),$$

where ${\mathit{\mu }}_C\left(V\right)$ denotes the De Blasi measure of weak non-compactness in the space $C[I,E]$ and ${\tilde{\mathrm{\Psi }}}_{\alpha }$ is given by (9).

Proof. 

Note, in view of Fact 1 ${\Im }_{a,g}^{\alpha ,\mu }v$ exists and it is weakly continuous function on I. Hence, $\mu \left({\Im }_{a,g}^{\alpha ,\mu }V\left(t\right)\right)$ makes sense. Now, define the function $G:I\times I\to {\mathbb{R}}^{+}$ by

$${\displaystyle G(t,s):=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\left\{\begin{array}{cc}{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right),\hfill & s\in [a,t],t>a\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.}$$

From the above definition, we have ${\Im }_a^{\alpha ,g}v\left(t\right)={\int }_a^{t}G(t,s)v\left(s\right)ds$ (with the Pettis integral in this formula). Applying the properties of this integral, for arbitrary Pettis integrable function $w\in P[I,E]$ and arbitrary point $t\in I$ we have

$${\int }_a^{t-\tau }w\left(s\right)ds+{\int }_{t-\tau }^{t}w\left(s\right)ds={\int }_a^{t}w\left(s\right)ds,\mathrm{for} \mathrm{some} 0<\tau <t+a.$$

As V is equicontinuous and by our assumptions on g we see, that $s\to G(t,s)v\left(s\right)$ is Pettis integrable and continuous on $[a,t]$, so the set $\left\{G(t,·)V(·)\right\}$ is Pettis uniformly integrable on I (PUI), so for any $v\in V$ and arbitrary $t\in I$ the set $\{\phi \left(G(t,·)v(·)\right):\phi \in E^{*},\parallel \phi \parallel \le 1\}$ is equi-integrable on $[a,t]$. Since ${\int }_{t-\tau }^{t}G(t,s)v\left(s\right)ds\in \tau ·\overline{conv}\{G(t,s)v\left(s\right):s\in [t-\tau ,t]\}$ and, by our assumptions, the last set is bounded, for any $\epsilon >0$ there exists (sufficiently small) $\tau$ such that

$$∥{\int }_{t-\tau }^{t}G(t,s)V\left(s\right)ds∥<\epsilon .$$

(39)

From this it follows that we can cover the set $\left\{{\int }_{t-\tau }^{t}G(t,s)v\left(s\right)ds:s\in [t-\tau ,t],v\in V\right\}$ by balls with radius less than $\epsilon$ and then by definition of the De Blasi measure of weak noncompactness $\mathit{\mu }$ we obtain

$$\mathit{\mu }\left(\left\{{\int }_{t-\tau }^{t}G(t,s)v\left(s\right)ds:s\in [t-\tau ,t],v\in V\right\}\right)<\epsilon .$$

Now, we need to estimate the set of integrals over $[a,t-\tau ]$. Define a function

$$v(·)=\mathit{\mu }\left(V\right(·\left)\right).$$

In view of Ambrosetti’s Lemma 2, v is a continuous function. Note, that from our assumption it follows that $s\to G(t,s)v\left(s\right)$ is continuous on $[a,t-\tau ]$, so is uniformly continuous.

Thus, there exists $\delta >0$ such that

$$\left|G(t,\eta )v\left(q\right)-G(t,s)v\left(s\right)\right|<\epsilon$$

(40)

provided that $|q-s|<\delta$ and $|\eta -s|<\delta$ with $\eta ,s,q\in [a,t-\tau ]$.

Now, let us consider a division of the interval $[a,t-\tau ]$ into n parts $a=t_0<t_1<\cdots <t_n=t-\tau$ such that $|t_i-t_{i-1}|<\delta$ for $i=1,2,\cdots ,n$. Put $T_i=[t_{i-1},t_i]$. As v is uniformly continuous, there exists $s_i\in T_i$ such that $v\left(s_i\right)=\mu \left(V\left(T_i\right)\right)$ ($i=1,2,\cdots ,n$).

Due to additivity of the integral

$$\begin{array}{ccc}& & \left\{{\int }_a^{t-\tau }G(t,s)x\left(s\right)ds:s\in [a,t-\tau ],x\in V\right\}\hfill \\ & \subset & \sum _{i=1}^n\left\{{\int }_{T_i}G(t,s)x\left(s\right)ds:s\in [a,t-\tau ],x\in V\right\},\hfill \end{array}$$

and consequently, by applying the mean value theorem for the Pettis integral

$${\int }_{T_i}G(t,s)V\left(s\right)ds\in meas\left(T_i\right)·\overline{\mathrm{conv}}\left\{G(t,s)V\left(s\right):s\in T_i\right\}.$$

Therefore, we have an estimation

$$\begin{array}{ccc}& & \mathit{\mu }\left(\left\{{\int }_a^{t-\tau }G(t,s)x\left(s\right)ds:s\in [a,t-\tau ],x\in V\right\}\right)\hfill \\ & \le & \sum _{i=1}^n\mathit{\mu }\left(\left\{{\int }_{T_i}G(t,s)x\left(s\right)ds:s\in [a,t-\tau ],x\in V\right\}\right)\hfill \\ & \le & \sum _{i=1}^{b}meas\left(T_i\right)·\mathit{\mu }\left(\overline{\mathrm{conv}}\left\{G(t,s)V\left(s\right):s\in T_i\right\}\right)\hfill \\ & \le & \sum _{i=1}^{b}meas\left(T_i\right)·\underset{s\in T_i}{max}G(t,s)·\mathit{\mu }V\left(T_i\right)\hfill \\ & \le & \sum _{i=1}^{b}meas\left(T_i\right)·G(t,t_i)·\mathit{\mu }V\left(T_i\right))\le \sum _{i=1}^{b}meas\left(T_i\right)·G(t,t_i)·v\left(s_i\right).\hfill \end{array}$$

Note that from (40) it follows that

$$\sum _{i=1}^{b}meas\left(T_i\right)·G(t,t_i)·v\left(s_i\right)\le {\int }_a^{t-\tau }G(t,s)v\left(s\right)ds+(t-\tau )·\epsilon .$$

Then,

$$\begin{array}{ccc}\hfill \left\{{\int }_a^{t}G(t,s)x\left(s\right)ds:s\in [a,t-\tau ],x\in V\right\}& \subset & \left\{{\int }_a^{t}G(t,s)x\left(s\right)ds:s\in [a,t-\tau ],x\in V\right\}\hfill \\ & +& \left\{{\int }_{t-\tau }^{t}G(t,s)x\left(s\right)ds:s\in [a,t-\tau ],x\in V\right\}\hfill \end{array}$$

and, consequently

$$\mathit{\mu }\left(\left\{{\int }_a^{t}G(t,s)x\left(s\right)ds:s\in [a,t],x\in V\right\}\right)\le {\int }_a^{t-\tau }G(t,s)v\left(s\right)ds+(t-\tau )·\epsilon +\epsilon .$$

As $\epsilon$ is arbitrarily small, we obtain

$$\mathit{\mu }\left(\left\{{\int }_a^{t}G(t,s)x\left(s\right)ds:s\in [a,t],x\in V\right\}\right)\le {\int }_a^{t-\tau }G(t,s)v\left(s\right)ds,$$

i.e., in terms of fractional integrals

$$\mathit{\mu }\left({\Im }_{a,g}^{\alpha ,\mu }V\left(t\right)\right)\le {\Im }_{a,g}^{\alpha ,\mu }\mu \left(V\left(t\right)\right).$$

It remains to prove the second estimation in our thesis. But it is immediate and follows the inequality proved in Theorem 2. □

Now, we can formulate the existence result for global solutions of our boundary value problem:

Theorem 4.

If the assumptions of Theorem 3 hold along with:

(1) 

For every $t\in I$, $f(t,·)$ is $ww$-sequentially continuous;

(2) 

For every $x\in C[I,E_w]$, $f(·,x(·\left)\right)\in P[I,E]$;

(3) 

For any $r>0$ and each $\phi \in E^{*}$ there exists an $L_{\psi }(I,\mathbb{R})$-integrable function $M_r^{\phi }:I\to {\mathbb{R}}^{+}$ such that $\left|\phi \left(f\right(t,x\left)\right)\right|\le M_r^{\phi }\left(t\right)$ for a.e. $t\in I$ and all $x\in C[I,E_w]$ whenever $∥x∥\le r$.

Moreover, there exists a continuous non-decreasing function $\Omega :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ and such that for all $\phi \in E^{*}$ with $∥\phi ∥\le 1$, ${∥M_r^{\phi }∥}_{\psi }<\Omega \left(r\right)$ and ${\int }_0^{\infty }\frac{dr}{{∥M_r^{\phi }∥}_{\psi }}=\infty$;

(4) 

For each function $x\in C[I,E_w]$

$$\mathit{\mu }\left(f\right(t,X)\le p(t)·\mathit{\mu }(X),for each bounded X\subset E and each t\in I,$$

where $p:I\to \mathbb{R}$ is continuous.

Then, there is $\rho >0$ such that for any $\lambda \in \mathbb{R}$ with $\left|\lambda \right|\le \rho$, the nonlinear n-point boundary value problem (1) has at least one pseudo-solution $u\in C[I,E_w]$.

Proof. 

Define an operator T on $C[I,E_w]$ generating the right-hand side of the integral Equation (34), i.e., of the form

$$\begin{array}{ccc}\hfill T\left(x\right)\left(t\right)& :=& \frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{c}_0\left(x\right)+\lambda {\Im }_{a,g}^{\alpha ,\mu }f(t,x\left(t\right))\hfill \\ & =& \frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{c}_0\left(x\right)+\lambda {\Im }_{a,g}^{1,\mu }{\Im }_{a,g}^{\alpha -1,\mu }f(t,x\left(t\right)),\hfill \end{array}$$

(41)

where

$$c_0\left(x\right)=\frac{\lambda }{\mathrm{\Lambda }}\left[\sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{\alpha +{\delta }_i,\mu }f({\xi }_i,x\left({\xi }_i\right))-{\Im }_{a,g}^{\alpha ,\mu }f(b,x\left(b\right))\right].$$

Firstly note that, for any $x\in C[I,E_w]$, we have (in view of Assumption (3)) that

$${\int }_a^b\psi \left(\frac{\left|\phi \right(f(t,x(t\left)\right)\left)\right|}{{∥M_r^{\phi }∥}_{\psi }}\right)dt\le {\int }_a^b\psi \left(\frac{M_r^{\phi }\left(t\right)}{{∥M_r^{\phi }∥}_{\psi }}\right)dt\le 1,$$

for any $r\ge ∥x∥$ and for every $\phi \in E^{*}$. Thus, $f(·,x\left((·)\right)\in {\mathcal{H}}_0^{\psi }\left(E\right)$ and ${∥f(·,x(·\left)\right)∥}_{\psi }\le {∥M_r^{\phi }∥}_{\psi }$. Reasoning as in the proof of Theorem 3, we know that the operator T makes sense.

I. Note that the operator T is well defined on $C[I,E_w]$. Indeed, by Theorem 2 and Lemma 1, we include that T maps $C[I,E_w]$ into $C[I,E_w]$.

Let us stress that Assumption (3) implies that all solutions are global, i.e., are defined on the interval I.

II. Let us construct an invariant set for T, which is required by Theorem 1. Recall some estimations proved by us. Let $x\in C[I,E_w]$. Then, by (13)

$$\begin{array}{ccc}\hfill {\Im }_{a,g}^{\alpha ,\mu }f(t_2,x\left(t_2\right))& -& {\Im }_{a,g}^{\alpha ,\mu }f(t_1,x\left(t_1\right))\hfill \\ & =& e^{-\mu g\left(t_2\right)}\left({\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t_2\right)}f(t_2,x\left(t_2\right))\right]-{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t_1\right)}f(t_2,x\left(t_2\right))\right]\right)\hfill \\ & +& {\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t_1\right)}f(t_1,x\left(t_1\right))\right]\left(e^{-\mu g\left(t_2\right)}-e^{-\mu g\left(t_1\right)}\right).\hfill \end{array}$$

As we will need to investigate continuity property of T, for any $\phi \in E^{*}$ with $\left|\phi \right|\le 1$ we need to estimate $\left|\phi \right({\Im }_{a,g}^{\alpha ,\mu }f(t_2,x\left(t_2\right))-{\Im }_{a,g}^{\alpha ,\mu }f(t_1,x\left(t_1\right))\left)\right|$:

$$\begin{array}{ccc}\hfill \left|\phi \right({\Im }_{a,g}^{\alpha ,\mu }f(t_2,x\left(t_2\right))& -& {\Im }_{a,g}^{\alpha ,\mu }f(t_1,x\left(t_1\right))\left)\right|\hfill \\ & \le & e^{-\mu \parallel g\left(a\right)\parallel }\phi ({\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t_2\right)}f(t_2,x\left(t_2\right))\right]-{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t_1\right)}f(t_2,x\left(t_2\right))\right])|\hfill \\ & +& \left(e^{-\mu g\left(t_2\right)}-e^{-\mu g\left(t_1\right)}\right)·e^{\mu \parallel g\parallel }·{\Im }_{a,g}^{\alpha ,0}{M}_r^{\phi }\left(t_1\right),\hfill \end{array}$$

(42)

where $r=\parallel x\parallel$. As $e^{-\mu \parallel g\left(a\right)\parallel }$ is a constant, let us estimate now the first term on the right-hand side of this inequality. For any $\phi \in E^{*}$ with $\left|\phi \right|\le 1$, let us consider functions under the integral sign in the form $[e^{\mu g(·)}·f(·,x(·))]\left(t\right)$, i.e.,

$$\begin{array}{ccc}& & \left|\phi \left({\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}f((·,x(·))\right]\left(t_2\right)-{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}f(·,x(·))\right]\left(t_1\right)\right)\right|\hfill \\ & =& \left|{\Im }_{a,g}^{\alpha ,0}\phi \left(\left[e^{\mu g(·)}f(·,x(·))\right]\right)\left(t_2\right)-{\Im }_{a,g}^{\alpha ,0}\phi \left(\left[e^{\mu g(·)}f(·,x(·))\right]\right)\left(t_1\right)\right|.\hfill \end{array}$$

By taking the supremum over all $\phi \in E^{*}$ with $\parallel \phi \parallel \le 1$, similarly as in (6) for $e^{\mu g(·)}f(·,x(·))$, we obtain

$$\begin{array}{ccc}& & \phi ({\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t_2\right)}f(t_2,x\left(t_2\right))\right]-{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t_1\right)}f(t_2,x\left(t_2\right))\right])\hfill \\ & \le & |{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t_2\right)}\left|\phi f(t_2,x\left(t_2\right))\right|\right]-{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g\left(t_1\right)}\left|\phi f(t_2,x\left(t_2\right))\right|\right]\left)\right|\hfill \\ & \le & \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\left({\int }_a^{\tau }\left|\frac{e^{\mu g\left(s\right)}}{{(g\left(t\right)-g\left(s\right))}^{1-\alpha }}-\frac{e^{\mu g\left(s\right)}}{{(g\left(\tau \right)-g\left(s\right))}^{1-\alpha }}\right|\left|\phi f(s,x\left(s\right))\right|g^{\prime }\left(s\right)ds\right.\hfill \\ \hfill & +& \left.{\int }_{\tau }^t\frac{e^{\mu g\left(s\right)}}{{(g\left(t\right)-g\left(s\right))}^{1-\alpha }}\left|\phi f(s,x\left(s\right))\right|g^{\prime }\left(s\right)ds\right)\hfill \\ & \le & \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\left({\int }_a^{\tau }\left|\frac{e^{\mu g\left(s\right)}}{{(g\left(t\right)-g\left(s\right))}^{1-\alpha }}-\frac{e^{\mu g\left(s\right)}}{{(g\left(\tau \right)-g\left(s\right))}^{1-\alpha }}\right|M_r^{\phi }\left(s\right)g^{\prime }\left(s\right)ds\right.\hfill \\ \hfill & +& \left.{\int }_{\tau }^t\frac{e^{\mu g\left(s\right)}}{{(g\left(t\right)-g\left(s\right))}^{1-\alpha }}{M}_r^{\phi }\left(s\right)g^{\prime }\left(s\right)ds\right)\hfill \\ & \le & \frac{2\parallel M_r^{\phi }{\parallel }_{\infty }{e}^{\mu ∥g∥}}{\mathrm{\Gamma }(1+\alpha )}{(g\left(t\right)-g\left(\tau \right))}^{\alpha }.\hfill \end{array}$$

Now, by estimating also the second term in the formula (42) and using the same method, we obtain

$$\begin{array}{ccc}\hfill & & ∥{\Im }_{a,g}^{\alpha ,\mu }f(t_2,x\left(t_2\right))-{\Im }_{a,g}^{\alpha ,\mu }f(t_1,x\left(t_1\right))∥\hfill \\ & \le & e^{-\mu g\left(a\right)}∥{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}{M}_r^{\phi }(·)\right]\left(t_2\right)-{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}{M}_r^{\phi }(·)\right]\left(t_1\right)∥\hfill \\ & +& ∥{\Im }_{a,g}^{\alpha ,0}\left[e^{\mu g(·)}{M}_r^{\phi }(·)\right]\left(t_1\right)∥\left|e^{-\mu g\left(t_2\right)}-e^{-\mu g\left(t_1\right)}\right|.\hfill \end{array}$$

(43)

Denote the right-hand side of this inequality by ${\mathrm{\omega }}_1(t_2,t_1)$. Clearly, by the properties of g, if $t_2\to t_1$, then ${\mathrm{\omega }}_1(t_1,t_2)\to 0$. As $T\left(x\right)\left(t\right):=\frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{c}_0\left(x\right)+\lambda {\Im }_{a,g}^{\alpha ,\mu }f(t,x\left(t\right))$ we need to investigate the continuity property of the first term. To do it, let us also consider

$${\mathrm{\omega }}_2(t_1,t_2)=\frac{e^{-\mu g\left(t_1\right)}{\left(g\left(t_1\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}-\frac{e^{-\mu g\left(t_2\right)}{\left(g\left(t_2\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}$$

and

$$\mathrm{\omega }(t_1,t_2)={\mathrm{\omega }}_2(t_1,t_2)\underset{x}{sup}\parallel c_0\left(x\right)\parallel +\left|\lambda \right|{\mathrm{\omega }}_1(t_1,t_2)=\mathrm{\omega }(a,\alpha ,\mu ,g,|t_2-t_1\left|\right),$$

but is not dependent on x and $\mathrm{\omega }\left(\tau \right)\to 0$ as $\tau \to 0^{+}$. The quantity $\parallel c_0\left(x\right)\parallel$ will be calculated later. Due to assumptions on g, we ensure that the function ${\Im }_{a,g}^{\alpha ,\mu }x$ is strongly continuous. By the definition of T, the function $\mathrm{\omega }$ is a modulus of continuity for T. It means we will restrict our attention to some ball (with radius $R_0$ which will be calculated later) and by recalling the fact that sets of functions sharing the same modulus of continuity are exactly equicontinuous families, we will define an invariant domain for T. Now, let us choose such a ball. We start with the estimation

$$∥{\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)∥=∥{\Im }_{a,g}^{\alpha ,\mu }x\left(t\right)-{\Im }_{a,g}^{\alpha ,\mu }x\left(a\right)∥\le \frac{4e^{\mu ∥g∥}∥g^{\prime }∥{\tilde{\mathrm{\Psi }}}_{\alpha }\left(∥g∥\right)}{\mathrm{\Gamma }\left(\alpha \right)}{∥\phi \left(x\right)∥}_{\psi }.$$

(44)

Define a convex and closed subset $Q\subset C[I,E_w]$ by

$$\mathcal{Q}=\left\{u\in [I,E_w]:u\left(a\right)=0,∥u∥\le R_0,\mathrm{and}∥u\left(t\right)-u\left(s\right)∥\le \mathrm{\omega }\left(\right|t-s\left|\right)\right\},$$

where $\left|\lambda \right|B(\alpha ,g,R_0,\mu ,\mathrm{\Lambda },{\delta }_i)+\left|\lambda \right|B_1(\alpha ,g,R_0,\mu )\le R_0$, which is satisfied for sufficiently small $\lambda$ (i.e., $\left|\lambda \right|\le \rho$ for some $\rho$) and for quantities

$$B(\alpha ,g,R_0,\mu ,\mathrm{\Lambda },{\delta }_i):=|\frac{1}{\mathrm{\Lambda }}|·4e^{\mu ∥g∥}∥g^{\prime }∥·\Omega \left(R_0\right)\left[\sum _{i=1}^{n-2}{\eta }_i\parallel \frac{{\tilde{\mathrm{\Psi }}}_{\alpha +{\delta }_i}\left(∥g∥\right)}{\mathrm{\Gamma }(\alpha +{\delta }_i)}+\frac{{\tilde{\mathrm{\Psi }}}_{\alpha }\left(∥g∥\right)}{\mathrm{\Gamma }\left(\alpha \right)}\right],$$

$$B_1(\alpha ,g,R_0,\mu ):=\frac{4e^{\mu ∥g∥}∥g^{\prime }∥{\tilde{\mathrm{\Psi }}}_{\alpha }\left(∥g∥\right)}{\mathrm{\Gamma }\left(\alpha \right)}\Omega \left(R_0\right).$$

We notice that by the properties of $\mathrm{\omega }$, the set $\mathcal{Q}$ is strongly equicontinuous as a subset of $C[I,E_w]$. For arbitrary $x\in \mathcal{Q}$ and $t,s\in I$, by (43), we immediately obtain $\parallel T\left(x\right)\left(t\right)-T\left(x\right)\left(s\right)\parallel \le \mathrm{\omega }\left(\right|t-s\left|\right)$. By applying (44) we can estimate

$$\begin{array}{ccc}\hfill \parallel T\left(x\right)\left(t\right)\parallel & \le & \parallel c_0\left(x\right)\parallel +\frac{4e^{\mu ∥g∥}∥g^{\prime }∥{\tilde{\mathrm{\Psi }}}_{\alpha }\left(∥g∥\right)}{\mathrm{\Gamma }\left(\alpha \right)}{∥M_{R_0}^{\phi }∥}_{\psi }\hfill \\ & \le & \parallel c_0\left(x\right)\parallel +\left|\lambda \right|\frac{4e^{\mu ∥g∥}∥g^{\prime }∥{\tilde{\mathrm{\Psi }}}_{\alpha }\left(∥g∥\right)}{\mathrm{\Gamma }\left(\alpha \right)}\Omega \left(R_0\right),\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill \parallel c_0\left(x\right)\parallel & =& ∥\frac{\lambda }{\mathrm{\Lambda }}\left[\sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{\alpha +{\delta }_i,\mu }f({\xi }_i,x\left({\xi }_i\right))-{\Im }_{a,g}^{\alpha ,\mu }f(b,x\left(b\right))\right]∥\hfill \\ & \le & |\frac{\lambda }{\mathrm{\Lambda }}|·\left[\sum _{i=1}^{n-2}{\eta }_i\parallel {\Im }_{a,g}^{\alpha +{\delta }_i,\mu }f({\xi }_i,x\left({\xi }_i\right))-{\Im }_{a,g}^{\alpha ,\mu }f(b,x\left(b\right))\parallel \parallel \right]\hfill \\ & \le & |\frac{\lambda }{\mathrm{\Lambda }}|·4e^{\mu ∥g∥}∥g^{\prime }∥·\Omega \left(R_0\right)\left[\sum _{i=1}^{n-2}{\eta }_i\parallel \frac{{\tilde{\mathrm{\Psi }}}_{\alpha +{\delta }_i}\left(∥g∥\right)}{\mathrm{\Gamma }(\alpha +{\delta }_i)}+\frac{{\tilde{\mathrm{\Psi }}}_{\alpha }\left(∥g∥\right)}{\mathrm{\Gamma }\left(\alpha \right)}\right]\hfill \\ & =& \left|\lambda \right|·B(\alpha ,g,R_0,\mu ,\mathrm{\Lambda },{\delta }_i).\hfill \end{array}$$

Thus,

$$\parallel T\left(x\right)\left(t\right)\parallel \le 4e^{\mu ∥g∥}∥g^{\prime }∥·\Omega \left(R_0\right)\left[\sum _{i=1}^{n-2}{\eta }_i\parallel \frac{{\tilde{\mathrm{\Psi }}}_{\alpha +{\delta }_i}\left(∥g∥\right)}{\mathrm{\Gamma }(\alpha +{\delta }_i)}+(1+|\frac{\lambda }{\mathrm{\Lambda }}\left|\right)\frac{{\tilde{\mathrm{\Psi }}}_{\alpha }\left(∥g∥\right)}{\mathrm{\Gamma }\left(\alpha \right)}\right]\le R_0$$

and consequently $T:\mathcal{Q}\to \mathcal{Q}$.

III. Now, we need to check the weak–weak sequentially continuity of T. Let $\left(u_n\right)$ be a sequence in $\mathcal{Q}$ such that $u_n\to u$ in the space $C[I,E_w]$. It is known, that weak convergence in $C[I,E_w]$ stands exactly for its boundedness and weak pointwise convergence (for any $t\in I$). The first condition is guaranteed by the definition of the set $\mathcal{Q}$.

Fix an arbitrary point $t\in I$. Consider now the operator T on the set $\mathcal{Q}$, and note that ${\Im }_{a,g}^{\alpha +{\delta }_i,\mu }f(t,u_n\left(t\right))$ satisfies all the assumptions of the Lebesgue dominated convergence theorem for the Pettis integral. Then, by assumption B) (1), we obtain weak convergence of ${\Im }_{a,g}^{\alpha +{\delta }_i,\mu }f(t,u_n\left(t\right))$ to ${\Im }_{a,g}^{\alpha +{\delta }_i,\mu }f(t,u\left(t\right))$. Moreover, as

$$c_0\left(u_n\right)=\frac{\lambda }{\mathrm{\Lambda }}\left[\sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{\alpha +{\delta }_i,\mu }f({\xi }_i,u_n\left({\xi }_i\right))-{\Im }_{a,g}^{\alpha ,\mu }f(b,u_n\left(b\right))\right],$$

then again by assumption B) (1) and arguing as above, we conclude that $c_0\left(u_n\right)$ converges weakly in $C[I,E_w]$ to $c_0\left(u\right)$. It implies that $\left(T{u}_n\right)\left(t\right)$ converges weakly to $\left(Tu\right)\left(t\right)$ in $(E,w)$ for each $t\in I$ (weakly pointwisely), so by the Dobrakov theorem we conclude that $T:\mathcal{Q}\to \mathcal{Q}$ is weakly–weakly sequentially continuous operator on the set $\mathcal{Q}$.

IV. Let us verify condition (3) in Theorem 1. Let V be a subset of $\mathcal{Q}$ satisfying $\overline{V}=\overline{conv}(\left(TV\right)\cup \{0\})$. Obviously, $V\left(t\right)\subset \overline{conv}(\left(TV\right)\left(t\right)\cup \{0\})$, $t\in I$. Since $T\left(\mathcal{Q}\right)$ is uniformly bounded and strongly equicontinuous in $C[I,E_w]$, it follows that V is also bounded and equicontinuous. In view of the Lemma 2, the function $v\left(t\right):=\mathit{\mu }\left(V\right(t\left)\right)$ is continuous on I, $V\left(t\right):=\{v(t):v\in V\}$ and

$$\begin{array}{ccc}\hfill T\left(V\right)\left(t\right)& =& \{T\left(v\right)\left(t\right):u\in V\}=\left\{\frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{c}_0\left(u\right)+\lambda {\Im }_{a,g}^{\alpha ,\mu }f(t,u\left(t\right)):u\in V\right\}\hfill \end{array}$$

We need to estimate now the measures of non-compactness of $\left\{{\Im }_{a,g}^{\alpha ,\mu }f(t,u\left(t\right)):u\in V\right\}$ and $\left\{\frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{c}_0\left(u\right):u\in V\right\}$. Put $v\left(s\right)=\mathit{\mu }\left(V\right(s\left)\right)$. Note, that by our assumption

$$s\to H\left(s\right):=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right)p\left(s\right)v\left(s\right)$$

is continuous on the compact interval $[a,t]$, and so is uniformly continuous.

Thus, there exists $\delta >0$ such that

$$\begin{array}{ccc}\hfill & & \left|\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right)p\left(s\right)v\left(q\right)\right.\hfill \\ & -& \left.\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right)p\left(s\right)\left(v\left(s\right)\right)\right|<\epsilon \hfill \end{array}$$

provided that $|q-s|<\delta$ and $|\eta -s|<\delta$ with $\eta ,s,q\in [a,t-\tau ]$. Divide the interval $a,t-\tau$ into n parts $a=t_0<t_1<\cdots <t_n=t-\tau$ such that $|t_i-t_{i-1}|<\delta$ for $i=1,2,\cdots ,n$. Put $T_i=[t_{i-1},t_i]$. As v is uniformly continuous, there exists $s_i\in T_i$ such that $v\left(s_i\right)=\mu \left(V\left(T_i\right)\right)$ ($i=1,2,\cdots ,n$).

As

$$\begin{array}{ccc}& & \left\{{\int }_a^{t-\tau }\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right)f(s,x\left(s\right))ds:s\in [a,t-\tau ],x\in V\right\}\hfill \\ & \subset & \sum _{i=1}^n\left\{{\int }_{T_i}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right)f(s,x\left(s\right))ds:s\in [a,t-\tau ],x\in V\right\},\hfill \end{array}$$

and from the mean value theorem for the Pettis integral

$$\begin{array}{ccc}& & {\int }_{T_i}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right)f(s,V\left(s\right))ds\hfill \\ & \in & meas\left(T_i\right)·\overline{\mathrm{conv}}\left\{\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right)f(s,V\left(s\right)):s\in T_i\right\}.\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}& & \mathit{\mu }\left(\left\{{\int }_a^{t-\tau }\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mathit{\mu }\left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right)f(s,x\left(s\right))ds:s\in [a,t-\tau ],x\in V\right\}\right)\hfill \\ & \le & \sum _{i=1}^n\mathit{\mu }\left(\left\{{\int }_{T_i}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right)f(s,x\left(s\right))ds:s\in [a,t-\tau ],x\in V\right\}\right)\hfill \\ & \le & \sum _{i=1}^{b}meas\left(T_i\right)·\mathit{\mu }\left(\overline{\mathrm{conv}}\left\{\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right)f(s,V\left(s\right)):s\in T_i\right\}\right)\hfill \\ & \le & \sum _{i=1}^{b}meas\left(T_i\right)·\underset{s\in T_i}{max}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right)·\mathit{\mu }\left(\{f(s,V\left(s\right)):s\in T_i\}\right)\hfill \\ & \le & \sum _{i=1}^{b}meas\left(T_i\right)·\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(t_i\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(t_i\right)\right)}{g}^{\prime }\left(t_i\right)·\underset{s\in T_i}{sup}p\left(s\right)·\mathit{\mu }\left(V\left(T_i\right)\right)\hfill \\ & \le & \sum _{i=1}^{b}meas\left(T_i\right)·\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(t_i\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(t_i\right)\right)}{g}^{\prime }\left(t_i\right)·\underset{s\in T_i}{sup}p\left(s\right)·v\left(s_i\right).\hfill \end{array}$$

Note that from (45), it follows that

$$\begin{array}{ccc}& & \sum _{i=1}^{b}meas\left(T_i\right)·\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(t_i\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(t_i\right)\right)}{g}^{\prime }\left(t_i\right)·\underset{s\in T_i}{sup}p\left(s\right)·v\left(s_i\right)\hfill \\ & \le & {\int }_a^{t-\tau }\sum _{i=1}^{b}meas\left(T_i\right)·\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(g\left(t\right)-g\left(t_i\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(t_i\right)\right)}{g}^{\prime }\left(t_i\right)·\underset{s\in T_i}{sup}p\left(s\right)·v\left(s\right)ds\hfill \\ & & +(t-\tau )·\epsilon .\hfill \end{array}$$

Finally, the above estimates give us

$$\begin{array}{ccc}& & \mathit{\mu }({\Im }_{a,g}^{\alpha ,\mu }f(t,V\left(t\right))\hfill \\ & \le & \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_a^t{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right)·\underset{s\in [a,t]}{sup}p\left(s\right)·v\left(s\right)ds+(t-\tau )·\epsilon +\epsilon .\hfill \end{array}$$

Since the last inequality is satisfied for any number $\epsilon >0$, we arrive at

$$\mathit{\mu }({\Im }_{a,g}^{\alpha ,\mu }f(t,V\left(t\right))\le \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_a^t{\left(g\left(t\right)-g\left(s\right)\right)}^{\alpha -1}{e}^{-\mu \left(g\left(t\right)-g\left(s\right)\right)}{g}^{\prime }\left(s\right)·\underset{s\in [a,t]}{sup}p\left(s\right)·v\left(s\right)ds.$$

Now, let us investigate $\mathit{\mu }\left(\frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{c}_0\left(V\right)\right)=\frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\mathit{\mu }\left(c_0\left(V\right)\right)$. Then, similar estimates together with Lemma 9 give us

$$\begin{array}{ccc}& & \mathit{\mu }\left(\frac{\lambda }{\mathrm{\Lambda }}\sum _{i=1}^{n-2}{\eta }_i{\Im }_{a,g}^{\alpha +{\delta }_i,\mu }f({\xi }_i,V\left({\xi }_i\right))-{\Im }_{a,g}^{\alpha ,\mu }f(b,V\left(b\right))\right)\hfill \\ & \le & |\frac{\lambda }{\mathrm{\Lambda }}|\left[\sum _{i=1}^{n-2}{\eta }_i+1\right]{\Im }_{a,g}^{\alpha ,\mu }\underset{s\in I}{sup}p\left(s\right)·\mathit{\mu }\left(V\left(I\right)\right)\hfill \end{array}$$

and hence

$$\mathit{\mu }\left(T\left(V\right)\left(t\right)\right)\le |\frac{\lambda }{\mathrm{\Lambda }}|\left[\sum _{i=1}^{n-2}{\eta }_i+1\right]{\Im }_{a,g}^{\alpha ,\mu }\underset{s\in I}{sup}p\left(s\right)·\mathit{\mu }\left(V\left(I\right)\right)+\left|\lambda \right|{\Im }_{a,g}^{\alpha ,\mu }\underset{s\in I}{sup}p\left(s\right)·\mu \left(V\left(t\right)\right).$$

Applying the relevant properties of measures of weak non-compactness and the definition of the set V, we obtain

$$\mathit{\mu }\left(V\right(t\left)\right)=\mathit{\mu }\left(conv\right(\{0\}\cup T\left(V\right)\left(t\right)\left)\right)=\mathit{\mu }\left(T\right(V\left)\right(t\left)\right)$$

and then

$$\begin{array}{ccc}\hfill \mathit{\mu }\left(V\right(t\left)\right)& =& \mathit{\mu }\left(T\left(V\right)\left(t\right)\right)\le \mathit{\mu }\left(\frac{e^{-\mu g\left(t\right)}{\left(g\left(t\right)\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{c}_0\left(V\right)\left(t\right)\right)+\left|\lambda \right|\mathit{\mu }\left({\Im }_{a,g}^{\alpha ,\mu }V\left(t\right)\right)\hfill \\ & \le & \frac{\lambda }{\mathrm{\Lambda }}|\left[\sum _{i=1}^{n-2}{\eta }_i+1\right]{\Im }_{a,g}^{\alpha ,\mu }\underset{s\in I}{sup}p\left(s\right)·\mathit{\mu }\left(V\left(I\right)\right)+\left|\lambda \right|{\Im }_{a,g}^{\alpha ,\mu }\underset{s\in I}{sup}p\left(s\right)·\mathit{\mu }\left(V\left(t\right)\right).\hfill \end{array}$$

$$v\left(t\right)\le {\mathit{\mu }}_c\left(V\right)·|\frac{\lambda }{\mathrm{\Lambda }}|\left[\sum _{i=1}^{n-2}{\eta }_i+1\right]{\Im }_{a,g}^{\alpha ,\mu }\underset{s\in I}{sup}p\left(s\right)+\left|\lambda \right|{\Im }_{a,g}^{\alpha ,\mu }\underset{s\in I}{sup}p\left(s\right)$$

and by taking the supremum over $t\in I$

$${\mathit{\mu }}_c\left(V\right)\le {\mathit{\mu }}_c\left(V\right)·|\frac{\lambda }{\mathrm{\Lambda }}|\left[\sum _{i=1}^{n-2}{\eta }_i+1\right]{\Im }_{a,g}^{\alpha ,\mu }\underset{s\in I}{sup}p\left(s\right)+\left|\lambda \right|{\Im }_{a,g}^{\alpha ,\mu }\underset{s\in I}{sup}p\left(s\right),$$

and using our assumptions, ${\mathit{\mu }}_c\left(V\right)=0$ and then V is relatively weakly compact subset of $C[I,E_w]$. Applying Theorem 1, we obtain the expected thesis.

Finally, Theorem 3, implies the existence of a fixed point for T, being also a pseudo-solution of the integral Equation (34). □

Remark 13.

Arguing similarly as in ([29], Theorem 5), it is possible to also consider the multivalued case of our main boundary value problem, but this is out of the planned scope of the paper since the study of BVP’s forms only an example of how the introduced generalized fractional calculus is fully applicable and generalizes previous approaches to the topic.

6. Conclusions

In this paper, we show that the tempered fractional calculus can, like the classical calculus of integer order, be extended to the class of vector functions and considered for problems whose assumptions are expressed in terms of the weak topology. The approach presented here using generalized integral operators with the Pettis integral of fractional order covers many known cases, including that for real-valued functions. It also allows interesting observations on the equivalence of differential and integral problems, or lack thereof, which underlies the application of such calculus to boundary problems. The paper is supplemented by a result for boundary value problems.