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Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems
Boundary Value Problems volume 2023, Article number: 57 (2023)
Abstract
In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjestype operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the wellknown properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals.
We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputotype fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch’s fixedpoint theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given.
1 Preliminaries
The domain of fractional calculus is a very rich field because of its applications, for instance, in wave propagation in viscoelastic horns, soundwave propagation or fractional models and controls (see [5, 16, 22]). There are several definitions for fractional integrals and for fractional derivatives [19, 36]. We are interested in the most general form of such operators. Till now, the most general known definition of the fractional operators seems to be the fractional integrals and derivatives of a Lebesgue function f with respect to another function g (see [36, Sect. 18.2], [19, Sect. 2.5] and [5]). However, let us mention that this definition allows us to operate only on realvalued functions. In the past decades, this general definition has proven its applicability in many and different natural situations, for instance, in [5], starting with the exponential growth model, the same problem was described by a fractional differential equation, and we shall see that the choice of the function g determines the accuracy of the model.
Our goal is to expand the applications of such an approach for vectorvalued functions. Recently, considerable attention has been paid to the theory of fractional calculus in abstract spaces, which is more complicated and different from the classical fractional calculus of realvalued functions. This is due to the fact that some of the longknown properties of the realvalued function do not carry over into arbitrary Banach spaces. For instance, the classical fundamental theorem of calculus in Banach spaces is more complicated than the standard one. In addition, the weak absolute continuity of Banachvalued functions does not necessary imply strong or everywhere weak differentiability.
The aim of this paper is twofold. On the one hand, we define and discuss the properties of the generalized form of the new fractional operators applied for the class of Pettis integrable functions that seems to be interesting in itself. On the other hand, we apply those results in order to ensure the existence of weakly continuous solutions for some boundary value problems of fractional order.
We should at least briefly recall why we discuss as one topic the fractional calculus with Orlicz spaces. This goes back to the origin of fractional calculus and fractional operators in function spaces. It is motivated by some applications to integral equations or partial differential equations [24, 27]. On the other hand, Pettis integrability is also strictly related to some weak integrability conditions in Orlicz spaces ([38], for instance).
However, our results complement some of those obtained in [1, 3, 4, 11, 12, 29–31, 35] or [39]), dealing with the properties of the fractional integral and differential operators when acting on the space of Pettis integrable functions.
Let us recall that a function \(\psi :{\mathbb{R} }^{+} \rightarrow { \mathbb{R} }^{+}\) is said to be a Young function if ψ is increasing, even, convex, and continuous with \(\psi (0)=0\) and \(\lim_{u\rightarrow \infty}\psi (u)=\infty \)). For any Young function ψ, the function \(\widetilde{\psi}:{\mathbb{R} }^{+} \rightarrow { \mathbb{R} }^{+}\) defined by \(\sup_{v\geq 0 }\{v u \psi (v)\}\) is called the Young complement of ψ and it is well known that ψ̃ is a Youngtype function as well.
The Orlicz space \(L_{\psi}=L_{\psi}([a,b],{\mathbb{R} })\) consists of all (classes of) measurable functions \(x:[a,b]\rightarrow {\mathbb{R} }\) for which
is finite (see, e.g., [20]). The particular choice \(\psi (u)= \psi _{p}(u) := \frac{1}{p} u^{p}\), \(p \in [1,\infty )\) leads to the 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 ψ, we have \(\psi (uv) \leq \psi (u) \psi (v) \) and \(\psi (\rho u) \leq \rho \psi (u)\) hold for any \(u,v \in {\mathbb{R} }\) and \(\rho \in [0,1]\). Also, for the nontrivial Young function ψ, \(L_{\infty }\subset L_{\psi} \subset L_{\psi}\). For further properties of Young functions and Orlicz spaces generated by such functions we refer the reader to [2, 20, 35].
In the forthcoming pages E will be considered as a Banach space with norm \(\Vert \cdot \Vert \) and with its dual space \(E^{\ast}\). Also, \(E_{w}\) denotes the space E when endowed with its weak topology \(\sigma (E,E^{\ast})\). Let \(C[I,E]\) denote the Banach space of (strongly) continuous functions \(x:I\rightarrow E\) endowed by the norm \(\Vert x \Vert _{0}=\sup_{t\in I} \Vert x(t) \Vert \). By \(C[I,E_{w}]\) we denote the Banach space of all weakly continuous functions \(x:I\rightarrow E \) with its weak topology (i.e., generated by continuous linear functionals on E).
Throughout this paper, we let g be a positive increasing function on an interval \(I := [a,b]\), having a positive continuous derivative, with \(g(a)=0\) (see, e.g., [19, Sect. 2.5] or [36, Sect. 18.2]).
In this paper, we will have one more important class of functions. Namely, we let \(\vartheta : {\mathbb{R} }^{+} \to {\mathbb{R} }^{+}\) be a Hölderian function, i.e., ϑ is increasing and continuous with \(\vartheta (0)=0\). The (generalized) Hölder space \(\mathcal{C}^{\vartheta}[I,E]\) consists, by definition, of all \(x \in C[I, E]\) satisfying
Equipped with the norm
the space \(\mathcal{C}_{g}^{\vartheta}[I,E]\) becomes a Banach space. Elements of \(\mathcal{C}_{g}^{\vartheta}[I,E]\) are called generalized Hölderian functions.
The particular choice \(g(t)=t\), \(\vartheta (t) = t^{\alpha}\), \(\alpha \in (0,1]\) leads, of course, to the classical Hölder space.
Let \(\mathcal{C}_{g}^{\vartheta}[I,E_{w}]\) denote the Banach space of generalized Hölderian functions \(x:I \to E\), with its weak topology (i.e., generated by continuous linear functionals on E).
Recall that the map \(T:X\rightarrow Y\), X and Y are Banach spaces and said to be weakly–weakly sequentially continuous (wwsequentially continuous) if and only it maps weakly convergent sequences \((x_{n})\) to \(x\in E\) into sequences (\(T(x_{n})\)) that are weakly convergent to \(T(x)\) in Y.
Definition 1
([13])
Let \({\mathcal{M}}_{E}\) be a family of all bounded subsets of E and \(B_{1}\) denotes the unit ball of E. The De Blasi measure of weak noncompactness is the mapping
defined by
For the properties of μ see [13]. The following important Ambrosettitype lemma will be used in the paper:
Lemma 1
([23])
Let \(V \subset C[I,E]\) be bounded and strongly equicontinuous. Then,

1.
\(t \mapsto \boldsymbol{\mu}(V(t)) \in C[I, {\mathbb{R} }^{+}]\), where \(V(t) := \{v(t): v \in V, t \in I\}\);

2.
\(\boldsymbol{\mu}_{C}(V)=\sup_{t\in I}\boldsymbol{\mu}(V(t))= \boldsymbol{\mu}(V(I))\),
where \(\boldsymbol{\mu}_{C}\) denotes the De Blasi measure of weak noncompactness in \(C[I,E]\).
For our purpose, we will need the following Mönch fixedpoint theorem whose foundations of use for the weak topology we can find in [6]
Theorem 1
([21])
Let \(\mathcal{Q}\) be a nonempty, closed, convex, and equicontinuous subset of a metrizable locally convex vector space \(C(I,E)\) such that \(0 \in \mathcal{Q}\). Suppose \(T: \mathcal{Q} \rightarrow \mathcal{Q}\) is weakly–weakly sequentially continuous. If the implication
holds for every subset \(V \subset \mathcal{Q}\), then the operator T has a fixed point in \(\mathcal{Q}\).
The following definition goes back to Pettis [28]
Definition 2
(Pettis integral)
A weakly measurable function \(x: I \rightarrow E\) is said to be Pettis integrable on I if

1.
x is Dunford integrable on I, that is, \(\varphi x \in L_{1}\) for every \(\varphi \in E^{\ast}\);

2.
for any measurable \(A\subset I\) there exists an element in E denoted by \(\int _{A} x(s) \,ds\) such that
$$ \varphi \biggl( \int _{A} x(s) \,ds \biggr) = \int _{A} \varphi x(s) \,ds \quad {\text{{for every }}} {\varphi \in E^{\ast}}. $$
By \(P[I,E]\) denote the space of Evalued Pettis integrable functions on I. In particular, the space \(P [I,{\mathbb{R} }] = L_{1}[I,{\mathbb{R} }]\). We need to introduce more function spaces. For convenience, we recall the following:
Definition 3
For any Young function ψ we define a class \(\mathbf{\mathcal{H}}^{\psi} (E)\) as
As its subspace let us consider
Moreover, the class \(\mathbf{\mathcal{H}}^{\psi}_{0}(E)\) (resp., \(\mathbf{\mathcal{\widetilde{H}}}^{\psi}_{0}(E)\)) is defined to be the subspace of \(\mathbf{\mathcal{H}}^{\psi} (E)\) (resp., \(\mathbf{\mathcal{\widetilde{H}}}^{\psi}(E)\)) composed of Pettis integrable functions on I, that is
In particular, the wellknown class \(\mathbf{\mathcal{H}}^{p}_{0} (E) \) denotes the class \(\mathbf{\mathcal{H}}^{\psi}_{0}(E)\) for the particular choice \(\psi \equiv \frac{\cdot ^{p}}{p}\).
Obviously, \(\mathbf{\mathcal{\widetilde{H}}}^{\psi}_{0} (E) \subseteq \mathbf{\mathcal{H}}^{\psi}_{0} (E) \subseteq \mathbf{\mathcal{H}}^{ \psi}(E)\) and \(\mathbf{\mathcal{\widetilde{H}}}^{\psi}_{0} (E) \equiv \mathbf{\mathcal{H}}^{\psi}_{0} (E)\) holds true whenever E is separable (cf. [28, Corollary 1.11]). Some special facts about these spaces are known (cf. [14, 28, 38]):
Proposition 1

(1)
If E is reflexive, then \(\mathbf{\mathcal{H}}^{1}(E) \equiv \mathbf{\mathcal{H}}^{1}_{0}(E)\).

(2)
For any Young function ψ with \(\lim_{u \to \infty}\psi (u)/u \to \infty \), \(\mathbf{\mathcal{\widetilde{H}}}^{\psi}(E) \subseteq \mathbf{\mathcal{H}}^{\psi}_{0}(E)\). In particular, \(\mathbf{\mathcal{\widetilde{H}}}^{p}(E) \subseteq \mathbf{\mathcal{H}}^{p}_{0}(E)\) 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, \(\mathbf{\mathcal{\widetilde{H}}}^{1}(E) \subseteq \mathbf{\mathcal{H}}^{1}_{0}(E)\) whenever E satisfies this additional condition.
Clearly, since the weak continuity implies a strong measurability (see [18, page 73]), in view of Proposition 1 it implies that:
Corollary 1
For any nontrivial Young function ψ the space \(C[I,E_{w}]\) is a proper subset of \(\mathbf{\mathcal{\widetilde{H}}}^{\psi}_{0} (E)\).
Let us stress that the connection between the Pettis integrability and Orlicz spaces is much deeper than presented in [38] (see [7]). In the following, we will integrate vectorvalued functions with respect to some realvalued ones. For this reason we recall the results that complement some of those from [28, 35], dealing with the integrability of Pettis integrable functions multiplied by realvalued ones.
Proposition 2
([11, Proposition 5])
If \(x\in \mathbf{\mathcal{H}}^{\psi}_{0}(E)\), then \(x(\cdot )y(\cdot )\in P[I,E]\) for every \(y \in L_{\tilde{\psi}}\).
Let us stress that y cannot be vector valued, unless the space E is a Banach algebra. Now, we should state an immediate, but important, consequence of Proposition 2:
Proposition 3
(cf. [28, Corollary 3.41])
If \(x \in P[I,E]\), then \(x(\cdot )y(\cdot )\in P[I,E]\) for every \(y \in L_{\infty}[I]\).
Let us recall necessary definitions and known facts about weaktype derivatives in Banach spaces. Let us collect all of them that are applied for problems described in the paper.
Definition 4
Consider a vectorvalued function \(x :I \rightarrow E\). If for every \(\varphi \in E^{*}\) functions φx are differentiable almost everywhere on I and if there exists a function \(y: I \rightarrow E \) such that for every \(\varphi \in E^{*}\) there exists a null set \(N(\varphi ) \subset I\) with
then the function x is said to be pseudodifferentiable on I.
In this above definition, y is called a pseudoderivative of x. If the null set independent of φ, then x is said to be a.e. weakly differentiable on I and y (in this case) is called a weak derivative of x and exists almost everywhere on I. In particular, when \(E={\mathbb{R} }\) it is clear that the pseudo and a.e. weak derivatives coincide with the classical derivatives of realvalued functions.
Let \(\mathfrak{D}_{p}\) denote the pseudodifferential operator (resp., \(\mathfrak{D}_{\omega}\) for the weak one). The best result for a descriptive definition of the Pettis integral is that given by Pettis in [28, Sect. 8] (see also [25, Theorem 5.1] and [18, 23]).
Lemma 2

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

(2)
A function \(x: I \to E\) is an indefinite Pettis integral if and only if x is weakly absolutely continuous and has a pseudoderivative \(\mathfrak{D}_{{p}} x\) on I. In this case, \(\mathfrak{D}_{{p}} x \in P[I,E]\) and
$$ x(t)=x(a)+ \int _{a}^{t} \mathfrak{D}_{{p}} x (s) \,ds, \quad t\in I. $$
Before embarking on the next section, we remark that it is natural to assume that the space E has total dual, i.e., a countable determining set. In fact, if E is separable, then both E and \(E^{\ast}\) have total dual, so even spaces like \(BV(I)\) or \(L_{\infty}(I)\) have this property. In this connections, all considered pseudoderivatives of a function from I to E, will be uniquely determined up to a set of measure zero. Deep results concerning this problem can be found in [26, Corollary 3.4, Theorem 3.6].
We also recall the following facts: for any continuous \(g: I \to {\mathbb{R} }\) having a positive, continuous derivative \(g'\) on I, Proposition 3 may be combined with Corollary 1 in order to assure that \(x(\cdot )g'(\cdot ) \in P[I,E]\) (resp. \(x( \cdot )g'(\cdot ) \in C[I,E_{w}]\)) holds true for every \(x \in \mathbf{\mathcal{H}}^{\psi}_{0} (E)\) (resp., \(x \in C[I,E_{w}]\)). From which, in view of Lemma 2, it follows that
Remark 1
Let us note that

The fact that the indefinite Pettis integral of a function \(x \in P[I,E]\) does not enjoy the strong property of being a.e. weakly differentiable (see [15]), tells us that \((\maltese )\) does not necessarily hold for arbitrary \(x \in P[I,E]\).

The formula \((\diamondsuit )\) is not uniquely determined unless E has total dual \(E^{\ast}\). Evidently, according to (e.g., [37, page 2] and [10]), it may happen that \((\frac{1}{g'(t)}\mathfrak{D}_{\omega} ) \Im ^{1,g}_{a} x=y\), with y being weakly equivalent to x (but they need not be necessarily a.e. equal).
2 Generalized fractional integrals
Various modifications and generalizations of classical fractional integration operators are known and are widely used both in theory and applications. In this section, we dwell on such modifications such as fractional integrals of a given function x with respect to another function g.
Definition 5
The generalized fractional (or gfractional) integral of a given function \(x: [a,b]\rightarrow E\) of order α is defined by
For completeness, we define \(\Im ^{\alpha ,g}_{a} x(a):= 0 \). In the preceding definition the sign “∫” stands for the Pettis integral (in particular, the Lebesgue integral when \(E={\mathbb{R} }\)).
It should be noted that, for the realvalued function \(x \in L_{1}[a,b]\), it is well known that (see, e.g., [5, 36]) \(\Im ^{\alpha ,g}_{a} x\) makes sense a.e. on I and \(\Im ^{\alpha ,g}_{a} \Im ^{ \beta ,g}_{a} x= \Im ^{ \beta ,g}_{a} \Im ^{\alpha ,g}_{a} x= \Im ^{\alpha +\beta ,g}_{a} x\) holds true for any \(\alpha , \beta >0\). We also remark that, in a special case \(g(t)=t\), \(t \in [a,b]\) or \(g(t)=\ln t\), \(t \in [1,e]\) we obtain two classical fractional integral operators: the RiemannLouville and the Hadamard ones.
Definition 5 allows us to unify different fractional integral for vectorvalued functions and consequently, in a unified manner, to solve some boundary value problems with different types of fractional integrals and derivatives. Clearly, it is not only a unification, we extend existing results too.
Example 2.1
Let \(\alpha >0\) and \(J \subset I\) be a set of positive measure. Consider the Banach space \(E=B[I]\) of bounded realvalued functions on I. Define a weakly measurable function \(x: I \to B[I]\) by
Obviously \(x \in P[I,B[I]]\). To see this, let us remark that any \(\varphi \in B^{\ast}[I] \) may be identified with a countable additive measure ζ defined on the σalgebra on I. More precisely, every bounded linear functional on \(B[I]\) is of the form \(x \longmapsto \int _{I} x(t) \,d\zeta \) for some countable additive measure ζ. Thus, for every measurable \(\Sigma \subset I\) we have
From which, by the definition of the Pettis integral, we conclude that \(x \in P[I,B[I]]\) as claimed. Now, we will show that \(\Im ^{\alpha ,g}_{a} x\) exists on I with \(\Im ^{\alpha ,g}_{a} x=\theta \): Evidently, for every measurable \(\Sigma \subset I\) we have
That is, by the definition of the Pettis integral, \(\Im ^{\alpha ,g}_{a} x\) exists on I and \(\Im ^{\alpha ,g}_{a} x=\theta \).
Remark 2
For any \(\alpha \geq 1\), \(\Im ^{\alpha ,g}_{a} x\) exists for any \(x\in \mathbf{\mathcal{H}}^{1}_{0} (E)\). This is a direct consequence of Proposition 3, as we obtain \(s \to (g(t)g(s))^{\alpha 1}g'(s)\in L_{\infty}[a,t]\) for a.e. \(t \in [a,b]\).
We sometimes considered some special cases of spaces E. Let us present one useful one:
Lemma 3
Let \(\alpha \in (0,1]\) and assume that E has no isomorphic copy of \(c_{0}\). Then, \(\Im ^{\alpha ,g}_{a}: \mathbf{\mathcal{\widetilde{H}}}^{1}_{0}(E) \to P[I,E]\).
Proof
Let \(x\in \mathbf{\mathcal{\widetilde{H}}}^{1}_{0}(E)\). By virtue of the fact that the strong measurability is preserved under a multiplication operation of functions (cf. e.g., [18]), the product \((g(t)  g(\cdot ))^{\alpha 1} g'(\cdot )x(\cdot ): [a, t] \to E\) is strongly measurable on \([0, t]\) for almost every \(t \in I\). Consequently, by Young’s inequality, it can be shown that for every \(\varphi \in E^{\ast}\), the realvalued function \(s \mapsto \varphi ((g(t)  g(s))^{\alpha 1} g'(s)x(s)) = (g(t)  g(s))^{ \alpha 1} g'(s) \varphi (x(s))\) is Lebesgue integrable on \([a, t]\), for almost every \(t \in I\). Hence, the existence of \(\Im ^{\alpha ,g}_{a} x\) follows from [17, Theorem 22].
Now, we proceed in order to show that \(\Im ^{\alpha ,g}_{a}: \mathbf{\mathcal{\widetilde{H}}}^{1}_{0}(E) \to P[I,E]\). To see this, let \(x\in \mathbf{\mathcal{\widetilde{H}}}^{1}_{0}(E)\), define \(y:= \Im ^{\alpha ,g}_{a} x \) and note that \(y \in \mathbf{\mathcal{H}}^{1}(E) \). Thus, for any interval \([c,d] \subseteq I\), and any \(\varphi \in E^{\ast}\) we have
where
Since \(x \in P[I,E]\), then owing to Proposition 3, we have that \(x(\cdot ) (g(c)g(\cdot ))^{\alpha} g'(s)\) and \(x(\cdot ) (g(b)g(\cdot ))^{\alpha} g'(\cdot )\) are Pettis integrable on I and so \(x_{[c,d]} \in E\). A combination of these results yields \(y \in \mathbf{\mathcal{H}}^{1}(E)\) and there exists an element \(x_{[c,d]} \in E\) such that \(\varphi ( x_{[c,d]})= \int _{c}^{d} \varphi (y(t)) \,dt\), for every \(\varphi \in E^{\ast}\) and any \([c,d] \subseteq I\). Since E has no copy of \(c_{0}\), it follows in view of [17, Theorem 23] that \(y \in P[I,E]\). The lemma is thus proved. □
In what follows, we outline and prove some aspects of a g–fractional integral in Banach spaces and weak topologies. The following theorem complements similar results in [32, Lemma 1] and [11, Theorem 2] dealing with the statements revealing how much the fractional integral \(\Im ^{\alpha ,g}_{a} x\) is “better”, in the sense of space inclusions, than the function x.
Theorem 2
Let \(\alpha \in (0,1]\). For any Young function ψ with its complementary Young function ψ̃ satisfying
the operator \(\Im ^{\alpha ,g}_{a}\) maps the space \(\mathbf{\mathcal{H}}^{\psi}_{0} (E)\) into the (generalized) Hölder space \(\mathcal{C}_{g}^{\widetilde{\Psi}_{\alpha}}[I,E_{w}]\). Also, for any \(x \in \mathbf{\mathcal{H}}^{\psi}_{0} (E) \) there is \(\varphi \in E^{\ast}\), with \(\Vert \varphi \Vert = 1\) such that
In particular, \(\Im ^{\alpha ,g}_{a}:C[I,E_{w}]\rightarrow \mathcal{C}_{g}^{ \widetilde{\Psi}_{\alpha}}[I,E_{w}]\). Here, \(\widetilde{\Psi}_{\alpha}: {\mathbb{R} }^{+} \to { \mathbb{R} }^{+} \) is defined as
To make the proof of Theorem 2 simpler we split it into several stages, providing the following lemmas:
Lemma 4
([11, Proposition 2])
For any \(\alpha \in (0,1]\), the function \(\widetilde{\Psi}_{\alpha} \) defined as in (5) is a Hölderiantype function, i.e., \(\widetilde{\Psi}_{\alpha}\) is well defined, increasing, and continuous with \(\widetilde{\Psi}_{\alpha}(0)=0\). In other words, the space \(\mathcal{C}_{g}^{\widetilde{\Psi}_{\alpha}}[I,E_{w}]\) is a Hölderiantype space.
Proof
It is clear that for any \(t>0\), the function
has a positive derivative for sufficiently large \(\sigma >0\) (because \(\widetilde{\psi}(u) \to 0\) as \(u \to 0\)). Consequently, for any \(t>0\), there is \(\sigma >0\) such that \(u_{t}(\sigma ) >0 \) and then for any \(t>0\) the set
Together with \(\widetilde{\Psi}_{\alpha}(0)=0\), this implies that \(\widetilde{\Psi}_{\alpha} \) is well defined on I. In view of the definition of \(\widetilde{\Psi}_{\alpha}\), for \(0 \leq t \leq s\) we have
Thus, we may put \(k= \widetilde{\Psi}_{\alpha}(s)\) in (5), which implies \(\widetilde{\Psi}_{\alpha}(t) \leq \widetilde{\Psi}_{\alpha}(s)\), as required for the monotonicity of \(\widetilde{\Psi}_{\alpha}\). Finally, the continuity of \(\widetilde{\Psi}_{\alpha}\) follows from the continuity and concavity of \(t \mapsto \int _{0}^{t} \widetilde{\psi}(s^{\alpha 1}) \,ds\). □
Lemma 5
Let \(\alpha \in (0,1]\). For any Young function ψ with its Young complement ψ̃ satisfying (4), the integral \(\Im ^{\alpha ,g}_{a} x\) exists (is convergent) for any \(x\in \mathbf{\mathcal{H}}^{\psi}_{0} (E)\). Moreover, it is true for every \(x \in \mathbf{\mathcal{\widetilde{H}}}^{\psi}(E)\) provided ψ satisfies the additional property that \(\lim_{u \to \infty}\psi (u)/u \to \infty \).
In particular, if E is reflexive (resp., weakly complete), \(\Im ^{\alpha ,g}_{a} x\), \(x\in \mathbf{\mathcal{H}}^{\psi} (E)\) (resp., \(x\in \mathbf{\mathcal{\widetilde{H}}}^{\psi} (E) \)) exists for any nontrivial Young function ψ.
Proof
First, let us define \(u:I\rightarrow {\mathbb{R} }^{+}\) by
and observe that for any \(t \in I\) the function
has a positive derivative for some sufficiently large \(\eta >0\) (because \(\widetilde{\psi}(u) \to 0\) as \(u \to 0\)). Consequently, for any \(t \in I\) there is a sufficiently large \(\eta >0\) such that \(u_{t}(\eta ) >0 \) and thus for any \(t \in I\)
This is in line with the following observations that they give:
hold for any \(k>0\), so \(u \in L_{\widetilde{\psi}}(I)\). The assertion of our lemma follows directly from Proposition 2.
Now, we claim that \(\Im ^{\alpha ,g}_{a} x\) exists for any \(x \in \mathbf{\mathcal{\widetilde{H}}}^{\psi}(E)\) with ψ satisfying the additional property \(\lim_{u \to \infty}\psi (u)/u \to \infty \). In view of the above observation, it follows from part (2) of Proposition 1.
Next, let us assume that E is weakly complete, \(x \in \mathbf{\mathcal{\widetilde{H}}}^{\psi}(E) \) for arbitrary ψ and note that in this case \(\mathbf{\mathcal{\widetilde{H}}}^{\psi}(E) \subset \mathbf{\mathcal{\widetilde{H}}}^{1}(E)\). Since the strong measurability is preserved under a multiplication operation, the pointwise product of strongly measurable functions \((g(t)g(\cdot ))^{\alpha 1}g'(\cdot )x(\cdot ): [a,t] \to E\) is strongly measurable on \([a,t]\), \(t \in I\). In view of Young’s inequality, we know that for every \(\varphi \in E^{\ast}\), the realvalued function, \(\varphi ((g(t)g(\cdot ))^{\alpha 1}g'(\cdot )x(\cdot ) ) = (g(t)g(\cdot ))^{\alpha 1}g'(\cdot )\varphi x(\cdot )\) is Lebesgue integrable on \([a, t]\) for every \(t \in I\). Hence, the result is a consequence of part (2) of Proposition 1.
Similarly, when E is reflexive, the result follows from part (1) of Proposition 1. In this case indeed, as for any nontrivial ψ we have \(\mathbf{\mathcal{H}}^{\psi} (E) \subseteq \mathbf{\mathcal{H}}^{1}(E) \). Consequently, for any \(x \in \mathbf{\mathcal{H}}^{\psi} (E)\) and every \(\varphi \in E^{\ast}\) the measurable realvalued function \(\varphi ((g(t)  g(\cdot ))^{\alpha 1}g'(\cdot )x(\cdot ) ) = (g(t)g(\cdot ))^{\alpha 1}g'(\cdot )\varphi x(\cdot )\) is Lebesgue integrable on \([a, t]\) for every \(t \in I\), and hence is weakly measurable. The fact that in reflexive spaces any weakly measurable \(u: I \to E\) is Pettis integrable if and only if \(\varphi u \in L_{1} \) holds for every \(\varphi \in E^{\ast}\) (cf. Lemma 1 part (1)), guarantees the existence of \(\Im ^{\alpha ,g}_{a} x\) on I. □
Remark 3
According to the assertion of Lemma 5, the function \((g(t)g(\cdot ))^{\alpha 1}g'(\cdot ) x(\cdot ) \in P[[a,t], E]\) for every \(t\in I\) and any \(x\in \mathbf{\mathcal{H}}^{\psi}_{0} (E)\). Consequently, accordingly to the definition of a Pettis integral for any \(t \in I\) there exists an element of E denoted by \(\Im ^{\alpha ,g}_{a} x(t)\) such that
holds true for every \(\varphi \in E^{\ast}\).
Remark 4
We should remark that, if \(\Im ^{\alpha ,g}_{a} x\) does not exist for some \(x \in \mathbf{\mathcal{H}}^{\psi}_{0} (E)\), then it cannot exist if we “enlarge” the space E into F. To see this, we argue by contradiction assuming that \(\Im ^{\alpha ,g}_{a} x\) (when we consider x as a function from \(\mathbf{\mathcal{H}}^{\psi}_{0} (F)\)) exists. In this case, for the particular choice for the functional \(\varphi \in F^{\ast}\) having \(\varphi _{E}=\theta \) we conclude, in view of (8) and \(x(I) \subseteq E\), that \(\varphi (\Im ^{\alpha ,g}_{a} x(t) )=\Im ^{\alpha ,g}_{a} \varphi (x(t) )=0\), from which \(\Im ^{\alpha ,g}_{a} x(t) \in E\). This would lead to a contradiction.
Remark 5
Let a Young function ψ be such that the integral in (4) is finite. For any \(\alpha \in (0,1)\), the assertion of Theorem 2 is still valid if at least one of the following cases holds true:

1.
\(x \in \mathbf{\mathcal{\widetilde{H}}}^{\psi}(E)\), where ψ satisfies the additional property \(\lim_{u \to \infty}\psi (u)/u \to \infty \);

2.
E is weakly complete and \(x \in \mathbf{\mathcal{\widetilde{H}}}^{\psi}(E)\);

3.
E is reflexive and \(x \in \mathbf{\mathcal{H}}^{\psi} (E)\).
Evidently, it follows from Theorem 2, as in view of Lemma 5, in all of the above cases we have \(\mathbf{\mathcal{\widetilde{H}}}^{\psi}(E) \subseteq \mathbf{\mathcal{H}}^{\psi} (E) \subseteq \mathbf{\mathcal{H}}^{\psi}_{0} (E)\).
We are now ready to provide the proof of Theorem 2.
Proof
of Theorem 2. Let \(a\leq t_{1}\leq t_{2}\leq b\) and \(x \in \mathbf{\mathcal{H}}^{\psi}_{0} (E) \). According to Lemma 5 and by the definition of the indefinite Pettis integral, we ensure that \(\Im ^{\alpha ,g}_{a} x\) is well defined. In view of Remark 3, it allows us to state the following chain of inequalities
where
and
We claim that \(h_{i}\in L_{\widetilde{\psi}}(I)\), \((i=1,2)\). Once our claim is established, in view of the Hölder inequality in Orlicz spaces, we conclude that
It remains to prove our claim by showing that \(h_{i}\in L_{\widetilde{\psi}}(I)\), \(i=1,2\). To see this, fix \(k>0\). An appropriate substitution, using some properties of Young functions, leads to the following estimation
In view of (6), the above observations guarantee the existence of \(k>0\) for which \(\int _{a}^{b} \widetilde{\psi} (\frac{h_{1}(s)}{k} ) \,ds\leq 1\). Then, we can conclude that \(h_{1}\in L_{\widetilde{\psi}}(I)\). Moreover, our definitions of \(\widetilde{\Psi}_{\alpha}\) and the norm in Orlicz spaces, along with the above observations, give us
Arguing similarly as above, we can show that
Thus, for any \(\varphi \in E^{\ast}\) equation (9) takes the form
This may be combined along with the Hahn–Banach theorem, in order to assure that
holds true for some \(\varphi \in E^{\ast}\) with \(\Vert \varphi \Vert =1\). Hence, \(\Im ^{\alpha ,g}_{a}: \mathbf{\mathcal{H}}^{\psi}_{0} (E) \rightarrow \mathcal{C}_{g}^{\widetilde{\Psi}_{\alpha}}[I,E_{w}]\). Also,
Moreover, in view of our definition \(\Im ^{\alpha ,g}_{a} x(a):= 0\), we observe that
We finally obtain
In this connection, the particular case follows from Corollary 1 and the theorem is then proved. □
Example 2.2
Let \(\alpha \in (0,1)\) and \(\psi (u)= \psi _{p}(u) := \frac{1}{p} u^{p}\), \(p \in (1,\infty )\). In this case, we have \(\tilde{\psi}_{p}= \psi _{\tilde{p}}\) with \(\frac{1}{p}+\frac{1}{\tilde{p}}=1\). It can be easily seen that (4) holds true if and only if \(p>\frac{1}{\alpha}\). From which we conclude that \(\Im ^{\alpha ,g}_{a}\) maps the Bochner space \(L_{p}[I,E]\), \(p>\frac{1}{\alpha}\) into the Hölder space \(\mathcal{C}_{g}^{\widetilde{\Psi}_{\alpha}}[I,E_{w}]\), where
For instance, in view of the above observation, \(\Im ^{\alpha ,g}_{a}: L_{2}[I, {\mathbb{R} }] \to \mathcal{C}^{\widetilde{\Psi}_{\alpha}}[I,{\mathbb{R} }]\) for \(\alpha \in (0.5,1)\) with \(\widetilde{\Psi}_{\alpha}(t)= \frac{t^{\alpha \frac{1}{2}}}{\sqrt{4\alpha 2}}\).
Remark 6
Theorem 2 may be combined with [11, Example 1] in order to assure the existence of a Young function ψ (for instance, \(\psi (u):= e^{u}u1\)) for which \(\Im ^{\alpha ,g}_{a}\) maps \(\mathbf{\mathcal{H}}^{\psi}_{0} (E)\) into \(\mathcal{C}_{g}^{\widetilde{\Psi}_{\alpha}}[I,I,E_{w}]\) “for all” \(\alpha \in (0,1]\). According to Example 2.2, this interesting phenomenon has no analog in the case of Lebesgue spaces \(L_{p}[I, {\mathbb{R} }]\)).
Example 2.3
Let \(\alpha >0\) and \(a,b\in {\mathbb{R} }^{+}\) such that \(ba=1\). Define a strongly measurable function \(x:[a,b] \to L_{2}[a,b]\) by
where \(\{e_{n}\}\) is an orthonormal system in \(L_{2}[a,b]\) and \(I_{n}'\)s are the pairwise disjoint subintervals of \([a,b]\) defined by \(I_{n}=(a+1/2^{n}, a+1/2^{n}+1/4^{n})\), \(n \in {\mathbb{N} }\). Since
holds true for every \(\varphi \in L_{2}[a,b]^{\ast}=L_{2}[a,b] \), we obtain \(\varphi x \in L_{2}[a,b]\) for every \(\varphi \in L_{2}[a,b]^{\ast}\). Hence, \(x \in P[[a,b], L_{2}[a,b]]\) (by applying Proposition 1). More precisely, \(x \in \mathbf{\mathcal{H}}^{\psi _{2}}_{0} (L_{2}[a,b])\). Since \(L_{2}[a,b]\) is reflexive, the integral \(\Im ^{\alpha ,g}_{a} x\) exists for any \(\alpha >0\) (cf. Remark 2 when \(\alpha \geq 1\) and Remark 5 when \(\alpha \in (0,1)\)). Moreover, in view of Example 2.2, we know that \(\Im ^{\alpha ,g}_{a} x \in \mathcal{C}_{g}^{\widetilde{\Psi}_{\alpha}}[[a,b], (L_{2}[a,b])_{\omega}]\), with \(\widetilde{\Psi}_{\alpha}(t)= \frac{t^{\alpha \frac{1}{2}}}{\sqrt{4\alpha 2}}\) holds for any \(\alpha \in (0.5,1)\).
Example 2.4
Let \(\alpha \in (0,1]\) and define \(x:[0,1] \to L_{1}[0,1]\) by
This function is weakly continuous on \(I = [0,1]\). Indeed, if \(\phi \in L_{\infty }\cong L_{1}^{*}\) corresponds to \(\varphi \in L_{1}^{*} \), then \(\varphi (x(t)) = \Im ^{ 1\alpha ,g}_{a} ( \frac{ \phi (t)}{g'(t) } )\). Since \(\Im ^{ 1\alpha ,g}_{a}\) maps \(C[I, {\mathbb{R} }]\) into itself, we can conclude that \(\varphi x \in C[I, {\mathbb{R} }]\) for every \(\varphi \in L_{1}^{*} \) that gives a reason to believe that x is weakly continuous on I. Consequently, in view of Theorem 2, it follows that \(\Im ^{\alpha ,g}_{a} x\) exists on I. In this context, we can show that
This is easy to demonstrate because, by letting \(\phi \in L_{\infty }\) corresponding to \(\varphi \in L_{1}^{*} \) and carrying out the necessary calculations using the substitution \(s=\frac{g(s)g(\xi )}{g(t)g(\xi )}\), it can be verified that
as needed for (12).
In view of the semigroup property of \(\Im ^{\alpha ,g}_{a}\) in Lebesgue spaces, an analogous reasoning as in [11, Lemma 2] gives us the following:
Lemma 6
Let \(\alpha , \beta \in (0,1]\). If \(x\in \mathbf{\mathcal{H}}^{\psi}_{0}(E)\), where ψ is a Young function with its complement ψ̃ satisfying
then
In particular, the property (14) holds true for every \(x \in C[I,E_{w}]\).
Let us investigate some important properties of generalized fractional integrals with Pettis integrals and measures of weak noncompactness. We need to prove a Goebel–Rzymowski lemma that is important in our considerations and very useful in many similar problems. We follow the idea from [9].
Lemma 7
Let μ be the De Blasi measure of weak noncompactness. For any \(\alpha \in (0,1]\), \(t \in I\) and any bounded strongly equicontinuous set \(V \subset C[I,E_{w}]\)
Proof
At the beginning, we note, in view of Theorem 2, that \(\Im ^{\alpha ,g}_{a} v\) exists and weakly continuous on I. Hence, \(\boldsymbol{\mu} ( \Im ^{\alpha ,g}_{a} V(t) )\) makes sense. Next, define a function \(G:I \times I \to {\mathbb{R} }^{+}\) by
From the above definition we have \(\Im ^{\alpha ,g}_{a} x(t) = \int _{a}^{t} G(t,s)x(s) \,ds\). From the properties of the Pettis integral for arbitrary \(w \in P[I,E]\) and \(t \in I\) we have
As V is equicontinuous, the set \(\{G(t,\cdot )V(\cdot ) \}\) is Pettis uniformly integrable on I, so for any \(x \in V\) the set \(\{ \varphi (G(t,\cdot )x(\cdot )) : \varphi \in E^{\ast}, \\varphi \ \leq 1 \}\) is equiintegrable. Then, for any \(\varepsilon > 0\) there exists (sufficiently small) τ such that
Thus, we can cover the set \(\{ \int _{t\tau}^{t} G(t,s) v(s) \,ds: s \in [t\tau ,t], v \in V \}\) by balls with radius less than ε and then
Now, let us estimate the set of integrals on \([a,t\tau ]\). Put \(v(\cdot ) = \boldsymbol{\mu} (V(\cdot ))\). In view of Lemma 1, v is a continuous function. Note that from our assumption it follows that \(s \to G(t,s)v(s)\) is continuous on \([a,t\tau ]\), and hence uniformly continuous.
Thus, there exists \(\delta > 0\) such that
provided that \(qs < \delta \) 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 \(t_{i}  t_{i1} < \delta \) for \(i=1,2,\ldots ,n\). Put \(T_{i} = [t_{i1},t_{i}]\). As v is uniformly continuous, there exists \(s_{i} \in T_{i}\) such that \(v(s_{i}) = \beta (V(T_{i}))\) (\(i=1,2, \ldots ,n\)).
As
by the mean value theorem for the Pettis integral
Hence,
Note that from (16) it follows that
Then,
and
As ε is arbitrarily small, we obtain
i.e.,
It remains to prove the second estimation. Let us observe that
As \(g(a) = 0\),
Thus, \(\Im ^{\alpha ,g}_{a} \boldsymbol{\mu}( V(t) ) \leq \frac{(g(t))^{\alpha}}{\alpha \cdot \Gamma (\alpha )}\cdot \boldsymbol{\mu}_{c}(V) \leq \frac{ \Vert g \Vert ^{\alpha}}{\Gamma (1 + \alpha )} \cdot \boldsymbol{\mu}_{c}(V) \). □
3 Generalized fractional derivatives
From now, the definitions of the gfractional derivatives of x become a natural requirement.
Definition 6
The g–Caputo fractionalpseudo (resp., weak) derivative of a given function x of order \(\alpha \in (m,m+1]\), \(m \in {\mathbb{N} }:= \{0,1,2, \ldots \}\) is defined by
Here, \(\mathfrak{\delta}_{{p}}\) and \(\mathfrak{\delta}_{\omega}\) are defined as
Remark 7
It is worthwhile to remark here that \(\frac{ d^{\alpha ,g}_{{p}}}{dt^{\alpha}} x\) (if exists), does not depend on the choice of the mth pseudoderivatives of x. Evidently, if \(\mathfrak{\delta}_{{p}}^{m} x=y_{1}\), \(\mathfrak{\delta}_{ {p}}^{m} x=y_{2}\), we know that \(y_{1}\), \(y_{2}\) are weakly equivalent on I. It follows that
Hence, \(\Im ^{m\alpha , g}_{a} y_{1}(t) =\Im ^{m\alpha , g}_{a} y_{2}\) as needed.
This is a good place to remark that the conditions required for the existence of gCaputo fractional derivative are very restrictive. A very rough condition that ensures the existence of \(\frac{ d^{\alpha ,g}_{p}}{dt^{\alpha}} x \) is that \(x \in AC^{m1}[[a,b], E_{w}]\). In other words, the g–Caputotype fractional derivative has the disadvantage that it completely loses its meaning if \(D^{m1} x\) fails to be (almost everywhere) differentiable on \([a,b]\). Unfortunately, even in the Hölder spaces, outside of the space of absolutely continuous functions, the gCaputotype fractional differential operator does not enjoy the “nice” behavior of being left inverse of the corresponding gfractional integral operator. In other words, outside of the space of absolutely continuous functions, the equivalence of the gfractional integral equations and the corresponding gCaputo fractional differential problem is no longer necessarily true even in the Hölder spaces. This goes back to the wellknown fact that the RiemannLouville fractional integral operator \(\Im _{0}^{\alpha ,t}\) is a continuous mapping from Hölder spaces “onto” Hölder spaces (which, of course, contains also continuous nowhere differentiable functions), see, e.g., [36, Theorem 13.13]. Indeed, in what follows, we will show that even in the context of realvalued Hölderian functions the converse implication from the fractional integral equations to the corresponding Caputotype differential form is no longer necessarily true.
To see this, let us consider a particular form of the fractional integral operator \(\Im ^{\beta ,g}_{a}\), \(\alpha \in (0,1)\) with \(g(t)=t\), \(t \in [0,1]\), \(E={\mathbb{R} }\). Let x be a Hölderian (but nowhere differentiable on \([0,1]\)) function of some critical order \(\gamma < 1\). According to [36, Theorem 13.13] we know that there is \(\alpha \in (0,1)\) depending only on γ and a Hölderian function \(y \notin AC[0,1]\) such that \(\Im _{0}^{\alpha ,t}y=x\). From this we can conclude that \(\frac{ d^{\alpha ,t}_{p}}{dt^{\alpha}} \Im _{0}^{\alpha ,t} y= \frac{ d^{\alpha ,t}_{p}}{dt^{\alpha}} x\) is “meaningless”. This gives a reason to believe that even on Hölder spaces (but out of the space of absolutely continuous functions), the operator \(\frac{d^{\alpha}}{d t^{\alpha}}\) has no left inverse of \(\Im _{0}^{\alpha ,t} y\) as required. For more examples revealing the lack of equivalence between differential and integral forms of the Caputotype fractional problems, we refer the reader to [12]. It will be clarified later how to avoid such a phenomenon (see formula \((\diamondsuit )\) and Lemma 8 below).
However, the following example shows that on the space \(C[I,E_{w}]\), but still outside of the space of weakly absolutely continuous functions, it is no longer necessarily true that \(\frac{ d^{\alpha ,g}_{p}}{dt^{\alpha}}\) is a left inverse of \(\Im _{a}^{\alpha ,g}\) for any \(\alpha >0\).
Example 3.1
Let \(\alpha \in (m,m+1]\), \(m\in {\mathbb{N} }\). Define \(x:[0,1] \to L_{1}[0,1]\) by
Reasoning as in Example 2.4, we can ensure that this function is weakly (but not weakly absolutely) continuous on I having a gCaputo fractional integral of order \(\alpha m \in (0,1]\) on I and that \(\Im ^{\alpha m,g}_{a} x(t)(\cdot )=\chi _{[a,t]}(\cdot )\). In this connection, in view of the continuity of x in Theorem 2 and Lemma 6, implies that
By the aid of (♢), it follows that \(\mathfrak{\delta}_{{p}}^{m} \Im ^{\alpha ,g}_{a} x= \chi _{[a,t]}\). Since \(\chi _{[a,t]}(\cdot )\), \(t \in I\) is weakly absolutely continuous and have no pseudo (so trivially no weak) derivatives on I (see [33, Theorem 3]), we conclude that the g–Caputo fractional pseudo (trivially weak) derivative is “meaningless”. Namely, \(\frac{ d^{\alpha ,g}_{p}}{dt^{\alpha}} \Im ^{\alpha ,g}_{a} x\neq x\) as required.
In order to avoid such a problem with the equivalence of the gCaputotype boundary value problem of fractional orders \(\alpha >1\) and the corresponding integral form, we are, similarly as in [12], going to modify (slightly) our definition of the g–Caputotype fractional differential operator into a more suitable one
Definition 7
The modified g–Caputo fractional pseudo (resp., weak) derivative “briefly MCFPD (resp., MCFWD)” of order \(m+\alpha \), \(m \in {\mathbb{N} }\), \(\alpha \in (0,1)\) applied to the function \(x \in P[I,E]\) is defined as
Obviously, Definition 7 coincides with the usual definition of the g–Caputotype fractional differential operators when \(m=0\). Also, unless the space E has total dual \(E^{\ast}\) (cf. [10]), a g–Caputo fractional pseudoderivative of x is not necessary uniquely determined.
In what follows we will show that the results obtained in Example 3.1 have no analog in the case of MCFPD whenever \(\alpha >1\). Evidently, arguing similarly as in [12, proof of Lemma 7], we can prove the following:
Lemma 8
Let \(\alpha >1\). Assume that \(\alpha =m+\eta \), where \(m\geq 1\) with some \(\eta \in (0,1)\). If ψ is a Young function with its complementary function ψ̃ satisfying
then \(\frac{d^{ \alpha ,g}_{{p}} }{d t^{ \alpha}} \Im ^{\alpha ,g}_{a}\) is well defined on \(\mathbf{\mathcal{H}}^{\psi}_{0}(E)\). If, additionally, the space E has total dual, then \(\frac{d^{ \alpha ,g}_{{p}} }{d t^{ \alpha}}\) is the leftinverse of \(\Im ^{\alpha ,g}_{a}\), where the fractional differential operator is taken in the sense of Definition 7.
Remark 8
Let us remark that, in view of \((\maltese )\), the assertion of Lemma 8 is still valid even in the case of applying the operator (MCFWD) provided \(x \in C[I,E_{w}]\).
The following example shows that our assumption that E has total dual is essential in Lemma 8 and cannot be omitted even if x is weakly absolutely continuous on I. Out of the context of such spaces we should assume instead that our derivatives should be of strongly bounded variation (cf. [26]).
Example 3.2
Let \(\alpha \in (m,m+1]\), \(m \geq 1\) and assume that \(B[I]\) is the Banach space of bounded realvalued functions on I. Define a Pettis integrable function \(x: I \to B[I]\) as in Example 2.1. The choice of this space is not accidental, because \(B[I]\) has no total dual.
Note that \(g'(\cdot ) x(\cdot ) \in P[I, B[I]]\) (cf. Proposition 3). Bearing in mind the existence of \(\Im ^{\alpha ,g}_{a} x\) on I and arguing similarly as in [12, proof of Lemma 7] there is no difficulty in proving that
On the one hand, reasoning as in Example 2.1, we know that
From which, by definition of \(\mathfrak{D}_{{p}}\), it can be easily seen that \(\mathfrak{D}_{{p}} \int _{a}^{t} g'(s) x(s) \,ds= \theta \) on I. On the other hand, in view of Lemma 2, we conclude that \(\mathfrak{D}_{{p}} \int _{a}^{t} g'(s) x(s) \,ds= g'(t) x(t)\) on I. In this connection, we deduce that the function \(t \mapsto \int _{a}^{t} g'(s) x(s) \,ds\) has two pseudoderivatives \(g'(t) x(t)\) and θ on I that differ on a set of positive measures. Consequently, we infer by the aid of (20) that on such a set \(\frac{d^{ \alpha ,g}_{{p}} }{d t^{ \alpha}} \Im ^{\alpha ,g}_{a} x \neq x\).
In the remaining part of this paper, all g–Caputo fractional pseudo (trivially weak) derivatives are taken in the sense of Definition 7.
Now, we are in the position to investigate the existence of solutions to the following g–Caputo fractional boundary value problem
combined with the nonlocal threepoint boundary conditions
Here, \(\frac{d^{ \alpha ,g}_{{p}} }{d t^{ \alpha}}\) denotes the g–Caputo fractional pseudodifferential operators defined as in (18). It is absolutely necessary to start from the definition of a solution of this problem. Let us introduce the following:
Definition 8
The function \(x \in C[I,E_{w}]\) is called a pseudo (resp., weak) solution to the problem ((21) and (22) if x admits a g–Caputo fractional pseudo (resp., weak) derivative of order \(\alpha \in (1,2)\) and satisfies (22) together with
and, respectively, for the weak derivative
Let us present some proposed integral form for the differential problem (21). Assuming very natural conditions, we always have the following relationship of their solutions:
Lemma 9
Let \(\alpha \in (1,2)\), \(\beta \in (0,1)\), \(p \in [0,\infty )\) and \(\xi \in I\) be such that \(\Vert g \Vert \neq p^{\frac{1}{\alpha 1}}g(\xi ) \). For any \(f\in P[I,E ]\), the integral equation modeled off the problem ((21) and (22)) in the form
where \(x=\Im ^{\beta ,g}_{a} u\) and
has a solution \(u \in C[I,E_{w}]\) provided \(x = \Im ^{\beta ,g}_{a} u\) is a solution of BVP (21) and (22).
Proof
Let \(x \in C[I,E_{w}]\) satisfy the problem ((21) and (22)) and define a function \(u := \frac{d^{ \beta ,g}_{{p}} }{d t^{ \beta}}x= \Im ^{1\beta ,g}_{a} \mathfrak{\delta}_{{p}} x\). By virtue of our boundary condition \(x(a)=0\), using Lemma 2 we arrive at \(x= \Im ^{\beta ,g}_{a} u\). Also, the boundary condition \(x(b)px(\xi )=c\) is transformed into \(\Im ^{\beta ,g}_{a} u(b)p\Im ^{\beta ,g}_{a} u(\xi )=c\). In this case, the differential equation (21) reads as
Now, in view of Definition 7 of MCFPD and letting
means that
Thus, “formally” we obtain
Operating by \(\Im ^{ \alpha ,g}_{a}\) yields
Now, differentiating \(\delta _{{p}}\) both sides twice, we arrive at
From which (still “formally”), we obtain
with some (presently unknown) quantities \(c_{0}, c_{1} \in E\). Since \(x(a)=\Im ^{\beta ,g}_{a} u(a)=0\) and \(\Im ^{\beta ,g}_{a} u(b)p\Im ^{\beta ,g}_{a} u(\xi )=c\), it can be easily seen that \(c_{0}=0\) and
Operating by \(\Im ^{1\beta ,g}_{a}\) on both sides of (27) yields
In this connection, we conclude that
Now, inserting \(c_{1}\) into (28) yields (“formally”) the integral equation. This completes the proof. □
We should answer the question when the two problems are equivalent. To do this we need to present a precise definition of the solutions for (23).
Definition 9
By a weak solution of (23) we mean a function \(u \in C[I,E_{w}]\) satisfying
Let us recall that if we are studying pseudosolutions, some negligible sets \(D_{\varphi}\), where the equation is not satisfied, are excluded and they are dependent on \(\varphi \in E^{\ast}\). Such a set does not affect the calculated fractional Pettis integrals (cf. Remark 7).
Since the space of all Pettis integrable functions is not complete, not all methods of the proofs of the existence of solutions to the integral equation (23) are allowed and we cannot follow many ideas taken from the case of the strong topology. We restrict our attention to the case of weakly continuous solutions of the mentioned integral equation and then to pseudosolutions of the problem (23).
Now, we are ready to present the following theorem that will allow us to introduce the assertions that provide conditions under which we ensure the existence of weakly continuous solutions to the integral equation (23).
Theorem 3
Let \(\beta \in (0,1)\), \(\alpha \in (1,2)\) such that \(\alpha \geq 1+\beta \). Assume that

A)
ψ is a Young function such that its complementary Young function ψ̃ satisfies
$$ \int _{0}^{t} \widetilde{\psi} \bigl(s^{\nu}\bigr) \,ds< \infty , \quad t >0, \nu := \max \{2+\beta  \alpha , \alpha \beta 1 \}; $$(29) 
B)
Let \(f:I \times E \times E \to E\) satisfy the following assumptions:

(1)
For every \(t \in I\), \(f(t,\cdot , \cdot )\) is wwsequentially continuous;

(2)
For every \(x, y \in C[I,E_{w}]\), \(f (\cdot ,x(\cdot ), y(\cdot ) ) \in P[I,E]\);

(3)
For any \(r>0\) and each \(\varphi \in E^{\ast}\) there exists an \(L_{{\psi}}(I,{{\mathbb{R} }})\)integrable function \(M_{r}^{\varphi}: I \to {\mathbb{R} }^{+}\) such that \( \varphi{(f(t,x,y))}  \leq M_{r}^{\varphi}(t)\) for a.e. \(t \in I\) and all \(x, y \in C[I,E_{w}] \) whenever \(\max \{ \Vert y \Vert , \Vert x \Vert \} \leq r\). Moreover, there exists a continuous nondecreasing function \(\Omega : {\mathbb{R} }^{+}\rightarrow { \mathbb{R} }^{+}\) and such that for all \(\varphi \in E^{*}\) with \(\Vert \varphi \Vert \leq 1\), \(\Vert M_{r}^{\varphi } \Vert _{\psi }<\Omega (r)\) and \(\int _{0}^{\infty } \frac{dr}{ \Vert M_{r}^{\varphi } \Vert _{\psi}} = \infty \);

(4)
There exists a positive constant k such that for arbitrary bounded sets \(B_{1}, B_{2} \subset E\), we have
$$ \boldsymbol{\mu}\bigl(f(I,B_{1},B_{2}) \bigr)\leq k \bigl[\boldsymbol{\mu}(B_{1}) + \boldsymbol{ \mu}(B_{2}) \bigr]. $$(30)

(1)
Then, there is \(\rho >0\) such that for any \(\lambda \in {\mathbb{R} }\) with \(\lambda  \leq \rho \), the integral equation (23) has at least one weak solution \(u \in C[I,E_{w}]\).
Remark 9

1.
In [34, Lemma 19] one can find some sufficient conditions to satisfy assumption B) (2).

2.
The integral in (29) is convergent, so in view of [11, Proposition 1], we also have
$$ \int _{0}^{t} \widetilde{\psi} \bigl(s^{\alpha \beta 2}\bigr) \,ds< \infty \quad \text{and}\quad \int _{0}^{t} \widetilde{\psi} \bigl(s^{1+\beta \alpha}\bigr) \,ds< \infty \quad \text{for any } t \in I. $$(31)
Before embarking on the proof of the above theorem, let us define a constant
Moreover, let us define a positive real number ρ by
where
In this case, for any \(\lambda \in {\mathbb{R} }\) with \(\lambda  < \rho \), we have
In this connection, reasoning as in [34, proof of inequality (44)], we can show that there exists \(R_{0}>0\) such that for any \(R > R_{0}\) we have
For brevity and to allow a generalization, let us keep in the following a symbol \(R_{0}\). We are assured that under our assumptions for sufficiently small λ we have global solutions, i.e., functions defined on the interval I.
Proof of Theorem 3
Define an operator T on \(C[I,E_{w}]\) generating the righthand side of the differential equation in BVP, i.e., of the form
where \(\beta \in (0, 1)\), \(\alpha \in (1,2)\), with \(\alpha \geq 1+\beta \), \(t \in I\) and
and \(c_{1}\) is defined by (24).
I. First, we note that the operators U and T are well defined on \(C[I,E_{w}]\). To see this, let us observe that for any \(u \in C[I,E_{w}]\), by Lemma 5\(\Im ^{\beta ,g}_{a} u\) is well defined and by Theorem 2, we have \(\Im ^{\beta ,g}_{a} u \in C[I,E_{w}] \).
Under assumption B) (2), for any \(u \in C[I,E_{w}]\) the superposition \(F(u) := f(\cdot , u(\cdot ), \Im ^{\beta ,g}_{a} u(\cdot ))\) is weakly measurable, Pettis integrable on I. Hence, in view of Remark 2, we obtain the existence of \(\Im ^{\alpha ,g}_{a} F(u)\) for any \(u \in C[I,E_{w}]\). This means that the element \(c_{1} \in E\) defined by (24) is well defined, so is the first component of T.
Moreover, by assumption B) (3) we have
for any \(r \geq \max \{ \Vert u \Vert , \Vert \Im ^{ \beta ,g}_{a} u \Vert \}\) and for every \(\varphi \in E^{\ast}\).
Then, \(F(u)(\cdot ) \in \mathbf{\mathcal{H}}^{\psi}_{0} (E)\) and \(\Vert Fu \Vert _{\psi}\leq \Vert M_{r}^{\varphi } \Vert _{\psi}\). Bearing in mind (29), it follows in view of Theorem 2 that U is a welldefined operator on \(C[I,E_{w}]\) with its values in \(C[I,E_{w}]\). In this connection, Corollary 1 and Proposition 2 yield that \(U u(\cdot )g'(\cdot ) \in P[I,E]\) holds true for any \(u \in C[I,E_{w}]\). Since \(\alpha \beta \geq 1\), in view of (31), by applying Lemma 6 and using Remark 2, it follows that \(\Im ^{\alpha \beta ,g}_{a} f(\cdot ,u(\cdot ), \Im ^{\beta ,g}_{a} u( \cdot )) = \Im ^{1,g}_{a} U u(\cdot ))\) and hence the operator T is defined on the space \(C[I,E_{w}]\).
Moreover, as \(\frac{c_{1} (g(t))^{\alpha \beta 1}}{ \Gamma (\alpha \beta )}\) is continuous with values in E and we just proved that \(U : C[I,E_{w}] \to C[I,E_{w}]\), then the same property holds true for T.
II. Now, let us construct an invariant domain for T, which is required by Theorem 1. Define a convex and closed subset \(Q \subset C[I,E_{w}]\) by
Lemma 4 implies that \(\mathcal{Q}\) is a strongly equicontinuous subset of \(C[I,E_{w}]\).
Observe that \(\beta \in (0, 1)\) and define \(r_{0} > R_{0}\) by \(r_{0} := \frac{R_{0}}{\Gamma (1+\beta )}\max \{1, \Vert g \Vert ^{\beta }\}\). For any \(u \in \mathcal{Q}\), we have \(\Vert u \Vert \leq r_{0}\) and \(\Vert \Im ^{\beta ,g}_{a}u \Vert \leq r_{0}\).
We proved in Theorem 2 that for any \(\theta \in (0,1]\) and \(x \in \mathbf{\mathcal{H}}^{\psi}_{0} (E) \) we have an estimation
for some \(\varphi \in E^{\ast}\) with \(\\varphi \ \leq 1\). Then, for \(\theta = \alpha  \beta  1\) this inequality is fulfilled. Take an arbitrary \(u \in \mathcal{Q}\). By applying the estimation from B) (3) and as \(M_{r_{0}}^{\varphi} \in L_{{\psi}}\), for any \(\varphi \in E^{\ast}\) we have
We can also estimate \(\varphi (c_{1})\)
By taking the supremum over all \(\varphi \in E^{\ast}\) with \(\Vert \varphi \Vert \leq 1\) in the above inequality and by applying the Hahn–Banach theorem we obtain
Moreover, again as a consequence of the Hahn–Banach theorem, for any \(u \in \mathcal{Q}\) and any \(t \in I\) there exists \(\varphi \in E^{\ast}\) with \(\Vert \varphi \Vert =1\) such that \(\Vert T (u)(t) \Vert = \varphi (T (u)(t))\). Using (36), there is no difficulty in showing that
Since \(r_{0}>R_{0}\), it follows in view of (33) that
Also, for any \(t,s \in I\) and any \(u \in \mathcal{Q}\), it can be easily seen that
III. Now, we need to prove that T is weakly–weakly sequentially continuous. Let \(\{u_{n}\}\) be a sequence in \(\mathcal{Q}\) and let \(u_{n} \rightarrow u\) in \(C[I,E_{w}]\). Recall, that weak convergence in \(C[I,E_{w}]\) means exactly its boundedness and weak pointwise convergence for any \(t \in I\). The first condition is assured by the definition of \(\mathcal{Q}\).
Fix an arbitrary \(t \in I\). Consider now the operator U and observe that
From the dominated convergence theorem for the Pettis integral applied to \(\Im ^{\beta ,g}_{a}\) we obtain convergence of \(\Im ^{\beta ,g}_{a} u_{n}(t)\) to \(\Im ^{\beta ,g}_{a} u(t))\). Hence, assumption B) (1) implies that the sequence \(f(t,u_{n}(t), \Im ^{\beta ,g}_{a} u_{n}(t))\) converges weakly to \(f(t,u(t), \Im ^{\beta ,g}_{a} u(t))\). This implies that \(\Im ^{1,g}_{a} U u_{n}(t) \to \Im ^{1,g}_{a} U u(t)\) and finally \((Tx_{n})(t)\) converges weakly to \((Tu)(t)\) in \((E,w)\) for each \(t\in I\), which means that \(T:\mathcal{Q}\rightarrow \mathcal{Q}\) is weakly–weakly sequentially continuous in \(\mathcal{Q}\).
IV. Let us verify condition (2) in Theorem 1.
Let V be a subset of \(\mathcal{Q}\) satisfying \(\bar{V}=\overline{conv}((TV)\cup \{0\})\). Obviously, \(V(t)\subset \overline{conv}((TV)(t)\cup \{0\})\), \(t\in I\). Since \(T (\mathcal{Q})\) is uniformly bounded and strongly equicontinuous in \(C[I,E_{w}] \), it follows that V is also bounded and equicontinuous. Taking into account our Lemma 1, the function \(v(t):= \boldsymbol{\mu}(V(t))\) is continuous on I, \(V(t) := \{ v(t): v \in V\}\) and
Arguing similarly as in [11, Step 3 of the proof of Theorem 3] (see also Lemma 7), we can show that
Also, by the aid of properties of μ (see [13, 21]), in view of Lemma 7, we obtain that \(\boldsymbol{\mu} ( \{ \Im ^{1,g}_{a} U V(t)\} \leq \Im ^{1,g}_{a} \boldsymbol{\mu}( \{ U V(t) \} \). Thus,
By applying assumption B) (4), we ensure that
An analogous reasoning leads to the estimate
Since
it follows in view of the definition of \(c_{1}\)
From the definition of the set V and by applying properties of measures of weak noncompactness we obtain
Hence, we can take the supremum over all \(t \in I\)
Taking into account that \(\lambda L \leq 1\), immediately, we obtain \(\boldsymbol{\mu}_{C} ( V ) = 0\), so V should be relatively weakly compact in \(C[I,E]\).
Finally, Theorem 1 implies that T has a fixed point being a pseudosolution to the integral equation (23). □
We point out that if E is reflexive then the implication (2) of Theorem 1 is automatically satisfied, as subsets of reflexive Banach spaces are weakly compact if and only they are weakly closed and norm bounded. In this situation, it is no longer necessary to assume any compactness hypothesis imposed on the nonlinearity of f to assure the existence of solutions to the fractional integral equation (23).
In addition to giving a conditions under which the integral equation (23) admits a solutions in the space \(C[I,E_{w}]\), Theorem 3 may be used to obtain a result concerning the existence of solutions to the boundary value problem (21) and (22).
Now, we are in the position to state and prove the following existence theorem:
Theorem 4
Let \(\beta \in (0,1)\), \(\alpha \in (1,2)\) such that \(\alpha \geq 1+\beta \). Assume that ψ is a Young function such that its complementary Young function ψ̃ satisfies
Assume that E has a total dual. If \(f:I\times E \rightarrow E\) is a function fulfilling all assumptions B) from Theorem 3, then the problem (21) and (22) admits at least one pseudosolution \(x \in C[I, E_{w}]\).
Proof
At the beginning, we note that if \(u \in C[I,E_{w}]\) solves the integral equation (23) then, obviously u is weakly absolutely continuous function having integrable pseudoderivative (cf. Lemma 2). Indeed, we have
Hence, by the definition of the pseudoderivative we have
for some \(p \in (1,\frac{1}{2+\beta \alpha})\). Now, since \(\Im ^{\alpha \beta 1,g}_{a} f(\cdot ,u(\cdot ), \Im ^{\beta ,g}_{a} u(\cdot )) \in C[I,E_{w}]\), it follows that
In view of Lemma 6 we conclude that
Consequently,
Seeing that \(u(a)=0\), we have, in view of Lemma 2, \(\Im ^{1,g}_{a} \mathfrak{\delta}_{{p}}u(t)=\int _{a}^{t} \mathfrak{D}_{{p}} u(s) \,ds=u(t)\), \(t \in I\). Now, let us define \(x:= \Im ^{\beta ,g}_{a} u \in C[I,E_{w}]\). Further,
Therefore, for any \(\varphi \in E^{\ast}\), by the commutative property of the gfractional integral operator, we have
Accordingly,
and
Thus, if \(u \in C[I,E_{w}]\) solves (23) then for all \(t \in I\) we obtain
Hence, by applying the above equality we obtain
By the definition of our generalized fractional derivative \(\frac{ d^{\alpha ,g}_{{p}}}{dt^{\alpha}}\) we infer
and by applying Lemma 8, we conclude that
On the other hand, there is no difficulty in showing that x satisfies
This means that x is a pseudosolution x of the problem ((21) and (22)). □
Some examples of the use of our theorem to the problem ((21) and (22)) can be found in [12, Sect. 4]. However, it is worth noting that:

1.
Our assumption that E has total dual is essential in Theorem 4 and cannot be omitted even if f is weakly absolutely continuous on I. Evidently, if we define \(f: I\times B[I]\times B[I] \to B[I]\) by
$$ f\bigl(t,x(t),y(t)\bigr) := \textstyle\begin{cases} \chi _{\{t\}}(\cdot ), & t \in J, x, y \in B[I], \\ \theta ,&t \notin J. \end{cases} $$Then, using similar arguments as in the proof of Theorem 4, by the aid of Example 3.2 we can show that
$$ \frac{ d^{\alpha ,g}_{{p}}}{dt^{\alpha}} x (t) \neq \lambda f \biggl( t, x(t), \frac{d^{\beta ,g}_{{p}} }{d t^{\alpha}} x(t) \biggr) $$holds true on a subset of I of positive measure.

2.
By virtue of the fact that the indefinite Pettis integral of a function \(f \in P[I,E]\) does not enjoy the strong property of being a.e. weakly differentiable, it is immediately clear that the result obtained in Theorem 4 has no analog if we replace \(\frac{d^{ \alpha ,g}_{{p}} }{d t^{ \alpha}}\) by \(\frac{d^{ \alpha ,g}_{\omega} }{d t^{ \alpha}}\).

3.
Arguing similarly as in the [11, Theorem 5], we are to consider the multivalued case of the problem ((21) and (22)).
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Salem, H.A.H., Cichoń, M. & Shammakh, W. Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems. Bound Value Probl 2023, 57 (2023). https://doi.org/10.1186/s1366102301745y
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DOI: https://doi.org/10.1186/s1366102301745y