A study of boundary value problem for generalized fractional differential inclusion via endpoint theory for weak contractions

This note is concerned with establishing the existence of solutions to a fractional differential inclusion of a ψ-Caputo-type with a nonlocal integral boundary condition. Using the concept of the endpoint theorem for φ-weak contractive maps, we investigate the existence of solutions to the proposed problem. An example is provided at the end to clarify the theoretical result.


Introduction
On different time ranges, fractional calculus has had great impact due to a diversity of applications that have contributed to several fields of technical sciences and engineering [1][2][3][4][5]. One of the principal options behind the popularity of the area is that fractionalorder differentiations and integrations are more beneficial tools in expressing real-world matters than the integer-order ones. Various studies in the literature, on distinct fractional operators such as the classical Riemann-Liouville, Caputo, Katugamploa, Hadamard, and Marchaud versions have shown versatility in modeling and control applications across various disciplines. However, such forms of fractional derivatives may not be able to explain the dynamic performance accurately, hence, many authors are found to be sorting out new fractional differentiations and integrations which have a kernel depending upon a function and this makes the range of definition expanded; see [6][7][8]. Furthermore, models based on these fractional operators provide excellent results to be compared with the integer-order differentiations [9][10][11][12].
Recently, the area of fractional-order differential inclusions has become mainly important as these equations were found to be of high importance in modeling stochastic and optimal controls problems [13]. By using techniques of nonlinear analysis the authors stud-ied different aspects such as establishing the existence and the uniqueness of solutions, the upper and lower solutions, and stability. We refer the reader to [14][15][16][17][18][19][20][21][22][23][24][25] for various qualitative studies.
Details from the historical attitude and recent improvements in the area are detailed in the monograph of Ahmad et al. [26] and the survey of Agarwal et al. [27].
In this paper, we deal with the following ψ-fractional differential inclusions: subject to ψ-boundary conditions of the form given continuous function, and λ i ∈ R, i = 0, 1, 2, . . . , m, are real constants such that θ will be determined later. We establish novel existence results of solutions for the above inclusion problem by using the endpoint theorem when the multivalued map is ϕ-weak contractive.
The result of the present paper unifies several classes of fractional differential inclusion with different boundary conditions. For example by taking ψ(y) = y in (1.1)-(1.2) the results agree for the classical Caputo fractional inclusions [28] with a combination of classical nonlocal Riemann-Liouville fractional and Neumann boundary conditions of the form: when ψ(y) = ln(y), the results agree with the Caputo-Hadamard fractional inclusions [29] equipped with classical fractional integral boundary conditions of Hadamard type of the form while the results for generalized Caputo fractional inclusions [30] with nonlocal Katugampola type integral boundary conditions follow by taking ψ(y) = y ρ /ρ. The paper is organized as follows. Section 2 recalls some basic and fundamental definitions and lemmas. In Sect. 3, we prove the existence of a solution to the proposed problem (1.1)-(1.2). An example is provided to demonstrate the main results in Sect. 4. [1,T] u(y) .

Preliminary results
which is supplied with the norm described by where ψ ∈ C m ([1, T], R), with ψ (y) > 0 on [1, T], and Now we introduce some notations and definitions of fractional calculus with respect to another function and give preliminary results that we will need in our proofs later.

Definition 2.1 ([28])
The ψ-fractional integration operator in the Riemann-Liouville sense of order σ > 0 with lower limit 1 for an integrable function g is defined by provided the integral exists.
Particularly, for 0 < σ < 1, we obtain We will investigate the existence of solutions to the problem (1.1)-(1.2) with the help of the following lemma.

Main results
We introduce in this section the function class Ψ of all mappings ϕ : for each z, w ∈ W, where P cl,bd (W) is a nonempty collection of all closed and bounded subsets of W, and H ρ (·, ·) denotes for the Hausdorff metric on P cl,bd given as  where ψ ∈ C 2 (J, R) such that ψ > 0 on [1, T].
We set an operator L : W → P(W) associated with the problem (1.1)-(1.2) as where S Z,u is the set of selections for Z by (H 3) There exists 0 < θ < 1, such that h(y, u)h(y,ū) ≤ θ |u -ū|.
If Z verifies the approximate endpoint property, then the inclusion problem (1.1)-(1.2) has a solution on [1, T], provided that Proof The proof will be given in two steps, where we show that L : W → P(W) given in (3.2) has an endpoint.
Step 1: L is closed multivalued of P(W). Let u n ∈ W such that u n → u, and (f n ) n≥1 ∈ L(u) be a sequence such that f n → f * whenever n → +∞. Then there exists a l n ∈ S Z,u n such that, for each y ∈ [1, T], we get Since Z has compact values, the sequence (l n ) n≥1 has a sub-sequence, still denoted by (l n ) n≥1 , which converges strongly to some l ∈ L 1 ([1, T], R), and hence l ∈ S Z,u . For every ν ∈ Z(y, u(y)), we have l n (y)l(y) ≤ l n (y)ν + νl(y) , which implies l n (y)l(y) ≤ H d Z(y, u n ), Z(y, u) ≤ Γ (σ + 1) (ψ(T)) σ u nuϕ u nu .
Since u nu → 0 then ϕ( u nu ) → 0 and h is a continuous function then, for each y ∈ [1, T], So f * ∈ L and L is closed multivalued.

An example
Let W = C( [1, e], R) be the space of all continuous functions defined on [1, e] and u ∈ W.
We define an operator L : W → P(W) L(u) = g ∈ W : there exists l ∈ S Z,u , g(y) = u(y), for all y ∈ Note that 0 is a unique endpoint of L, i.e. L(0) = {0}, which implies that sup u∈L(0) u = 0, thus inf u∈W sup g∈L(u) ϑg = 0. The operator L as a consequence has the approximate endpoint property. Therefore all conditions of Theorem 3.4 are satisfied, then the inclusion problem (4.1) has at least one solution on [1, e].

Conclusion
In the present work, the endpoint theorem for ϕ-weak contractive maps was used to establish the existence results of solutions for fractional differential inclusion which involves the ψ-Caputo fractional derivative. Systems of fractional differential inclusions with the ψ-Caputo derivative provide more adaptable models, in the sense that by a proper choice of the function ψ, hidden features of real-world phenomena could be extracted. An illustrative example is presented to point out the applicability of our main results. Our results are not only new in the given configuration but also correspond to some new results associated with the specific choice of the function ψ involved in the given problem.