A New Class of ψ -Caputo Fractional Differential Equations and Inclusion

In the present research work, we investigate the existence of a solution for new boundary value problems involving fractional diﬀerential equations with ψ -Caputo fractional derivative supplemented with nonlocal multipoint, Riemann–Stieltjes integral and ψ -Riemann–Liouville fractional integral operator of order c boundary conditions. Also, we study the existence result for the inclusion case. Our results are based on the modern tools of the ﬁxed-point theory. To illustrate our results, we provide examples.


Introduction
Fractional calculus has played a very important role in different areas of research (see [1,2] and the references cited therein). Consequently, fractional differential equations have grasped the interest of many researchers working in diverse applications [3][4][5][6]. Recently, several researchers have tried to propose different types of fractional operators that deal with derivatives and integrals of arbitrary orders and their applications. For instance, Kilbas et al. in [2] introduced the properties of fractional integrals and fractional derivatives concerning another function. Some generalized fractional integral and differential operators and their properties were introduced by Agrawal in [7]. Very recently, Almeida in [8] presented a new type of fractional differentiation operator, the so-called ψ-Caputo fractional operator, and extended work of Caputo [2,9]. Almeida et al. in [10,11] investigated the existence and uniqueness of the results of nonlinear fractional differential equations involving a Caputo-type fractional derivative with respect to another function, employing the fixed-point theorem and Picard iteration method. Numerous interesting results concerning the existence, uniqueness, and stability of initial value problems and boundary value problems for fractional differential equations with ψ-Caputo fractional derivatives by applying different types of fixed-point techniques were obtained by Abdo et al. [12,13], Vivek et al. [14], and Wahash et al. [15]. An important application that is controlled by the theory of ψ-fractional differentiation can be found in [16].
In this paper, we investigate a new boundary value problem of fractional differential equations supplemented with nonlocal multipoint, Riemann-Stieltjes integral fractional boundary conditions involving Riemann-Liouville fractional integral operator of order c > 0 with respect to function ψ given by the form where c D α 0 + ,ψ and c D and A i (·), (i � 1, 2) is a function of bounded variation. a i (i � 1, 2, 3) is a real constant, and μ i , η i (i � 1, . . . , m) are positive constants.
We also study the corresponding inclusion problem that is given by where F: C([0, 1], R) ⟶ P(R) is a multivalued function, where P(R) is the family of all subsets of R and the other quantities are the same as defined in problem (1).
Notice that this Riemann-Stieltjes integral fractional boundary conditions arise in manifold applications of computational fluid dynamics, distribution methods, and so forth (for example, see [17,18]). is paper is organized as follows. In Section 2, we recall some preliminary results and some related definitions. In Section 3, we discuss the existence results of solutions by relying on Krasnoselskii fixed-point theorem and Leray-Schauder nonlinear alternative. Also, we present an example. Finally, we describe the inclusion case and deduce the existence of solutions by applying Krasnoselskii's multivalued fixed-point theorem in Section 4.

Preliminaries
For the convenience of the reader, we present here some necessary basic definitions, lemmas, and results which are used throughout this paper [2,8,10,12,19,20].
endowed with the norm‖f‖ C n c,ε � n−1 k�0 ‖c k f‖ C + ‖c n f‖ C ε,ρ . e convention C n c,0 [a, b] � C n c [a, b] endowed with the norm ‖f‖ C n c � n k�0 ‖c k f‖ C is used.

Definition 2.
Let α > 0, h be an integrable function defined on [a, b], and ψ ∈ C n [a, b] be an increasing differentiable function such that ψ ]. e left-sided ψ-Riemann-Liouville fractional integral of order α of a function h is given by ]. e leftsided ψ-Riemann-Liouville fractional derivative of order α of a function h is defined by where n � [α] + 1 and [α] denote the integer part of the real number α.
e left-sided ψ-Caputo fractional derivative of order α of a function h is defined by where ψ (s)ds.  (1) relies on the following fixed-point theorems [21,22].
Theorem 1 (Krasnoselskii's fixed-point theorem). Let p be a closed, convex, bounded, and nonempty subset of a Banach space X. Let T 1 and T 2 be operators such that (i) T 1 (u 1 ) + T 2 (u 2 ) belong to p whenever u 1 , u 2 ∈ p (ii) T 1 is compact and T 2 is a contraction mapping en, there exist u 0 ∈ p such that u 0 � T 1 (u 0 ) + T 2 (u 0 ). Theorem 2 (Leray-Schauder fixed-point theorem). Let C be a closed and convex subset of a Banach space E and U be an open subset of C with 0 ∈ U. Suppose that V: U ⟶ C is a continuous, compact (that is, V(U) is a relatively compact subset of C ) map. en, either For computational convenience, we set the following: Lemma 3. Let y, g ∈ C([0, 1], R); then, the linear ψ−fractional differential equation has a solution u(t) on [0, 1] given by Proof. We apply ψ-Riemann-Liouville fractional integral of order α to both sides of the linear ψ−fractional differential equation: We obtain Next, applying ψ-Riemann-Liouville fractional integral of order β to both sides of (14), we obtain where c 1 and c 2 are arbitrary constants. By Definition 1, general solution (15) can be written as Using the boundary condition Applying the operator I c 0 + ,ψ , c > 0, on equation (17), we obtain

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Using the boundary condition Inserting the value of c 2 in (17) yields solution (12). e converse follows by direct computation.

Main Results
In this section, we prove the existence of solutions of problem (1). We shall assume that f and h are in the Banach Here, we define an operator T: U ⟶ U associated with problem (1) by erefore, problem (1) has a solution if and only if the operator T has a fixed point.
For computational convenience, we introduce the notations Now, we will state and prove the existence result via Krasnoselskii's fixed-point theorem.
Proof. For a positive number ϵ, consider B ϵ � u ∈ U: { ‖u‖⩽ϵ},where ϵ ≥ ‖M‖G + G 1 , and we split T into two operators T 1 and T 2 where T � T 1 + T 2 , on the bounded set B ϵ by For any u ∈ B ϵ , by using (H2), we have

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Hence, T 1 u + T 2 u ∈ B ϵ . Next, we show that T 2 is a contraction mapping. Let u, u ∈ R and t ∈ [0, 1], so by using (H1), we have By assumption ρG 2 + G 3 < 1, we obtain that T 2 is a contraction mapping. Since f and h are continuous functions, we have T 1 as continuous. Also, T 1 is uniformly bounded on B ϵ as Finally, we prove the compactness of the operator T 1 .

Existence Results for Inclusion Case
In this section, we extend the results to cover the inclusion problem and prove the existence of solutions for problem (2) by applying the fixed-point theorem [23]. We recall some basic notations for the inclusion case [24][25][26][27][28][29][30].
For each y ∈ C([0, 1], R), define the set of selections of F by Let (X, d) be a metric space induced from the normed space (X; ‖ · ‖). We have H d : P(X) × P(X) ⟶ R ∪ ∞ { } given by For convenience, we denote Our result is based on the following fixed-point theorem.

Theorem 5.
Let U and U be, respectively, the open and closed subsets of Banach space X, such that 0 ∈ U; let χ 1 (u): U ⟶ P cp,cv (X) be multivalued and χ 2 (u): U ⟶ X be single-valued such that χ 1 (U) + χ 2 (U) is bounded. Suppose that (a) χ 2 is a contraction with a contraction k < (1/2) (b) χ 1 is u.s.c and compact en, either (i) the operator inclusion λx ∈ χ 1 x + χ 2 x has a solution for λ � 1 or (ii) there is an element u ∈zU such that λu ∈ χ 1 u + χ 2 u for some λ > 1, where zU is the boundary of U Theorem 6. Assume that where G and G 1 are defined in (21) and (22), respectively. en, problem (2) has at least one solution on } be an open set in U. Define the multivalued operator χ 1 : D ⟶ P(U) by Journal of Mathematics 11 and define the single-valued operator χ 2 : D ⟶ U by (52) Observe that χ � χ 1 + χ 2 , and it is given by Indeed, if z ∈ χ(u), then there exists f ∈ S F,u , such that S F,u � f ∈ L 1 [0, 1], R + : f(t) ∈ F(t, u(t)), for a.e. t ∈ [0, 1] .
We will show that the maps χ 1 and χ 2 satisfy the hypotheses of eorem 5. is will be done in several steps.