Study of a Class of Generalized Multiterm Fractional Differential Equations with Generalized Fractional Integral Boundary Conditions

&e aim of this work is to study the new generalized fractional differential equations involving generalized multiterms and equipped with multipoint boundary conditions. &e nonlinear term is taken from Orlicz space. &e existence and uniqueness results, with the help of contemporary tools of fixed point theory, are investigated. &e Ulam stability of our problem is also presented. &e obtained results are well illustrated by examples.

Furthermore, there has been a significant development in fractional derivatives and integrals due to the necessity of a better model for real phenomena. In [16], Katugampola suggested a new fractional integral that combines the Riemann-Liouville and Hadamard integrals into a single integral. In 2015, Caputo and Fabrizio [17] defined the socalled Caputo-Fabrizio fractional derivatives by imposing a nonsingular exponential kernel. Later, the exponential kernel was replaced by the Mittag-Leffler function [18,19]. Another definition of generalized proportional fractional derivatives-generated Caputo and Riemann-Liouville involving exponential functions in their kernels was introduced by Jarad et al. [20]. Another new type of new fractional derivative can be found in [21]. An application on such derivatives can be seen in [22]. More recently, generalized fractional derivatives that contain kernels depending on the function on the space of absolutely continuous functions were investigated [23]. Following the above points, many authors established various generalized fractional boundary value problems (GFBVPs) (see, for example, [24][25][26][27]). It is worth to mention that some authors worked on the existence of solutions and studied the stability analysis for nonlinear singular fractional differential equations. For more details, we refer to [28][29][30].
On the other hand, Birnbaum and Orlicz [31] proposed a more general setting of function spaces in 1931, which were called Orlicz spaces. ese spaces are the generalization of the classical Lebesgue spaces L p (1 < p < ∞), where the kernel is given by a convex function instead of x p . Some results that deal with differential equations in the framework of Orlicz spaces can be found in [32,33].
In this paper, we investigate the boundary value problem as follows: where C D α,ρ 0 + is the generalized fractional derivative of Caputo type of order α, 3 < α ≤ 4, I β,ρ 0 denotes the Katugampola-type fractional integral of order 0 < β < 1, f is defined on an Orlicz space L F ([0, 1]), and δ i ∈ R, (i � 1, . . . , m). e rest of the paper is organized as follows. In Section 2, we will briefly recall some preliminary materials related to our problem. In Section 3, we discuss the existence and uniqueness results of GFBVP using some fixed point theorems and support the obtained results by using examples to well illustrate. e Ulam stability of our problem is given in Section 4.

Preliminaries
For convenience of the reader, we present some basic definitions about generalized fractional calculus theory, which can be found in [16,34,35]. Also, we introduce some necessary concepts for Orlicz spaces which are used throughout this paper. For more details about Orlicz spaces, one can see [36,37].
be a right continuous, monotone, increasing function with the following: en, the function defined by is called N− function. Alternatively, the function F is an N− function iff F is continuous, even, and convex with the following: Definition 2. For an N-function, we define where Q − 1 is the right inverse of the right derivative of F and is called the complementary of F and it satisfies the following condition: Remark 1. Note that the function F * is also N-function and the complementary pairs F and F * satisfy the following Young inequality: is called a Young function if it is convex and satisfies the conditions is space endowed with the Luxemburg norm, i.e., and the pair (L F ([0, 1]), ‖u‖ F ) is a Banach space.

Definition 5 (Katugampola generalized fractional integral).
e generalized fractional integral of order α ∈ C with R(α) > 0 and ρ > 0 for − ∞ < a < t < ∞ is defined by where u belongs to the space X p c (a, b), which denotes the space of all complex-valued Lebesgue measurable functions f on [a, b] for which ‖f‖ X p c < ∞, where the norm is defined by Note that integral (8) is called the left-sided generalized fractional integral. 2 Discrete Dynamics in Nature and Society Definition 6 (Katugampola generalized Caputo fractional derivative). e left generalized fractional derivative of order for 0 ≤ ϵ ≤ 1, endowed with the norms ‖f‖ C n δ � n k�0 ‖δ k f‖ C and ‖f‖ C n δ,ϵ � n− 1 k�0 ‖δ k f‖ C + ‖δ n f‖ C ϵ,ρ , respectively. Next, we recall some basic properties of generalized fractional integral and derivative [39].
Our main results are based on utilizing the following fixed point theorems.
Theorem 3 (Schaefer's fixed point theorem [40]). Let X be a Banach space. Assume that L: X ⟶ X is a completely continuous operator and the set T � u ∈ X: u � { λLu, 0 < λ < 1} is bounded. en, L has a fixed point in X.
Theorem 4 (Krasnoselskii's fixed point theorem [40]). Let N be a closed, convex, bounded, and nonempty subset of a Banach space X. Let T 1 and T 2 be operators such that

is a compact and continuous and T 2 is a contraction mapping
en, there exists u 0 ∈ N such that u 0 � T 1 (u 0 ) + T 2 (u 0 ). For computational convenience, we introduce the following notations: Lemma 2. Let h ∈ L F ([0, 1]) and ψ be given by (15). en, the solution of the GFBVP C D α,ρ is given by Proof. By using Lemma 1, we obtain where C i ∈ R, (i � 0, 1, 2, 3) are the arbitrary constants. Using the boundary conditions Discrete Dynamics in Nature and Society Substituting the values of C 0 , C 1 , C 2 , and C 3 in (18), we get (17).

Main Results
In this section, we prove the existence and uniqueness of solutions of GFBVP (1). We assume that f belongs to an Relative to GFBVP (1), in view of Lemma 2, we define an operator L: X ⟶ X as Notice that GFBVP (1) has solutions if and only if the operator L has fixed points. e following result plays a major role in our analysis.

Lemma 3. Let F be a Young function which has a Young
for α ∈ (3, 4] and β ∈ (0, 1). en, the operator L exists and is well defined.
Proof. Let u ∈ X. Define a function as follows: At the beginning, we will show that ϱ 1 ∈ L F * [0, 1]. By using appropriate substitution and properties of the Young functions, one obtains by the assumption of the theorem, and we get Similarly, setting Discrete Dynamics in Nature and Society where where g: R + ⟶ R + is a continuous and increasing function with g(0) � 0. Using the Hölder inequality, we have en, 0 < |t ρ − τ ρ | < ϵ and by the continuity of g, we see that Lu is continuous, which completes the proof.
Our first existence result relies on Schaefer's fixed point theorem.
□ Discrete Dynamics in Nature and Society en, the GFBVP (1) has at least one solution on [0, 1].

Proof.
e proof proceeds in few steps as follows: Step 1: we will prove the operator L maps bounded sets into bounded sets in C([0, 1], R). Let E ⊂ X be a bounded set. en, for all u ∈ E, by using the assumption (H 1 ), for t ∈ [0, 1], we are able to obtain us, we obtain 6 Discrete Dynamics in Nature and Society Now, with some efforts of computation, we have Discrete Dynamics in Nature and Society 7 erefore, we have In a similar way, we arrive at Step 2: from (32)-(36), we can immediately get the operator L: X ⟶ X to map bounded sets into bounded sets. Let 0 < t 1 < t 2 < 1, and for all u ∈ E, we get us, we have Similarly, it can be easily shown that 8 Discrete Dynamics in Nature and Society e right side of (38)-(41) goes to zero as (t 2 − t 1 ) ⟶ 0. In view of steps I and II, it follows that by Arzela-Ascoli theorem, the sets L(u): erefore, L(E) is a relatively compact set in X. We consider the set T � u ∈ X; u � λLu, 0 < λ < 1 { }. en, T is bounded. Indeed, let u ∈ T. So, u � λLu, 0 < λ < 1, for any t ∈ [0, 1], and it follows that us, all the hypotheses of eorem 3 are satisfied. erefore, we can conclude that the operator L has at least one fixed point. Hence, the GFBVP (1) has at least one solution on [0, 1].
For computation convenience, we set Now, we make use of eorem 4 to prove the existence of solutions of GFBVP (1). □ Theorem 6. Let f: [0, 1] × R 4 ⟶ R be a continuous function such that the following assumptions hold: for all t ∈ [0, 1], u, v, p, q, w, p, q, w ∈ R and L > 0, Proof. We define B r � u ∈ X: ‖u‖ ≤ r { }, where r⩾‖ϕ‖(σ+ KV). We split the operator L defined by (21) on B r as L � L 1 + L 2 , where L 1 and L 2 are given by Discrete Dynamics in Nature and Society For u, v ∈ B r , we find that Consequently, we obtain which shows that L 1 (u) + L 2 (u) ∈ B r . In what follows, we prove that L 2 is a contraction. Let u, v ∈ B r and for all t ∈ [0, 1], we get With the same process, one has (53) Hence, we get It remains to show that L 1 is continuous and compact. e continuity of the function f implies that the operator L 1 is continuous. To achieve the compactness of the operator L 1 , we first prove that L 1 is uniformly bounded on B r as follows: So, Now, let sup t∈[0,1]×B 4 r |f(t, u, p, q, w)| � η. Consequently, for t 1 , t 2 ∈ [0, 1], t 1 < t 2 , we have Furthermore, erefore, as (t 2 − t 1 ) ⟶ 0, the right-hand sides of the above inequalities tend to zero independently of u ∈ B r . us, L 1 is equicontinuous and so it is relatively compact on B r . According to the Arzela-Ascoli theorem, the operator L 1 is compact. By using eorem 4, there exists at least one solution of GFBVP (1) on [0, 1].
In the next result, we prove the existence of solutions for GFBVP (1) by applying Banach's fixed point theorem. □ Theorem 7. Assume that the condition (H 2 ) holds. en, GFBVP (1) has a unique solution on [0, 1], provided that where A � σ + KV and V, K, and σ are given by (43)-(45).
Proof. Consider the operator L: X ⟶ X defined by We erefore, Similarly, By taking the norm from t ∈ [0, 1], it yields ‖Lu‖ ≤ r, which shows that L maps B r into itself. In order to show that the operator L is a contraction, let u, v ∈ C([0, 1], R). en, for each t ∈ [0, 1], we obtain Discrete Dynamics in Nature and Society Similarly, Consequently, us, in view of condition (59), it follows that the operator L is a contraction. Hence, the operator L has a unique fixed point which corresponds to a unique solution of GFBVP (1).

Ulam Stability
In this section, we develop the criteria for Ulam stability of GFBVP (1) by means of its equivalent integral equation. For simplicity, we set where v ∈ X and f: [0, 1] × R 4 ⟶ R is a continuous function. Next, we define a continuous nonlinear operator F: X ⟶ X as For definitions of Ulam-Hyers, generalized Ulam-Hyers and Ulam-Hyers-Rassias stability, and we refer to [41].
there exists a solution u ∈ X of (1) satisfying the inequality where ϵ 1 is a positive real number depending on ϵ.
Definition 8. GFBVP (1) is generalized Ulam-Hyers stable if there exists Θ ∈ C(R + , R + ) such that, for each solution v ∈ X of (1), there exists a solution u ∈ X of (1) with Definition 9. GFBVP (1) is Ulam-Hyers-Rassias stable with respect to Ψ ∈ C([0, 1], R) if there exists a real number b > 0 such that, for each ϵ > 0 and for each solution v ∈ X of (1), we can find a solution u ∈ X of (1) satisfying the inequality where ϵ 1 is a positive real number depending on ϵ.

en, GFBVP (1) is Ulam-Hyers-Rassias stable with respect to Ψ.
Proof. Similar to the discussion in the proof of eorem 8, we have with ϵ 1 � ϵ/(1 − LA), and we get the desired result.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.

Authors' Contributions
Wafa Shammakh, Hadeel Z. Alzumi, and Zahra Albarqi contributed to the design and implementation of the research, to the analysis of the results, and to the writing of the manuscript.