On Coupled Systems for Hilfer Fractional Differential Equations with Nonlocal Integral Boundary Conditions

Department of Mechanical Engineering Technology, College of Industrial Engineering Technology, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, &ailand Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematices, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, &ailand


Introduction
e theory of fractional differential equations has been widely used in pure mathematics and applications in the fields of physics, biology, and engineering. ere are many interesting results for qualitative analysis and applications. We refer the interested reader, in fractional calculus, to the classical reference texts such as [1][2][3][4][5][6][7]. In the literature, there exist several different definitions of fractional integrals and derivatives, and the most popular of them are Riemann-Liouville, Caputo, and other less-known such as Hadamard fractional derivative and the Erdeyl-Kober fractional derivative. A generalization of derivatives of both Riemann-Liouville and Caputo was given by Hilfer in [8] as where n − 1 < α < n, 0 ≤ β ≤ 1, t > a ≥ 0, and D n � (d n /dt n ).
He named it as generalized fractional derivative of order α and a type β. Many authors call it the Hilfer fractional derivative. We notice that when β � 0, the Hilfer fractional derivative corresponds to the Riemann-Liouville fractional derivative: H D α,0 u(t) � D n I n− α u(t).
When β � 1, the Hilfer fractional derivative corresponds to the Caputo fractional derivative: Such derivative interpolates between the Riemann-Liouville and Caputo derivative. Some properties and applications of the Hilfer derivative are given in [9,10] and the references cited therein.
To the best of our knowledge, there is no work carried out on systems of boundary value problems with Hilfer fractional derivative in the literature. is paper come to fill this gap. us, the objective of the present work is to introduce a new class of coupled systems of Hilfer-type fractional differential equations with nonlocal integral boundary conditions and develop the existence and uniqueness of solutions. In precise terms, we consider the following coupled system: where H D α,β and H D α 1 ,β 1 are the Hilfer fractional derivatives of orders α and α 1 , 1 < α, α 1 < 2, and parameters β and β 1 , respectively, 0 ≤ β, β 1 ≤ 1, and I φ i , I ψ j are the Riemann-Liouville fractional integrals of order φ i > 0 and ψ j > 0, respectively, the points . . , n are given real constants. e paper is organized as follows. We present our main results in Section 3, by applying Leray-Schauder alternative, Krasnoselskii's fixed point theorem, and Banach's contraction mapping principle, while Section 2 contains some preliminary concepts related to our problem. Examples are constructed to illustrate the main results.

Preliminaries
In this section, we introduce some notations and definitions of fractional calculus and present preliminary results needed in our proofs later [2,5].

Definition 1.
e Riemann-Liouville fractional integral of order α > 0 of a continuous function u: [a, ∞) ⟶ R is defined by provided the right-hand side exists on (a, ∞).

Definition 2.
e Riemann-Liouville fractional derivative of order α > 0 of a continuous function is defined by where n � [α] + 1, [α] denotes the integer part of real number α, provided the right-hand side is pointwise defined on (a, ∞).

Definition 3.
e Caputo fractional derivative of order α > 0 of a continuous function is defined by provided the right-hand side is pointwise defined on (a, ∞).

en, the system
is equivalent to the following integral equations: 2 Journal of Mathematics Proof. Operating fractional integral I α on both sides of the first equation in (9) and using Lemma 1, we obtain en, where By a similar way, we obtain By setting from the boundary conditions x(a) � 0 and y(a) � 0, we obtain c 2 � 0 and d 2 � 0. en, we obtain Substituting the values of c 1 and d 1 in (18), we obtain solutions (10) and (11). e converse follows by direct computation. is completes the proof.

Main Results
. It is obvious that the product Journal of Mathematics In view of Lemma 2, we define two operators K: where where For computational convenience, we set Banach's contraction mapping principle is applied in the first result to prove existence and uniqueness of solutions of system (4).
(32) Journal of Mathematics For (x, y) ∈ B r , we have (33) Hence, Similarly, we have and hence (36) Consequently, it follows that which implies KB r ⊂ B r . Next, we will show that the operator K is a contraction mapping. For any (x 1 , y 1 ), (x 2 , y 2 ) ∈ X × Y, we obtain Journal of Mathematics erefore, we obtain the following inequality: In addition, we also obtain From (39) and (40), it yields As (M 1 + M 3 )(ℓ 1 + ℓ 2 ) + (M 2 + M 4 )(n 1 + n 2 ) < 1, therefore, K is a contraction operator. By Banach's fixed point theorem, the operator K has a unique fixed point, which is the unique solution of (4) on [a, b]. e proof is completed. Now, we prove our second existence result via Leray-Schauder alternative. Lemma 3 (Leray-Schauder alternative, see [19]). Let F: E ⟶ E be a completely continuous operator. Let en, either the set ξ(F) is unbounded, or F has at least one fixed point.

Theorem 2.
Assume that there exist real constants u i , v i ≥ 0 for i � 1, 2 and u 0 , v 0 > 0 such that, for any Proof. By continuity of the functions f and g on [a, b] × R × R, the operator K is continuous. We will show that the operator K: X × Y ⟶ X × Y is completely continuous. Let Φ ⊂ X × Y be bounded.
en, there exist positive constants L 1 and L 2 such that |f(t, x, y)| ≤ L 1 , |g(t, x, y)| ≤ L 2 , ∀(x, y) ∈ Φ. (44) en, for any (x, y) ∈ Φ, we have 6 Journal of Mathematics which yields Similarly, we obtain that Hence, from the above inequalities, we obtain that the set KΦ is uniformly bounded. Next, we are going to prove that the set KΦ is equicontinuous. For any (x, y) ∈ Φ and τ 1 , τ 2 ∈ [a, b] such that τ 1 < τ 2 , we have Journal of Mathematics erefore, we obtain Analogously, we can obtain the following inequality: Hence, the set KΦ is equicontinuous. By applying the Arzelá-Ascoli theorem, the set KΦ is relative compact which implies that the operator K is completely continuous. Lastly, (51) en, we obtain which imply that us, we obtain where which shows that the set ξ is bounded. erefore, by applying Lemma 3, the operator K has at least one fixed point. erefore, we deduce that problem (4) has at least one solution on [a, b]. e proof is complete. e last existence theorem is based on Krasnoselskii's fixed point theorem.
Lemma 4 (Krasnoselskii's fixed point theorem, see [20]). Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A and B be operators such that (i) Ax + By ∈ M, where x, y ∈ M, (ii) A is compact and continuous, and (iii) B is a contraction mapping. en, there exists z ∈ M such that z � Az + Bz. Theorem 3. Assume that f, g: [a, b] × R × R ⟶ R are continuous functions satisfying assumption (H 1 ) in eorem 1. In addition, we suppose and there exist two positive constants P, Q such that, for all t ∈ [a, b] and x i , y i ∈ R, i � 1, 2, f t, x 1 , x 2 ≤ P, g t, x 1 , x 2 ≤ Q. (55) then problem (4) has at least one solution on [a, b].