Existence results for a self-adjoint coupled system of nonlinear second-order ordinary differential inclusions with nonlocal integral boundary conditions

A coupled system of nonlinear self-adjoint second-order ordinary differential inclusions supplemented with nonlocal nonseparated coupled integral boundary conditions on an arbitrary domain is studied. The existence results for convex and nonconvex valued maps involved in the given problem are proved by applying the nonlinear alternative of Leray–Schauder for multivalued maps and Covitz–Nadler’s fixed point theorem for contractive multivalued maps, respectively. Illustrative examples for the obtained results are presented. The paper concludes with some interesting observations.


Introduction
Inspired by the work of Bitsadze and Samarskii [1] on nonlocal elliptic boundary value problems, Il'in and Moiseev [2,3] initiated the study of nonlocal boundary value problems for second-order ordinary differential equations. Nonlocal boundary value problems constitute an important area of research as such problems find their applications in chemical engineering, thermo-elasticity, underground water flow, and population dynamics; for details and examples, see [4,5]. For a variety of interesting results on nonlocal boundary value problems, we refer the reader to the works [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and the references cited therein. Self-adjoint differential equations are found to be of great interest in certain disciplines, for example, see [22][23][24][25]. In [26], a self-adjoint coupled system of nonlinear ordinary differential equations with nonlocal multi-point boundary conditions was studied. In a recent article [27], the authors established existence results for a self-adjoint coupled system of nonlinear second-order ordinary differential equations complemented with nonlocal nonseparated integral boundary conditions. The aim of the present paper is to consider and investigate the existence of solutions for the multivalued case of the problem discussed in [27]. In precise terms, we consider a selfadjoint coupled system of second-order ordinary differential inclusions on an arbitrary domain, subject to the nonlocal nonseparated integral coupled boundary conditions given by (q(t)v (t)) ∈ μ 2 G(t, u(t), v(t)), t ∈ [a, b], where a < η < ξ < b, p, q ∈ C([a, b], R + ), α i , β i , λ i ∈ R + , i = 1, 2, 3, 4, μ j ∈ R + , j = 1, 2, and F, G : [a, b] × R × R − → P(R) are given multivalued maps, P(R) is the family of all nonempty subsets of R.
We establish existence criteria for the solutions of problem (1.1) for convex and nonconvex valued multivalued maps F and G by applying the nonlinear alternative of Leray-Schauder for multivalued maps in the convex case and Covitz and Nadler's fixed point theorem for contractive multivalued maps in the nonconvex case, respectively. The tools of the fixed point theory employed in our analysis are standard, however their application to problem (1.1) is new. We emphasize that the results derived in this paper are new and enrich the literature on self-adjoint multivalued nonlocal boundary value problems.
The rest of the paper is organized as follows. We present background material about multivalued analysis in Sect. 2, while the main results are presented in Sect. 3. Numerical examples illustrating the obtained results are constructed in Sect. 4.

Preliminaries
We begin this section by reviewing some basic definitions, lemmas, and theorems on multivalued maps from [28,29] which are related to the study of problem (1.1).
Let (X , · ) be a normed space. We denote the classes of all closed, bounded, compact, and compact and convex sets in X by P cl , P b , P cp , and P cp,c , respectively.
A multivalued map F : X → P(X ) is (a) convex (closed) valued if F(x) is convex (closed) for all x ∈ X ; (b) upper semicontinuous (u.s.c.) on X if for each x 0 ∈ X , the set F(x 0 ) is a nonempty closed subset of X , and if for each open set N of X containing F(x 0 ), there exists an open neighborhood N 0 of x 0 such that F(N 0 ) ⊆ N ; (c) bounded on bounded sets if F(B) = x∈B F(x) is bounded in X for all B ∈ P b (X ) (i.e. sup x∈B {sup{|y| : y ∈ F(x)}} < ∞); (d) completely continuous if F(B) is relatively compact for every B ∈ P b (X ). F has a fixed point if there is x ∈ X such that x ∈ F(x).
A multivalued map F : W → P cl (R) is said to be measurable if, for every b ∈ R, the function t − → d(b, F(t)) = inf{|b -c| : c ∈ F(t)} is measurable. We define the graph of F to be the set Fr(F) = {(x, y) ∈ X × Y , y ∈ F(x)}. The fixed point set of the multivalued operator F will be denoted by FixF.
Remark 2.1 (The relationship between closed graphs and upper-semicontinuity) If F : X → P cl (X ) is u.s.c., then Fr(F) is a closed subset of X × Y i.e. for every sequence {x n } n∈N ⊂ X and {y n } n∈N ⊂ X , if when n → ∞, x n → x * , y n → y * , and y n ∈ F(x n ), then y * ∈ F(x * ). Conversely, if F is completely continuous and has a closed graph, then it is upper semi-continuous. (t, u, v) is upper semicontinuous for almost all t ∈ [a, b]; Further, a Carathéodory function F is called L 1 -Carathéodory if (iii) for each ρ > 0, there exists ρ ∈ L 1 ([a, b], R + ) such that

Definition 2.2 A multivalued map
for all u , v ≤ ρ and for a. e. t ∈ [a, b].
Let us now recall the following lemma from [27]. can be expressed by the formulas: Let us consider the set of selection functions F and G for each (u, v) ∈ F × F defined by and S G, (u,v) Define the operators 1 , 2 : where Next, we introduce an operator : , where 1 and 2 are defined by (2.5) and (2.6) respectively. For the sake of computational convenience, we set the notation where 2p , 3 3

The Carathéodory case
To prove our first existence result for multivalued problem (1.1), we need the following known results.
where E i (i = 1, 2) are given in (2.7). Then problem (1.1) has at least one solution on [a, b].
Proof Consider the operators 1 , 2 : F × F → P(F × F) defined by (2.5) and (2.6) respectively. It follows from assumption (H 1 ) that the sets S F, (u,v) and S G, (u,v) are nonempty for each (u, v) ∈ F × F . Then, forf ∈ S F, (u,v) ,ĝ ∈ S G, (u,v) and ∀(u, v) ∈ F × F , we have Now, we will verify that the operator satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. In the first step, we show that Let 0 ≤ ω ≤ 1. Then, for each t ∈ [0, 1], we have (u,v) are convex valued as F and G are convex valued maps, therefore Then, for t ∈ [a, b], we have 3 3 Similarly, we can obtain that Thus, we get where D i (i = 1, . . . , 4) are defined by (2.8). In consequence, we have × p 2 ψ 2 ν * + φ 2 ν * → 0 as t 2 → t 1 independent of (u, v).
Analogously, it can be shown that Obviously, the right-hand sides of the above inequalities tend to zero independently of (u, v) ∈ B ν * as t 2t 1 − → 0. Therefore, the operator (u, v) is equicontinuous, and hence we deduce that (u, v) : F × F → P(F × F) is completely continuous by the Arzelá-Ascoli theorem.
In the next step, we show that (u, v) is upper semicontinuous. Instead it will be established that (u, v) has a closed graph in view of the fact that a completely continuous operator is upper semicontinuous if it has a closed graph.
Consider the continuous linear operators 1 , 2 : given by From Lemma 3.1, we know that ( 1 , 2 ) • (S F , S G ) is a closed graph operator. Moreover, we which leads to the conclusion that (h k ,h k ) ∈ (u * , v * ).

Finally, we show that there exists an open set
Using the arguments employed in the second step, we find that Then we have where E i , i = 1, 2, are given by (2.7). Consequently, we have According to (H 3 ), there exists N such that (u, v) = N . Let us set Observe that the operator :Ū − → P cp,cv (F ) × P cp,cv (F ) is completely continuous and upper semicontinuous. From the choice of U, there is no (u, v) ∈ ∂U such that (u, v) ∈ (u, v) for some ∈ (0, 1). Therefore, by the nonlinear alternative of Leray-Schauder type (Lemma 3.2), we deduce that has a fixed point (u, v) ∈Ū which is a solution of problem (1.1).

The Lipschitz case
The forthcoming result is based on the fixed point theorem for contraction multivalued operators due to Covitz and Nadler [32], which is stated below.   d(a, b) and d(a, B) = inf b∈B d(a, b). Then (P b,cl (X), H d ) is a metric space and (P cl (X), H d ) is a generalized metric space (see [33]).  d(0, G(t, 0, 0)) ≤ B 2 (t). Then the boundary value problem (1.1) has at least one solution on [a, b] Proof Consider the multivalued map : F × F → P(F × F) defined at the beginning of the proof of Theorem 3.3. Observe that the fixed points of (u, v) are solutions of problem (1.1).
Notice that the sets S F, (u,v) and S G, (u,v) are nonempty, and consequently = ∅ for each (u, v) ∈ F × F . Then, by assumption (H 5 ), the multivalued maps F(·, (u, v)) and G(·, (u, v)) are measurable, and thus admit measurable selections. Now we shall show that the operator (u, v) satisfies the hypothesis of Lemma 4.1.
η a s a 1 q(z) dz ds Since F and G have compact values, we pass onto subsequences (if necessary) to get that f k andĝ k converge tof andĝ in L 1 ([a, b], R) respectively. Thenf ∈ S F, (u,v) andĝ ∈ S G, (u,v) , and for each t ∈ [a, b], we have Therefore (u, v) ∈ , and hence (u, v) is closed. Next, we show that is a contraction on P cl (F ) × P cl (F ), that is, there exists a positive number γ < 1 such that (u,v) such that, for each t ∈ [a, b], we obtain By (H 6 ), we have that and So there existθ f ∈ F(t, u(t), v(t)) andθ g ∈ G(t, u(t), v(t)) such that and Since the multivalued operators W 1 (t) ∩ F(t, u(t)v(t)) and W 2 (t) ∩ G(t, u(t), v(t)) are measurable, there exist functionsf 2 (t),ĝ 2 (t) which are measurable selections for W 1 and W 2 .

Conclusions
We have developed the existence theory for a self-adjoint coupled system of nonlinear second-order ordinary differential inclusions supplemented with nonlocal integral multistrip coupled boundary conditions on an arbitrary domain. Our study includes the cases of convex as well as nonconvex multivalued maps. Nonlinear alternative of Leray-Schauder type for multivalued maps and Covitz and Nadler's fixed point theorem for contractive multivalued maps are applied to prove the main results. Numerical examples are constructed for the illustration of the obtained results. Our results are new in the given configuration and enrich the related literature. Moreover, several new results can be recorded as special cases of the present work by fixing the parameters appearing in the system. For example, we obtain the existence results for an antiperiodic multivalued boundary value problem of self-adjoint coupled second-order ordinary differential inclusions by fixing α i = 1, β i = 1, λ i = 0, i = 1, 2, 3, 4, in the results of this paper, which are indeed new.