Abstract
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.
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1 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 [6–21] and the references cited therein. Self-adjoint differential equations are found to be of great interest in certain disciplines, for example, see [22–25]. 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 self-adjoint coupled system of second-order ordinary differential inclusions on an arbitrary domain, subject to the nonlocal nonseparated integral coupled boundary conditions given by
where \(a<\eta <\xi <b\), \(p, q \in C([a,b], \mathbb{R}^{+})\), \(\alpha _{i}\), \(\beta _{i}\), \(\lambda _{i} \in \mathbb{R}^{+}\), \(i =1, 2, 3, 4\), \(\mu _{j} \in \mathbb{R}^{+}\), \(j=1, 2\), and \(F,G:[a,b]\times \mathbb{R} \times \mathbb{R} \longrightarrow \mathcal{P}(\mathbb{R})\) are given multivalued maps, \(\mathcal{P}(\mathbb{R})\) is the family of all nonempty subsets of \(\mathbb{R}\).
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.
2 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 \((\mathcal{X}, \|\cdot \|)\) be a normed space. We denote the classes of all closed, bounded, compact, and compact and convex sets in \(\mathcal{X}\) by \({\mathcal{P}}_{cl}\), \({\mathcal{P}}_{b}\), \({\mathcal{P}}_{cp}\), and \({\mathcal{P}}_{cp,c}\), respectively.
A multivalued map \(F : \mathcal{X} \to {\mathcal{P}}(\mathcal{X})\) is (a) convex (closed) valued if \(F(x)\) is convex (closed) for all \(x \in \mathcal{X}\); (b) upper semicontinuous (u.s.c.) on \(\mathcal{X}\) if for each \(x_{0} \in \mathcal{X}\), the set \(F(x_{0})\) is a nonempty closed subset of \(\mathcal{X}\), and if for each open set \(\mathcal{N}\) of \(\mathcal{X}\) containing \(F(x_{0})\), there exists an open neighborhood \(\mathcal{N}_{0}\) of \(x_{0}\) such that \(F(\mathcal{N}_{0}) \subseteq \mathcal{N}\); (c) bounded on bounded sets if \(F(\mathbb{B}) = \bigcup_{x \in \mathbb{B}}F(x)\) is bounded in \(\mathcal{X}\) for all \(\mathbb{B} \in {\mathcal {P}}_{b}(\mathcal{X})\) (i.e. \(\sup_{x \in \mathbb{B}}\{\sup \{|y| : y \in F(x)\}\} < \infty )\); (d) completely continuous if \(F(\mathbb{B})\) is relatively compact for every \(\mathbb{B} \in {\mathcal {P}}_{b}(\mathcal{X})\). F has a fixed point if there is \(x\in \mathcal{X}\) such that \(x \in F(x)\).
A multivalued map \(F : W \to {\mathcal {P}}_{cl}({\mathbb{R}})\) is said to be measurable if, for every \(b \in {\mathbb{R}}\), the function \(t \longmapsto d(b,F(t)) = \inf \{|b-c|: c \in F(t)\}\) is measurable. We define the graph of F to be the set \({\mathit{{Fr}}}(F)=\{(x,y)\in X \times Y, y\in F(x)\}\). 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 \(F : \mathcal{X} \to \mathcal{P}_{cl}(\mathcal{X})\) is u.s.c., then \({\mathit{{Fr}}}(F)\) is a closed subset of \(X \times Y \) i.e. for every sequence \(\{x_{n}\}_{n \in \mathbb{N}} \subset \mathcal{X}\) and \(\{y_{n}\}_{n \in \mathbb{N}} \subset \mathcal{X}\), if when \(n \to \infty \), \(x_{n} \to x_{*}\), \(y_{n} \to y_{*}\), and \(y_{n} \in F(x_{n})\), then \(y_{*} \in F(x_{*})\). Conversely, if F is completely continuous and has a closed graph, then it is upper semi-continuous.
Definition 2.2
A multivalued map \(F : [a,b] \times \mathbb{R}^{2} \to {\mathcal{P}}(\mathbb{R})\) is said to be Carathéodory if
-
(i)
\(t \longmapsto F(t,u,v)\) is measurable for each \(u,v \in \mathbb{R}\);
-
(ii)
\((u,v) \longmapsto F(t,u,v)\) is upper semicontinuous for almost all \(t\in [a,b]\);
Further, a Carathéodory function F is called \(L^{1}\)-Carathéodory if
-
(iii)
for each \(\rho > 0\), there exists \(\Omega _{\rho} \in L^{1}([a,b],\mathbb{R}^{+})\) such that
$$ \bigl\Vert F (t,u,v) \bigr\Vert = \sup \bigl\{ \vert x \vert : x \in F (t, u,v) \bigr\} \le \Omega _{\rho} (t)$$for all \(\|u\| ,\|v\| \le \rho \) and for a. e. \(t \in [a,b]\).
Definition 2.3
A function \((u,v) \in \mathcal{F} \times \mathcal{F}\), where \(\mathcal{F}= C^{2}([a,b],\mathbb{R})\), is a solution of the self-adjoint coupled system (1.1) if it satisfies the coupled conditions of (1.1) and there exist functions \(\hat{f},\hat{g}\in L^{1} ([a,b],\mathbb{R})\) such that \(\hat{f}(t) \in F (t,u(t),v(t))\), \(\hat{g}(t) \in G(t,u(t),v(t))\) a.e on \([a,b]\).
Let us now recall the following lemma from [27].
Lemma 2.4
For \(f_{1},g_{1} \in C([a,b], {\mathbb{R}})\) and \(R\neq 0\), \(E\neq 0\), the solution of the linear system
can be expressed by the formulas:
and
where
Let us consider the set of selection functions F and G for each \((u,v) \in \mathcal{F} \times \mathcal{F}\) defined by
and
Define the operators \(\Theta _{1},\Theta _{2}:\mathcal{F} \times \mathcal{F} \to { \mathcal {P}}(\mathcal{F} \times \mathcal{F})\) by
and
where
and
Next, we introduce an operator \(\Theta : \mathcal{F} \times \mathcal{F} \to {\mathcal {P}}(\mathcal{F} \times \mathcal{F})\) as
where \(\Theta _{1}\) and \(\Theta _{2}\) are defined by (2.5) and (2.6) respectively.
For the sake of computational convenience, we set the notation
where
3 The Carathéodory case
To prove our first existence result for multivalued problem (1.1), we need the following known results.
Lemma 3.1
([30])
Let X be a Banach space. Let \(F : [a, b] \times \mathbb{R}^{2} \to {\mathcal {P}}_{cp,c}(\mathbb{R})\) be an \(L^{1}\)- Carathéodory multivalued map, and let φ be a linear continuous mapping from \(L^{1}([a,b],\mathbb{R})\) to \(C([a,b],\mathbb{R})\). Then the operator
is a closed graph operator in \(C([a,b],\mathbb{R}) \times C([a,b],\mathbb{R})\).
Lemma 3.2
(Nonlinear alternative for Kakutani maps [31])
Let \(\mathcal{S}\) be a Banach space, \(\mathcal{S}_{1}\) be a closed convex subset of \(\mathcal{S}\), U be an open subset of \(\mathcal{S}_{1}\), and \(0\in U\). Suppose that \(F: \overline{U}\to {\mathcal {P}}_{c,cv}(\mathcal{S}_{1})\) is an upper semicontinuous compact map; here \({\mathcal {P}}_{c,cv}(\mathcal{S}_{1})\) denotes the family of nonempty, compact convex subsets of \(\mathcal{S}_{1}\). Then either
-
(i)
F has a fixed point in U̅ or
-
(ii)
there are \(u\in \partial U\) and \(\lambda \in (0,1)\) with \(u\in \lambda F(u)\).
Now we are in a position to present our first main result.
Theorem 3.3
Assume that
- \((H_{1})\):
-
\(F,G:[a,b]\times \mathbb{R}^{2}\longrightarrow \mathcal{P}(\mathbb{R})\) are Carathéodory possessing compact and convex values;
- \((H_{2})\):
-
There exist continuous nondecreasing functions \(\psi _{1},\psi _{2},\phi _{1},\phi _{2}:[0,\infty )\longrightarrow (0, \infty )\) such that
$$ \bigl\Vert F(t,u,v) \bigr\Vert _{\mathcal{P}}:=\sup \bigl\{ \vert \hat{f} \vert :\hat{f}\in F(t,u,v) \bigr\} \leqslant p_{1}(t) \bigl[\psi _{1} \bigl( \Vert u \Vert \bigr) +\phi _{1} \bigl( \Vert v \Vert \bigr) \bigr]$$and
$$ \bigl\Vert G(t,u,v) \bigr\Vert _{\mathcal{P}}:=\sup \bigl\{ \vert \hat{g} \vert :\hat{g}\in G(t,u,v) \bigr\} \leqslant p_{2}(t) \bigl[\psi _{2} \bigl( \Vert u \Vert \bigr) +\phi _{2} \bigl( \Vert v \Vert \bigr) \bigr]$$for each \((t,u,v)\in [a,b]\times \mathbb{R}^{2}\), where \(p_{1},p_{2}\in C([a,b],\mathbb{R}^{+})\);
- \((H_{3})\):
-
There exists a constant \(N>0\) such that
$$ \frac{N}{\mathcal{E}_{1} \Vert p_{1} \Vert [\psi _{1}(N)+\phi _{1}(N)]+\mathcal{E}_{2} \Vert p_{2} \Vert [\psi _{2}(N)+\phi _{2}(N)]}>1,$$where \(\mathcal{E}_{i}\) (\(i=1,2\)) are given in (2.7).
Then problem (1.1) has at least one solution on \([a,b]\).
Proof
Consider the operators \(\Theta _{1},\Theta _{2}:\mathcal{F} \times \mathcal{F} \to { \mathcal {P}}(\mathcal{F} \times \mathcal{F})\) 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 \((u,v) \in \mathcal{F} \times \mathcal{F}\). Then, for \(\hat{f} \in S_{F,(u,v)}\), \(\hat{g} \in S_{G,(u,v)}\) and \(\forall (u,v) \in \mathcal{F} \times \mathcal{F}\), we have
and
where \(h_{1}\in \Theta _{1}(u,v)\), \(h_{2}\in \Theta _{2}(u,v)\), and hence \((h_{1},h_{2}) \in \Theta (u,v)\).
Now, we will verify that the operator Θ satisfies the assumptions of the nonlinear alternative of Leray–Schauder type. In the first step, we show that \(\Theta (u,v)\) is convex valued for each \((u,v) \in \mathcal{F} \times \mathcal{F}\). Let \((h_{i}, \tilde{h_{i}})\in (\Theta _{1},\Theta _{2})\), \(i=1,2\). Then there exist \(\hat{f}_{i}\in S_{F,(u,v)}\), \(\hat{g}_{i}\in S_{G,(u,v)}\), \(i=1,2\), such that, for each \(t \in [a,b]\), we have
and
Let \(0 \le \omega \le 1\). Then, for each \(t \in [0,1]\), we have
and
Since \(S_{F,(u,v)}\), \(S_{G,(u,v)}\) are convex valued as F and G are convex valued maps, therefore \(\omega h_{1}+(1-\omega )h_{2} \in \Theta _{1}\), \(\omega \tilde{h_{1}}+(1- \omega )\tilde{h_{2}} \in \Theta _{2} \), and hence \(\omega ( h_{1},\tilde{h_{1}})+(1-\omega )(h_{2},\tilde{h_{2}}) \in \Theta \).
Now, we show that Θ maps bounded sets into bounded sets in \(\mathcal{F} \times \mathcal{F}\). For a positive number \(\nu ^{*}\), let \(B_{\nu ^{*}} = \{(u,v) \in \mathcal{F} \times \mathcal{F}: \|(u,v)\| \le \nu ^{*} \}\) be a bounded set in \(\mathcal{F} \times \mathcal{F}\). Then, for each \(h_{i} \in \Theta _{i}\) (\(i=1,2\)), \((u,v)\in B_{ \nu ^{*}}\), there exist \(\hat{f} \in S_{F,(u,v)}\), \(\hat{g} \in S_{G,(u,v)}\) such that
and
Then, for \(t\in [a,b]\), we have
Similarly, we can obtain that
Thus, we get
where \(\mathcal{D}_{i}\) (\(i=1,\ldots ,4\)) are defined by (2.8). In consequence, we have
where \(\mathcal{E}_{i}\), \(i=1,2\), are defined in (2.7).
Next, we verify that \(\Theta (u,v)\) is equicontinuous. Let \(t_{1}, t_{2} \in [a,b]\) with \(t_{1}< t_{2}\). Then, for \(\hat{f} \in S_{F,(u,v)}\), \(\hat{g} \in S_{G,(u,v)}\), we get
Analogously, it can be shown that
Obviously, the right-hand sides of the above inequalities tend to zero independently of \((u,v)\in B_{\nu ^{*}}\) as \(t_{2}-t_{1}\longrightarrow 0\). Therefore, the operator \(\Theta (u,v)\) is equicontinuous, and hence we deduce that \(\Theta (u,v):\mathcal{F} \times \mathcal{F} \to {\mathcal {P}}( \mathcal{F} \times \mathcal{F})\) is completely continuous by the Arzelá–Ascoli theorem.
In the next step, we show that \(\Theta (u,v)\) is upper semicontinuous. Instead it will be established that \(\Theta (u,v)\) has a closed graph in view of the fact that a completely continuous operator is upper semicontinuous if it has a closed graph. Let \(( u_{k},v_{k})\longrightarrow ( u_{\ast},v_{\ast})\) and \((h_{k} ,\tilde{h_{k}})\in \Theta (u_{k},v_{k})\) and \((h_{k} ,\tilde{h_{k}})\longrightarrow ( h_{\ast},\tilde{h_{\ast}}) \). Then we have to show that \(( h_{\ast},\tilde{h_{\ast}}) \in \Theta (u_{\ast},v_{\ast})\). Associated with \((h_{k} ,\tilde{h_{k}}) \in \Theta (u_{k},v_{k})\) and \(\hat{f}_{k} \in S_{F,(u,v)}\), \(\hat{g}_{k} \in S_{G,(u,v)}\), for each \(t \in [a,b]\), we have
and
Consider the continuous linear operators \(\Psi _{1},\Psi _{2}:L^{1}([a,b],\mathcal{F}\times \mathcal{F}) \longrightarrow C([a,b],\mathcal{F}\times \mathcal{F}) \) given by
and
From Lemma 3.1, we know that \((\Psi _{1},\Psi _{2}) \circ (S_{F},S_{G})\) is a closed graph operator. Moreover, we have \((h_{k},\tilde{h_{k}}) \in (\Psi _{1},\Psi _{2}) \circ (S_{F,(u_{k},v_{k})},S_{G,(u_{k},v_{k})})\) for all k. Since \((u_{k},v_{k}) \longrightarrow (u_{\ast},v_{\ast})\), \((h_{k}, \tilde{h_{k}}) \longrightarrow (h_{\ast},\tilde{h_{\ast}})\), it follows that \(\hat{f}_{\ast }\in S_{F,(u,v)}\), \(\hat{g}_{\ast }\in S_{G,(u,v)}\) such that
and
which leads to the conclusion that \((h_{k},\tilde{h_{k}})\in \Theta (u_{\ast},v_{\ast})\).
Finally, we show that there exists an open set \(U\subseteq \mathcal{F} \times \mathcal{F} \to {\mathcal {P}}( \mathcal{F} \times \mathcal{F})\) with \((u,v) \notin \epsilon \Theta (u,v)\) for any \(\epsilon \in (0,1)\) and all \((u,v) \in \partial U \). Let \(\epsilon \in (0,1)\) and \((u,v) \in \epsilon \Theta (u,v)\). Then there exist \(\hat{f}\in S_{F},_{(u,v)}\) and \(\hat{g}\in S_{G},_{(u,v)}\) such that, for \(t \in [a,b]\), we have
and
Using the arguments employed in the second step, we find that
and
Then we have
where \(\mathcal{E}_{i}\), \(i=1,2 \), are given by (2.7). Consequently, we have
According to \((H_{3})\), there exists N such that \(\|(u,v)\|\neq N \). Let us set
Observe that the operator \(\Theta :\bar{U}\longrightarrow \mathcal{P}_{cp,cv}(\mathcal{F}) \times \mathcal{P}_{cp,cv}(\mathcal{F}) \) is completely continuous and upper semicontinuous. From the choice of U, there is no \((u,v)\in \partial U \) such that \((u,v) \in \epsilon \Theta (u,v)\) for some \(\epsilon \in (0,1)\). Therefore, by the nonlinear alternative of Leray–Schauder type (Lemma 3.2), we deduce that Θ has a fixed point \((u,v)\in \bar{U}\) which is a solution of problem (1.1). □
4 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.
Lemma 4.1
(Covitz and Nadler)
Let \((X,d)\) be a complete metric space. If \(G : X \to P_{cl}(X)\) is a contraction, then \(\mathit{Fix} G \ne \emptyset \).
Remark 4.2
Let \((X,d)\) be a metric space induced from the normed space \((X; \|\cdot \|)\). Consider \(H_{d} :{\mathcal{P}}(X) \times {\mathcal{P}}(X) \to \mathbb{R} \cup \{\infty \}\) given by
where \(d(A,b) = \inf_{a\in A}d(a,b)\) and \(d(a,B) = \inf_{b\in B}d(a,b)\). Then \((P_{b,cl}(X), H_{d})\) is a metric space and \((P_{cl}(X), H_{d})\) is a generalized metric space (see [33]).
Theorem 4.3
Assume that the following conditions hold:
- \((H_{5})\):
-
\(F,G : [a,b] \times \mathbb{R}^{2} \to {\mathcal{P}}_{cp}(\mathbb{R})\) are such that \(F(\cdot ,u,v), G(\cdot ,u,v) : [a,b] \to {\mathcal{P}}_{cp}( \mathbb{R})\) are measurable for each \(u,v\in \mathbb{R}\);
- \((H_{6})\):
-
For almost all \(t \in [a,b]\) and \(u,v,\bar{u},\bar{v} \in \mathbb{R}\) with \(\mathcal{B}_{1},\mathcal{B}_{2} \in C([a,b], \mathbb{R}^{+})\),
$$\begin{aligned}& H_{d}(F(t,u,v), F(t,\bar{u},\bar{v})\le \mathcal{B}_{1}(t) \bigl( \vert u-\bar{u} \vert + \vert v- \bar{v} \vert \bigr), \\& H_{d}(G(t,u,v), G(t,\bar{u},\bar{v})\le \mathcal{B}_{2}(t) \bigl( \vert u- \bar{u} \vert + \vert v-\bar{v} \vert \bigr), \end{aligned}$$and \(d(0,F(t,0,0))\le \mathcal{B}_{1}(t)\), \(d(0,G(t,0,0))\le \mathcal{B}_{2}(t)\).
Then the boundary value problem (1.1) has at least one solution on \([a, b]\) if \(\mathcal{E}_{1}\|\mathcal{B}_{1}\|+\mathcal{E}_{2}\|\mathcal{B}_{2} \|<1\), where \(\mathcal{E}_{1}\), \(\mathcal{E}_{2} \) are given in (2.7).
Proof
Consider the multivalued map \(\Theta : \mathcal{F} \times \mathcal{F} \to {\mathcal {P}}(\mathcal{F} \times \mathcal{F})\) defined at the beginning of the proof of Theorem 3.3. Observe that the fixed points of \(\Theta (u,v)\) are solutions of problem (1.1).
Notice that the sets \(S_{F,(u,v)}\) and \(S_{G,(u,v)}\) are nonempty, and consequently \(\Theta \ne \emptyset \) for each \((u,v)\in \mathcal{F} \times \mathcal{F}\). Then, by assumption \((H_{5})\), the multivalued maps \(F(\cdot , (u,v))\) and \(G(\cdot , (u,v))\) are measurable, and thus admit measurable selections.
Now we shall show that the operator \(\Theta (u,v)\) satisfies the hypothesis of Lemma 4.1. Firstly, we verify that \(\Theta (u,v)\in {\mathcal{P}}_{cl}(\mathcal{F}) \times {\mathcal{P}}_{cl}( \mathcal{F}) \) for each \((u,v) \in \mathcal{F} \times \mathcal{F}\). Let \((h_{k},\tilde{h}_{k}) \in \Theta (u_{k},v_{k})\) such that \((h_{k},\tilde{h}_{k})\) converges to \((h,\tilde{h})\) as \(k \to \infty \) in \(\mathcal{F} \times \mathcal{F}\). So \((h,\tilde{h}) \in \mathcal{F} \times \mathcal{F}\), and there exist \(\hat{f}_{k} \in S_{F,(u_{k},v_{k})}\) and \(\hat{g}_{k} \in S_{G,(u_{k},v_{k})}\) such that, for each \(t \in [a,b]\), we have
and
Since F and G have compact values, we pass onto subsequences (if necessary) to get that \(\hat{f}_{k}\) and \(\hat{g}_{k}\) converge to f̂ and ĝ in \(L^{1}([a,b],\mathbb{R})\) respectively. Then \(\hat{f}\in S_{F,(u,v)}\) and \(\hat{g}\in S_{G,(u,v)}\), and for each \(t \in [a,b]\), we have
and
Therefore \((u,v) \in \Theta \), and hence \(\Theta (u,v)\) is closed.
Next, we show that Θ is a contraction on \({\mathcal{P}}_{cl}(\mathcal{F}) \times {\mathcal{P}}_{cl}( \mathcal{F})\), that is, there exists a positive number \(\gamma <1\) such that
Let \(( u,\bar{u}),(v,\bar{v}) \in \mathcal{F} \times \mathcal{F}\) and \((h_{1},\tilde{h_{1}}) \in \Theta (u,v)\). Then there exist \(\hat{f}_{1}(t) \in S_{F,(u,v)}\) and \(\hat{g}_{1}(t) \in S_{G,(u,v)}\) such that, for each \(t \in [a,b]\), we obtain
and
By \((H_{6})\), we have that
and
So there exist \(\hat{\vartheta _{f}} \in F(t,u(t),v(t))\) and \(\hat{\vartheta _{g}} \in G(t,u(t),v(t))\) such that
Define \(W_{1},W_{2} : [a,b] \to \mathcal{P}(\mathbb{R})\) by
and
Since the multivalued operators \(W_{1}(t)\cap F(t,u(t)v(t))\) and \(W_{2}(t)\cap G(t,u(t),v(t))\) are measurable, there exist functions \(\hat{f}_{2}(t)\), \(\hat{g}_{2}(t)\) which are measurable selections for \(W_{1}\) and \(W_{2}\). Thus \(\hat{f}_{2}(t) \in F(t,u(t),v(t))\), \(\hat{g}_{2}(t) \in G(t,u(t),v(t))\), and for each \(t \in [a,b]\), we have
and
For each \(t \in [a,b]\), let us define
and
Then
which implies that
In a similar manner, one can establish that
In consequence, we get
Similarly, by interchanging the roles of \((u,v)\) and \((\bar{u},\bar{v})\), we can obtain that
Therefore, it follows by the assumption \(\mathcal{E}_{1} \|\mathcal{B}_{1}\|+\mathcal{E}_{2}\|\mathcal{B}_{2} \|<1\) that Θ is a contraction, So, by Lemma 4.1, Θ has a fixed point \((u,v)\), which is a solution of problem (1.1). The proof is finished. □
5 Examples
Example 5.1
Consider the following self-adjoint coupled system of second-order ordinary differential inclusions with boundary conditions:
Here, \(p(t)= 1/(t+13)\), \(q(t)=8/(4t^{2}+2t+12)\), \(\mu _{1}=3/36\), \(\mu _{2}=2/93\), \(a=0\), \(b=3\), \(\eta =1/2\), \(\xi =5/2\), \(\lambda _{1}=1/7\), \(\lambda _{2}=2/7\), \(\lambda _{3}=3/7\), \(\lambda _{4}=4/7\), \(\alpha _{1}=7/3\), \(\alpha _{2}=5/3\), \(\alpha _{3}=4/3\), \(\alpha _{4}=1\), \(\beta _{1}=1/9\), \(\beta _{2}=2/9\), \(\beta _{3}=3/9\), \(\beta _{4}=4/9 \), and \(F(t,u,v)\), \(G(t,u,v)\) will be fixed later.
Using the given data, we find that \(|R|\approx 1.323129\neq 0\), \(|E|\approx 115.6354\neq 0\) (R and E are given in (2.4)), \(\bar{p}\approx 0.0625\), \(\bar{q}=0.148148\), \(\mathcal{D}_{1}\approx 17.1389708\), \(\mathcal{D}_{2}\approx 0.06036034\), \(\mathcal{D}_{3}\approx 38.2023705\), \(\mathcal{D}_{4}\approx 4.565128967\), \(\mathcal{E}_{1}\approx 17.19933114\), and \(\mathcal{E}_{2}\approx 42.76749946\) (p̄, q̄ and \(\mathcal{D}_{i}\) (\(i=1,\dots ,4\)) are defined in (2.8), while \(\mathcal{E}_{1}\), \(\mathcal{E}_{2}\) are given in (2.7)).
For illustration of Theorem 3.3, we choose
and
For \(f \in F\), we have
and for \(g \in G\), we have
Thus
and
with \(p_{1}(t)=\frac{t}{108t^{2}+32}\), \(p_{2}(t)= \frac{\cos ^{2} (\pi t)}{t^{3}+120}\), \(\psi _{1}(\|u\|)=\phi _{1}(\|v\|)= \psi _{2}(\|u\|) =\phi _{2}(\|v\|)=1\). Furthermore, it is found that \(N >N_{1}\), where \(N_{1}=0.81272506\) (N is given in \((H_{3})\)). Clearly, all the hypotheses of Theorem 3.3 are satisfied. Thus, there exists at least one solution for problem (5.1) on \([0,3]\).
Example 5.2
Consider the following boundary value problem of self-adjoint coupled second-order ordinary differential inclusions:
where \(p(t)= 1/(t^{2}+2)\), \(q(t)=2/(t+6)\), \(\mu _{1}=1/16\), \(\mu _{2}=3/43\), \(a=0\), \(b=2\), \(\eta =1/4\), \(\xi =1\), \(\lambda _{1}=2/3\), \(\lambda _{2}=1\), \(\lambda _{3}=4/3\), \(\lambda _{4}=5/3\), \(\alpha _{1}=1/2\), \(\alpha _{2}=1\), \(\alpha _{3}=5/8\), \(\alpha _{4}=4/7\), \(\beta _{1}=2\), \(\beta _{2}=1/6\), \(\beta _{3}=1/5\), \(\beta _{4}=3/5 \), and \(F(t,u,v)\), \(G(t,u,v)\) will be fixed later.
Using the given values, it is found that \(|R|\approx 3.083\neq 0\), \(|E|\approx 8.506200\neq 0\) (R and E are given in (2.4)), \(\bar{p}\approx 0.16\), \(\bar{q}=0.25\), \(\mathcal{D}_{1}\approx 6.31401038\), \(\mathcal{D}_{2}\approx 0.72123977\), \(\mathcal{D}_{3}\approx 12.94560512\), \(\mathcal{D}_{4}\approx 3.23687872\), \(\mathcal{E}_{1}\approx 7.035250153\), and \(\mathcal{E}_{2}\approx 16.18248385\) (p̄, q̄ and \(\mathcal{D}_{i}\) (\(i=1,\dots ,4\)) are defined in (2.8), while \(\mathcal{E}_{1}\), \(\mathcal{E}_{2}\) are given in (2.7)).
For illustrating Theorem 4.3, we take the following multivalued maps \(F, G:[0,2]\times \mathbb{R} \to \mathcal{P}(\mathbb{R})\):
Letting \(\mathcal{B}_{1}(t)=\frac{1}{3t+160}\) and \(\mathcal{B}_{2}(t)=\frac{1}{t^{2}+188}\), we find that \(H_{d}(F(t,u,v),F(t,\bar{u},\bar{v}))\le \mathcal{B}_{1}(t)(|u- \bar{u}|+|v-\bar{v}|)\) and \(H_{d}(G(t,u,v),G(t,\bar{u},\bar{v}))\le \mathcal{B}_{2}(t)(|u- \bar{u}|+|v-\bar{v}|)\). Clearly, \(d(0,F(t,0,0))=\frac{1}{190}\le \mathcal{B}_{1}(t)\) and \(d(0,G(t,0,0))=\frac{1}{200}\le \mathcal{B}_{2}(t)\) for almost all \(t \in [0,2]\). Moreover, \(\|\mathcal{B}_{1}\|=1/160\) and \(\|\mathcal{B}_{2}\|=1/188\) and \(\mathcal{E}_{1}\|\mathcal{B}_{1}\|+\mathcal{E}_{2}\|\mathcal{B}_{2} \| \approx 0.1300473552 < 1\). Thus all the assumptions of Theorem 4.3 hold true. Therefore, by conclusion of Theorem 4.3, problem (5.2) with F, G given by (5.3) has at least one solution on \([0,2]\).
6 Conclusions
We have developed the existence theory for a self-adjoint coupled system of nonlinear second-order ordinary differential inclusions supplemented with nonlocal integral multi-strip 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 \(\alpha _{i}=1\), \(\beta _{i}=1\), \(\lambda _{i}=0\), \(i =1, 2, 3, 4\), in the results of this paper, which are indeed new.
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Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. (KEP-MSc-53-130-1443). The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors also thank the reviewers for their constructive remarks on their work.
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This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. KEP-MSc-53-130-1443.
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Each of the authors, BA, AA, SKN, and AAl contributed equally to each part of this work. All authors read and approved the final manuscript.
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Ahmad, B., Almalki, A., Ntouyas, S.K. et al. Existence results for a self-adjoint coupled system of nonlinear second-order ordinary differential inclusions with nonlocal integral boundary conditions. J Inequal Appl 2022, 111 (2022). https://doi.org/10.1186/s13660-022-02846-5
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DOI: https://doi.org/10.1186/s13660-022-02846-5