Nonzero Solutions for Nonlinear Systems of Fourth-Order Boundary Value Problems

Abstract

This study is devoted to the investigation of nonlinear systems of fourth-order boundary value problems. Namely, using some techniques from matrix analysis and ordinary differential equations, a Lyapunov-type inequality providing a necessary condition for the existence of nonzero solutions is obtained. Next, an estimate involving generalized eigenvalues is derived as an application of our main result.

1. Introduction

In this study, we investigate the system of differential equationssubjected to the boundary conditionswhere and are the continuous functions with . Clearly, is a trivial solution to (1) and (2). The aim of this study is to obtain necessary conditions for the existence of nonzero solutions to the considered problem. Namely, we establish new Lyapunov-type inequalities [1] for (1) and (2) under reasonable conditions on the nonlinearities and . Our approach is based essentially on matrix analysis and some arguments from ordinary differential equations.

Fourth-order differential equations are useful in modeling many phenomena from physics ([2–7] and the references therein), which makes the study of such equations particularly interesting. In the literature, several contributions have been devoted to the investigation of sufficient conditions ensuring the existence of solutions to fourth-order boundary value problems ([2, 8–19] and the references therein). The study of necessary conditions for the existence of nontrivial solutions to fourth-order differential equations via Lyapunov-type inequalities has been investigated by some authors [20–22]. For instance, in [20], among other results, it was shown that, if is a nontrivial solution towhere is continuous, thenwhere .

For recent contributions related to Lyapunov-type inequalities, see e.g., [23–29] and the references therein.

This study is organized as follows. The next section is devoted to some preliminaries. In Section 3, the obtained results as well as their proofs are presented. Finally, some applications to generalized eigenvalue problems are given in Section 4.

2. Some Preliminaries

First, we fix some notations. We denote by the partial order in the Euclidean space defined asfor every .

We denote by the set of square matrices having nonnegative coefficients, i.e.,

For , the trace of is denoted by , the determinant of is denoted by , and the spectral radius of is denoted by , i.e.,where are the complex eigenvalues of .

We equip with the norm defined aswhere is the Euclidean norm in .

The following lemmas will be useful later.

Lemma 1. Let withwhere is the zero vector in . Then, .

Proof. The result follows from the fact thatis a normal cone in with normal constant equal to 1 (e.g., [30]).

Lemma 2 (See e.g., [31]). Let . If , then

Lemma 3 (See [25]). Let . Then,

Lemma 4 (See [12]). Let be a solution towhere . Then,where

Lemma 5. For all ,

Proof. Fix . Since is nondecreasing in , we deduce thatSimilarly, it can be easily shown that is nondecreasing in , which yieldsCombining (17) with (18), for all , we obtainHence, (16) is proved.

Throughout this study, we denote by the norm in defined as

3. Results and Proofs

We investigate (1) and (2) under the following assumptions: (A1) are continuous (A2) are continuous (A3) (where 0 is the zero function) (A4) For all ,where are the continuous functions.

By solution to (1) and (2), we mean a pair of functions , , satisfying (1) and the initial conditions (2). A solution to (1) and (2) is said to be nontrivial, if .

For , let

Theorem 1. If (1) and (2) have a nontrivial solution, then

Proof. Let be a nontrivial solution to (1) and (2), and suppose thatBy Lemma 4, is a nontrivial solution to the system of integral equations:Using (A4) and (16), for all , we obtainwhich leads toSimilarly, by (A4) and (16), we obtainCombining (27) with (28), we deduce thatwhere andNext, using Lemma 3 and (24), we deduce thatOn the other hand, using Lemma 1 and (29), we obtainSince is nontrivial, then , and the above inequality leads toBut by Lemma 2 and (31), we know thatwhich contradicts (33). This proves (23).

Next, we discuss some particular cases of Theorem 1.

3.1. Nonlinearities Involving Trigonometric Functions

Consider the system of differential equationsunder the boundary conditions (2), where . Observe that (35) is a particular case of (1) with

Obviously, and satisfy (A3). Moreover, for all ,

Then, (A4) is satisfied with

Hence, by Theorem 1, we deduce the following.

Corollary 1. If (35) and (2) have a nontrivial solution, then

3.2. Nonlocal Source Terms

Consider the system of differential equationsunder the boundary condition (16), where and . Problem (40) is a particular case of (1) withfor all . Obviously, and satisfy (A3). Moreover, for all ,and similarly

Then, (A4) is satisfied withThen, by Theorem 1, we deduce the following.

Corollary 2. If (39) and (2) have a nontrivial solution, then

Example 1. Consider the system of differential equation (35) with and for all . Elementary calculations show thatHence, by Corollary 1, we deduce that (35) and (2) have no nontrivial solution.

4. Generalized Eigenvalues Problems

We say that is a generalized eigenvalue of the system of differential equationssubjected to the boundary condition (2), if (2) and (44) admit a nonzero solution . Notice that (44) is a particular case of (1) with

Moreover, (A2)–(A4) are satisfied with

Hence, by Theorem 1, if is a generalized eigenvalue of (2) and (44), then

On the other hand, for , we have

Hence, (50) reduces toTherefore, the following result follows.

Corollary 3. If is a generalized eigenvalue of (43) and (2), then

5. Conclusion

Using some techniques from matrix analysis and ordinary differential equations, a necessary condition for the existence of nonzero solutions to (1) and (2) is obtained (Theorem 1). As particular cases of (1), we discussed nonlinearities involving trigonometric functions (Corollary 1) and nonlocal source terms (Corollary 2). Finally, we applied our main result to obtain an estimate involving generalized eigenvalues (Corollary 3).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

The third author was supported by Researchers Supporting Project number, King Saud University, Riyadh, Saudi Arabia (RSP-2021/4).