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

)is 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.

is study is organized as follows. e 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.

Some Preliminaries
First, we fix some notations. We denote by ≤ R 2 the partial order in the Euclidean space R 2 defined as We denote by M + 2 the set of square matrices having nonnegative coefficients, i.e., For C ∈ M + 2 , the trace of C is denoted by Trace(C), the determinant of C is denoted by det(C), and the spectral radius of C is denoted by ρ C , i.e., where λ i (C) are the complex eigenvalues of C.
where ‖ · ‖ 2 is the Euclidean norm in R 2 . e following lemmas will be useful later.
where 0 → is the zero vector in Proof. e result follows from the fact that is a normal cone in R 2 with normal constant equal to 1 (e.g., [30]).

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Next, we discuss some particular cases of eorem 1.

Nonlinearities Involving Trigonometric Functions.
Consider the system of differential equations (36)

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 ( eorem 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
e authors declare that there are no conflicts of interest.