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.
Highlights
We investigate the system of differential equations
Fourth-order differential equations are useful in modeling many phenomena from physics ([2,3,4,5,6,7] and the references therein), which makes the study of such equations interesting
Several contributions have been devoted to the investigation of sufficient conditions ensuring the existence of solutions to fourth-order boundary value problems ([2, 8,9,10,11,12,13,14,15,16,17,18,19] and the references therein). e 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 [20,21,22]
Summary
Where ρ, σ: [0, 1] ⟶ R and μ, ξ: [0, 1] × C([0, 1]) × C([0, 1]) ⟶ R are the continuous functions with μ(·, 0, 0) ξ(·, 0, 0) ≡ 0. E aim of this study is to obtain necessary conditions for the existence of nonzero solutions to the considered problem. Several contributions have been devoted to the investigation of sufficient conditions ensuring the existence of solutions to fourth-order boundary value problems ([2, 8,9,10,11,12,13,14,15,16,17,18,19] and the references therein). E 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 [20,21,22].
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