Abstract
The aim of this paper is to study a fourth-order separated boundary value problem with the right-hand side function satisfying one-sided Nagumo-type condition. By making a series of a priori estimates and applying lower and upper functions techniques and Leray-Schauder degree theory, the authors obtain the existence and location result of solutions to the problem.
Highlights
In this paper we apply the lower and upper functions method to study the fourth-order nonlinear equation u 4 t f t, u t, u t, u t, u t, 0 < t < 1, 1.1 with f : 0, 1 × Ê4 → Ê being a continuous function. This equation can be used to model the deformations of an elastic beam, and the type of boundary conditions considered depends on how the beam is supported at the two endpoints 1, 2
We apply lower and upper functions technique and topological degree method to prove the existence of solutions by making a priori estimates for the third derivative of all solutions of problems 1.1 and 1.2
Observe that the estimation R depends only on the functions hE, γ2, Γ2, and ρ and it does not depend on the boundary conditions
Summary
By making a series of a priori estimates and applying lower and upper functions techniques and Leray-Schauder degree theory, the authors obtain the existence and location result of solutions to the problem
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