Existence Results for A Perturbed Fourth-Order Equation

Mohammad Reza Heidari Tavani

doi:10.22342/jims.23.2.498.55-65

Abstract

‎The existence of at least three weak solutions for a class of perturbed‎‎fourth-order problems with a perturbed nonlinear term is investigated‎. ‎Our‎‎approach is based on variational methods and critical point theory‎.

Highlights

Consider the following fourth-order problem u(iυ)(x) = λf (x, u(x)) + h(u(x)), x ∈ [0, 1], u(0) = u (0) = 0, (1)u (1) = 0, u (1) = μ g(u(1)), where λ and μ are a positive parameters, f : [0, 1] × R → R is L1-Caratheodory function, g : R → R is a continuous function and h : R → R is a Lipschitz continuous function with the Lipschitz constant 0 < L < 1, i.e.,|h(t1) − h(t2)| ≤ L|t1 − t2|for every t1, t2 ∈ R, and h(0) = 0
An elastic beam of length d = 1, which is clamped at its left side x = 0, and resting on a kind of elastic bearing at its right side x = 1 which is given by μg
Classical bending theory of elastic beams are very important in engineering sciences and so many studies have been done on a variety of problems like this

Summary

Introduction

For example in [12], authors considered iterative solutions for problem (1) in the case of λ = μ = 1 with nonlinear boundary conditions. In the present paper, using one kind of three critical points theorem obtained in [5] which we recall , we establish the existence of at least three weak solutions for the problem (1). For example this theorem used to ensure the existence of at least three solutions for perturbed boundary value problems in the papers [2, 3, 7, 10].

Results

Conclusion