Existence and Multiplicity of Positive Solutions of a Nonlinear Discrete Fourth‐Order Boundary Value Problem

Ruyun Ma,Yanqiong Lu

doi:10.1155/2012/918082

Abstract

we show the existence and multiplicity of positive solutions of the nonlinear discrete fourth‐order boundary value problem Δ4u(t − 2) = λh(t)f(u(t)), t ∈ 𝕋2, u(1) = u(T + 1) = Δ2u(0) = Δ2u(T) = 0, where λ > 0, h : 𝕋2 → (0, ∞) is continuous, and f : ℝ → [0, ∞) is continuous, T > 4, 𝕋2 = {2,3, …, T}. The main tool is the Dancer′s global bifurcation theorem.

Highlights

It’s well known that the fourth order boundary value problem u t f t, u t, t ∈ 0, 1, 1.1u0 u1 u 0 u 1 0 can describe the stationary states of the deflection of an elastic beam with both ends hinged, it models a rotating shaft
Ma and Xu 15 applied the fixed point theorem in cones to obtain some results on the existence of generalized positive solutions
It is the purpose of this paper to show some new results on the existence and multiplicity of generalized positive solutions of 1.4, 1.8 by Dancer’s global bifurcation theorem

Summary

Introduction

It’s well known that the fourth order boundary value problem u t f t, u t , t ∈ 0, 1 , 1.1. Ma and Xu 15 applied the fixed point theorem in cones to obtain some results on the existence of generalized positive solutions. It is the purpose of this paper to show some new results on the existence and multiplicity of generalized positive solutions of 1.4 , 1.8 by Dancer’s global bifurcation theorem. The rest of the paper is organized as follows: in Section 2, we present the form of the Green’s function of 1.4 , 1.8 and its properties, and we enunciate the Dancer’s global bifurcation theorem 16, Corollary 15.2. For other results on the existence and multiplicity of positive solutions and nodal solutions for fourth-order boundary value problems based on bifurcation techniques, see 17– 21

Preliminaries and Dancer’s Global Bifurcation Theorem

Proof of the Main Results

Some Examples