Bifurcation and positive solutions of a nonlinear fourth-order dynamic boundary value problem on time scales

Hua Luo

doi:10.1186/1687-1847-2013-64

Abstract

This paper discusses the spectrum properties of a linear fourth-order dynamic boundary value problem on time scales and obtains the existence result of positive solutions to a nonlinear fourth-order dynamic boundary value problem. The key condition which makes nonlinear problem have at least one positive solution is related to the first eigenvalue of the associated linear problem. The proof of the main result is based upon the Krein-Rutman theorem and the global bifurcation techniques on time scales.

Highlights

In, Luo and Ma [ ] considered the second-order dynamic boundary value problem on time scales u (t) + fuσ (t) =, t ∈ [, ]T, ( . )u( ) = u( ) =, where f ∈ C([, ∞), (, ∞))
Wang and Sun [ ] and Luo and Gao [ ] applied the Schauder fixed point theorem to show the existence of positive solutions of a fourth-order dynamic boundary value problem under different boundary value conditions
We have proved the following

Summary

Introduction

In , Luo and Ma [ ] considered the second-order dynamic boundary value problem on time scales u (t) + fuσ (t) = , t ∈ [ , ]T, We have a natural question if we could establish some optimal results for the fourth-order dynamic boundary value problem u (t) = f t, u(t), u (t) , t ∈ , ρ (T) T, u( ) = u σ (T) = u ( ) = u (T) = , where T is a time scale and , T ∈ T, < ρ(T), f : [ , ρ(T)]T × [ , ∞) × (–∞, ] → [ , ∞) is continuous. Wang and Sun [ ] and Luo and Gao [ ] applied the Schauder fixed point theorem to show the existence of positive solutions of a fourth-order dynamic boundary value problem under different boundary value conditions.

And we can easily check that

Since yn is bounded in