Existence of Nontrivial Solutions and Sign-Changing Solutions for Nonlinear Dynamic Equations on Time Scales

Huiye Xu

doi:10.1155/2011/604170

Abstract

By using the topological degree theory and the fixed point index theory, the existence of nontrivial solutions and sign-changing solutions for a class of boundary value problem on time scales is obtained. We should point out that the integral operator corresponding to above boundary value problem is not assumed to be a cone mapping.

Highlights

Let Ì be a time scale which has the subspace topology inherited from the standard topology on Ê
We study the existence of nontrivial solutions and sign-changing solutions of the following problem on time scales:
The existence of the sign-changing solutions of the above dynamic equations is being raised by an ever-increasing number of researchers; it is difficult to deal with sign-changing solution problems by the cone mappings theory

Summary

Introduction

Let Ì be a time scale which has the subspace topology inherited from the standard topology on Ê. We note that Li et al 10 were concerned with the existence of positive solutions for problem 1.1 under some conditions concerning the first eigenvalue corresponding to the relevant linear operator Their main results improved and generalized ones in 5–9, 11. The existence of the sign-changing solutions of the above dynamic equations is being raised by an ever-increasing number of researchers; it is difficult to deal with sign-changing solution problems by the cone mappings theory Stimulated by these works, in , Sun and Liu used the lattice structure to present some methods of computation of the topological degree for the operator that is quasiadditive on lattice.

Preliminaries and Some Lemmas

14 It follows that

Conclusion