Study on the Existence of Sign-Changing Solutions of Case Theory Based a Class of Differential Equations Boundary-Value Problems

Hongwei Ji

doi:10.4236/apm.2017.712042

Abstract

By using the fixed point theorem under the case structure, we study the existence of sign-changing solutions of A class of second-order differential equations three-point boundary-value problems, and a positive solution and a negative solution are obtained respectively, so as to popularize and improve some results that have been known.

Highlights

Open AccessThe existence of nonlinear three-point boundary-value problems has been studied [1]-[6], and the existence of sign-changing solutions is obtained
Most studies were focused on the cone fixed point index theory [7] [8] [9], just a few took use of case theory to study the topological degree of non-cone mapping and the calculation of fixed point index, and the case theory was combined with the topological degree theory to study the sign-changing solutions
[10] [11] have given the method of calculating the topological degree under the case structure, and taken use of the fixed point theorem of non-cone mapping to study the existence of nontrivial solutions for the nonlinear Sturm-Liouville problems

Summary

Introduction

The existence of nonlinear three-point boundary-value problems has been studied [1]-[6], and the existence of sign-changing solutions is obtained. [10] [11] have given the method of calculating the topological degree under the case structure, and taken use of the fixed point theorem of non-cone mapping to study the existence of nontrivial solutions for the nonlinear Sturm-Liouville problems. Boundary-value problem (1) is equivalent to Hammerstein nonlinear integral equation hereunder. After the necessary preparation, when the non-linear term f is under certain assumptions, we get the existence of sign-changing solution of such boundary-value problems. [8], we can see that we generalize and improve the nonlinear term f , and remove the conditions of strictly increasing function, and the method is different from Ref.

Knowledge

Results

Conclusion