Multiple Solutions of Boundary Value Problems fornth-Order Singular Nonlinear Integrodifferential Equations in Abstract Spaces

Yanlai Chen,Baoxia Qin,Tingqiu Cao

doi:10.1155/2015/736139

Abstract

The authors discuss multiple solutions for thenth-order singular boundary value problems of nonlinear integrodifferential equations in Banach spaces by means of the fixed point theorem of cone expansion and compression. An example for infinite system of scalar third-order singular nonlinear integrodifferential equations is offered.

Highlights

Singular nonlinear boundary value problems of the ordinary differential equations appeared frequently in applications
We show that the operator A defined by (39) is reasonable for u ∈ QR \ Qr with any R > r > 0
P is a normal cone in E and the normal constant N = 1

Summary

Introduction

Singular nonlinear boundary value problems of the ordinary differential equations appeared frequently in applications. In Chen [9], the boundary value problems of a class of nth-order nonlinear integrodifferential equations of mixed type in Banach space are considered, and the existence of three solutions is obtained by using the fixed point index theory. Such equations do not have singular nonlinear terms. Through finding the relations from ‖u‖c to ‖u(n−2)‖c (u belongs to the special cone), we triumphantly overcome the singularity and use the fixed point theorem of cone expansion and compression directly to obtain the existence of multiple solutions for singular boundary value problems of nonlinear integrodifferential equations in Banach spaces. We consider the following singular boundary value problem (SBVP for short) for an nth-order nonlinear integrodifferential equations in Banach spaces E:. A map u ∈ Cn−2[J, E] ∩ Cn[J󸀠, E] is called a solution of SBVP (2) if it satisfies (2)

Preliminaries and Several Lemmas

Main Results and an Example

Conclusion