A generalization of the compression cone method for integral operators with changing sign kernel functions

Ankai Liu,Wenying Feng

doi:10.1186/s13661-019-1186-0

Abstract

In this paper, a new class of order cones in the space of continuous functions is introduced. The result unifies some previous work in studying the existence of solutions for differential equations using the compression cone techniques and fixed point theorems. It is shown that the method is more adaptable, particularly in dealing with changing sign Green’s functions. Applications are illustrated by examples. Limitations of such a new method are also discussed.

Highlights

1 Introduction Recently, it has been shown that the following Hammerstein integral equation has important applications in the rapidly developing field of machine learning [2]: T
Convergence of the system is governed by fixed points of the corresponding integral operator
It is known that equation (1) can be seen as an inverse of a differential equation subject to certain boundary conditions

Summary

Introduction

It has been shown that the following Hammerstein integral equation has important applications in the rapidly developing field of machine learning [2]: T. To construct the cone in a space such as C[0, T], usually a positive Green’s function for the BVP is required to ensure a positive kernel for the integral operator. It leads to existence of positive solutions for the original BVP [12, 17, 21, 23, 24, 26]. A bounded linear functional L is used to define a new type of cones in dealing with changing sign Green’s functions for differential equations:. The sub-linear and super-linear cases are discussed along with examples to illustrate their applications

Main result Consider the existence of a fixed point for the integral equation

Sub-linear and super-linear case