Multiple Positive Solutions for First-Order Impulsive Integrodifferential Equations on the Half Line in Banach Spaces

Dajun Guo

doi:10.1155/2011/634612

Abstract

The author discusses the multiple positive solutions for an infinite boundary value problem of first-order impulsive superlinear integrodifferential equations on the half line in a Banach space by means of the fixed point theorem of cone expansion and compression with norm type.

Highlights

Let E be a real Banach space and P a cone in E which defines a partial ordering in E by x ≤ y if and only if y − x ∈ P
The author discusses the multiple positive solutions for an infinite boundary value problem of first-order impulsive superlinear integrodifferential equations on the half line in a Banach space by means of the fixed point theorem of cone expansion and compression with norm type
K ∈ C D, R, D { t, s ∈ J × J : t ≥ s}, H ∈ C J × J, R, R denotes the set of all nonnegative numbers

Summary

Introduction

Let E be a real Banach space and P a cone in E which defines a partial ordering in E by x ≤ y if and only if y − x ∈ P. In paper 2 , we considered the infinite boundary value problem IBVP for first-order impulsive nonlinear integrodifferential equation of mixed type on the half line in E:. By using the fixed point index theory, we discussed the multiple positive solutions of IBVP 1.1. In this paper, we discuss the multiple positive solutions of an infinite three-point boundary value problem which includes IBVP 1.1 as a special case for superlinear case by means of different method, that is, by using the fixed point theorem of cone expansion and compression with norm type, which was established by the author in 3 see 1 , and the key point is to introduce a new cone Q. Consider the infinite three-point boundary value problem for first-order impulsive nonlinear integrodifferential equation of mixed type on the half line in E:. BPC J, P and Q are two cones in space BPC J, E and Q ⊂ BPC J, P . u ∈ BPC J, P ∩ C1 J , E is called a positive solution of IBVP 1.5 if u t > θ for t ∈ J and u t satisfies 1.5

Several Lemmas

Main Theorems

Conclusion