Positive Solutions for Nonlinear Integro-Differential Equations of Mixed Type in Banach Spaces

Yan Sun

doi:10.1155/2013/787038

Abstract

We establish some new existence theorems on the positive solutions for nonlinear integro-differential equations which do not possess any monotone properties in ordered Banach spaces by means of Banach contraction mapping principle and cone theory based on some new comparison results.

Highlights

A cone P ⊂ E is said to be normal if there exists a positive constant υ such that ‖x+y‖ ≥ υ, ∀x, y ∈ P, ‖x‖ = 1, ‖y‖ = 1
In this paper, we consider the existence of the unique positive solution and at least one positive solution for the following initial value problem (IVP) of the nonlinear integrodifferential equation of mixed type in ordered Banach spaces E: u󸀠󸀠 (t) = g (t, u (t), (Tu) (t), (Su) (t)), ∀t ∈ I, (1)u (0) = u0, u󸀠 (0) = u1, where I = [0, a] (a > 0), g(t, u(t), (Tu)(t), (Su)(t)) = f(t, u(t), u(t), (Tu)(t), (Su)(t)), u0, u1 ∈ E, f ∈ C[I × P × P × P × P, P], and f is nonincreasing with the second variable u(t), with f is nondecreasing with the third variable u(t)
Let E be a Banach space and H ⊂ C[I, E] if H is a countable set of strongly measurable functions y : I → E such that there exists φ ∈ L[I, R+] such that ‖y(t)‖ ≤ φ(t), t ∈ I, y ∈ H

Summary

Introduction

A cone P ⊂ E is said to be normal if there exists a positive constant υ such that ‖x+y‖ ≥ υ, ∀x, y ∈ P, ‖x‖ = 1, ‖y‖ = 1. A positive cone P is called normal if and only if there exists N > 0 such that 0 ≤ x ≤ y implies ‖x‖ ≤ N‖y‖.

Objectives

Results

Conclusion