Nonlinear Sum Operator Equations with a Parameter and Application to Second-Order Three-Point BVPs

Wen-Xia Wang,Piao-Piao Shi,Xi-Lan Liu

doi:10.1155/2014/259016

Abstract

A class of nonlinear sum operator equations with a parameter on order Banach spaces were considered. The existence and uniqueness of positive solutions for this kind of operator equations and the dependence of solutions on the parameter have been obtained by using the properties of cone and nonlinear analysis methods. The critical value of the parameter was estimated. Further, the application to some nonlinear three-point boundary value problems was given to show the significance of the discussion.

Highlights

Introduction and PreliminariesThe aim of this paper is to investigate the existence and uniqueness of positive solution for the following operator equations: L (λ, x) = x, (1)where L(λ, x) = Ax + λBx, A is an operator with concavity, B is a pseudo subhomogeneous operator, and λ is a parameter
We introduce definitions of pseudo subhomogeneous operator and pseudo generalized α-concave operator
We assume that E is a real Banach space, P is a normal cone in E with the normal constant N, e > θ, and A, B : P → P are increasing operators

Summary

Introduction and Preliminaries

The aim of this paper is to investigate the existence and uniqueness of positive solution for the following operator equations:. By using degreetheoretic arguments, Gupta [2] obtained the existence and uniqueness theorems for the following three-point boundary value problem: x󸀠󸀠 (t) = f (t, x (t) , x󸀠 (t)) − e (t) , t ∈ (0, 1) , (3). It is well known that fixed point theory is an effective tool in the treatment of existence results of boundary value problems for nonlinear differential equations. We introduce definitions of pseudo subhomogeneous operator and pseudo generalized α-concave operator. An increasing pseudo subhomogeneous operator and an increasing pseudo generalized α-concave operator may have no fixed point in Pe. For example, Let E = C[0, 1], P = {x ∈ E | x(t) ≥ 0, t ∈ [0, 1]}, e(t) ≡ 1, and T1x(t) = (1 − t)x(t)/(1 + x(t)).

Positive Solutions of Operator Equation

Three-Point Nonlinear Boundary Value Problem