New Fixed Point Theorems of Mixed Monotone Operators and Applications to Singular Boundary Value Problems on Time Scales

Huiye Xu

doi:10.1155/2011/567054

Abstract

Some new existence and uniqueness theorems of fixed points of mixed monotone operators are obtained, and then they are applied to a nonlinear singular second-order three-point boundary value problem on time scales. We prove the existence and uniqueness of a positive solution for the above problem which cannot be solved by using previously available methods.

Highlights

The study of mixed monotone operators has been a matter of discussion since they were introduced by Guo and Lakshmikantham 1 in 1987, because it has important theoretical meaning and wide applications in microeconomics, the nuclear industry, and so on see 1–4
We extend the main results of 9 to mixed monotone operators
Without demanding compactness and continuity conditions and the existence of upper and lower solutions, we study the existence, uniqueness, and iterative convergence of fixed points of a class of mixed monotone operators

Summary

Introduction

The study of mixed monotone operators has been a matter of discussion since they were introduced by Guo and Lakshmikantham 1 in 1987, because it has important theoretical meaning and wide applications in microeconomics, the nuclear industry, and so on see 1–4. Without demanding compactness and continuity conditions and the existence of upper and lower solutions, we study the existence, uniqueness, and iterative convergence of fixed points of a class of mixed monotone operators. We apply these results to the following singular second-order three-point boundary value problem on time scales:. We note that there is no result on the uniqueness of solutions and convergence of the iterative sequences for singular boundary value problems on time scales. In our abstract results on mixed monotone operators, since the compactness and continuity conditions are not required, they can be directly applied to singular boundary value problem 1.1. Our main result generalizes and improves Theorem 2.3 in 18

Preliminaries and Abstract Theorems

10 Define a cone by