Positive Solutions for Impulsive Differential Equations with Mixed Monotonicity and Optimal Control

Lingling Zhang,Rui Guo,Noriaki Yamazaki

doi:10.1155/2014/974968

Abstract

We consider positive solutions and optimal control problem for a second order impulsive differential equation with mixed monotone terms. Firstly, by using a fixed point theorem of mixed monotone operator, we study positive solutions of the boundary value problem for impulsive differential equations with mixed monotone terms, and sufficient conditions for existence and uniqueness of positive solutions will be established. Also, we study positive solutions of the initial value problem for our system. Moreover, we investigate the control problem of positive solutions to our equations, and then, we prove the existence of an optimal control and its stability. In addition, related examples will be given for illustrations.

Highlights

By using a fixed point theorem of mixed monotone operator, we study positive solutions of the boundary value problem for impulsive differential equations with mixed monotone terms, and sufficient conditions for existence and uniqueness of positive solutions will be established
Mixed monotone operators have been introduced by Guo and Lakshmikantham [1] in 1987
By using a fixed point theorem of mixed monotone operator, we study the existence and uniqueness of positive solutions to the boundary value problem of impulsive differential equations with mixed monotone terms:

Summary

Introduction

Mixed monotone operators have been introduced by Guo and Lakshmikantham [1] in 1987. Recently, many authors have investigated those kinds of operators in Banach spaces and obtained a lot of interesting and important results (see [2,3,4,5,6,7,8,9]). By using a fixed point theorem of mixed monotone operator, we study the existence and uniqueness of positive solutions to the boundary value problem of impulsive differential equations with mixed monotone terms:. The theory on mixed monotone operators has attracted much attention and has been widely studied, such as Guo and Lakshmikantham [1] have applied the monotone iterative technique to discuss an initial value problem of differential equations without impulse: u󸀠 (t) = f (t, u (t) , u (t)) , t ∈ [0, a] , (6) They obtained the existence of the coupled quasisolutions by mixed monotone sequence of coupled quasi upper and lower solutions. With a fixed point theorem of generalized concave operator, the authors [19] have studied the optimal control problem of positive solutions to the following second order impulsive differential equation:.

A Fixed Point Theorem of Mixed Monotone Operator

Conclusion