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Abstract and Applied Analysis
Volume 2014, Article ID 974968, 11 pages
Research Article

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

1Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China
2Department of Mathematics, Kanagawa University, Yokohama 221-8686, Japan

Received 7 January 2014; Accepted 13 June 2014; Published 7 July 2014

Academic Editor: Yonghong Wu

Copyright © 2014 Lingling Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.