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

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.