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Periodic boundary value problems for systems of first order impulsive differential equations. (English) Zbl 0732.34040
We discuss the periodic boundary value problem for a system of impulsive differential equations in which impulses occur at fixed times and prove, with the help of upper and lower solutions, that the problem has a solution lying between the upper and lower solutions. We also consider the theory of impulsive differential inequalities relative to periodic boundary value problems. We note that the method of upper and lower solutions has been utilized to study various initial and boundary value problems for differential equations without impulses. However, these discussions depend on abstract methods. Our proofs employ simple new ideas and avoids abstract arguments. Our approach is entirely based on calculus methods and it is therefore easy to understand the basic ideas involved, and at the same time, achieve the same goals as that of abstract methods. Moreover, our methods provide new proofs even to the same problems for systems of differential equations.
MSC:
34C25Periodic solutions of ODE
34A40Differential inequalities (ODE)
34A37Differential equations with impulses
34B15Nonlinear boundary value problems for ODE