Boundary value problems for a class of impulsive functional equations. (English) Zbl 1142.34362

Nieto, Juan J.; Rodríguez-López, Rosana

Comput. Math. Appl. 55, No. 12, 2715-2731 (2008).

Summary: This paper is related to the existence and approximation of solutions for impulsive functional differential equations with periodic boundary conditions. We study the existence and approximation of extremal solutions to different types of functional differential equations with impulses at fixed times, by the use of the monotone method. Some of the options included in this formulation are differential equations with maximum and integro-differential equations. In this paper, we also prove that the Lipschitzian character of the function which introduces the functional dependence in a differential equation is not a necessary condition for the development of the monotone iterative technique to obtain a solution and to approximate the extremal solutions to the equation in a given functional interval. The corresponding results are established for the impulsive case. The general formulation includes several types of functional dependence (delay equations, equations with maxima, integro-differential equations). Finally, we consider the case of functional dependence which is given by nonincreasing and bounded functions.

MSC:

34K13	Periodic solutions to functional-differential equations
34K45	Functional-differential equations with impulses

Keywords:

impulsive functional differential equation; periodic boundary value problem; lower and upper solutions; comparison results; monotone iterative technique

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