zbMATH — the first resource for mathematics

Periodic boundary value problems for a class of second-order impulsive integro-differential equations in Banach spaces. (English) Zbl 0889.45016
The paper deals with a periodic boundary value problem for a second order nonlinear integro-differential equation with impulses at fixed moments in a Banach space.
The concepts of lower and upper solutions for this problem are introduced and then the monotone iterative method is used to construct a couple of monotone sequences converging to the extremal solutions of the problem in a sector defined by a lower solution $$u_{0}$$ and an upper solution $$v_{0}$$, with $$u_{0}\leq v_{0}$$.
For it, the authors first study a periodic boundary value problem for a linear impulsive integro-differential equation and prove a comparison result.
Reviewer: Eduardo Liz (Vigo)

MSC:
 45N05 Abstract integral equations, integral equations in abstract spaces 45J05 Integro-ordinary differential equations
Full Text:
References:
 [1] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones, (1988), Academic Press New York · Zbl 0661.47045 [2] Guo, D.; Liu, X., Extremal solutions of nonlinear impulsive integro-differential equations in Banach spaces, J. Math. Anal. Appl., 177, 538-552, (1993) · Zbl 0787.45008 [3] Du, Y., Fixed points of increasing operators in ordered Banach spaces and applications, Appl. Anal., 38, 1-20, (1990) · Zbl 0671.47054 [4] Liu, X., Monotone iterative technique for impulsive differential equations in a Banach space, J. Math. Phy. Sci., 24, 183-191, (1990) · Zbl 0717.34068 [5] Liu, X.; Guo, D., Initial value problems for first order impulsive integro-differential equations in Banach spaces, Comm. Appl. Nonl. Anal., 2, 65-83, (1995) · Zbl 0858.34068
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.