Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2005; 42(2): 255-268

Printed March 1, 2005

Copyright © The Korean Mathematical Society.

Periodic solutions in nonlinear neutral difference equations with functional delay

Mariette R. Maroun and Youssef N. Raffoul

Baylor University, University of Dayton

Abstract

We use Krasnoselskii's fixed point theorem to show that the nonlinear neutral difference equation with delay \begin{equation} x(t+1) = a(t)x(t)+ c(t)\Delta x(t-g(t))+ q\big(t, x(t), x(t-g(t)\big)\nonumber \end{equation} has a periodic solution. To apply Krasnoselskii's fixed point theorem, one would need to construct two mappings; one is contraction and the other is compact. Also, by making use of the variation of parameters techniques we are able, using the contraction mapping principle, to show that the periodic solution is unique.

Keywords: Krasnoselski, contraction, nonlinear neutral difference equation, periodic solutions, unique solution

MSC numbers: 39A10, 39A12