Pergamon zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Computers Math. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO Applic. Vol. 28, No. 1-3, pp. 185-189, 1994 Copyright@1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0898-1221(94)00106-5 0898-1221/94 $7.00 + 0.00 zyxwvutsrq Monotone Flows and Fixed Points for Dynamic Systems on Time Scales V. LAKSHMIKANTHAM Florida Institute of Technology, Department of Applied Mathematics Melbourne, FL 32901 U.S.A. B. KAYMAK~ALAN* Middle East Technical University, Department of Mathematics Ankara 06531, Turkey Abstract-utilizing dynamic systems on time scales, the theory of monotone flows and fixed points is considered, which unifies the theory of continuous and discrete dynamic systems. zyxwvutsrqponmlkjihgfe 1. INTRODUCTION Recently initiated theory of dynamic systems on time scales (closed subsets of reals) provides the framework to handle both continuous and discrete dynamical systems simultaneously so as to bring out better insight and understanding of the subtle differences of these two systems. To describe such a theory in a unified way, Aulbach and Hilger [1,2] developed necessary calculus of functions on time scales and discussed certain basic results. In a sequence of papers, Kay- makqalan [3-61 investigated the fundamental theory of such dynamic systems including theory of dynamic inequalities, existence of extremal solutions, comparison results, Lyapunov-like methods and monotone iterative technique. In [7], Gouze and Hadeler generalize classical results on error bounds for fixed points and interpret within the framework of the theory of dynamical systems, as a result of monotone flows. Such results are also extended to differential equations. They stress that a decomposition of a given function with required monotone properties is not the essential requirement but the decomposition together with the existence of a suitable initial state is. In this paper, we shall utilize the dynamic systems on time scales to unify the theory of monotone flows and fixed points for discrete as well as continuous dynamical systems. For this purpose, we shall first prove needed inequality theory for dynamic systems on time scale and then apply the theory developed to study monotone flows. Our results show a better understanding of the differences between discrete and continuous dynamical systems since they are both embedded in one framework. 2. PRELIMINARIES Let T be a time scale (any closed subset of R with order and topological structure in a canonical way) with to > 0 as a minimal element and no maximal element. Since a time scale T may or may not be connected, we need the concept of jump operators. *This work was partially supported by the Scientific and Technical Research Council of Turkey (TBAG-C2). Typeset by &S-w 185 brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector