PERIODIC BOUNDARY VALUE PROBLEMS ON TIME SCALES ´ PETR STEHLIK Received 20 April 2004 We extend the results concerning periodic boundary value problems from the continuous calculus to time scales. First we use the Schauder fixed point theorem and the concept of lower and upper solutions to prove the existence of the solutions and then we investigate a monotone iterative method which could generate some of them. Since this method does not work on each time scale, a condition containing a Lipschitz constant of right-hand side function and the supremum of the graininess function is introduced. 1. Introduction We recall the basic. | PERIODIC BOUNDARY VALUE PROBLEMS ON TIME SCALES PETR STEHLÍK Received 20 April 2004 We extend the results concerning periodic boundary value problems from the continuous calculus to time scales. First we use the Schauder fixed point theorem and the concept of lower and upper solutions to prove the existence of the solutions and then we investigate a monotone iterative method which could generate some of them. Since this method does not work on each time scale a condition containing a Lipschitz constant of right-hand side function and the supremum of the graininess function is introduced. 1. Introduction We recall the basic definitions concerning calculus on time scales. Further details can be found in the survey monography upon this topic 2 its second part 3 or in the paper 6 . Time scale T is an arbitrary nonempty closed subset of the real numbers R. The natural numbers N the integers Z or the union of intervals 0 1 u 2 3 are examples of time scales. For t e T we define the forward jump operator ơ T - T and the backward jump operator p T - T by Ơ t inf s e T s t p t sup s e T s t where we put inf 0 sup T and sup 0 inf T. We say that a point t e T is right scattered left scattered right dense left dense if Ơ t t p t t Ơ t t p t t respectively. A point t e T is isolated if it is right scattered and left scattered. A point t e T is dense if it is right dense and left dense. Finally we define the forward graininess function p T 0 00 by p t Ơ t - t. Similarly we define the backward graininess function V T 0 00 by v t t - p t . Copyright 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005 1 2005 81-92 DOI 82 Periodic boundary value problems on time scales We define the function fa t f ơ t and the modified time scale TK as follows if T has a left-scattered maximum m then TK T m otherwise TK T. We endow T with the topology inherited from R. The continuity of the function f T R is defined in the usual manner. The .
