Báo Cáo Báo cáo hóa học PERIODIC BOUNDARY VALUE PROBLEMS ON TIME SCALES ´ PETR STEHLIK

ERIODIC 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 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] ∪ [2,3] are examples of time scales.