Let T be a periodic time scale. We use a fixed point theorem due to Krasnosel'skiȋ to show that the nonlinear neutral dynamic system with delayhas a periodic solution. We assume that k is a fixed constant if T = R and is a multiple of the period of T if T = R. Under a slightly more stringent inequality we show that the periodic solution is unique using the contraction mapping principle.

In this study, we employ the fixed point theorem of Krasnoselskii and the concepts of separate and large contractions to show the existence of a periodic solution of a highly nonlinear delay differential equation. Also, we give a classification theorem providing sufficient conditions for an operator to be a large contraction, and hence, a separate contraction. Finally, under slightly different conditions, we obtain the existence of a positive periodic solution.

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We introduce the concept of 'shift operators' in order to establish sufficient conditions for the existence of the resolvent for the Volterra integral equationon time scales. The paper will serve as the foundation for future research on the qualitative analysis of solutions of Volterra integral equations on time scales, using the notion of the resolvent.2000 Mathematics subject classification: primary 45D05; secondary 39A12.

We investigate the exponential stability of the zero solution to a system of dynamic equations on time scales. We do this by defining appropriate Lyapunov-type functions and then formulate certain inequalities on these functions. Several examples are given.

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