ABSTRACT. By means of the shift operators we introduce a new periodicity concept on time scales. This new approach will enable researchers to investigate periodicity notion on a large class of time scales whose members may not satisfy the condition there exists a P > 0 such that t ± P ∈ T for all t ∈ T, which is being currently used. Therefore, the results of this paper open an avenue for the investigation of periodic solutions of q-difference equations and more.

No abstract

No abstract

No abstract

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.

No abstract

Using the topological degree method and Schaefer's fixed point theorem, we deduce the existence of periodic solutions of nonlinear system of integro-dynamic equations on periodic time scales. Furthermore, we provide several applications to scalar equations, in which we develop a time scale analog of Lyapunov's direct method and prove an analog of Sobolev's inequality on time scales to arrive at a priori bound on all periodic solutions. Therefore, we improve and generalize the corresponding results in Burton et al. (Ann Mat Pura Appl 161:271-283, 1992)

Abstract. We introduce the principal matrix solution Z(t, s) of the linear Volterratype vector integro-dynamic equationand prove that it is the unique matrix solution ofWe also show that the solution ofis unique and given by the variation of parameters formula2010 Mathematics Subject Classification. 34N05, 45D05, 39A13, 45J05.