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.

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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.

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