Volterra integral equations on time scales: Basic qualitative and quantitative results with applications to initial value problems on unbounded domains

Abstract This article introduces the basic qualitative and basic quantitative theory of Volterra integral equations on time scales and thus may be considered as a foundation for future advanced studies in the field. New sufficient conditions are introduced that guarantee: existence; uniqueness; approximation; boundedness and certain growth rates of solutions to both linear and nonlinear problems. The main techniques employed are contemporary components of nonlinear analysis, including: the fixed–point theorems of Banach and Schäfer; Picard iterations; inequality theory on time scales; and a novel definition of measuring distance in metric spaces and normed spaces. As an application of the new findings, we present some results concerning nonlinear initial value problems for dynamic, differential and difference equations on unbounded domains. We also present some suggestions concerning open problems and possible directions for further work.