Basic existence, uniqueness and approximation results for positive solutions to nonlinear dynamic equations on time scales

Abstract

This paper focuses on the qualitative and quantitative properties of solutions to certain nonlinear dynamic equations on time scales. We present some new sufficient conditions under which these general equations admit a unique, positive solution. These positive (and hence non-oscillatory) solutions: extend across unbounded intervals; and tend to a finite limit as the independent variable becomes large and positive. Our methods include: Banach’s fixed-point theorem, including the method of Picard iterations; and weighted norms and metrics in the time scale setting. Due to the wide-ranging nature of dynamic equations on time scales our results are novel: for ordinary differential equations; for difference equations; for combinations of the two areas; and for general time scales — this is demonstrated via some examples. Furthermore, we state an open problem of interest.