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In this article, we study the existence of solutions to first order non-linear initial value problems within the field of "dynamic equations on time scales". We employ the method of upper and lower solutions and Schauder's fixed point theorem. We also provide sufficient conditions under which the upper and lower solutions converge uniformly to a solution. Some examples are given to illustrate the new results. [ABSTRACT FROM AUTHOR]

Abstract: This article investigates both basic qualitative and basic quantitative properties of solutions to first- and higher-order dynamic equations on time scales and thus provides a foundation and framework for future advanced nonlinear studies in the field. Particular focus lies in the: existence; uniqueness; dependency; approximation; and explicit representation, of solutions to nonlinear initial value problems. The main tools used are from modern areas of nonlinear analysis, including: the fixed-point theorems of Banach and Schäfer; the method of successive approximations; a novel definition of measuring distance in metric spaces and normed spaces; and a “separation” of variables technique is introduced to the general time scale setting. The new results compliment and extend those of Stefan Hilger’s seminal paper of 1990. As an application of the new results we present and analyse a simple model from economics, known as the Keynesian–Cross model with “lagged” income, in the general time scale environment. Ideas suggesting further applications and possible new directions for the novel results are also presented. [Copyright &y& Elsevier]