Abstract
Abstract In this paper, some new Gronwall-Bellman-type nonlinear dynamic inequalities containing integration on infinite intervals on time scales are established, which provides new bounds on unknown functions and can be used as a handy tool in the qualitative analysis of solutions of certain dynamic equations on time scales. MSC:26E70, 26D15, 26D10.
Highlights
In the analysis of solutions of certain differential, integral and difference equations, if the solutions are unknown, it is necessary to make estimate for their bounds
Gronwall-Bellman inequality [, ] and its various generalizations which provide explicit bounds for solutions of differential, integral and difference equations have proved to be of particular importance in this aspect, and much effort has been made to establish such inequalities over the years
Initiated the theory of time scales as a theory capable to contain both difference and differential calculus in a consistent way, many authors have expounded on various aspects of the theory of dynamic equations on time scales
Summary
In the analysis of solutions of certain differential, integral and difference equations, if the solutions are unknown, it is necessary to make estimate for their bounds. We will establish some new Gronwall-Bellman type nonlinear dynamic inequalities containing integration on infinite intervals on time scales, which provide new bounds on unknown functions in some certain dynamic equations on time scales. The following two theorems include some important properties for delta derivative, nabla derivative, and the Cauchy integral on time scales.
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