Abstract
The main aim of this paper is to establish some new explicit bounds of solutions of a certain class of nonlinear dynamic inequalities (with and without delays) of Gronwall-Bellman type on a time scale T which is unbounded above. These on the one hand generalize and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of delay dynamic equations on time scales. Some examples are considered to demonstrate the applications of the main results.
Highlights
In 1919 Thomas Gronwall [8] proved that if β and u are real-valued continuous functions defined on J, where J is an interval in R, t0 ∈ J, and u is differentiable in the interior J0 of J, u′(t) ≤ β(t)u(t), for t ∈ J0, (1)
T0 s where J is an interval in R, t0 ∈ J, and α, β, u ∈ C(J, R+)
Since (14) provides an explicit bound to the unknown function u(t) and a tool to the study of many qualitative as well as quantitative properties of solutions of dynamic equations, it has become one of the very few classic and most influential results in the theory and applications of dynamic inequalities
Summary
The generalizations of (11) on time scales has been studied in [17, 19] and some explicit upper bounds of the unknown function are obtained. The Gronwall-Bellman dynamic inequality, which is the time scale version of (3) has been proved in [5, Theorem 6.4].
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