Some dynamic inequalities on time scales and their applications

Abstract

This paper is devoted to the study of Gronwall-Bellman-type inequalities on an arbitrary time scale T. We investigate some new explicit bounds of a certain class of nonlinear retarded dynamic inequalities of Gronwall-Bellman type on time scales. These inequalities extend some known dynamic inequalities on time scales. We also generalize and unify some continuous inequalities and their corresponding discrete analogues. To illustrate the benefits of our work, we present some applications of these results. The main results will be proved by using some analysis techniques and a simple consequence of the Keller's chain rule on time scales.

On some generalizations of nonlinear dynamic inequalities on time scales and their applications

“…for all t ∈ T 0 , where ζ and ξ are defined as in (22) and (23) respectively. Now a suitable application of Lemma 3.4 to (27) with v(t 0 ) = 0, yields (28) v(t) ≤ t t0 e ζ (t, σ(s))ξ(s)∆s.…”

“…The book on the subject of time scales by Bohner and Peterson [14] summarizes and organizes much of time scale calculus. During the past decade a number of dynamic inequalities have been established by some authors which are motivated by some applications, for example, when studying the behavior of solutions of certain class of dynamic equations on a time scale T. We refer the reader to [5,8,9,14,16,22,23,24,25,26,35,36,37,38,48,49,50] for contributions, and the references cited therein.…”

Delay dynamic double integral inequalities on time scales with applications

“…Recently, to dealt with the difficulties encountered in the solutions of differential and difference equations, the Gronwall-Bellman integral inequality on time scales has become one of the most important topics in mathematics research. A lot of work has been done in this area [103][104][105][106]. To discuss the abstract analysis of the solutions of certain types of dynamic equations, Gronwall-Bellman delay type inequality on time scales can play a significant role, but it still cannot deal with the abstract analysis for the solutions of some more general differential and difference equations.…”

On some generalizations of dynamic Opial-type inequalities on time scales

“…During the last decade, many inequalities have been studied by different authors (see [1,2,[10][11][12][13][14]26] and the references cited therein).…”