Estimates of Certain Integral Inequalities on Time Scales

Abstract

The main objective of this paper is to establish explicit bounds on certain integral inequalities on time scales, which can be used as tools in the study of certain classes of integral equations on time scales. Some applications of our results are also given.

On Stability and Stabilization of Perturbed Time Scale Systems with Gronwall Inequalities

“…The following lemma is useful in our main results and its proof in [30] uses the time scale version of the inequality obtained by Sansone and Conti [29, p. 86]. By using Comparison Theorem 2.1 and Gronwall's inequality 2.2, we get it.…”

“…At first, one of the methods of the perturbation theory was referred to integral inequalities to quest some type of stability. In the last few years, the search was directed to the time scale integral inequalities by using diverse techniques and some significant results were obtained (see [8,30,24,25]). We are interested in further generalizing some nonlinear dynamic integral estimations.…”

Nonoscillation of all solutions of a higher order nonlinear delay dynamic equation on time scales

“…The motivation for the form of the inequality comes from the recent papers of Guo et al [13] and Ferreira and Torres [9]. Other similar inequalities on time scales can be found in [1,17,19]. (H 1 , R + ), and p ∈ C(R + , R + ) with p nondecreasing and p(u) > 0 for u > 0.…”

Integral inequalities and stability of an equilibrium state on a time scale

“…With rare exceptions (see [4]), the main integral inequality used on a time scale is the Gronwall inequality (see [3]) or some of its modifications. Lemmas 4 and 5 of the present paper enable one to extend the range of applicability of integral inequalities on a time scale in the process of analysis of solutions of dynamic equations.…”

“…[4],Theorem 3.5). Let functions u, p : T → R + be r d-continuous on T, let a functionf : T → R + be Δ-differentiable on T, and let f t Δ ( ) ≥ 0. all t ∈T ,where the function e t a p ( , ) , a ∈T , is a solution of the initial-value problemx t Δ ( ) = p t x t ( ) ( ) ,x a ( ) = 1.…”