Some new generalized retarded inequalities for discontinuous functions and their applications

Zhaowen Zheng,Xin Gao,Jing Shao

doi:10.1186/s13660-015-0943-6

Zhaowen Zheng, Xin Gao + Show 1 more

Open Access

https://doi.org/10.1186/s13660-015-0943-6

Copy DOI

Abstract

In this paper, some new generalized retarded inequalities for discontinuous functions are discussed, which are effective in dealing with the qualitative theory of some impulsive differential equations and impulsive integral equations. Compared with some existing integral inequalities, these estimations can be used as tools in the study of differential-integral equations with impulsive conditions.

Highlights

1 Introduction In analyzing the impulsive phenomenon of a physical system governed by certain differential and integral equations, one often needs some kinds of inequalities, such as Gronwalllike inequalities; these inequalities and their various linear and nonlinear generalizations are crucial in the discussion of the existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential and integral equations
In [ ], Lipovan studied the inequality with delay (b(t) ≤ t, b(t) → ∞ as t → ∞)
In, Borysenko [ ] obtained the explicit bound to the unknown function of the following integral inequality with impulsive effect: t u(t) ≤ a(t) + f (s)u(s) ds +

Summary

Introduction

In analyzing the impulsive phenomenon of a physical system governed by certain differential and integral equations, one often needs some kinds of inequalities, such as Gronwalllike inequalities; these inequalities and their various linear and nonlinear generalizations are crucial in the discussion of the existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential and integral equations (see [ – ] and references therein). In , Borysenko [ ] obtained the explicit bound to the unknown function of the following integral inequality with impulsive effect:. Zheng et al Journal of Inequalities and Applications (2016) 2016:7 in , Iovane [ ] studied the following integral inequalities:. Αiur(ti – ) , t

We can obtain

Then t

Consider q d