A class of retarded nonlinear integral inequalities and its application in nonlinear differential-integral equation

Wang

Description

In this paper, we discuss a class of retarded nonlinear integral inequalities and give an upper bound estimation of an unknown function by the integral inequality technique. This estimation can be used as a tool in the study of differential-integral equations with the initial conditions. MSC: 26D10, 26D15, 26D20, 34A12, 34A40. In this paper, we discuss a class of retarded nonlinear integral inequalities and give an upper bound estimation of an unknown function by the integral inequality technique. This estimation can be used as a tool in the study of differential-integral equations with the initial conditions. MSC: 26D10, 26D15, 26D20, 34A12, 34A40.

Sujets

• Estimation

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 3
• Langue: English

Extrait

WangJournal of Inequalities and Applications2012,2012:154 http://www.journalofinequalitiesandapplications.com/content/2012/1/154

R E S E A R C HOpen Access A class of retarded nonlinear integral inequalities and its application in nonlinear differential-integral equation

* Wu-Sheng Wang

* Correspondence: [e-mail] Department of Mathematics, Hechi University, Yizhou, Guangxi 546300, P.R. China

Abstract In this paper, we discuss a class of retarded nonlinear integral inequalities and give an upper bound estimation of an unknown function by the integral inequality technique. This estimation can be used as a tool in the study of differential-integral equations with the initial conditions. MSC:26D10; 26D15; 26D20; 34A12; 34A40 Keywords:integral inequality; analysis technique; retarded differential-integral equation; estimation

1 Introduction Gronwall-Bellman inequalities [, ] can be used as important tools in the study of exis- tence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential equations, integral equations, and integral-differential equations. There can be found a lot of generalizations of Gronwall-Bellman inequalities in various cases from literature (e.g., [–]).

Lemma (Abdeldaim and Yakout [])We assume that u(t)and f(t)are nonnegative real- valued continuous functions defined on I= [,∞)and they satisfy the inequality        s t t p+p pp p u(t)≤u  +f(s)u(s)ds+ f(s)u(s)u(s) +f(λ)u(λ)dλds, (.)   for all t∈I, where u  > and p∈(, )are constants. Then   t p+  u(t)≤u+ (.)  f(s)D  (s)ds,∀t∈I, p+   where p     –p t s –p +p  –p –p D  (t) =β(t)u+ f(s)exp–f(λ)dλds, (.)   +p  p   t andβ(t) =exp(f(s)ds), for all t∈I.  ©2012 Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.