Some retarded nonlinear integral inequalities and their applications in retarded differential equations

Wang

Description

In this article, we discuss some generalized retarded nonlinear integral inequalities, which not only include nonlinear compound function of unknown function but also include retard items, and give upper bound estimation of the unknown function by integral inequality technique. This estimation can be used as tool in the study of differential equations with the initial conditions. 2000 MSC : 26D10; 26D15; 26D20; 34A12; 34A40.

Sujets

• Estimation

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 22
• Langue: English

Extrait

WangJournal of Inequalities and Applications2012,2012:75 http://www.journalofinequalitiesandapplications.com/content/2012/1/75

R E S E A R C H

Open Access

Some retarded nonlinear integral inequalities their applications in retarded differential equations

WuSheng Wang

Correspondence: wang4896@126. com Department of Mathematics, Hechi University, Yizhou, Guangxi 546300, P. R. China

and

Abstract In this article, we discuss some generalized retarded nonlinear integral inequalities, which not only include nonlinear compound function of unknown function but also include retard items, and give upper bound estimation of the unknown function by integral inequality technique. This estimation can be used as tool in the study of differential equations with the initial conditions. 2000 MSC: 26D10; 26D15; 26D20; 34A12; 34A40. Keywords:integral inequality, integral inequality technique, Retarded differential equation, estimation

1 Introduction GronwallBellman inequalities [1,2] and their various generalizations can be used important tools in the study of existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential equations, integral equations, and inte graldifferential equations. Lemma 1(Gronwall [1]).Let u(t)be a continuous function defined on the interval[a, a+h],a, h are nonnegative constants and t    0≤u(t)≤bu(s) +a ds,t∈[a,a+h] (1:1)

Then, 0≤u(t)≤ahexp(bh),∀tÎ[a,a+h]. Lemma 2(Bellman [2]).Let f, uÎC([0,h], [0,∞)),h, c are positive constants. If u satisfy the inequality t  u(t)≤c+f(s)u(s)ds,t∈[0,h] (1:2)

 t Then, u(t)≤cexpf(s)ds,t∈[0,h . 0 1 Lemma 3(Lipovan [3]).Let u, fÎC([t 0 ,T),R + ).Further, letaÎC([t 0 ,T),[t 0 ,T))be nondecreasing witha(t)≤t on[t 0 ,T),and let c be a nonnegative constant. Then the inequality

© 2012 Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution 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.