Some generalized nonlinear retarded integral inequalities with applications

Wang

Description

In this article we discuss some new generalized nonlinear Gronwall-Bellman-Type integral inequalities with two variables, which include a non-constant term outside the integrals. We use our result to deal with the estimate on the solutions of partial differential equations with the initial and boundary conditions. Mathematics Subject Classification 2000 : 26D10; 26D15; 26D20; 34A40.

Sujets

• Boundary value problem
• Boundedness
• Uniqueness quantification

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 5
• Langue: English

Extrait

WangJournal of Inequalities and Applications2012,2012:31 http://www.journalofinequalitiesandapplications.com/content/2012/1/31

R E S E A R C HOpen Access Some generalized nonlinear retarded integral inequalities with applications WuSheng Wang

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

Abstract In this article we discuss some new generalized nonlinear GronwallBellmanType integral inequalities with two variables, which include a nonconstant term outside the integrals. We use our result to deal with the estimate on the solutions of partial differential equations with the initial and boundary conditions. Mathematics Subject Classification 2000: 26D10; 26D15; 26D20; 34A40. Keywords:integral inequality, integral inequality technique, boundary value problem, boundedness, uniqueness

1 Introduction Various generalizations of Gronwall inequality [1,2] are fundamental tools in the study of existence, uniqueness, boundedness, stability and other qualitative properties of solu tions of differential equations, integral equations, and differentialintegral equations. There are a lot of articles investigating its generalizations such as [323]. Recently, Pach patte [19] provided the explicit estimations of following integral inequalities:

and

α i (t) n   p p u(t)≤c+p[a i (s)u(s) +b i (s)u(s)]ds, i=1 α i (t 0 ) α i (t) n   p u(t)≤c+p[a i (s)u(s)w(u(s)) +b i (s)u(s)]ds, i=1 α i (t 0 )

α i (x)β i (y) n    p p u(x,y)≤c+p[a i (s,t)u(s,t) +b i (s,t)u(s,t)]dtds, i=1 α i (x 0 )β i (y 0 ) α i (x)β i (y) n    p u(x,y)≤c+p[a i (s,t)u(s,t)w(u(s,t)) +b i (s,t)u(s,t)]dtds, i=1 α i (x 0 )β i (y 0 )

© 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.