Some new nonlinear integral inequalities and their applications in the qualitative analysis of differential equations

Zheng Bin, Feng, Feng Qinghua

Description

In this paper, some new nonlinear integral inequalities are established, which provide a handy tool for analyzing the global existence and boundedness of solutions of differential and integral equations. The established results generalize the main results in Sun (J. Math. Anal. Appl. 301 , 265-275, 2005), Ferreira and Torres (Appl. Math. Lett. 22 , 876-881, 2009), Xu and Sun (Appl. Math. Comput. 182, 1260-1266, 2006) and Li et al. (J. Math. Anal. Appl. 372 , 339-349 2010). MSC 2010 : 26D15; 26D10

Sujets

• Integral equation
• Differential equation
• Boundedness

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 6
• Langue: English

Extrait

Zheng and FengJournal of Inequalities and Applications2011,2011:20 http://www.journalofinequalitiesandapplications.com/content/2011/1/20

R E S E A R C HOpen Access Some new nonlinear integral inequalities and their applications in the qualitative analysis of differential equations 1* 1,2 Bin Zhengand Qinghua Feng

* Correspondence: [e-mail] 1 School of Science, Shandong University of Technology, Zibo, Shandong 255049, China Full list of author information is available at the end of the article

Abstract In this paper, some new nonlinear integral inequalities are established, which provide a handy tool for analyzing the global existence and boundedness of solutions of differential and integral equations. The established results generalize the main results in Sun (J. Math. Anal. Appl.301, 265275, 2005), Ferreira and Torres (Appl. Math. Lett. 22, 876881, 2009), Xu and Sun (Appl. Math. Comput. 182, 12601266, 2006) and Li et al. (J. Math. Anal. Appl.372, 339349 2010). MSC 2010: 26D15; 26D10 Keywords:integral inequality, global existence, integral equation, differential equa tion, bounded

1 Introduction During the past decades, with the development of the theory of differential and integral equations, a lot of integral inequalities, for example [112], have been discovered, which play an important role in the research of boundedness, global existence, stability of solutions of differential and integral equations. In [9], the following two theorems for retarded integral inequalities were established. Theorem A:R + = [0,∞). Letu,f,gbe nondecreasing continuous functions defined onR + and letcbe a nonnegative constant. Moreover, letωÎC(R + ,R + ) be nondecreas 1 ing withω(u)>0 on (0,∞) andaÎC(R + ,R + ) be nondecreasing witha(t)≤tonR + . m,nare constants, andm > n >0. If  αt m m m nn m−n u(t)≤c+ [f(s)u(s)ω(u(s)) +g(s)u(s)]ds,t∈R + m−n

then fortÎ[0,ξ]   α(t)α(t) 1 −1 m−n u(t)≤ {Ω[Ω(c+g(s)ds) +f(s)ds]}

 r 1  1 (r) =d where 1 ,r> 0,Ωis the inverse ofΩ,Ω(∞) =∞, andξÎR + 1 m−n ωs   α(t)α(t) −1 is chosen so that (c+g(s)ds) +f(s)ds∈Dom(Ω .

© 2011 Zheng and Feng; 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.