Some New Nonlinear Weakly Singular Integral Inequalities of Wendroff Type with Applications
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Research Article Some New Nonlinear Weakly Singular Integral Inequalities of Wendroff Type with Applications Wing-Sum Cheung,1 Qing-Hua Ma,2 and Shiojenn Tseng3 1
Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong Faculty of Information Science and Technology, Guangdong University of Foreign Studies, Guangzhou 510420, China 3 Department of Mathematics, Tamkang University, Tamsui, Taipei 25137, Taiwan 2
Correspondence should be addressed to Wing-Sum Cheung, [email protected] Received 20 March 2008; Accepted 26 August 2008 Recommended by Sever Dragomir Some new weakly singular integral inequalities of Wendroff type are established, which generalized some known weakly singular inequalities for functions in two variables and can be used in the analysis of various problems in the theory of certain classes of integral equations and evolution equations. Application examples are also given. Copyright q 2008 Wing-Sum Cheung et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction In the study of differential and integral equations, one often deals with certain integral inequalities. The Gronwall-Bellman inequality and its various linear and nonlinear generalizations are crucial in the discussion of existence, uniqueness, continuation, boundedness, oscillation, and stability properties of solutions. The literature on such inequalities and their applications is vast; see 1–6, and the references are given therein. Usually, the integrals concerning such inequalities have regular or continuous kernels, but some problems arising from theoretical or practical phenomena require us to solve integral inequalities with singular kernels. For example, Henry 7 used this type of integral inequalities to prove global existence and exponential decay results for a parabolic Cauchy problem; Sano and Kunimatsu 8 gave a sufficient condition for stabilization of semilinear parabolic distributed systems by making use of a modification of Henry-type inequalities; Ye et al. 9 proved a generalization of this type of inequalities and used it to study the dependence of the solution on the order and the initial condition of a fractional differential equation. All such inequalities are proved by an iteration argument, and the estimation formulas are expressed by a complicated power series which are sometimes not very convenient for applications. To avoid this shortcoming, Medved˘ 10 presented a new method for studying Henry-type inequalities and established
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Journal of Inequalities and Applications
explicit bounds with relatively simple formulas which are similar to the classic Gronwall˘ Bellman inequalities. Very recently, Ma and Pe˘cari´c 11 used a modification of Medved’s method to study certain class of nonlinear inequalities of Henry-type, which generalized some known results and were used as handy and effective tools in the study of
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