Báo cáo hoa học: "Research Article An Extension to Nonlinear Sum-Difference Inequality and Applications"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article An Extension to Nonlinear Sum-Difference Inequality and Applications | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 486895 17 pages doi 2009 486895 Research Article An Extension to Nonlinear Sum-Difference Inequality and Applications Wu-Sheng Wang1 and Xiaoliang Zhou2 1 Department of Mathematics Hechi University Yizhou Guangxi 546300 China 2 Department of Mathematics Guangdong Ocean University Zhanjiang 524088 China Correspondence should be addressed to Xiaoliang Zhou zjhdzxl@ Received 31 March 2009 Revised 31 March 2009 Accepted 17 May 2009 Recommended by Martin J. Bohner We establish a general form of sum-difference inequality in two variables which includes both more than two distinct nonlinear sums without an assumption of monotonicity and a nonconstant term outside the sums. We employ a technique of monotonization and use a property of stronger monotonicity to give an estimate for the unknown function. Our result enables us to solve those discrete inequalities considered in the work of . Cheung 2006 . Furthermore we apply our result to a boundary value problem of a partial difference equation for boundedness uniqueness and continuous dependence. Copyright 2009 . Wang and X. Zhou. 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 Gronwall-Bellman inequality 1 2 is a fundamental tool in the study of existence uniqueness boundedness stability invariant manifolds and other qualitative properties of solutions of differential equations and integral equation. There are a lot of papers investigating them such as 3-15 . Along with the development of the theory of integral inequalities and the theory of difference equations more attentions are paid to some discrete versions of Bellman-Gronwall type inequalities . 16-18 . Starting from the basic form n-1 u n a n f s u s s 0 discussed in 19 .

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