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Báo cáo hóa học: " Research Article Estimation on Certain Nonlinear Discrete Inequality and Applications to Boundary Value Problem"

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 Estimation on Certain Nonlinear Discrete Inequality and Applications to Boundary Value Problem | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 708587 8 pages doi 10.1155 2009 708587 Research Article Estimation on Certain Nonlinear Discrete Inequality and Applications to Boundary Value Problem Wu-Sheng Wang Department of Mathematics Hechi University Guangxi Yizhou 546300 China Correspondence should be addressed to Wu-Sheng Wang wang4896@126.com Received 1 November 2008 Accepted 14 January 2009 Recommended by John Graef We investigate certain sum-difference inequalities in two variables which provide explicit bounds on unknown functions. Our result enables us to solve those discrete inequalities considered by Sheng and Li 2008 . Furthermore we apply our result to a boundary value problem of a partial difference equation for estimation. Copyright 2009 Wu-Sheng Wang. 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 Various generalizations of the Gronwall inequality 1 2 are fundamental tools 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 3-8 . 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 Gronwall-Bellman-type inequalities such as 9-11 . Some recent works can be found for example in 12-17 and some references therein. We first introduce two lemmas which are useful in our main result. Lemma 1.1 the Bernoulli inequality 18 . Let 0 a 1 and z -1 then 1 z a 1 az. Lemma 1.2 see 19 . Assume that u n a n b n are nonnegative functions and a n is nonincreasing for all natural numbers if for all natural numbers u n afn b s u s 1.1 s n 1 2 Advances in Difference Equations .