TAILIEUCHUNG - Báo cáo hóa học: " Research Article Stability of a Jensen Type Logarithmic Functional Equation on Restricted Domains and Its Asymptotic Behaviors"

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 Stability of a Jensen Type Logarithmic Functional Equation on Restricted Domains and Its Asymptotic Behaviors | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 432796 13 pages doi 2010 432796 Research Article Stability of a Jensen Type Logarithmic Functional Equation on Restricted Domains and Its Asymptotic Behaviors Jae-Young Chung Department of Mathematics Kunsan National University Kunsan 573-701 Republic of Korea Correspondence should be addressed to Jae-Young Chung jychung@ Received 28 June 2010 Revised 30 October 2010 Accepted 25 December 2010 Academic Editor Roderick Melnik Copyright 2010 Jae-Young Chung. 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. Let R be the set of positive real numbers B a Banach space f R B and e 0 p q P Q e R with pqPQ 0. We prove the Hyers-Ulam stability of the Jensen type logarithmic functional inequality f fxpyq - Pf x - Qf y e in restricted domains of the form x y x 0 y 0 xkys d for fixed k s e R with k 0 or s 0 and d 0. As consequences of the results we obtain asymptotic behaviors of the inequality as xkys OT. 1. Introduction The stability problems of functional equations have been originated by Ulam in 1940 see 1 . One of the first assertions to be obtained is the following result essentially due to Hyers 2 that gives an answer for the question of Ulam. Theorem . Suppose that S is an additive semigroup B is a Banach space e 0 and f S B satisfies the inequality f x y - f x - fy II e for all x y e S. Then there exists a unique function A S B satisfying A x y A x Ay 2 Advances in Difference Equations for which f x - A x e for all X e S. In 1950-1951 this result was generalized by the authors Aoki 3 and Bourgin 4 5 . Unfortunately no results appeared until 1978 when Th. M. Rassias generalized the Hyers result to a new approximately linear mappings 6 . Following the Rassias result a great number of the .

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