tailieunhanh - báo cáo hóa học:" Research Article Stability of a Mixed Type Functional Equation on Multi-Banach Spaces: A Fixed Point Approach"

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 Mixed Type Functional Equation on Multi-Banach Spaces: A Fixed Point Approach | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 283827 9 pages doi 2010 283827 Research Article Stability of a Mixed Type Functional Equation on Multi-Banach Spaces A Fixed Point Approach Liguang Wang Bo Liu and Ran Bai School of Mathematical Sciences Qufu Normal University Qufu 273165 China Correspondence should be addressed to Liguang Wang wangliguang0510@ Received 11 December 2009 Accepted 29 March 2010 Academic Editor Marlene Frigon Copyright 2010 Liguang Wang 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. Using fixed point methods we prove the Hyers-Ulam-Rassias stability of a mixed type functional equation on multi-Banach spaces. 1. Introduction and Preliminaries The stability problem of functional equations originated from a question of Ulam 1 concerning the stability of group homomorphisms. Hyers 2 gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers s theorem was generalized by Aoki 3 for additive mappings and by Rassias 4 for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias has provided a lot of influence in the development of what we call generalized Hyers-Ulam-Rassias stability of functional equations. In 1990 Rassias 5 asked whether such a theorem can also be proved for p 1. In 1991 Gajda 6 gave an affirmative solution to this question when p 1 but it was proved by Gajda 6 and Rassias and Semrl 7 that one cannot prove an analogous theorem when p 1. In 1994 a generalization was obtained by Gavruta 8 who replaced the bound x p llynp by a general control function ộ x y . Beginning around 1980 the stability problems of several functional equations and approximate homomorphisms have been extensively investigated by a number of authors and there are many .