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 A Fixed Point Approach to the Stability of the Functional Equation f x y F f x ,f y | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 912046 8 pages doi 2009 912046 Research Article A Fixed Point Approach to the Stability of the Functional Equation f x y F f x f y Soon-Mo Jung1 and Seungwook Min2 1 Mathematics Section College of Science and Technology Hongik University Jochiwon 339-701 South Korea 2 Division of Computer Science Sangmyung University 7 Hongji-dong Jongno-gu Seoul 110-743 South Korea Correspondence should be addressed to Soon-Mo Jung smjung@ Received 20 July 2009 Revised 20 September 2009 Accepted 30 September 2009 Recommended by Massimo Furi By applying the fixed point method we will prove the Hyers-Ulam-Rassias stability of the functional equation f x y F f x f y under some additional assumptions on the function F and spaces involved. Copyright 2009 . Jung and S. Min. 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 1940 Ulam 1 gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms Let G1 be a group and let G2 be a metric group with the metric d v . Given e 0 does there exist a Ỗ 0 such that if a function h G1 G2 satisfies the inequality d h xy h x h y Ỗ for all x y e G1 then there exists a homomorphism H G1 G2 with d h x H x e for all x e G1 The case of approximately additive functions was solved by Hyers 2 under the assumption that G1 and G2 are Banach spaces. Indeed he proved that each solution of the inequality f x y - f x - f y e for all x and y can be approximated by an exact solution say an additive function. Rassias 3 attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows IIAx
