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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 671514,

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 671514, 11 pages doi:10.1155/2011/671514 Research Article Fixed Points and Random Stability of a Generalized Apollonius Type Quadratic Functional Equation Min June Kim,1 Seung Won Schin,1 Dohyeong Ki,1 Jaewon Chang,1 and Ji-Hye Kim2 1 2 Mathematics Branch, Seoul Science High School, Seoul 110-530, Republic of Korea Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea Correspondence should be addressed to Ji-Hye Kim, saharin@hanyang.ac.kr Received 23 December 2010; Revised 28 February 2011; Accepted 28 February 2011 Academic Editor: Yeol J. Cho Copyright q 2011 Min June Kim et al | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 671514 11 pages doi 10.1155 2011 671514 Research Article Fixed Points and Random Stability of a Generalized Apollonius Type Quadratic Functional Equation Min June Kim 1 Seung Won Schin 1 Dohyeong Ki 1 Jaewon Chang 1 and Ji-Hye Kim2 1 Mathematics Branch Seoul Science High School Seoul 110-530 Republic of Korea 2 Department of Mathematics Research Institute for Natural Sciences Hanyang University Seoul 133-791 Republic of Korea Correspondence should be addressed to Ji-Hye Kim saharin@hanyang.ac.kr Received 23 December 2010 Revised 28 February 2011 Accepted 28 February 2011 Academic Editor Yeol J. Cho Copyright 2011 Min June Kim 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 the fixed-point method we prove the generalized Hyers-Ulam stability of a generalized Apollonius type quadratic functional equation in random Banach spaces. 1. Introduction The stability problem of functional equations was 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 theorem was generalized by Aoki 3 for additive mappings and by Th. M. Rassias 4 for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias 4 has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Gavruta 5 by replacing the unbounded Cauchy difference by a general control function in the spirit of the Th. M. Rassias approach. On the other hand in 1982-1998 J. M. Rassias generalized the Hyers stability result by presenting a weaker condition controlled by a product of .