Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 309026, 11 pages doi: Research Article Generalized Hyers-Ulam Stability of the Pexiderized Cauchy Functional Equation in Non-Archimedean Spaces Abbas Najati1 and Yeol Je Cho2 1 Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran 2 Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea Correspondence should be addressed to Yeol Je Cho, yjcho@ Received 22 October 2010; Accepted 8 March 2011 Academic Editor: Jong Kim Copyright q 2011 A. Najati and Y. J. Cho. This is an open access article distributed under. | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 309026 11 pages doi 2011 309026 Research Article Generalized Hyers-Ulam Stability of the Pexiderized Cauchy Functional Equation in Non-Archimedean Spaces Abbas Najati1 and Yeol Je Cho2 1 Department of Mathematics Faculty of Sciences University ofMohaghegh Ardabili Ardabil 56199-11367 Iran 2 Department of Mathematics Education and the RINS Gyeongsang National University Jinju 660-701 Republic of Korea Correspondence should be addressed to YeolJe Cho yjcho@ Received 22 October 2010 Accepted 8 March 2011 Academic Editor Jong Kim Copyright 2011 A. Najati and Y. J. Cho. 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. We prove the generalized Hyers-Ulam stability of the Pexiderized Cauchy functional equation f x y g x h y in non-Archimedean spaces. 1. Introduction The stability problem of functional equations was originated from a question of Ulam 1 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 In other words we are looking for situations when the homomorphisms are stable that is if a mapping is almost a homomorphism then there exists a true homomorphism near it. If we turn our attention to the case of functional equations we can ask the following question. When the solutions of an equation differing slightly from a given one must be close to the true solution of the given equation. For Banach spaces the Ulam problem was first solved by Hyers 2 in 1941 which states that if Ỗ 0 and f X Y is a mapping where X Y are Banach spaces such that llf
