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Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài : Orthogonal Stability of an Additive-Quadratic Functional Equation | Park Fixed Point Theory and Applications 2011 2011 66 http www.fixedpointtheoryandapplications.eom content 2011 1 66 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Orthogonal Stability of an Additive-Quadratic Functional Equation Choonkil Park Correspondence baak@hanyang.ac. kr Department of Mathematics Research Institute for Natural Sciences Hanyang University Seoul 133-791 Republic of Korea Springer Abstract Using the fixed point method and using the direct method we prove the Hyers-Ulam stability of an orthogonally additive-quadratic functional equation in orthogonality spaces. 2010 Mathematics Subject Classification Primary 39B55 47H10 39B52 46H25. Keywords Hyers-Ulam stability fixed point orthogonally additive-quadratic functional equation orthogonality space 1. Introduction and Preliminaries Assume that X is a real inner product space and f X R is a solution of the orthogonally Cauchy functional equationfx y fx fy x y 0. By the Pythagorean theorem fx x 2 is a solution of the conditional equation. Of course this function does not satisfy the additivity equation everywhere. Thus orthogonally Cauchy equation is not equivalent to the classic Cauchy equation on the whole inner product space. Pinsker 1 characterized orthogonally additive functionals on an inner product space when the orthogonality is the ordinary one in such spaces. Sundaresan 2 generalized this result to arbitrary Banach spaces equipped with the Birkhoff-James orthogonality. The orthogonally Cauchy functional equation f x y f x f y x y in which is an abstract orthogonality relation was first investigated by Gudder and Strawther 3 . They defined by a system consisting of five axioms and described the general semi-continuous real-valued solution of conditional Cauchy functional equation. In 1985 Ratz 4 introduced a new definition of orthogonality by using more restrictive axioms than of Gudder and Strawther. Moreover he investigated the structure of orthogonally .
