Nonlinear approximation of an ACQ-functional equation in nan-spaces
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RESEARCH
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Nonlinear approximation of an ACQ-functional equation in nan-spaces Hassan Azadi Kenary1, Jung Rye Lee2* and Choonkil Park3 * Correspondence: [email protected]. kr 2 Department of Mathematics, Daejin University, Kyeonggi 487711, Korea Full list of author information is available at the end of the article
Abstract In this paper, using the fixed point and direct methods, we prove the generalized Hyers-Ulam stability of an additive-cubic-quartic functional equation in NAN-spaces. Mathematics Subject Classification (2010) 39B52·47H10·26E30·46S10·47S10 Keywords: generalized Hyers-Ulam stability, non-Archimedean normed space, fixed point method
1. Introduction and preliminaries A classical question in the theory of functional equations is the following: “When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?” If the problem accepts a solution, we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940. In the next year, Hyers [2] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In 1978, Rassias [3] proved a generalization of the Hyers’ theorem for additive mappings. The result of Rassias has provided a lot of influence during the last three decades in the development of a generalization of the Hyers-Ulam stability concept. This new concept is known as generalized Hyers-Ulam stability or Hyers-UlamRassias stability of functional equations (see [4-8]). Furthermore, in 1994, a generalization of the Rassias’ theorem was obtained by Găvruta [9] by replacing the bound ε(|| x||p + ||y||p) by a general control function (x, y). The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y)
is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. In 1983, a generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [10] for mappings f : X ® Y, where X is a normed space and Y is a Banach space. In 1984, Cholewa [11] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group and, in 2002, Czerwik [12] proved the generalized Hyers-Ulam stability of the quadratic functional equation.
© 2011 Azadi Kenary et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Azadi Kenary et al. Fixed Point Theory and Applications 2011, 2011:60 http://www.fixedpointtheoryandapplications.com/content/2011/1/60
The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [13-32]). In
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