Stability of general multi-Euler-Lagrange quadratic functional equations in non-Archimedean fuzzy normed spaces
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RESEARCH
Open Access
Stability of general multi-Euler-Lagrange quadratic functional equations in non-Archimedean fuzzy normed spaces Tian Zhou Xu1* and John Michael Rassias2 *
Correspondence: [email protected] 1 School of Mathematics, Beijing Institute of Technology, Beijing, 100081, P.R. China Full list of author information is available at the end of the article
Abstract In this paper we prove the generalized Hyers-Ulam stability of the system defining general Euler-Lagrange quadratic mappings in non-Archimedean fuzzy normed spaces over a field with valuation using the direct and the fixed point methods. MSC: 39B82; 39B52; 46H25 Keywords: stability of general multi-Euler-Lagrange quadratic functional equation; direct method; fixed point method; non-Archimedean fuzzy normed space
1 Introduction Let K be a field. A valuation mapping on K is a function | · | : K → R such that for any r, s ∈ K the following conditions are satisfied: (i) |r| ≥ and equality holds if and only if r = ; (ii) |rs| = |r| · |s|; (iii) |r + s| ≤ |r| + |s|. A field endowed with a valuation mapping will be called a valued field. The usual absolute values of R and C are examples of valuations. A trivial example of a non-Archimedean valuation is the function | · | taking everything except for into and || = . In the following we will assume that | · | is non-trivial, i.e., there is an r ∈ K such that |r | = , . If the condition (iii) in the definition of a valuation mapping is replaced with a strong triangle inequality (ultrametric): |r + s| ≤ max{|r|, |s|}, then the valuation | · | is said to be non-Archimedean. In any non-Archimedean field we have || = | – | = and |n| ≤ for all n ∈ N. Throughout this paper, we assume that K is a valued field, X and Y are vector spaces over K, a, b ∈ K are fixed with λ := a + b = , (λ := a = , if a = b) and n is a positive integer. Moreover, N stands for the set of all positive integers and R (respectively, Q) denotes the set of all reals (respectively, rationals). A mapping f : X n → Y is called a general multi-Euler-Lagrange quadratic mapping if it satisfies the general Euler-Lagrange quadratic equations in each of their n arguments: f x , . . . , xi– , axi + bxi , xi+ , . . . , xn + f x , . . . , xi– , bxi – axi , xi+ , . . . , xn = a + b f (x , . . . , xn ) + f x , . . . , xi– , xi , xi+ , . . . , xn
(.)
© 2012 Xu and Rassias; 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.
Xu and Rassias Advances in Difference Equations 2012, 2012:119 http://www.advancesindifferenceequations.com/content/2012/1/119
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for all i = , . . . , n and all x , . . . , xi– , xi , xi , xi+ , . . . , xn ∈ X . Letting xi = xi = in (.), we get f (x , . . . , xi– , , xi+ , . . . , xn ) = . Putting xi = in (
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