Stability Problem of Ulam for Euler-Lagrange Quadratic Mappings

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Research Article Stability Problem of Ulam for Euler-Lagrange Quadratic Mappings Hark-Mahn Kim, John Michael Rassias, and Young-Sun Cho Received 26 May 2007; Revised 9 August 2007; Accepted 9 November 2007 Recommended by Ondrej Dosly

We solve the generalized Hyers-Ulam stability problem for multidimensional EulerLagrange quadratic mappings which extend the original Euler-Lagrange quadratic mappings. Copyright © 2007 Hark-Mahn 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. 1. Introduction In 1940, Ulam [1] proposed, at the University of Wisconsin, the following problem: “give conditions in order for a linear mapping near an approximately linear mapping to exist.” In 1968, Ulam proposed the general Ulam stability problem: “when is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?” The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus the stability question of functional equations is “how do the solutions of the inequality diﬀer from those of the given functional equation?” If the answer is aﬃrmative, we would say that the equation is stable. In 1978, Gruber [2] remarked that Ulam problem is of particular interest in probability theory and in the case of functional equations of diﬀerent types. We wish to note that stability properties of diﬀerent functional equations can have applications to unrelated fields. For instance, Zhou [3] used a stability property of the functional equation f (x − y) + f (x + y) = 2 f (x) to prove a conjecture of Z. Ditzian about the relationship between the smoothness of a mapping and the degree of its approximation by the associated Bernstein polynomials. Above all, Ulam problem for ε-additive mappings f : E1 →E2 between Banach spaces, that is, f (x + y) − f (x) − f (y) ≤ ε for all x, y ∈ E1 , was solved by Hyers [4] and then generalized by Th. M. Rassias [5] and G˘avrut¸a [6] who permitted the Cauchy diﬀerence

2

Journal of Inequalities and Applications

to become unbounded. 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. A large list of references can be found, for example, in [7–9] and references therein. We note that J. M. Rassias introduced the Euler-Lagrange quadratic mappings, motivated from the following pertinent algebraic equation a1 x1 + a2 x2 2 + a2 x1 − a1 x2 2 = a2 + a2 x1 2 + x2 2 . 1

2

(1.1)

Thus the second author of this paper introduced and investigated the stability problem of Ulam for the relative Euler-Lagrange functional equation

f a1 x1 + a2 x2 + f a2 x1 − a1 x2 = a21 + a22

f x1 + f x2

(1

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