On the stability of mappings and an answer to a problem

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Abstract
The main purpose of this paper is to prove a theorem concerning the HYERS-ULAM stability of mappings, which gives a generalization of the results from [1] and [3J. It also answers a. problem posed by TH.M.RASSIAS [3]. The question concerning the stability of mappings has been originally raised by S.M.ULAM [4]. The first answer was given in 1941 by D. H.HYERS (see [2] for a research survey of the development of the subject). In this paper we provide a generalization of a theorem of TH.M.RASSIAS [3] concerning the HYERS-ULAM stability of mappings and we also answer a problem that TH.M.RASSIAS posed in [3].
Using the triangle inequality and the last two relations, it follows: We will prove by mathematical induction after k the following inequality: Indeed, for k = 2 and k = 3 we have the relation (4) and, respectively (5). Suppose (6) true for k and let us prove it for k + 1. We replace y by kx in (2) and we obtain:~f Hence, it follows using (6) for the last inequality. So, relation (6) is true for any k ~ 2, integer. Dividing (6) by k we obtain: k-l l. ,.
f(x)~ 0 3 A 3 1 k 0 3 C 6 ( x , mx). We see that for n == 1 we have (7). We suppose (8) true for n and we will prove it for n -t-1.We replace :r by Aj: in (8)  Therefore, the sequence {f(knx) kn}n~|N* is a fundamental sequence. Because X is a BANACH space it follows that there exists . ~x G, denoted by T(~), so T : G 2014~ X and we claim that T is an additive mapping. From (2) we have + kny)kny), Vx, y E G.
Taking the limit as n --~ co we obtain: T(x + y) -T(x) -T(y)~ ~ l i m 1 k n 0 3 C 6 ( k n x , k n y ) = 0 using the relation (1). This implies T(x + y) = T(x + r(t/), ~x, y E G. To prove that (3) holds, we take the limit as nx in (8)  in [3] .
We prove that the best possible value of A: is 2. Set R(p) = 2 2 -2 p a n d Q ( k , p ) = k . s ( k , p ) k -k p , k > 2.
Thus, (9) is proved. This last result gives an answer to a problem that was posed by TH.M. RASSIAS in 1991.