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On the Hyers-Ulam stability of the functional equations that have the quadratic property. (English) Zbl 0928.39013
Let \(X\) be a real normed linear space and \(Y\) be a real Banach space. The Hyers-Ulam stability of the quadratic functional equation \[ f(x+y)+ f(x-y)=2f(x) +2f(y),\quad x,y\in X\tag{1} \] for \(f:X\to Y\) on the restricted domain \(\| x\|+\| y\|>d\) for a \(d>0\) is investigated. Furthermore, the Hyers-Ulam stability of another quadratic functional equation \[ f(x+y+z)+ f(x)+f(y)+f(z)= f(x+y)+f(y+z)+ f(z+x),\quad x,y,z\in X,\tag{2} \] with condition \(\| f(x)+f(-x) \|\leq\gamma\) for a \(\gamma>0\) and \(x\in X\) or \(\| f(x)-f(-x) \|\leq\gamma\) for a \(\gamma>0\) and \(x\in X\) is treated, first on the whole domain and then on the restricted domain \(\| x \|+\| y\|+ \| z\|>d\). The results are applied to the study of the asymptotic behaviour of equation (1) and (2).
Reviewer: M.C.Zdun (Kraków)

39B72 Systems of functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges
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[1] Aczél, J.; Dhombres, J., Functional equations in several variables, (1989), Cambridge Univ. Press · Zbl 0685.39006
[2] Borelli, C.; Forti, G.L., On a general hyers – ulam stability result, Internat. J. math. math. sci., 18, 229-236, (1995) · Zbl 0826.39009
[3] Cholewa, P.W., Remarks on the stability of functional equations, Aequationes math., 27, 76-86, (1984) · Zbl 0549.39006
[4] Czerwik, S., On the stability of the quadratic mappings in normed spaces, Abh. math. sem. univ. Hamburg, 62, 59-64, (1992) · Zbl 0779.39003
[5] Drljević, H., On the stability of the functional quadratic onA, Publ. inst. math. (beograd), 36, 111-118, (1984) · Zbl 0598.65029
[6] Fenyö, I., On an inequality of P. W. cholewa, () · Zbl 0636.39006
[7] Forti, G.L., Hyers – ulam stability of functional equations in several variables, Aequationes math., 50, 143-190, (1995) · Zbl 0836.39007
[8] Ger, R., On functional inequalities stemming from stability questions, () · Zbl 0770.39007
[9] Hyers, D.H., On the stability of the linear functional equation, Proc. nat. acad. sci. U.S.A., 27, 222-224, (1941) · Zbl 0061.26403
[10] Hyers, D.H.; Rassias, Th.M., Approximate homomorphisms, Aequationes math., 44, 125-153, (1992) · Zbl 0806.47056
[11] Kannappan, Pl., Quadratic functional equation and inner product spaces, Results math., 27, 368-372, (1995) · Zbl 0836.39006
[12] Parnami, J.C.; Vasudeva, H.L., On Jensen’s functional equation, Aequationes math., 43, 211-218, (1992) · Zbl 0755.39008
[13] Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. amer. math. soc., 72, 297-300, (1978) · Zbl 0398.47040
[14] Skof, F., Sull’approssimazione delle applicazioni localmente δ-additive, Atti accad. sci. Torino cl. sci. fis. mat. natur., 117, 377-389, (1983)
[15] Skof, F., Proprietà locali e approssimazione di operatori, Rend. sem. mat. fis. milano, 53, 113-129, (1983)
[16] Skof, F., Approssimazione di funzioni δ-quadratiche su dominio restretto, Atti accad. sci. Torino cl. sci. fis. mat. natur., 118, 58-70, (1984)
[17] Ulam, S.M., Problems in modern mathematics, (1960), Wiley New York · Zbl 0137.24201
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