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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)

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