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Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings. (English) Zbl 0928.39014
Let $$X$$ be a normed linear space and $$Y$$ be a real Banach space. The author considers the Ulam stability problem for nonlinear Euler-Lagrange quadratic mappings, that is for the mappings $$Q:X\to Y$$ satisfying the following system of functional equations $m_1m_2 Q(a_1x_1+ a_2 x_2) +Q(m_2a_2x_1- m_1a_1x_2)= (m_1a_1^2+ m_2a_2^2)\bigl[ m_2 Q(x_1)+ m_1Q(x_2) \bigr],$ for all vectors $$(x_1,x_2)\in X^2$$ and any fixed pair $$(a_1, a_2)$$ of reals $$a_i$$ and any fixed pair $$(m_1,m_2)$$ of positive reals $$m_i$$ $$(i=1,2)$$ and $m^2_1m_2Q(a_1x)+ m_1Q(m_2a_2x) =m^2_0m_2 Q\left({m_1\over m_0} a_1x\right)+ m_0^2m_1Q \left({m_2\over m_0} a_2x\right),$ with $$m_0={m_1 m_2+1 \over m_0}$$ for all $$x\in X$$ and fixed above reals $$a_i$$ and positive reals $$m_i$$. The problem is solved under some special assumptions on the mapping $$Q$$. The author solves the problem separately for the two cases $${m_1a^2_1 +m_2a^2_2\over m_0}= :m>1$$ and $$m<1$$.
Reviewer: M.C.Zdun (Kraków)

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B72 Systems of functional equations and inequalities 39B52 Functional equations for functions with more general domains and/or ranges
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##### References:
 [1] Forti, G.L., Hyers – ulam stability of functional equations in several variables, Aequationes math., 50, 143-190, (1995) · Zbl 0836.39007 [2] Gruber, P.M., Stability of isometries, Trans. amer. math. soc., 245, 263-277, (1978) · Zbl 0393.41020 [3] Hyers, D.H., The stability of homomorphisms and related topics, Global analysis—analysis on manifolds, Teubner-texte math., 57, (1983), Teubner Leipzig, p. 140-153 · Zbl 0517.22001 [4] Rassias, J.M., On approximation of approximately linear mappings by linear mappings, J. funct. anal., 46, 126-130, (1982) · Zbl 0482.47033 [5] Rassias, J.M., On approximation of approximately linear mappings by linear mappings, Bull. sci. math., 108, 445-446, (1984) · Zbl 0599.47106 [6] Rassias, J.M., Solution of a problem of Ulam, J. approx. theory, 57, 268-273, (1989) · Zbl 0672.41027 [7] Rassias, J.M., Complete solution of the multi-dimensional problem of Ulam, Discuss. math., 14, 101-107, (1994) · Zbl 0819.39012 [8] Rassias, J.M., Solution of a stability problem of Ulam, Discuss. math., 12, 95-103, (1992) · Zbl 0779.47005 [9] Rassias, J.M., On the stability of the euler – lagrange functional equation, Chinese J. math., 20, 185-190, (1992) · Zbl 0753.39003 [10] Rassias, J.M., On the stability of the non-linear euler – lagrange functional equation in real normed linear spaces, J. math. phys. sci., 28, 231-235, (1994) · Zbl 0840.46024 [11] Rassias, J.M., On the stability of the multi-dimensional non-linear euler – lagrange functional equation, Geometry, analysis and mechanics, (1994), World Scientific Singapore, p. 275-285 · Zbl 0842.39013 [12] Rassias, J.M., On the stability of the general euler – lagrange functional equation, Demonstratio math., 29, 755-766, (1996) · Zbl 0884.47040 [13] Szekelyhidi, L., Note on Hyers’ theorem, C. R. math. rep. acad. sci. Canada, 8, 127-129, (1986) · Zbl 0604.39007 [14] Ulam, S.M., A collection of mathematical problems, (1968), Interscience New York, p. 63 · Zbl 0086.24101
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