Ulam stability problem for quadratic mappings of Euler–Lagrange. (English) Zbl 1074.39027

The authors consider a generalization of the quadratic functional equation. Let \({\mathcal A}\) and \({\mathcal B}\) be linear spaces and \(a,b\in\mathbb{N}\). A mapping \(Q:{\mathcal A}\to {\mathcal B}\) satisfying the equation \[ (a+b)aQ(x)+(a+b)bQ(y)=Q(ax+by)+abQ(x-y),\;\;\;x,y\in {\mathcal A} \] is called the {quadratic mapping of Euler–Lagrange}.
The aim of the paper is to prove the stability of the above equation. It is shown that, with \({\mathcal B}\) being a normed space, for \(f:{\mathcal A}\to {\mathcal B}\) satisfying the inequality \[ \| (a+b)aQ(x)+(a+b)bQ(y)-Q(ax+by)-abQ(x-y)\| \leq\varphi(x,y),\;\;\;x,y\in {\mathcal A} \] with the approximate reminder \(\varphi\) satisfying some assumptions, there exists a unique quadratic mapping of Euler-Lagrange \(Q:{\mathcal A}\to{\mathcal B}\) which is, in some sense, close to \(f\). Additionally, similar results are given for mappings between Banach modules.


39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
Full Text: DOI


[1] Aczél, J.; Dhombres, J., Functional equations in several variables, (1989), Cambridge University Press Cambridge · Zbl 0685.39006
[2] Cholewa, P.W., Remarks on the stability of functional equations, Aequationes math., 27, 76-86, (1984) · Zbl 0549.39006
[3] Czerwik, S., On the stability of the quadratic mapping in normed spaces, Abh. math. sem. univ. Hamburg, 62, 59-64, (1992) · Zbl 0779.39003
[4] Czerwik, S., The stability of the quadratic functional equation, (), 81-91 · Zbl 0844.39008
[5] Gaˇvruta, P., A generalization of the hyers – ulam – rassias stability of approximately additive mappings, J. math. anal. appl., 184, 431-436, (1994) · Zbl 0818.46043
[6] Gruber, P.M., Stability of isometries, Trans. amer. math. soc., 245, 263-277, (1978) · Zbl 0393.41020
[7] Hyers, D.H., On the stability of the linear functional equation, Proc. natl. acad. sci., 27, 222-224, (1941) · Zbl 0061.26403
[8] Hyers, D.H.; Isac, G.; Rassias, Th.M., Stability of functional equations in several variables, (1998), Birkhäuser Basel · Zbl 0894.39012
[9] Hyers, D.H.; Rassias, Th.M., Approximate homomorphisms, Aequationes math., 44, 125-153, (1992) · Zbl 0806.47056
[10] Jun, K.W.; Lee, Y.H., On the hyers – ulam – rassias stability of a pexiderized quadratic inequality, Math. inequal. appl., 4, 1, 93-118, (2001) · Zbl 0976.39031
[11] Rassias, J.M., On the stability of the euler – lagrange functional equation, Chinese J. math., 20, 185-190, (1992) · Zbl 0753.39003
[12] Rassias, J.M., Solution of the Ulam stability problem for euler – lagrange quadratic mappings, J. math. anal. appl., 220, 613-639, (1998) · Zbl 0928.39014
[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] Rassias, Th.M., On the stability of functional equations in Banach spaces, J. math. anal. appl., 251, 264-284, (2000) · Zbl 0964.39026
[15] Ulam, S.M., A collection of mathematical problems, (1960), Interscience New York · Zbl 0086.24101
[16] Ding-Xuan Zhou, On a conjecture of Z. Ditzian, J. approx. theory, 69, 167-172, (1992) · Zbl 0755.41029
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