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On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces. (English) Zbl 1117.39017

A quasi-norm is a real-valued function on a vector space \(X\) satisfying the following: (1) \(\| x\| = 0\) if and only if \(x =0\); (2) \(\| \lambda x\| = | \lambda| \cdot \| x\| \) for all scalars \(\lambda\) and all \(x\in X\); (3) There is a constant \(K \geq 1\) such that \(\| x+y\| \leq K(\| x\| + \| y\| )\) for all \(x, y \in X\). Then the pair \((X, \| \cdot \| )\) is said to be a quasi-normed space. A quasi-norm \(\| \cdot \| \) is called a \(p\)-norm \((0 < p \leq 1)\) if \(\| x+y\| ^p \leq \| x\| ^p + \| y\| ^p \quad (x, y \in X)\). By the Aoki-Rolewicz theorem [see S. Rolewicz, Metric linear spaces. 2nd ed. Mathematics and its applications (East European Series), 20. Dordrecht- Boston-Lancaster: D. Reidel Publishing Company, Warszawa: PWN-Polish Scientific Publishers. (1985; Zbl 0573.46001)]), each quasi-norm is equivalent to some \(p\)-norm. Since it is much easier to work with \(p\)-norms than quasi-norms, henceforth the authors restrict their attention mainly to \(p\)-norms.
The functional equation
\[ f(ax+y) + f(x+ay) = (a+1)(a-1)^2[f(x)+ f(y)] + a(a+1)f(x+y) \]
is called the Euler-Lagrange type cubic functional equation. The authors prove the stability of this equation for fixed integers \(a\) with \(a \neq 0, \pm 1\) in the framework of quasi-Banach spaces by using the direct method.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
46B03 Isomorphic theory (including renorming) of Banach spaces
46B20 Geometry and structure of normed linear spaces
39B62 Functional inequalities, including subadditivity, convexity, etc.

Citations:

Zbl 0573.46001
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References:

[1] Aczél, J.; Dhombres, J., Functional Equations in Several Variables (1989), Cambridge Univ. Press · Zbl 0685.39006
[2] Bae, J. H.; Park, W. G., Generalized Jensen’s functional equations and approximate algebra homomorphisms, Bull. Korean Math. Soc., 39, 401-410 (2002) · Zbl 1017.39012
[3] Benyamini, Y.; Lindenstrauss, J., Geometric Nonlinear Functional Analysis, vol. 1, Colloq. Publ., vol. 48 (2000), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0946.46002
[4] Czerwik, S., On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62, 59-64 (1992) · Zbl 0779.39003
[5] Czerwik, S., The stability of the quadratic functional equation, (Rassias, Th. M.; Tabor, J., Stability of Mappings of Hyers-Ulam Type (1994), Hadronic Press: Hadronic Press Florida), 81-91 · Zbl 0844.39008
[6] Gajda, Z., On stability of additive mappings, Int. J. Math. Math. Sci., 14, 431-434 (1991) · Zbl 0739.39013
[7] Gǎvruta, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184, 431-436 (1994) · Zbl 0818.46043
[8] Hyers, D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27, 222-224 (1941) · Zbl 0061.26403
[9] Hyers, D. H.; Isac, G.; Rassias, Th. M., Stability of Functional Equations in Several Variables (1998), Birkhäuser: Birkhäuser Basel · Zbl 0894.39012
[10] Hyers, D. H.; Rassias, Th. M., Approximate homomorphisms, Aequationes Math., 44, 125-153 (1992) · Zbl 0806.47056
[11] 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
[12] Jun, K. W.; Kim, H. M., The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl., 274, 867-878 (2002) · Zbl 1021.39014
[13] Jun, K. W.; Kim, H. M.; Chang, I. S., On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation, J. Comput. Anal. Appl., 7, 21-33 (2005) · Zbl 1087.39029
[14] Jung, S. M., On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl., 222, 126-137 (1998) · Zbl 0928.39013
[15] Rassias, J. M., On the stability of the Euler-Lagrange functional equation, Chinese J. Math., 20, 185-190 (1992) · Zbl 0753.39003
[16] 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
[17] Rassias, J. M., Solution of the Ulam stability problem for cubic mappings, Glasnik Matem., 36, 63-72 (2001) · Zbl 0984.39014
[18] Rassias, Th. M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300 (1978) · Zbl 0398.47040
[19] Rassias, Th. M., On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251, 264-284 (2000) · Zbl 0964.39026
[20] Rassias, Th. M., Functional Equations, Inequalities and Applications (2003), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht · Zbl 1047.39001
[21] Rolewicz, S., Metric Linear Spaces (1984), Reidel/PWN: Reidel/PWN Dordrecht/Warsaw · Zbl 0577.46040
[22] Skof, F., Proprietà locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53, 113-129 (1983)
[23] Tabor, J., Stability of the Cauchy functional equation in quasi-Banach spaces, Ann. Polon. Math., 83, 243-255 (2004) · Zbl 1101.39021
[24] Ulam, S. M., Problems in Modern Mathematics (1964), Wiley: Wiley New York, (Chapter VI) · Zbl 0137.24201
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