# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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) $\Vert x\Vert = 0$ if and only if $x =0$; (2) $\Vert \lambda x\Vert = \vert \lambda\vert \cdot \Vert x\Vert$ for all scalars $\lambda$ and all $x\in X$; (3) There is a constant $K \geq 1$ such that $\Vert x+y\Vert \leq K(\Vert x\Vert + \Vert y\Vert )$ for all $x, y \in X$. Then the pair $(X, \Vert \cdot \Vert )$ is said to be a quasi-normed space. A quasi-norm $\Vert \cdot \Vert$ is called a $p$-norm $(0 < p \leq 1)$ if $\Vert x+y\Vert ^p \leq \Vert x\Vert ^p + \Vert y\Vert ^p \quad (x, y \in X)$. By the Aoki-Rolewicz theorem [see {\it 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 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. (functional equations)
Full Text:
##### References:
 [1] Aczél, J.; Dhombres, J.: Functional equations in several variables. (1989) · 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. 48 (2000) · 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. Stability of mappings of Hyers -- Ulam type, 81-91 (1994) · 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) · Zbl 0907.39025 [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, No. 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) · Zbl 1047.39001 [21] Rolewicz, S.: Metric linear spaces. (1984) · Zbl 0526.49018 [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) · Zbl 0137.24201