Cubic Functional Equations on Restricted Domains of Lebesgue Measure Zero

Choi, Chang-Kwon ; Chung, Jaeyoung ; Ju, Yumin ; Rassias, John

Canadian mathematical bulletin, Tome 60 (2017) no. 1, pp. 95-103

Résumé

Let $X$ be a real normed space, $Y$ a Banach space, and $f\,:\,X\,\to \,Y$ . We prove theUlam–Hyers stability theorem for the cubic functional equation $$f\left( 2x\,+\,y \right)\,+\,f\left( 2x\,-\,y \right)\,-\,2f\left( x\,+\,y \right)\,-\,2f\left( x\,-\,y \right)\,-\,12f\left( x \right)\,=\,0$$ in restricted domains. As an application we consider a measure zero stability problem of the inequality $$\left\| f\left( 2x\,+\,y \right)\,+\,f\left( 2x\,-\,y \right)\,-\,2f\left( x\,+\,y \right)\,-\,2f\left( x\,-\,y \right)\,-\,12f\left( x \right) \right\|\,\le \,\varepsilon$$ for all $\left( x,\,y \right)$ in $\Gamma \,\subset \,{{\mathbb{R}}^{2}}$ of Lebesgue measure 0.

Détail

DOI : 10.4153/CMB-2016-041-4

Mots-clés : 39B82, Baire category theorem, cubic functional equation, first category, Lebesgue measure, Ulam-Hyers stability

Bibliographie

[1] [1] Alsina, C. and Garcia-Roig, J. L., On a conditional Cauchy equation on rhombuses. In: Functional analysis, approximation theory and numerical analysis, World Scientific, River Edge, NJ, 1994. Google Scholar

[2] [2] Batko, B., Stability of an alternative functional equation. J. Math. Anal. Appl. 339(2008), no. 1, 303–311. http://dx.doi.Org/10.101 6/j.jmaa.2007.07.001 Google Scholar

[3] [3] Batko, B., On approximation of approximate solutions ofDhombres’ equation. J. Math. Anal. Appl. 340(2008), no. 1, 424–432. http://dx.doi.Org/10.101 6/j.jmaa.2007.08.009 Google Scholar

[4] [4] Brzdek, J., On the quotient stability of a family of functional equations. Nonlinear Anal. 71(2009), no. 10, 4396–4404. http://dx.doi.Org/10.1016/j.na.2009.02.123 Google Scholar

[5] [5] Brzdek, J., On a method of proving the Hyers-Ulam stability of functional equations on restricted domains. Aust. J. Math. Anal. Appl. 6(2009), 1–10. Google Scholar

[6] [6] Bahyrycz, A. and Brzdek, J., On solutions of the d'Alembert equation on a restricted domain. Aequationes Math. 85(2013), no. 1-2, 169–183. http://dx.doi.Org/10.1007/s00010-012-0162-x Google Scholar

[7] [7] Brzdçk, J. and Sikorska, J., A conditional exponential functional equation and its stability. Nonlinear Anal. 72(2010), no. 6, 2923–2934. http://dx.doi.Org/10.101 6/j.na.2009.11.036 Google Scholar

[8] [8] Chung, J., Stability of functional equations on restricted domains in a group and their asymptotic behaviors. Comput. Math. Appl. 60(2010), no. 9, 2653–2665. http://dx.doi.Org/1 0.101 6/j.camwa.2O10.09.003 Google Scholar

[9] [9] Chung, J. and Rassias, J. M., Quadratic functional equations in a set of Lebesgue measure zero. J. Math. Anal. Appl. 419(2014), no. 2, 1065–1075. http://dx.doi.Org/10.101 6/j.jmaa.2O14.05.032 Google Scholar

[10] [10] Ger, R. and Sikorska, J., On the Cauchy equation on spheres. Ann. Math. Sil. 11(1997), 89–99. Google Scholar

[11] [11] Hyers, D. H., On the stability of the linear functional equations. Proc. Nat. Acad. Sci. USA 27(1941), 222–224. http://dx.doi.Org/10.1073/pnas.27.4.222 Google Scholar

[12] [12] Jun, K.-W. and Kim, H.-M., The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 274(2002), no. 2, 867–878. http://dx.doi.Org/10.1016/S0022-247X(02)00415-8 Google Scholar

[13] [13] Jung, S.-M., Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis. Springer Optimization and Its Applications, 48, Springer, New York, 2011. http://dx.doi.Org/10.1007/978-1-4419-9637-4 Google Scholar

[14] [14] Jung, S.-M., On the Hyers-Ulam stability of the functional equations that have the quadratic property. J. Math. Anal. Appl. 222(1998), no. 1, 126–137. http://dx.doi.Org/1 0.1006/jmaa.1 998.591 6 Google Scholar

[15] [15] Kuczma, M., Functional Equations on restricted domains. Aequationes Math. 18(1978), no. 1-2, 1–34. http://dx.doi.Org/10.1007/BF01844065 Google Scholar

[16] [16] Oxtoby, J. C., Measure and category. Graduate Texts in Mathematics, 2, Springer-Verlag, New York, 1980. Google Scholar

[17] [17] Rassias, J. M. and Rassias, M. J., On the Ulam stability of Jensen and Jensen type mappings on restricted domains. J. Math. Anal. Appl. 281(2003), no. 2, 516–524. http://dx.doi.Org/10.1016/S0022-247X(03)00136-7 Google Scholar

[18] [18] Rassias, J. M., On the Ulam stability of mixed type mappings on restricted domains. J. Math. Anal. Appl. 281(2002), no. 2, 747–762. http://dx.doi.Org/10.101 6/S0022-247X(02)00439-0 Google Scholar

[19] [19] Sikorska, J., On two conditional Vexider functional equations and their stabilities. Nonlinear Anal. 70(2009), no. 7, 2673–2684. http://dx.doi.Org/10.1016/j.na.2008.03.054 Google Scholar

[20] [20] Skof, F., Sull'approssimazione dette applicazioni localmente S-additive. Atii Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 117(1983), no. 4-6, 377–389. Google Scholar