Abstract
We prove that a set-valued function satisfying some functional inclusions admits, in appropriate conditions, a unique selection satisfying the corresponding functional equation. As a consequence we obtain the result on the Hyers–Ulam stability of that functional equation.
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References
Aczél J.: Lectures on Functional Equations and Their Applications. Academic Press, New York (1966)
Brzdȩk J.: On a method of proving the Hyers–Ulam stability of functional equations on restricted domains. AJMAA 6, 1–10 (2009)
Brzdȩk J., Popa D., Xu B.: Selections of set-valued maps satisfying a linear inclusions in single variable via Hyers–Ulam stability. Nonlinear Anal. 74, 324–330 (2011)
Forti G.L.: Hyers–Ulam stability of functional equations in several variables. Aequ. Math. 50, 143–190 (1995)
Forti G.L.: Comments on the core of the direct method for proving Hyers–Ulam stability of functional equations. J. Math. Anal. Appl. 295, 127–133 (2004)
Gajda Z., Ger R.: Subadditive multifunctions and Hyers–Ulam stability. Numer. Math. 80, 281–291 (1987)
Hyers D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)
Hyers D.H., Isac G., Rassias Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Boston (1998)
Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Uniwersytet Śla̧ski , Katowice, PWN, Warsaw (1985)
Nikodem K., Popa D.: On selections of general linear inclusions. Publ. Math. Debrecen 75, 239–249 (2009)
Páles Z.: Generalized stability of the Cauchy functional equation. Aequ. Math. 56, 222–232 (1998)
Páles Z.: Hyers–Ulam stability of the Cauchy functional equation on square-symmetric grupoids. Publ. Math. Debrecen 58, 651–666 (2001)
Popa D.: A stability result for a general linear inclusion. Nonlinear Funct. Anal. Appl. 3, 405–414 (2004)
Popa D.: Functional inclusions on square-symmetric grupoids and Hyers–Ulam stability. Math. Inequal. Appl. 7, 419–428 (2004)
Popa D.: A property of a functional inclusion connected with Hyers–Ulam stability. J. Math. Inequal. 4, 591–598 (2009)
Rassias Th.M.: On the stability of linear mappings in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)
Smajdor W.: Superadditive set-valued functions. Glas. Mat. 21, 343–348 (1986)
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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Piszczek, M. The properties of functional inclusions and Hyers–Ulam stability. Aequat. Math. 85, 111–118 (2013). https://doi.org/10.1007/s00010-012-0119-0
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DOI: https://doi.org/10.1007/s00010-012-0119-0