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The properties of functional inclusions and Hyers-Ulam stability. (English) Zbl 1271.39031
Let $$Y$$ be a normed space over $$\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$$, let $$K$$ be a set and let $$n(Y):=2^Y\setminus\{\emptyset\}$$. Furthermore assume that $$F: K\to n(Y)$$, $$\psi: Y\to Y$$, $$a: K\to K$$ are given functions and that $$\lambda\in(0,1)$$. Under the hypotheses that $$\psi$$ is $$\lambda$$-Lipschitz and that $$\sup\{\text{diam}(F(x))\mid x\in K\}$$ is finite, the author among others proves:
(1)
If $$Y$$ is complete and if $$\psi(F(a(x)))\subseteq F(x)$$ for all $$x\in K$$, then there is a unique $$f: K\to K$$ such that $$f(x)\in\overline{F(x)}$$ for all $$x\in K$$ and such that $$\psi\circ f\circ a=f$$.
(2)
If $$F(x)\subseteq\psi(F(a(x)))$$ for all $$x$$ then there is some $$f: K\to K$$ such that $$F(x)=\{f(x)\}$$ for all $$x$$. Moreover, this function $$f$$ satisfies $$\psi\circ f\circ a=f$$.
The (general) results are applied to prove some stability results.
Reviewer’s remark: The condition that $$\psi$$ is $$\lambda$$-Lipschitz is formulated in terms of a two-place function $$d$$ which seems to be undefined. From the context it should become clear that $$d(x,y)=\| x-y\|$$. But probably some of the results, especially those cited above, could be proved for (general) metric spaces $$Y$$.

MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B05 General theory of functional equations and inequalities 54C60 Set-valued maps in general topology 54C65 Selections in general topology 39B52 Functional equations for functions with more general domains and/or ranges
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References:
 [1] Aczél J.: Lectures on Functional Equations and Their Applications. Academic Press, New York (1966) · Zbl 0139.09301 [2] Brzdek J.: On a method of proving the Hyers–Ulam stability of functional equations on restricted domains. AJMAA 6, 1–10 (2009) [3] Brzdek 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) · Zbl 1205.39025 · doi:10.1016/j.na.2010.08.047 [4] Forti G.L.: Hyers–Ulam stability of functional equations in several variables. Aequ. Math. 50, 143–190 (1995) · Zbl 0836.39007 · doi:10.1007/BF01831117 [5] 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) · Zbl 1052.39031 · doi:10.1016/j.jmaa.2004.03.011 [6] Gajda Z., Ger R.: Subadditive multifunctions and Hyers–Ulam stability. Numer. Math. 80, 281–291 (1987) · Zbl 0639.39014 [7] Hyers D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941) · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222 [8] Hyers D.H., Isac G., Rassias Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Boston (1998) · Zbl 0907.39025 [9] Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Uniwersytet Ślaski , Katowice, PWN, Warsaw (1985) · Zbl 0555.39004 [10] Nikodem K., Popa D.: On selections of general linear inclusions. Publ. Math. Debrecen 75, 239–249 (2009) · Zbl 1212.39041 [11] Páles Z.: Generalized stability of the Cauchy functional equation. Aequ. Math. 56, 222–232 (1998) · Zbl 0922.39008 · doi:10.1007/s000100050058 [12] Páles Z.: Hyers–Ulam stability of the Cauchy functional equation on square-symmetric grupoids. Publ. Math. Debrecen 58, 651–666 (2001) · Zbl 0980.39022 [13] Popa D.: A stability result for a general linear inclusion. Nonlinear Funct. Anal. Appl. 3, 405–414 (2004) · Zbl 1067.39043 [14] Popa D.: Functional inclusions on square-symmetric grupoids and Hyers–Ulam stability. Math. Inequal. Appl. 7, 419–428 (2004) · Zbl 1058.39026 [15] Popa D.: A property of a functional inclusion connected with Hyers–Ulam stability. J. Math. Inequal. 4, 591–598 (2009) · Zbl 1189.39032 · doi:10.7153/jmi-03-57 [16] Rassias Th.M.: On the stability of linear mappings in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978) · Zbl 0398.47040 · doi:10.1090/S0002-9939-1978-0507327-1 [17] Smajdor W.: Superadditive set-valued functions. Glas. Mat. 21, 343–348 (1986) · Zbl 0617.26010
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