# zbMATH — the first resource for mathematics

On selections of set-valued inclusions in a single variable with applications to several variables. (English) Zbl 1277.39032
Suppose that $$(Y,d)$$ is a metric space and $$n(Y)$$ denotes the family of all nonempty subsets of $$Y$$ and $$\delta(A)=\sup\{d(x,y): x,y \in A\}$$ is the diameter of $$A\subset Y$$. Suppose that $$K$$ is a nonempty set and $$F: Y \to n(Y), \quad \psi:Y\to Y,\quad a:K\to K$$ are maps and $$\lambda \in (0, \infty)$$ such that $$d(\psi(x), \psi(y))\leq \lambda d(x,y)$$ for $$x, y\in Y$$, and $$\lim_{n\to \infty} {\lambda}^n\delta (F(a^n(x)))=0$$ for each $$x\in K$$. The author proves that
(1) If $$Y$$ is complete and $$\Psi(F(a(x)))\subset F(x)$$ for each $$x\in K$$, then $$\lim_{n\to \infty} cl \psi^n\circ F \circ a^n(x)=f(x)$$ exists and $$f$$ is a unique selection of the multifunction $$clF$$ such that $$\psi \circ f \circ a=f$$.
(2) If $$F(x)\subset \psi(F(a(x)))$$ for each $$x\in K$$, then $$F$$ is a single-valued function and $$\psi \circ F \circ a=F$$.

##### MSC:
 39B05 General theory of functional equations and inequalities 39B82 Stability, separation, extension, and related topics for functional equations 54C60 Set-valued maps in general topology 54C65 Selections in general topology
##### Keywords:
set-valued map; selection; inclusion; metric space
Full Text:
##### References:
  Aczél J.: Lectures on Functional Equations and Their Applications. Academic Press, New York (1966) · Zbl 0139.09301  Brzdak 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  Forti G.L.: Hyers–Ulam stability of functional equations in several variables. Aequationes Math. 50, 143–190 (1995) · Zbl 0836.39007 · doi:10.1007/BF01831117  Gajda Z., Ger R.: Subadditive multifunctions and Hyers–Ulam stability. Numer. Math. 80, 281–291 (1987) · Zbl 0639.39014  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  Hyers D.H., Isac G., Rassias Th.M.: Stability of functional equations in several variables. Birkhäuser, Boston (1998) · Zbl 0907.39025  Kuczma M.: An introduction to the theory of functional equations and inequalities. Polish Scientific Publishers and Silesian University, Warszawa (1985) · Zbl 0555.39004  Nikodem, K.: K-convex and K-concave set-valued function. Zeszyty Naukowe Politech. Łódz. Mat. 559 (Rozprawy Nauk. 144) (1989) · Zbl 0712.39027  Nikodem K., Popa D.: On selections of general linear inclusions. Publ. Math. Debrecen 75, 239–249 (2009) · Zbl 1212.39041  Páles Z.: Generalized stability of the Cauchy functional equation. Aequationes Math. 56, 222–232 (1998) · Zbl 0922.39008 · doi:10.1007/s000100050058  Páles Z.: Hyers–Ulam stability of the Cauchy functional equation on square-symmetric grupoids. Publ. Math. Debrecen 58, 651–666 (2001) · Zbl 0980.39022  Piszczek, M.: The properties of functional inclusions and Hyers–Ulam stability. Aequationes Math. (2012). doi: 10.1007/s00010-012-0119-0 · Zbl 1271.39031  Popa D.: A stability result for a general linear inclusion. Nonlinear Funct. Anal. Appl. 3, 405–414 (2004) · Zbl 1067.39043  Popa D.: Functional inclusions on square-symetric grupoid and Hyers–Ulam stability. Math. Inequal. Appl. 7, 419–428 (2004) · Zbl 1058.39026  Popa D.: A property of a functional incusion connected with Hyers–Ulam stability. J. Math. Inequal. 4, 591–598 (2009) · Zbl 1189.39032 · doi:10.7153/jmi-03-57  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  Rådström H.: An embeldding theorem for spaces of convex sets. Proc. Am. Math. Soc. 3, 165–169 (1952) · Zbl 0046.33304 · doi:10.2307/2032477  Smajdor W.: Superadditive set-valued functions. Glas. Mat. 21, 343–348 (1986) · Zbl 0617.26010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.