The fixed point alternative and Hyers-Ulam stability of generalized additive set-valued functional equations
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RESEARCH
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The fixed point alternative and Hyers-Ulam stability of generalized additive set-valued functional equations Sun Young Jang* *
Correspondence: [email protected] Department of Mathematics, University of Ulsan, Ulsan, 680-749, Korea
Abstract We define generalized additive set-valued functional equations, which are related with the following generalized additive functional equations: l–1 ) + f (xl ), f (x1 + · · · + xl ) = (l – 1)f ( x1 +···+x l–1 f
x + ··· + x x + ··· + x + x x + ··· + x 1 l–1 1 l–2 l 2 l + xl + f + xl–1 + · · · + f + x1 l–1 l–1 l–1 = 2 f (x1 ) + f (x2 ) + · · · + f (xl )
for a fixed integer l with l > 1, and they prove the Hyers-Ulam stability of the generalized additive set-valued functional equations by using the fixed point method. MSC: Primary 39B52; 54C60; 91B44 Keywords: Hyers-Ulam stability; generalized additive set-valued functional equation; closed and convex set; cone
1 Introduction and preliminaries After the pioneering papers were written by Aumann [] and Debreu [], set-valued functions in Banach spaces have been developed in the last decades. We can refer to the papers by Arrow and Debreu [], McKenzie [], the monographs by Hindenbrand [], Aubin and Frankowska [], Castaing and Valadier [], Klein and Thompson [] and the survey by Hess []. The theory of set-valued functions has been much related with the control theory and the mathematical economics. Let Y be a Banach space. We define the following: Y : the set of all subsets of Y ; Cb (Y ): the set of all closed bounded subsets of Y ; Cc (Y ): the set of all closed convex subsets of Y ; Ccb (Y ): the set of all closed convex bounded subsets of Y ; Ccc (Y ): the set of all closed compact subsets of Y . We can consider the addition and the scalar multiplication on Y as follows: C + C = x + x : x ∈ C, x ∈ C ,
λC = {λx : x ∈ C},
©2014 Jang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Jang Advances in Difference Equations 2014, 2014:127 http://www.advancesindifferenceequations.com/content/2014/1/127
where C, C ∈ Y and λ ∈ R. Further, if C, C ∈ Cc (Y ), then we denote by C ⊕ C = C + C. We can easily check that λC + λC = λ C + C ,
(λ + μ)C ⊆ λC + μC,
where C, C ∈ Y and λ, μ ∈ R. Furthermore, when C is convex, we obtain (λ + μ)C = λC + μC for all λ, μ ∈ R+ . For a given set C ∈ Y , the distance function d(· , C) and the support function s(· , C) are, respectively, defined by d(x, C) = inf x – y : y ∈ C ,
s x∗ , C = sup x∗ , x : x ∈ C ,
x ∈ Y, x∗ ∈ Y ∗ .
For every pair C, C ∈ Cb (Y ), we define the Hausdorff distance between C and C by h C, C = inf λ > : C ⊆ C + λBY , C ⊆ C + λBY , where BY is the closed unit ball in Y . The following proposition is related with some properties of the Hausdorff dis
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