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Hyers-Ulam stability of a generalized additive set-valued functional equation
Journal of Inequalities and Applications volume 2013, Article number: 101 (2013)
In this paper, we define a generalized additive set-valued functional equation, which is related to the following generalized additive functional equation:
for a fixed integer l with , and prove the Hyers-Ulam stability of the generalized additive set-valued functional equation.
MSC:39B52, 54C60, 91B44.
1 Introduction and preliminaries
The theory of set-valued functions has been much related to the control theory and the mathematical economics. After the pioneering papers 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 stability problem of functional equations originated from a question of Ulam  concerning the stability of group homomorphisms. Hyers  gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki  for additive mappings and by Th.M. Rassias  for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by Găvruta  by replacing the unbounded Cauchy difference with a general control function in the spirit of Th.M. Rassias’ approach. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [15–17]).
Let Y be a Banach space. We define the following:
: the set of all subsets of Y;
: the set of all closed bounded subsets of Y;
: the set of all closed convex subsets of Y;
: the set of all closed convex bounded subsets of Y.
On we consider the addition and the scalar multiplication as follows:
where and . Further, if , then we denote .
It is easy to check that
Furthermore, when C is convex, we obtain for all .
For a given set , the distance function and the support function are respectively defined by
For every pair , we define the Hausdorff distance between C and by
where is the closed unit ball in Y.
The following proposition reveals some properties of the Hausdorff distance.
Proposition 1.1 For every and , the following properties hold:
Lemma 1.2 
Let be the Banach space of continuous real-valued functions on endowed with the uniform norm . Then the mapping , given by , satisfies the following properties:
is closed in
for all and all .
Let be a set-valued function from a complete finite measure space into . Then f is Debreu integrable if the composition is Bochner integrable (see ). In this case, the Debreu integral of f in Ω is the unique element such that is the Bochner integral of . The set of Debreu integrable functions from Ω to will be denoted by . Furthermore, on , we define for all . Then we obtain that is an abelian semigroup.
In this paper, we define a generalized additive set-valued functional equation and prove the Hyers-Ulam stability of the generalized additive set-valued functional equation.
Throughout this paper, let X be a real vector space and Y be a Banach space.
2 Stability of a generalized additive set-valued functional equation
Definition 2.1 Let . The generalized additive set-valued functional equation is defined by
for all . Every solution of the generalized additive set-valued functional equation is called a generalized additive set-valued mapping.
Note that there are some examples in .
Theorem 2.2 Let be a function such that
for all . Suppose that is a mapping satisfying
for all . Then there exists a unique generalized additive set-valued mapping such that
for all .
Proof Let in (3). Since is convex, we get
and if we replace x by , in (5), then we obtain
for all integers n, m with . It follows from (2) and (6) that is a Cauchy sequence in .
Let for each . Then we claim that A is a generalized additive set-valued mapping. Note that
Since , we have
which tends to zero as . So, A is a generalized additive set-valued mapping. Letting and passing the limit in (6), we get the inequality (4).
Now, let be another generalized additive set-valued mapping satisfying (1) and (4). So,
which tends to zero as for all . So, we can conclude that for all , which proves the uniqueness of A, as desired. □
Corollary 2.3 Let and be real numbers, and let X be a real normed space. Suppose that is a mapping satisfying
for all . Then there exists a unique generalized additive set-valued mapping satisfying
for all .
The proof follows from Theorem 2.2 by taking
for all . □
Theorem 2.4 Let be a function such that
for all . Suppose that is a mapping satisfying (3). Then there exists a unique generalized additive set-valued mapping such that
for all .
Proof It follows from (5) that
for all .
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 2.5 Let and be real numbers, and let X be a real normed space. Suppose that is a mapping satisfying (7). Then there exists a unique generalized additive set-valued mapping satisfying
for all .
The proof follows from Theorem 2.4 by taking
for all . □
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SYJ was supported by the 2012 Research Fund of University of Ulsan and had written this paper during visiting the Research Institute of Mathematics, Seoul National University. CP was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299).
The authors declare that they have no competing interests.
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
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Jang, S.Y., Park, C. & Cho, Y. Hyers-Ulam stability of a generalized additive set-valued functional equation. J Inequal Appl 2013, 101 (2013). https://doi.org/10.1186/1029-242X-2013-101
- Hyers-Ulam stability
- generalized additive set-valued functional equation
- closed and convex set