Hyers-Ulam stability of a generalized additive set-valued functional equation
© Jang et al.; licensee Springer. 2013
Received: 9 November 2012
Accepted: 24 February 2013
Published: 13 March 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.
where and . Further, if , then we denote .
Furthermore, when C is convex, we obtain for all .
where is the closed unit ball in Y.
The following proposition reveals some properties of the Hausdorff distance.
Lemma 1.2 
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
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 .
for all .
for all integers n, m with . It follows from (2) and (6) that is a Cauchy sequence in .
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).
which tends to zero as for all . So, we can conclude that for all , which proves the uniqueness of A, as desired. □
for all .
for all . □
for all .
for all .
The rest of the proof is similar to the proof of Theorem 2.2. □
for all .
for all . □
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).
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