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HyersUlam stability of a generalized additive setvalued functional equation
Journal of Inequalities and Applications volume 2013, Article number: 101 (2013)
Abstract
In this paper, we define a generalized additive setvalued functional equation, which is related to the following generalized additive functional equation:
for a fixed integer l with $l>1$, and prove the HyersUlam stability of the generalized additive setvalued functional equation.
MSC:39B52, 54C60, 91B44.
1 Introduction and preliminaries
The theory of setvalued functions has been much related to the control theory and the mathematical economics. After the pioneering papers written by Aumann [1] and Debreu [2], setvalued functions in Banach spaces have been developed in the last decades. We can refer to the papers by Arrow and Debreu [3], McKenzie [4], the monographs by Hindenbrand [5], Aubin and Frankowska [6], Castaing and Valadier [7], Klein and Thompson [8] and the survey by Hess [9].
The stability problem of functional equations originated from a question of Ulam [10] concerning the stability of group homomorphisms. Hyers [11] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [12] for additive mappings and by Th.M. Rassias [13] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by Găvruta [14] 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:
${2}^{Y}$: the set of all subsets of Y;
${C}_{b}(Y)$: the set of all closed bounded subsets of Y;
${C}_{c}(Y)$: the set of all closed convex subsets of Y;
${C}_{cb}(Y)$: the set of all closed convex bounded subsets of Y.
On ${2}^{Y}$ we consider the addition and the scalar multiplication as follows:
where $C,{C}^{\prime}\in {2}^{Y}$ and $\lambda \in \mathbb{R}$. Further, if $C,{C}^{\prime}\in {C}_{c}(Y)$, then we denote $C\oplus {C}^{\prime}=\overline{C+{C}^{\prime}}$.
It is easy to check that
Furthermore, when C is convex, we obtain $(\lambda +\mu )C=\lambda C+\mu C$ for all $\lambda ,\mu \in {\mathbb{R}}^{+}$.
For a given set $C\in {2}^{Y}$, the distance function $d(\cdot ,C)$ and the support function $s(\cdot ,C)$ are respectively defined by
For every pair $C,{C}^{\prime}\in {C}_{b}(Y)$, we define the Hausdorff distance between C and ${C}^{\prime}$ by
where ${B}_{Y}$ is the closed unit ball in Y.
The following proposition reveals some properties of the Hausdorff distance.
Proposition 1.1 For every $C,{C}^{\prime},K,{K}^{\prime}\in {C}_{cb}(Y)$ and $\lambda >0$, the following properties hold:

(a)
$h(C\oplus {C}^{\prime},K\oplus {K}^{\prime})\le h(C,K)+h({C}^{\prime},{K}^{\prime})$;

(b)
$h(\lambda C,\lambda K)=\lambda h(C,K)$.
Let $({C}_{cb}(Y),\oplus ,h)$ be endowed with the Hausdorff distance h. Since Y is a Banach space, $({C}_{cb}(Y),\oplus ,h)$ is a complete metric semigroup (see [7]). Debreu [2] proved that $({C}_{cb}(Y),\oplus ,h)$ is isometrically embedded in a Banach space as follows.
Lemma 1.2 [2]
Let $C({B}_{{Y}^{\ast}})$ be the Banach space of continuous realvalued functions on ${B}_{{Y}^{\ast}}$ endowed with the uniform norm ${\parallel \cdot \parallel}_{u}$. Then the mapping $j:({C}_{cb}(Y),\oplus ,h)\to C({B}_{{Y}^{\ast}})$, given by $j(A)=s(\cdot ,A)$, satisfies the following properties:

(a)
$j(A\oplus B)=j(A)+j(B)$;

(b)
$j(\lambda A)=\lambda j(A)$;

(c)
$h(A,B)={\parallel j(A)j(B)\parallel}_{u}$;

(d)
$j({C}_{cb}(Y))$ is closed in $C({B}_{{Y}^{\ast}})$
for all $A,B\in {C}_{cb}(Y)$ and all $\lambda \ge 0$.
Let $f:\mathrm{\Omega}\to ({C}_{cb}(Y),h)$ be a setvalued function from a complete finite measure space $(\mathrm{\Omega},\mathrm{\Sigma},\nu )$ into ${C}_{cb}(Y)$. Then f is Debreu integrable if the composition $j\circ f$ is Bochner integrable (see [18]). In this case, the Debreu integral of f in Ω is the unique element $(D){\int}_{\mathrm{\Omega}}f\phantom{\rule{0.2em}{0ex}}d\nu \in {C}_{cb}(Y)$ such that $j((D){\int}_{\mathrm{\Omega}}f\phantom{\rule{0.2em}{0ex}}d\nu )$ is the Bochner integral of $j\circ f$. The set of Debreu integrable functions from Ω to ${C}_{cb}(Y)$ will be denoted by $D(\mathrm{\Omega},{C}_{cb}(Y))$. Furthermore, on $D(\mathrm{\Omega},{C}_{cb}(Y))$, we define $(f+g)(\omega )=f(\omega )\oplus g(\omega )$ for all $f,g\in D(\mathrm{\Omega},{C}_{cb}(Y))$. Then we obtain that $((\mathrm{\Omega},{C}_{cb}(Y)),+)$ is an abelian semigroup.
Setvalued functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [19–27]).
In this paper, we define a generalized additive setvalued functional equation and prove the HyersUlam stability of the generalized additive setvalued functional equation.
Throughout this paper, let X be a real vector space and Y be a Banach space.
2 Stability of a generalized additive setvalued functional equation
Definition 2.1 Let $f:X\to {C}_{cb}(Y)$. The generalized additive setvalued functional equation is defined by
for all ${x}_{1},\dots ,{x}_{l}\in X$. Every solution of the generalized additive setvalued functional equation is called a generalized additive setvalued mapping.
Note that there are some examples in [28].
Theorem 2.2 Let $\phi :{X}^{l}\to [0,\mathrm{\infty})$ be a function such that
for all ${x}_{1},\dots ,{x}_{l}\in X$. Suppose that $f:X\to ({C}_{cb}(Y),h)$ is a mapping satisfying
for all ${x}_{1},\dots ,{x}_{l}\in X$. Then there exists a unique generalized additive setvalued mapping $A:X\to ({C}_{cb}(Y),h)$ such that
for all $x\in X$.
Proof Let ${x}_{1}=\cdots ={x}_{l}=x$ in (3). Since $f(x)$ is convex, we get
and if we replace x by ${l}^{n}x$, $n\in \mathbb{N}$ in (5), then we obtain
and
So,
for all integers n, m with $n\ge m$. It follows from (2) and (6) that $\{\frac{f({l}^{n}x)}{{l}^{n}}\}$ is a Cauchy sequence in $({C}_{cb}(Y),h)$.
Let $A(x)={lim}_{n\to \mathrm{\infty}}\frac{f({l}^{n}x)}{{l}^{n}}$ for each $x\in X$. Then we claim that A is a generalized additive setvalued mapping. Note that
Since $h(A\oplus B,C\oplus D)\le h(A,C)+h(B,D)$, we have
which tends to zero as $n\to \mathrm{\infty}$. So, A is a generalized additive setvalued mapping. Letting $m=0$ and passing the limit $m\to \mathrm{\infty}$ in (6), we get the inequality (4).
Now, let $T:X\to ({C}_{cb}(Y),h)$ be another generalized additive setvalued mapping satisfying (1) and (4). So,
which tends to zero as $n\to \mathrm{\infty}$ for all $x\in X$. So, we can conclude that $A(x)=T(x)$ for all $x\in X$, which proves the uniqueness of A, as desired. □
Corollary 2.3 Let $1>p>0$ and $\theta \ge 0$ be real numbers, and let X be a real normed space. Suppose that $f:X\to ({C}_{cb}(Y),h)$ is a mapping satisfying
for all ${x}_{1},\dots ,{x}_{l}\in X$. Then there exists a unique generalized additive setvalued mapping $A:X\to Y$ satisfying
for all $x\in X$.
Proof
The proof follows from Theorem 2.2 by taking
for all ${x}_{1},\dots ,{x}_{l}\in X$. □
Theorem 2.4 Let $\phi :{X}^{l}\to [0,\mathrm{\infty})$ be a function such that
for all ${x}_{1},\dots ,{x}_{l}\in X$. Suppose that $f:X\to ({C}_{cb}(Y),h)$ is a mapping satisfying (3). Then there exists a unique generalized additive setvalued mapping $A:X\to ({C}_{cb}(Y),h)$ such that
for all $x\in X$.
Proof It follows from (5) that
for all $x\in X$.
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 2.5 Let $p>1$ and $\theta \ge 0$ be real numbers, and let X be a real normed space. Suppose that $f:X\to ({C}_{cb}(Y),h)$ is a mapping satisfying (7). Then there exists a unique generalized additive setvalued mapping $A:X\to Y$ satisfying
for all $x\in X$.
Proof
The proof follows from Theorem 2.4 by taking
for all ${x}_{1},\dots ,{x}_{l}\in X$. □
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Acknowledgements
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 (NRF2012R1A1A2004299).
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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. HyersUlam stability of a generalized additive setvalued functional equation. J Inequal Appl 2013, 101 (2013). https://doi.org/10.1186/1029242X2013101
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Keywords
 HyersUlam stability
 generalized additive setvalued functional equation
 closed and convex set
 cone