- Research Article
- Open access
- Published:
Fuzzy Stability of Quadratic Functional Equations
Advances in Difference Equations volume 2010, Article number: 412160 (2010)
Abstract
The fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces have been investigated by Moslehian et al. In this paper, we prove the generalized Hyers-Ulam stability of the following quadratic functional equations and
in fuzzy Banach spaces.
1. Introduction and Preliminaries
Katsaras [1] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [2–4]. In particular, Bag and Samanta [5], following Cheng and Mordeson [6], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [7]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [8].
We use the definition of fuzzy normed spaces given in [5, 9, 10] to investigate a fuzzy version of the generalized Hyers-Ulam stability for the quadratic functional equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ2_HTML.gif)
in the fuzzy normed vector space setting, where are nonzero real numbers with
.
Definition 1.1 (see [5, 9, 10]).
Let be a real vector space. A function
is called a fuzzy norm on
if, for all
and all
,
for
,
if and only if
for all
,
if
,
,
is a nondecreasing function of
and
,
for ,
is continuous on
.
The pair is called a fuzzy normed vector space.
The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [9, 10].
Definition 1.2 (see [5, 9, 10]).
Let be a fuzzy normed vector space. A sequence
in
is said to be convergent or converges if there exists an
such that
for all
. In this case,
is called the limit of the sequence
and we denote it by
-
.
Definition 1.3 (see [5, 9, 10]).
Let be a fuzzy normed vector space. A sequence
in
is called Cauchy if for each
and each
there exists an
such that, for all
and all
, we have
.
It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.
We say that a mapping between fuzzy normed vector spaces
and
is continuous at a point
if, for each sequence
converging to
in
, the sequence
converges to
. If
is continuous at each
, then
is said to be continuous on
(see [8]).
The stability problem of functional equations is originated from a question of Ulam [11] concerning the stability of group homomorphisms. Hyers [12] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [13] for additive mappings and by Th. M. Rassias [14] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [14] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [15] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.
A square norm on an inner product space satisfies the parallelogram equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ3_HTML.gif)
The functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ4_HTML.gif)
is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [16] for mappings , where
is a normed space and
is a Banach space. Cholewa [17] noticed that the theorem of Skof is still true if the relevant domain
is replaced by an Abelian group. In [18], Czerwik proved the generalized Hyers-Ulam stability of the quadratic functional equation. During the last two decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [19–31]).
This paper is organized as follows. In Section 2, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.1) in fuzzy Banach spaces. In Section 3, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.2) in fuzzy Banach spaces.
Throughout this paper, assume that is a vector space and that
is a fuzzy Banach space. Let
be nonzero real numbers with
.
2. Generalized Hyers-Ulam Stability of the Quadratic Functional Equation (1.1)
In this section, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.1) in fuzzy Banach spaces.
Theorem 2.1.
Let be a function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ5_HTML.gif)
for all . Let
be a mapping with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ6_HTML.gif)
uniformly on . Then
-
exists for each
and defines a quadratic mapping
such that if for some
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ7_HTML.gif)
for all , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ8_HTML.gif)
for all .
Furthermore, the quadratic mapping is a unique mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ9_HTML.gif)
uniformly on .
Proof.
For a given , by (2.2), we can find some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ10_HTML.gif)
for all . By induction on
, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ11_HTML.gif)
for all , all
and all
.
Letting in (2.6), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ12_HTML.gif)
for all and all
. So we get (2.7) for
.
Assume that (2.7) holds for . Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ13_HTML.gif)
This completes the induction argument. Letting and replacing
and
by
and
in (2.7), respectively, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ14_HTML.gif)
for all integers .
It follows from (2.1) and the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ15_HTML.gif)
that for a given there is an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ16_HTML.gif)
for all and
. Now we deduce from (2.10) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ17_HTML.gif)
for all and all
. Thus the sequence
is Cauchy in
. Since
is a fuzzy Banach space, the sequence
converges to some
. So we can define a mapping
by
-
; namely, for each
and
,
.
Let . Fix
and
. Since
, there is an
such that
for all
. Hence for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ18_HTML.gif)
The first four terms on the right-hand side of the above inequality tend to 1 as , and the fifth term is greater than
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ19_HTML.gif)
which is greater than or equal to . Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ20_HTML.gif)
for all . Since
for all
, by
,
for all
. Thus the mapping
is quadratic, that is,
for all
.
Now let, for some positive and
, (2.3) hold. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ21_HTML.gif)
for all . Let
. By the same reasoning as in the beginning of the proof, one can deduce from (2.3) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ22_HTML.gif)
for all positive integers . Let
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ23_HTML.gif)
Combining (2.18) and (2.19) and the fact that , we observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ24_HTML.gif)
for large enough . Thanks to the continuity of the function
, we see that
. Letting
, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ25_HTML.gif)
To end the proof, it remains to prove the uniqueness assertion. Let be another quadratic mapping satisfying (2.5). Fix
. Given that
, by (2.5) for
and
, we can find some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ26_HTML.gif)
for all and all
. Fix some
and find some integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ27_HTML.gif)
for all . Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ28_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ29_HTML.gif)
It follows that for all
. Thus
for all
.
Corollary 2.2.
Let and let
be a real number with
. Let
be a mapping with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ30_HTML.gif)
uniformly on . Then
-
exists for each
and defines a quadratic mapping
such that if for some
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ31_HTML.gif)
for all , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ32_HTML.gif)
for all .
Furthermore, the quadratic mapping is a unique mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ33_HTML.gif)
uniformly on .
Proof.
Define and apply Theorem 2.1 to get the result.
Similarly, we can obtain the following. We will omit the proof.
Theorem 2.3.
Let be a function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ34_HTML.gif)
for all . Let
be a mapping satisfying (2.2) and
. Then
-
exists for each
and defines a quadratic mapping
such that if for some
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ35_HTML.gif)
for all , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ36_HTML.gif)
for all .
Furthermore, the quadratic mapping is a unique mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ37_HTML.gif)
uniformly on .
Corollary 2.4.
Let and let
be a real number with
. Let
be a mapping satisfying (2.26) and
. Then
-
exists for each
and defines a quadratic mapping
such that if for some
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ38_HTML.gif)
for all , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ39_HTML.gif)
for all .
Furthermore, the quadratic mapping is a unique mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ40_HTML.gif)
uniformly on .
Proof.
Define and apply Theorem 2.3 to get the result.
3. Generalized Hyers-Ulam Stability of the Quadratic Functional Equation (1.2)
In this section, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.2) in fuzzy Banach spaces.
Lemma 3.1.
Let and
be real vector spaces. If a mapping
satisfies
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ41_HTML.gif)
for all , then the mapping
is quadratic, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ42_HTML.gif)
for all .
Proof.
Assume that satisfies (3.1).
Letting in (3.1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ43_HTML.gif)
for all .
Letting in (3.1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ44_HTML.gif)
for all . Replacing
by
in (3.4), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ45_HTML.gif)
for all . It follows from (3.4) and (3.5) that
for all
. So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ46_HTML.gif)
for all . Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ47_HTML.gif)
for all . Replacing
and
by
and
in (3.7), respectively, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ48_HTML.gif)
for all , as desired.
Theorem 3.2.
Let be a function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ49_HTML.gif)
for all . Let
be a mapping with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ50_HTML.gif)
uniformly on . Then
-
exists for each
and defines a quadratic mapping
such that if for some
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ51_HTML.gif)
for all , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ52_HTML.gif)
for all .
Furthermore, the quadratic mapping is a unique mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ53_HTML.gif)
uniformly on .
Proof.
For a given , by (3.10), we can find some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ54_HTML.gif)
for all . By induction on
, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ55_HTML.gif)
for all , all
, and all
.
Letting in (3.14), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ56_HTML.gif)
for all and all
. So we get (3.15) for
.
Assume that (3.15) holds for . Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ57_HTML.gif)
This completes the induction argument. Letting and replacing
and
by
and
in (3.15), respectively, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ58_HTML.gif)
for all integers .
It follows from (3.9) and the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ59_HTML.gif)
that for a given there is an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ60_HTML.gif)
for all and
. Now we deduce from (3.18) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ61_HTML.gif)
for each and all
. Thus the sequence
is Cauchy in
. Since
is a fuzzy Banach space, the sequence
converges to some
. So we can define a mapping
by
-
; namely, for each
and
,
.
Let . Fix
and
. Since
, there is an
such that
for all
. Hence for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ62_HTML.gif)
The first four terms on the right-hand side of the above inequality tend to 1 as , and the fifth term is greater than
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ63_HTML.gif)
which is greater than or equal to . Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ64_HTML.gif)
for all . Since
for all
, by
,
for all
. By Lemma 3.1, the mapping
is quadratic.
Now let, for some positive and
, (3.18) hold. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ65_HTML.gif)
for all . Let
. By the same reasoning as in the beginning of the proof, one can deduce from (3.18) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ66_HTML.gif)
for all positive integers . Let
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ67_HTML.gif)
Combining (3.26) and (3.27) and the fact that , we observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ68_HTML.gif)
for large enough . Thanks to the continuity of the function
, we see that
. Letting
, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ69_HTML.gif)
To end the proof, it remains to prove the uniqueness assertion. Let be another quadratic mapping satisfying (3.1) and (3.13). Fix
. Given that
, by (3.13) for
and
, we can find some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ70_HTML.gif)
for all and all
. Fix some
and find some integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ71_HTML.gif)
for all . Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ72_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ73_HTML.gif)
It follows that for all
. Thus
for all
.
Corollary 3.3.
Let and let
be a real number with
if
and with
if
. Let
be a mapping with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ74_HTML.gif)
uniformly on . Then
-
exists for each
and defines a quadratic mapping
such that if for some
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ75_HTML.gif)
for all , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ76_HTML.gif)
for all .
Furthermore, the quadratic mapping is a unique mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F412160/MediaObjects/13662_2010_Article_1287_Equ77_HTML.gif)
uniformly on .
Proof.
Define and apply Theorem 3.2 to get the result.
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Lee, J., Jang, SY., Park, C. et al. Fuzzy Stability of Quadratic Functional Equations. Adv Differ Equ 2010, 412160 (2010). https://doi.org/10.1155/2010/412160
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DOI: https://doi.org/10.1155/2010/412160