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Stability of functional equation obtained through a fixed-point alternative in intuitionistic fuzzy normed spaces
Advances in Difference Equations volume 2012, Article number: 141 (2012)
Abstract
In this paper, we determine the stability of a generalized Hyers-Ulam-Rassias-type theorem concerning the additive functional equation in the framework of intuitionistic fuzzy normed spaces through the fixed-point alternative. Further, we prove some stability results of an additive functional equation in this setup through the direct method.
MSC:39B52, 39B82, 46S40.
1 Introduction
The study of the stability problem of functional equations originated from a question of S.M. Ulam [30] concerning the stability of group homomorphisms.
Let be a group and be a metric group with the metric . Given , does there exist a such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all ?
If the answer is affirmative, we would say the equation of homomorphism is stable. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Hyers [5] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers theorem was generalized by Aoki [3] for additive mappings and by Rassias [26] for linear mappings by considering an unbounded Cauchy difference for all and . Following the same approach as Rassias, Gajda [7] gave an affirmative solution of this problem for and also proved that it is possible to solve the Rassias-type theorem for . A further generalization was obtained by Gǎvruta [8], who replaced by a general control function . The paper of Rassias has significantly influenced the development of what we now call the Hyers-Ulam-Rassias stability of functional equations. Since then, several stability problems for various functional equations have been investigated in [1, 2, 6, 9–12, 27, 32]. Quite recently, the stability problem for the Pexiderized quadratic functional equation, Jensen functional equation, cubic functional equation, functional equations associated with inner product spaces, and a mixed type additive-cubic functional equation were considered in [15, 17, 21, 29], and [31], respectively, in the intuitionistic fuzzy normed spaces; while the idea of intuitionistic fuzzy normed space was introduced in [28], and further studied in [18–20, 22–24, 34] to deal with some summability problems.
In 2003, Radu [25] proposed that the fixed-point alternative method is very useful for obtaining the solution of the Ulam problem and obtained the stability of the Cauchy functional equation in Banach spaces through the fixed-point method. Since then, several stability problems of this concept have been established by various authors, e.g., [13, 14, 16, 33] and references therein.
The aim of this paper is to present a relationship between three various disciplines: the theory of fuzzy spaces, the theory of functional equations, and fixed-point theory. We determine the stability of the additive functional equation
in the setting of intuitionistic fuzzy normed spaces by using the fixed-point alternative theorem. Also, we investigate the stability of this functional equation through the direct method.
2 Definitions, notations and preliminary results
In this section, we recall some notations, basic definitions, and preliminary results used in this paper.
A binary operation is said to be a continuous t-norm if it satisfies the following conditions:
-
(a)
* is associative and commutative,
-
(b)
* is continuous,
-
(c)
for all ,
-
(d)
whenever and for each .
A binary operation is said to be a continuous t-conorm if it satisfies the following conditions:
(a′) ♢ is associative and commutative,
(b′) ♢ is continuous,
(c′) for all ,
(d′) whenever and for each .
Using the notions of continuous t-norm and t-conorm, Saadati and Park [28] have recently introduced the concepts of intuitionistic fuzzy normed space and defined convergence and Cauchy sequences in this setting as follows.
Definition 2.1 The five-tuple is said to be an intuitionistic fuzzy normed spaces (for short, IFN-Spaces) if X is a vector space, * is a continuous t-norm, ♢ is a continuous t-conorm, and μ, ν are fuzzy sets on satisfying the following conditions. For every and
-
(i)
,
-
(ii)
,
-
(iii)
if and only if ,
-
(iv)
for each ,
-
(v)
,
-
(vi)
is continuous,
-
(vii)
and ,
-
(viii)
,
-
(ix)
if and only if ,
-
(x)
for each ,
-
(xi)
,
-
(xii)
is continuous,
-
(xiii)
and .
In this case, is called an intuitionistic fuzzy norm. For simplicity in notation, we denote the intuitionistic fuzzy normed spaces by instead of . For example, let be a normed space, and let and for all . For all and every , consider
Then is an intuitionistic fuzzy normed space.
Definition 2.2 Let be an intuitionistic fuzzy normed space. Then a sequence is said to be
-
(i)
convergent to with respect to the intuitionistic fuzzy norm if, for every and , there exists such that and for all . In this case, we write or as .
-
(ii)
Cauchy sequence with respect to the intuitionistic fuzzy norm if, for every and , there exists such that and for all . IFN-space is said to be complete if every Cauchy sequence in is convergent in IFN-space. In this case, is called intuitionistic fuzzy Banach space.
Remark 2.3 Let be a real normed linear space,
for all and . Then if and only if .
Recall the following results related to the concept of fixed point.
Theorem 2.4 (Banach’s contraction principle)
Let be a complete generalized metric space and consider a mapping be a strictly contractive mapping, that is,
for some (Lipschitz constant) . Then
-
(i)
The mapping J has one and only one fixed point ;
-
(ii)
The fixed-point is globally attractive, that is,
for any starting point ;
-
(iii)
One has the following estimation inequalities for all and :
(2.1)(2.2)(2.3)
Theorem 2.5 (The alternative of fixed point [4])
Suppose we are given a complete generalized metric space and a strictly contractive mapping , with Lipschitz constant L. Then, for each given element , either
or
for some natural number . Moreover, if the second alternative holds then
-
(i)
The sequence is convergent to a fixed point of J;
-
(ii)
is the unique fixed point of J in the set ;
-
(iii)
, .
3 Stability of the additive functional equation through the fixed-point alternative
Using the fixed point alternative, here we can prove the stability of the Hyers-Ulam-Rassias-type theorem in IFN-spaces. First, we prove the following lemma which will be used in our main result.
Lemma 3.1 Let X be a linear space, be an IFN-space and be a function. Consider a set and define
for all , and . Then is a complete generalized metric on G.
Proof Let , and . Then, for all and , we have
and
Therefore,
and
for each and . Thus, , which is a triangle inequality for . The rest of the conditions follow directly from the definition. □
Theorem 3.2 Let X be a linear space and f be a mapping from X to an intuitionistic fuzzy Banach space . Suppose that is a function such that
for all and . If holds for some real number α with then there exists a unique additive mapping such that ,
for all and .
Proof Putting and in (3.2). Then for and
Replacing x by , we get
Consider the set and the mapping d defined on by
for all and . It is known that is a complete generalized metric on G by Lemma 3.1. Now we consider the linear mapping such that for all . Let be such that . Then, for all and , we have
Using the hypothesis of the function φ and a mapping J, we obtain
this implies
and similarly
for all and . From above, we conclude that implies . Hence,
for all . Using the hypothesis of the function φ and from (3.4), we have
for all , and . Replacing t by in (3.5), we get
for all , and . It follows that
Using the fixed-point alternative we deduce the existence of a fixed point of J, that is, the existence of a mapping such that
for all . The mapping T is a unique fixed point of J in the set . It follows that T is the unique fixed point of J with the property that there exists such that
for all and . Moreover, we have as which implies
for all . Also implies . This means that (3.3) holds. For all and , write
Letting in (3.6) and using (3.2), we get
Similarly, we obtain
for all and . Thus, the mapping T satisfies (1.1) and so it is additive. □
Corollary 3.3 Let X be a normed linear space and be an intuitionistic fuzzy Banach space. Let θ be a positive real number and r is a real number with . If a mapping satisfies the conditions
for all and , then there exists a unique additive mapping such that ,
for all and .
Proof Taking in Theorem 3.2, for all , and choosing , we get the desired result. □
Theorem 3.4 Let X be a linear space and be a function such that there exists with for all and . Suppose f is a mapping from X to an intuitionistic fuzzy Banach space satisfying (3.2). Then there exists a unique additive mapping such that ,
for all and .
Proof Consider a complete generalized metric space same as in the proof of Theorem 3.2. We define a linear mapping such that
for all . Indeed, for given g and h in G, . Then
for all and . By the given hypothesis and using (3.8), we have
and
for all and . This means that . Thus, for all g and h in G. It follows from (3.2) that
for all and . From the definition of complete generalized metric space, we have . Using the fixed-point alternative, we deduce the existence of a fixed point of J, that is, the existence of a mapping such that for all . Moreover, we have which implies
for all . Also implies . The rest of the proof can be done by the same way as in Theorem 3.2. □
Corollary 3.5 Let X be a normed linear space and be an intuitionistic fuzzy Banach space. Let θ be a positive real number and r is a real number with . If a mapping satisfies the conditions,
for all and , then there exists a unique additive mapping such that ,
for all and .
Proof Taking in Theorem 3.4, for all , and choosing , we get the desired result. □
4 Stability of the additive functional equation through the direct method
In this section, we deal with the stability results concerning the additive functional equation via direct method in intuitionistic fuzzy normed spaces.
Theorem 4.1 Let X be a linear space and be an IFN-space. Suppose that is a function such that for some real number α with
for all and . Let f be a mapping from X to an intuitionistic fuzzy Banach space such that
for all and . Then there exists a unique additive mapping such that
for all and .
Proof Put and in (4.2). Then for all and
Replacing x by in (4.4) and using (4.1), we obtain
and
for all , and an integer . By replacing , we get
It follows from
and (4.5) that
for all , and , where and . Replacing x by in the last inequalities, we have
whence
for all , , and . Hence,
Since , we have . This shows that is a Cauchy sequence in an intuitionistic fuzzy Banach space and so it converges to some point . Thus, we define a mapping such that
Hence, for all and , we have
Moreover, if we put in (4.6), we get
for all , and . Therefore,
and
for all , and . Letting in the above inequalities, we obtain
Hence, T satisfies (4.3). Let . Then
and by using (4.2)
Letting in (4.7) and (4.8), we get
Similarly, we obtain
for all and . This means that T satisfies (1.1) and so it is additive. To prove the uniqueness of T, assume that S be another additive mapping from X into Y, which satisfies (4.3). For , clearly and for all n. It follows from (4.3) that
and similarly
We see that the right-hand side of (4.9) and (4.10) tending to 1 and 0, respectively, as . Therefore, and for all and . Hence, . □
Corollary 4.2 Let X be a normed linear space and be an intuitionistic fuzzy Banach space. Let θ be a positive real number and r is a real number with . If a mapping satisfies the conditions,
for all and , then there exists a unique additive mapping such that
for all and .
Proof Taking in Theorem 4.1, for all , and choosing , we get the desired result. □
Theorem 4.3 Let X be a linear space and be an IFN-space. Suppose that is a function such that for some real number α with
for all and . Let an intuitionistic fuzzy Banach space and a map satisfies (4.2). Then there exists a unique additive mapping such that
for all and .
Proof From (4.4), it is easy to see that
for all and . Replacing x by , we get
It follows that, for all and , we have
Proceeding the same lines as in the proof of Theorem 4.1, we get
for all , and . Thus,
Rest of the proof can be done by the same way as in Theorem 4.1. □
Corollary 4.4 Let X be a normed linear space and be an intuitionistic fuzzy Banach space. Let θ be a positive real number and r is a real number with . If a mapping satisfies the conditions
for all and , then there exists a unique additive mapping such that
for all and .
Proof Taking in Theorem 4.3, for all , and choosing , we get the desired result. □
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The authors would like to thank the anonymous reviewers for their valuable comments.
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Mohiuddine, S., Alghamdi, M.A. Stability of functional equation obtained through a fixed-point alternative in intuitionistic fuzzy normed spaces. Adv Differ Equ 2012, 141 (2012). https://doi.org/10.1186/1687-1847-2012-141
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DOI: https://doi.org/10.1186/1687-1847-2012-141
Keywords
- t-norm
- t-conorm
- additive functional equation
- intuitionistic fuzzy normed space
- fixed point