Abstract
In intuitionistic fuzzy normed spaces, we investigate some stability results for the functional equation which is said to be a functional equation associated with inner products space.
1. Introduction and Preliminaries
The aim of this article is to prove an intuitionistic fuzzy version of the Hyers-Ulam-Rassias stability for the functional equation: which is said to be a functional equation associated with inner product spaces. It was shown by Rassias [1] that the norm defined over a real vector space is induced by an inner product if and only if for a fixed integer it follows for all . Interesting new results concerning functional equations associated with inner product spaces have recently been obtained by Park et al. [2, 3] and Najati and Rassias [4] as well as for the fuzzy stability of a functional equation associated with inner product spaces [5].
Stability problem of a functional equation was first posed by Ulam [6] which was answered by Hyers [7] on approximately additive mappings and then generalized by Aoki [8] and Rassias [9] for additive mappings and linear mappings, respectively. Later there have been proved several new results on stability of various classes of functional equations in the Hyers-Ulam sense (cf. the following books and papers [10–18] and the references cited therein), as well as various fuzzy stability results concerning Cauchy, Jensen, quadratic and cubic functional equations (cf. [19–22]). Furthermore some stability results concerning Jensen, cubic, mixed-type additive and cubic functional equations were investigated (cf. [23–26]) in the spirit of intuitionistic fuzzy normed spaces, while the idea of intuitionistic fuzzy normed space was introduced in [27] and further studied in [28–35].
In this section, we recall some notations and basic definitions used in this paper as follows.
Definition 1.1 (cf. [36]). A binary operation is said to be a continuous -norm if it satisfies the following conditions:(a) is commutative and associative, (b) is continuous,(c) for all , (d) whenever and for each .
Definition 1.2 (cf. [36]). A binary operation is said to be a continuous -conorm if it satisfies the following conditions:(a’) is commutative and associative, (b’) is continuous,(c’) for all , (d’) whenever and for each .
Using the notions of continuous -norm and continuous -conorm, Saadati and Park [27] have recently introduced the concept of intuitionistic fuzzy normed spaces as follows.
Definition 1.3. The five-tuple is said to be an intuitionistic fuzzy normed space (for short, IFNS) if is a vector space, is a continuous -norm, is a continuous -conorm, and 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.
Example 1.4 (cf. [37]). Letbe a normed space, andfor all . For all and everyand, consider Then is an IFNS.
The concepts of convergence and Cauchy sequences in an intuitionistic fuzzy normed space are studied in [27].
Let be an IFNS. Then, a sequence is said to be intuitionistic fuzzy convergent to if, for every and , there exists such that and for all . In this case we write . The sequence is said to be intuitionistic fuzzy Cauchy sequence if, for every and , there exists such that and for all . is said to be complete if every intuitionistic fuzzy Cauchy sequence in is intuitionistic fuzzy convergent in .
2. Intuitionistic Fuzzy Stability
Throughout this section, assume that , , and are linear space, IFNS, and intuitionistic fuzzy Banach space, respectively. For convenience, we use the following abbreviation for a given function :
We begin with the Hyers-Ulam-Rassias type theorem in IFNS for the functional (1.1) which is said to be a functional equation associated with inner product spaces.
Theorem 2.1. Let be a function such that for some real number with . Suppose that an even function with satisfies the inequality for all and all . Then there exists a unique quadratic function such that for all and , where
Proof. Put in (2.2), and, using the evenness of , we obtain
for all and . Interchanging with in (2.5) and using the evenness of , we get
for all and . It follows from (2.5) and (2.6) that
for all and . Putting in (2.2) and using the evenness of , we obtain
for all and . Hence, we obtain from (2.7) and (2.8) that
for all and . So
for all and . Putting in (2.2), we obtain
for all and . It follows from (2.10) and (2.11) that
for all and . Letting in (2.8) and replacing by in the obtained inequality, we get
for all and . It follows from (2.10), (2.11), (2.12), and (2.13) that
for all and . Applying (2.12) and (2.14), we obtain
for all and . Setting in (2.2), we obtain
for all and . It follows from (2.15) and (2.16) that
It follows that
Define
Then, by our assumption,
Replacing by in (2.18) and applying (2.20), we get
Thus for each , we have
where . Let and be given. Since and , there exists some such that and . Since , there is some such that for each . It follows that
for all . This shows that the sequence is Cauchy in . Since is intuitionistic fuzzy Banach space, converges to some point . Thus, we can define a mapping such that . Moreover, if we put in (2.22), we get
Thus,
Now, we will show that is quadratic. Setting and in (2.2), we obtain
for all and . Letting in (2.26), we obtain
for all and all . This means that satisfies the functional (1.1) and so it is quadratic (see Lemma 2.2 of [4]).
Next, we approximate the difference between and in intuitionistic fuzzy sense. By (2.25), we have
for every and large enough . To prove the uniqueness of , assume that is another quadratic mapping from to , which satisfies the required inequality. Then, for each and ,
Since and are quadratic, we have
for all and . Since and , we
Therefore and for all and . Hence, for all . This completes the proof of the theorem.
Theorem 2.2. Let be a function such that for some real number with . Suppose that an odd function satisfies the inequality for all and all . Then there exists a unique additive function such that for all and , where
Proof. Put in (2.32) and using the oddness of , we obtain
for all and . Interchanging with in (2.35) and using the oddness of , we get
for all and . It follows from (2.35) and (2.36) that
for all and . Setting in (2.32) and using the oddness of , we get
for all and . It follows from (2.37) and (2.38) that
for all and . Putting in (2.32), we obtain
for all and . It follows from (2.39) and (2.40) that
for all and . Replacing and by and in (2.41); respectively, we have
It follows that
Define
Then by the assumption
Replacing by in (2.43) and using (2.45), we obtain
Thus, for each , we have
where , . Let and be given. Since and , there exists some such that and . Since , there is some such that for each . It follows that
for all . This shows that the sequence is Cauchy in . Since is intuitionistic fuzzy Banach space, converges to some point . Thus, we can define a mapping such that . Moreover, if we put in (2.47), we get
Thus,
Next we will show that is additive. Putting and in (2.32), we obtain
for all and . Letting in (2.51), we obtain
for all and all . This means that satisfies the functional (1.1), and so it is additive (see Lemma 2.1 of [4]).
Now, we approximate the difference between and in intuitionistic fuzzy sense. For every , and sufficiently large , by (2.50), we have
To prove the uniqueness of , assume that is another additive mapping from to , which satisfies the required inequality. Then, for each and ,
Therefore, by the additivity of and , we have
for all and . Since and , we get
Therefore, and for all and . Hence, for all . This completes the proof of the theorem.
Theorem 2.3. Let be a function such that for some real number with . Suppose that a function with satisfies the inequality for all and all . Then there exists a unique quadratic function and a unique additive function such that for all and , where
Proof. Passing to the even part and odd part of , we deduce from (2.57) that On the other hand, Applying the proofs of Theorems 2.1 and 2.2, we get a unique quadratic function and a unique additive function satisfying Also, Therefore, This completes the proof of the theorem.
Acknowledgment
The authors are very grateful to the referees for their helpful comments and suggestions.