Browder's Fixed Point Theorem and Some Interesting Results in Intuitionistic Fuzzy Normed Spaces

Abstract

We define and study Browder's fixed point theorem and relation between an intuitionistic fuzzy convex normed space and a strong intuitionistic fuzzy uniformly convex normed space. Also, we give an example to show that uniformly convex normed space does not imply strongly intuitionistic fuzzy uniformly convex.

1. Introduction

In recent years, the fuzzy theory has emerged as the most active area of research in many branches of mathematics and engineering. Fuzzy set theory is a powerful hand set for modelling uncertainty and vagueness in various problems arising the field of science and engineering. Now a large number of research papers appear by using the concept of fuzzy set/numbers, and fuzzification of many classical theories has also been made. It has also very useful applications in various fields, for example, nonlinear operator [1], stability problem [2, 3], and so forth. The fuzzy topology [4–8] proves to be a very useful tool to deal with such situations where the use of classical theories breaks down. One of the most important problems in fuzzy topology is to obtain an appropriate concept of an intuitionistic fuzzy metric space and an intuitionistic fuzzy normed space. These problems have been investigated by Park [9] and Saadati and Park [10], respectively, and they introduced and studied a notion of an intuitionistic fuzzy normed space. The topic of fuzzy topology has important applications as quantum particle physics. On the other hand, these problems are also important in modified fuzzy spaces [11–14].

There are many situations where the norm of a vector is not possible to find and the concept of intuitionistic fuzzy norm [10, 15–17] seems to be more suitable in such cases, that is, we can deal with such situations by modelling the inexactness through the intuitionistic fuzzy norm.

Schauder [18] introduced the fixed point theorem, and since then several generalizations of this concept have been investigated by various authors, namely, Kirk [19], Baillon [20], Browder [21, 22] and many others. Recently, fuzzy version of various fixed point theorems was discussed in [18, 23–28] and also its relations were investigated in [7, 29].

Quite recently the concepts of -intuitionistic fuzzy compact set and strongly intuitionistic fuzzy uniformly convex normed space are studied, and Schauder fixed point theorem in intuitionistic fuzzy normed space is proved in [30]. As a consequence of Theorem   4.1 [30] and Browder's theorems [26, 27] in crisp normed linear space we have Browder's theorems in intuitionistic fuzzy normed space. Also we give relation between an intuitionistic fuzzy uniformly convex normed space and a strongly intuitionistic fuzzy uniformly convex normed space. Furthermore, we construct an example to show that intuitionistic fuzzy uniformly convex normed space does not imply strongly intuitionistic fuzzy uniformly convex.

Definition 1.1.

A binary operation is said to be a continuous -norm if it satisfies the following conditions:

(a) is associative and commutative; (b) is continuous; (c) for all ; (d) whenever and for each .

Definition 1.2.

A binary operation is said to be a continuous -conorm if it satisfies the following conditions:

is associative and commutative; is continuous; for all ; whenever and for each . Using the notions of continuous -norm and -conorm, Saadati and Park [10] have recently introduced the concept of intuitionistic fuzzy normed space as follows.

Definition 1.3.

The five-tuple is said to be intuitionistic fuzzy normed spaces (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 and (xiii) and . In this case is called an intuitionistic fuzzy norm.

The concepts of convergence and Cauchy sequences in an intuitionistic fuzzy normed space are studied in [10].

Definition 1.4.

Let be an IFNS. Then, a sequence is said to be intuitionistic fuzzy convergent to if and for all . In this case we write or as .

Definition 1.5.

Let be an IFNS. Then, is said to be intuitionistic fuzzy Cauchy sequence if and for all and .

Definition 1.6.

Let be an IFNS. Then is said to be complete if every intuitionistic fuzzy Cauchy sequence in is intuitionistic fuzzy convergent in .

Definition 1.7 (see [2]).

Let be an IFNS with the condition

(1.1)

Let , for all . Then is an ascending family of norms on . These norms are called -norms on corresponding to intuitionistic fuzzy norm .

It is easy to see the following.

Proposition 1.8.

Let be an IFNS satisfying (1.1), and let be a sequence in . Then and if and only if for all .

Proposition 1.9 (see [25]).

Let be an IFNS satisfying (1.1). Then a subset of is -intuitionistic fuzzy bounded if and only if is bounded with respect to , for all , where denotes the -norm of .

Recently, [1, 29] introduced the concept of strong and weak intuitionistic fuzzy continuity as well as strong and weak intuitionistic fuzzy convergent.

Some notations and results which will be used in this paper [25] are given below.

(i)The linear space and is an intuitionistic fuzzy normed space with respect to the intuitionistic fuzzy norm defined as

(1.2)

satisfying (1.1) condition and , for all .

(ii)If is a intuitionistic fuzzy normed space, then denotes the set of all linear functions from to which are bounded as linear operators from to , where and denote -norms of and , respectively.

(iii) is clearly the first dual space of for .

Definition 1.10.

Let be an intuitionistic fuzzy normed space satisfying (1.1), and let be the intuitionistic fuzzy normed space defined in the above remark. A sequence is said to be -intuitionistic fuzzy weakly convergent and converges to if for all and for all (dual space with respect to ), , that is

(1.3)

for all.

Definition 1.11 (see [25]).

Let be an IFNS. A mapping is said to be intuitionistic nonexpansive if

(1.4)

for all , for all .

2. Browder's Theorems and Some Results in IFNS

In this section, we discuss the idea of fuzzy type of some Browder's fixed point theorems in intuitionistic fuzzy normed space by [26, 27]. As a consequence of Theorem 4.1 [30] and Browder's theorems [26, 27] in crisp normed linear space we have Browder's theorems in intuitionistic fuzzy normed space.

Theorem 2.1.

Let be a nonempty -intuitionistic fuzzy weakly compact convex subset of a strong intuitionistic fuzzy uniformly convex normed space satisfying (1.1). Then every intuitionistic fuzzy nonexpansive mapping has a fixed point.

Theorem 2.2.

Let be a strong intuitionistic fuzzy uniformly convex normed space satisfying (1.1). Let be an -intuitionistic fuzzy bounded and -intuitionistic fuzzy closed convex subset of , and let be an intuitionistic fuzzy nonexpansive mapping of into . Suppose that for , as strongly intuitionistic fuzzy convergent while as -intuitionistic fuzzy weakly convergent. Then .

Now, we give relation between a intuitionistic fuzzy uniformly convex normed space and a strongly intuitionistic fuzzy uniformly convex normed space.

Proposition 2.3.

If is a strong intuitionistic fuzzy uniformly convex normed space, then it is a intuitionistic fuzzy uniformly convex normed space.

Proof.

Recall that is said to be strong intuitionistic fuzzy uniformly convex if for each , there exists such that . For all , where

(2.1)

and together with

(2.2)

and also is said to be intuitionistic fuzzy uniformly convex by Definition 1.7 if for each , there exists such that

(2.3)

for all , where

(2.4)

Suppose that is strong intuitionistic fuzzy uniformly convex normed space. Choose and such that . Now, implies that

(2.5)

for all .

Case 2.4.

If for some (say),

(2.6)

then

(2.7)

Case 2.5.

If for some (say), implies that there exists a sequence with , and with , , for all

(2.8)

For (2.7), (2.8) we get that implies and . Similarly, implies and . Now, implies

(2.9)

there exists (say) such that

(2.10)

and . We claim that

(2.11)

If possible suppose that and implies that there exists a sequence with such that and with such that and , for all and , for all and

(2.12)

for all and

(2.13)

for all

(2.14)

, a contradiction. Thus

(2.15)

Now from above we get that, implies

(2.16)

(2.17)

such that the idea of strong uniform convexity of

(2.18)

(2.19)

for all

(2.20)

for all

(2.21)

Now for any positive number (say) we get

(2.22)

Thus,

(2.23)

implies that

(2.24)

for all

(2.25)

for all

(2.26)

for all

(2.27)

for all

(2.28)

for all

(2.29)

for all

(2.30)

From (2.21), (2.30), we get that, implies , for any , for any that is . Hence, we have . So is intuitionistic fuzzy uniformly convex.

Remark 2.6.

However, the reverse of this Proposition 2.3 is untrue. This fact can be seen in the following example.

Let ( is the set of all real numbers). Suggest two norms on as the following: and , where . Clearly , for all , and it can be easily verified that is uniformly convex with regard to . Define a function and by

(2.31)

It can be easily verified that is a intuitionistic fuzzy norm on . Now,

(2.32)

Thus is intuitionistic fuzzy uniformly convex normed space. So, since this is not uniformly convex with regard to , therefore there exists such that for all we have such that,

(2.33)

Now, and . Reversely, and and , for . Hence

(2.34)

Again,

(2.35)

So, and . We suggest that and . If ,

, , , for , , for , for ; however this is a contradiction that is

(2.36)

Otherwise , , , and thus

(2.37)

By (2.36) and (2.37) we get

(2.38)

Again

(2.39)

Hence with (2.35), (2.36), (2.37), (2.38), and (2.39) we have

(2.40)

but

(2.41)

Moreover but that is . However is not a uniformly convex intuitionistic fuzzy normed space.

3. Conclusion

We studied here the concept of intuitionistic fuzzy normed space as an extension of the fuzzy normed space, which provides a larger setting to deal with the uncertainly and vagueness in natural problems arising in many branches of science and engineering. In this new setup we established Browder's fixed point theorem and some interesting results in intuitionistic fuzzy normed space which could be very useful tools in the development of fuzzy set theory.