Fuzzy 2 Metric Spaces Homework

Institute of Mathematics, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China

Copyright © 2012 Zhenhua Jiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The author extends two fixed point theorems (due to Gregori, Sapena, and Žikić, resp.) in fuzzy metric spaces to intuitionistic fuzzy metric spaces.

1. Introduction

In this paper, we pay our attention to the fixed point theory on intuitionistic fuzzy metric spaces. Since Zadeh [1] introduced the theory of fuzzy sets, many authors have studied the character of fuzzy metric spaces in different ways [2–5]. Among others, fixed point theorem was an important subject. Gregori and Sapena [6] investigated fixed point theorems for fuzzy contractive mappings defined on fuzzy metric spaces. Recently, Žikić [7] proved a fixed point theorem for mappings on fuzzy metric space which improved the result of Gregori and Sapena. As further development, Atanassov [8] introduced and studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets, and later there has been much progress in the study of intuitionistic fuzzy sets [9, 10]. Using the idea of intuitionistic sets, Park [11] defined the notion of intuitionistic fuzzy metric spaces with the help of continuous -norms and continuous -conorms as a generalization of fuzzy metric space. Recently, several authors studied the structure of intuitionistic fuzzy metric spaces and fixed point theorems for the mappings defined on intuitionistic fuzzy metric spaces. We refer the reader to [11–13] for further details. In this paper, we will prove the following two fixed point theorems.

The first theorem extends Gregori-Sapena's fixed point theorem [6] in fuzzy metric spaces to complete intuitionistic fuzzy metric spaces. As preparation, we recall the definition of -increasing sequence [6]. A sequence of positive real numbers is said to be an -increasing sequence if there exists such that , for all .

Theorem 1.1. Let be a complete intuitionistic fuzzy metric space such that for every -increasing sequence and arbitrary , hold.
Let and be a mapping satisfying and for all . Then has a unique fixed point.

The second theorem extends Žikić’s fixed point theorem [7] in fuzzy metric space to intuitionistic fuzzy metric space.

Theorem 1.2. Let be a complete intuitionistic fuzzy metric space such that for some and , hold.
Let and be a mapping satisfying and for all . Then has a unique fixed point.

2. Basic Notions and Preliminary Results

For the sake of completeness, in this section we will recall some definitions and preliminaries on intuitionistic fuzzy metric spaces.

Definition 2.1 (see [14]). Let be a nonempty fixed set. An intuitionistic fuzzy set is an object having the form where the functions and denote the degree of membership and the degree of nonmembership of each element to the set , respectively, and for each .

For developing intuitionistic fuzzy topological spaces, in [10], Çoker introduced the intuitionistic fuzzy sets and in as follows.

Definition 2.2 (see [10]). and .

By Definition 2.2, Çoker defined the notion of intuitionistic fuzzy topological spaces.

Definition 2.3 (see [10]). An intuitionistic fuzzy topology on a nonempty set is a family of intuitionistic fuzzy sets in satisfying the following axioms:
(T1); (T2) for any ; (T3) for any arbitrary family .
In this case, the pair is called an intuitionistic fuzzy topological space.

Definition 2.4 (see [15]). A binary operation is a continuous -norm (triangular norm) if satisfies the following conditions: is associative and commutative; is continuous; for all ; whenever and , and .
By this definition, it is easy to see that . According to condition (a), the following product is well defined: , and we will denote it by .

Definition 2.5 (see [15]). A binary operation is a continuous -conorm (triangular conorm) if satisfies the following conditions: (e) is associative and commutative; (f) is continuous; (g) for all ; (h) whenever and , and .
By this definition, it is easy to see that . According to condition (e), the following product is well defined: , and we also denote this product by .

Remark 2.6. The origin of concepts of -norms and related -conorms was in the theory of statistical metric spaces in the work of Menger [5]. These concepts are known as the axiomatic skeletons that we use for characterizing fuzzy intersections and unions, respectively. Basic examples of -norms are and , and basic examples of -conorms are and .

Definition 2.7 (see [13]). A 5-tuple

Только лишь мошонка. Офицер гордо кивнул: - Да. Когда церковь получит все останки этого великого человека, она причислит его к лику святых и разместит отдельные части его тела в разных соборах, чтобы все могли проникнуться их величием.

- А у вас здесь… - Беккер не сдержал смешка.

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