Abstract

The aim of this paper is to introduce and study the fuzzy neighborhood, the limit fuzzy number, the convergent fuzzy sequence, the bounded fuzzy sequence, and the Cauchy fuzzy sequence on the base which is adopted by Abdul Hameed (every real number is replaced by a fuzzy number (either triangular fuzzy number or singleton fuzzy set (fuzzy point))). And then, we will consider that some results respect effect of the upper sequence on the convergent fuzzy sequence, the bounded fuzzy sequence, and the Cauchy fuzzy sequence.

1. Introduction

Zadeh [1] introduced the concept of fuzzy set in 1965. Kramosil and Michálek [2] introduced the concept of fuzzy metric space using continuous -norms in 1975.

Matloka [3] introduced bounded and convergent sequences of fuzzy numbers and studied some of their properties in 1986. Sequences of fuzzy numbers also were discussed by Nanda [4], Kwon [5], Esi [6], and many others.

In 2010, Al-Tai defined the fuzzy metric space, the fuzzy sequence, and many other concepts, on other base, that the family of fuzzy real numbers is . is the family of fuzzy integer numbers, where every is a singleton fuzzy set (fuzzy point) (see [7–12]). are the family of fuzzy rational numbers and the family of fuzzy irrational numbers, respectively, where every or is a triangular fuzzy number [13] and by using the representation Theorem (the resolution principle) [14].

In this research, we will consider the fuzzy neighborhood, the limit fuzzy number, the convergent fuzzy sequence, the bounded fuzzy sequence, and the Cauchy fuzzy sequence, and then we will introduce some results about properties of the above concepts on the base which depended by Al-Tai to define the fuzzy metric space.

Definition 1.1. Let be a fuzzy metric space. If , then a neighborhood of is a family of fuzzy numbers consisting of all fuzzy numbers , such that . The fuzzy number will be called the fuzzy radius of .

Definition 1.2. A fuzzy number is a limit fuzzy number of the family of the fuzzy numbers , if every fuzzy neighborhood of contains a fuzzy number , such that .

Definition 1.3 (Al-Tai [13]). If we have the family of the fuzzy numbers and the family of the fuzzy natural numbers , the fuzzy sequence in is a fuzzy function from to .
A fuzzy sequence consists of all ordered tuples (sequences) at degree , for all ; is its -cut, . The fuzzy value of the fuzzy function at is denoted by .

Definition 1.4. A fuzzy sequence in a fuzzy metric space is said to be convergent, if there is a fuzzy number , such that for every , there is a fuzzy natural number , with , which implies that
That is, for every -cut of , there is a natural number , with , such that where -cut of , -cut of , , and , for all . will be called the fuzzy limit of , and we write
If is not convergent at any, then is said to be divergent.

Definition 1.5. The family of the fuzzy numbers in the fuzzy metric space is bounded, if there is a fuzzy real number and a fuzzy number , such that for all . That is, if -cut of and -cut of , then where -cut of all , for all .

Definition 1.6. The fuzzy sequence in a fuzzy metric space is said to be bounded if its fuzzy range is bounded.

Definition 1.7. A fuzzy sequence in a fuzzy metric space is said to be a Cauchy fuzzy sequence if for every , there is a fuzzy natural number , such that whenever and . In other words, every -cut of is a Cauchy sequence; that is, for every -cut of , there is a natural number , such that for every , where -cut of , -cut of , , , and , for all .

Examples 1.8. (1) converges to , since for all -cut of , for all , for all , for all .
Therefore,
(2) converges to , since for all -cut of , for all , for all , for all .
Therefore,
(3) converges to , where -cut of is .
And -cut of is
Since -cut of , for all , for all , for all .
Therefore,
(4) converges to .
(5) is divergent and unbounded.
(6) is divergent, since for all -cut of is divergent. And it is bounded.

Theorem 1.9. Let be a fuzzy sequence in the fuzzy metric space . If the upper sequence of the -cut of is bounded for all , then is bounded.

Proof. Let , be a supremum of a sequence , for all . We have from the hypothesis, the upper sequence is bounded for all ; that is, there is a real number -cut of , such that for all , , for all .
When , where . That is, there is a real number -cut of , such that for all , , for all .
When , where . That is, there is a real number -cut of , such that for all , , for all .
And so on, when , where . That is, there is a real number -cut of  , such that for all , for all . By Definitions 1.5 and 1.6, is bounded.

Theorem 1.10. Let be a fuzzy sequence in the fuzzy metric space . If the upper sequence of the -cut of is bounded for all , then is convergent.

Proof. Suppose that the upper sequence of the -cut of is bounded for all . By Theorem 1.9, is bounded; that is, every sequence , of the -cut of is bounded for all , but every sequence , of the -cut of is monotonic, for all ; then, every sequence , of the -cut of is convergent, for all (see [15, Theorem 3.14]). By Definition 1.4, is convergent.

Theorem 1.11. Let be a fuzzy sequence in the fuzzy metric space . If the upper sequence of the -cut of is convergent for all , then is convergent.

Proof. Suppose that the upper sequence of the -cut of is convergent for all , so it is bounded (see [15, Theorem 3.2-c]). By Theorem 1.10, is convergent.

Theorem 1.12. Every convergent fuzzy sequence in a fuzzy metric space is bounded.

Proof. Let be a convergent fuzzy sequence in a fuzzy metric space . By Definition 1.4, every sequence , of the -cut of is convergent for all ; that is, the upper sequence of is bounded for all (see [15, Theorem 3.2-c]). By Theorem 1.9, is bounded.

Theorem 1.13. Every bounded fuzzy sequence in the fuzzy metric space is convergent.

Proof. Let be a bounded fuzzy sequence in a . By Definition 1.5, we get that the upper sequence of is bounded for all . By Theorem 1.10, is convergent.

Theorem 1.14. Let be a fuzzy sequence in a fuzzy metric space ; then converges to if and only if every fuzzy neighborhood of contains all but finitely many of the terms of .

Proof. Suppose that is a fuzzy neighborhood of . For , , , imply . We have from the hypothesis converges to , so for same , there exists , such that , implying . That is, .
Conversely, suppose that every fuzzy neighborhood of contains all but finitely many of . Fix , and let be the set of all , such that .
By assumption, there exists corresponding to , such that if . Thus if . Hence converges to .

Theorem 1.15. In any fuzzy metric space , every convergent fuzzy sequence is a Cauchy fuzzy sequence.

Proof. Let be a convergent fuzzy sequence in a fuzzy metric space . By Definition 1.4, every sequence , of the -cut of is convergent for all ; then every sequence , of the -cut of is a Cauchy sequence for all (see [15, Theorem 3.11-a]); by Definition 1.7, is a Cauchy fuzzy sequence.

Theorem 1.16. Let be a fuzzy sequence in the fuzzy metric space . If the upper sequence of the -cut of is bounded for all ,then is a Cauchy fuzzy sequence.

Proof. Suppose that the upper sequence of the -cut of is bounded for all . By Theorem 1.10, is convergent. By Theorem 1.15, is a Cauchy fuzzy sequence.

Theorem 1.17. In the fuzzy metric space , every bounded fuzzy sequence is a Cauchy fuzzy sequence.

Proof. Let be a bounded fuzzy sequence in . By Theorem 1.13, is a convergent fuzzy sequence. By Theorem 1.15, is a Cauchy fuzzy sequence.

Theorem 1.18. Let be a fuzzy sequence in the fuzzy metric space . If the upper sequence of the -cut of is convergent for all , then is a bounded fuzzy sequence.

Proof. Suppose that the upper sequence of the -cut of is convergent for all . By Theorem 1.11, is a convergent fuzzy sequence. By Theorem 1.12, is a bounded fuzzy sequence.

Theorem 1.19. Let be a fuzzy sequence in the fuzzy metric space . If the upper sequence of the -cut of is convergent for all , then is a Cauchy fuzzy sequence.

Proof. Suppose that the upper sequence of the -cut of is convergent for all . By Theorem 1.11, is a convergent fuzzy sequence. By Theorem 1.15, is a Cauchy fuzzy sequence.