Degree representation of Banach contraction principle in fuzzymetric spaces 1

Abstract

In this paper, a new approach to Banach contraction principle in fuzzy metric space is provided. The concepts of convergent sequence, Cauchy sequence, complete metric space, contractive map and existence of fixed points, are all defined with some degrees. Adopting these degree concepts, a degree representation of Banach contradiction principle in fuzzy metric space is given by means of an inequality.

Keywords

Fixed point theorem fuzzy metric space Banach contraction principle

1 Introduction

Since Zadeh [29] introduced fuzzy set theory, metric spaces have been generalized to the fuzzy case. In [11], Kramosil and Michalek firstly proposed the notion of fuzzy metric (KM-metric, in short) spaces. Later, George and Veeramani [2, 3] modified the notion of KM-metric spaces by using continuous t-norm and generalized the concept of a probabilistic metric space to a fuzzy situation (called GV-metric space, in short). In the sense of KM-metric and GV-metric, Yue [27] discussed its induced fuzzifying topology and fuzzifying uniformities. As a generalization of KM-metric spaces and partial metric spaces [13], Yue and Gu [28] defined fuzzy partial (pseudo-)metric spaces. In a different direction, Shi introduced pointwise metric spaces [21] and (L, M)-fuzzy metric spaces [22], and discussed their relations with fuzzy topologies. Afterwards, Pang proposed the notion of pointwise pseudo-metric chains to characterize (L, M)-fuzzy pseudo-metric spaces [16] and introduced the compactness theory in (L, M)-fuzzy pseudo-metric spaces [17]. Recently, Xiu and Pang [26] introduced the concepts ofL-partial pseudo-quasi-metric spaces, which can be considered as generalizations of pointwise pseudo-quasi-metric spaces and partial pseudo-quasi-metric spaces.

Fixed point theory is a very useful tool in solving a variety of problems in control theory, economic theory, nonlinear analysis and global analysis. The Banach contraction principle in complete metric spaces is the most famous and one of the most versatile elementary results in fixed point theory. With the development of fuzzy set theory, fixed point theorems are investigated extensively in fuzzy metric spaces. In [24], Vasuki gave a common fixed point theorem in a fuzzy metric space and as a special case, the Banach contraction principle in a fuzzy metric space is obtained. For a detailed survey, we refer to [1 , 20] and the references therein. In these literatures, there is a common character that should be mentioned. Namely, the metric spaces, no matter KM-metric space or GV-metric space, are both fuzzy, but the related concepts in these fuzzy metric spaces are crisp, i.e., not fuzzy. As we all know, the essence of fuzzy set theory, to some extent, is as follows:

The membership of an element in a set should be endowed with some degree, which is called membership degree;

Every mathematical object should be defined in a degree representation.

In this sense, fixed point theorems in fuzzy metric spaces should also be equipped with a degree-formed description.

The aim of this paper is to define the fuzzy counterparts of several basic concepts in metric spaces and to give a degree representation of the Banach contraction principle in fuzzy metric spaces.

2 Preliminaries

In this section, we mainly present several notions with respect to metric spaces and fuzzy metric spaces. For details, we refer to [2 , 24].

Definition 2.1. A map d : X × X ⟶ [0, + ∞) is called a metric on X if it satisfies:

d (x, y) =0 if and only if x = y;

d (x, y) = d (y, x);

d (x, y) ⩽ d (x, z) + d (z, y).

For a metric d on X, the pair (X, d) is called a metric space.

Definition 2.2. A map f : (X, d_X) ⟶ (Y, d_Y) is called contractive if there exists a constant k ∈ (0, 1) such that d_Y (f (x) , f (y)) ⩽ k · d_X (x, y).

Definition 2.3. Suppose (X, d) is a metric space, x ∈ X and {x_n} is a sequence in X.

{x_n} is said to be convergent to x if there exists $N \in ℕ$ such that d (x_n, x) < ɛ when n > N for all ɛ > 0.

{x_n} in X is called a Cauchy sequence if there exists $N \in ℕ$ such that d (x_n, x_m) < ɛ when n, m > N for all ɛ > 0.

(X, d) is said to be complete if every Cauchy sequence is convergent.

Definition 2.4. A binary operation ∗ : [0, 1] × [0, 1] ⟶ [0, 1] is called a (left-) continuous t-norm if it satisfies:

∗ is associative and commutative;

a ∗ 1 = a for all a ∈ [0, 1];

a ∗ b ⩽ c ∗ e whenever a ⩽ c and b ⩽ e;

∗ is (left-) continuous.

For each (left-) continuous t-norm ∗, the implication → can be determined by a → b = ⋁ {c ∈ [0, 1] ∣ a ∗ c ⩽ b} and we have $(ADJ) a * b ⩽ c \Leftrightarrow a ⩽ b \to c .$

Remark 2.5. The three most commonly used continuous t-norm are the minimum, denoted by ∧, the usual product, denoted by · and the Lukasievicz t-norm, denoted by ⋄ (a ⋄ b = max {0, a + b - 1}). All these three t-norms are idempotent, we assume that t-norm ∗ is also idempotent in this paper.

Definition 2.6. A map M : X × X × [0, + ∞) ⟶ [0, 1] is called a fuzzy metric if it satisfies:

M (x, y, 0) =0;

M (x, y, t) =1 for all t > 0 if and only if x = y;

M (x, y, t) = M (y, x, t);

M (x, y, t) ∗ M (y, z, s) ⩽ M (x, z, t + s) for all x, y, z ∈ X and t, s > 0;

M (x, y, ·) : [0, + ∞) ⟶ [0, 1] is left continuous;

$lim_{t \to \infty} M (x, y, t) = 1$ for all x, y ∈ X.

The pair (X, M) is called a fuzzy metric space. Usually, this kind of fuzzy metric is called a KM metric.

Lemma 2.7. Let (X, M, ∗) be a fuzzy metric space. Then M (x, y, ·) is non-decreasing for all x, y ∈ X.

3 Main results

Recall that the value M (x, y, t) in Definition 2.6 can be thought as the degree of the nearness between x and y with respect to t. To some extent, this is exactly the essence of fuzzy mathematics. Namely, every concept should be endowed with some degree. In this section, we will define the fuzzy counterparts of several concepts in metric spaces with some degree. Moreover, we will describe the Banach contraction principle in this manner in the framework of fuzzy metric spaces.

Definition 3.1. Let (X, M, ∗) be a fuzzy metric space, x ∈ X and {x_n} be a sequence in X. The degree to which {x_n} converges to x is defined by $Conv ({x_{n}}, x) = ⋀_{ɛ > 0} ⋁_{N \in ℕ} ⋀_{n > N} M (x_{n}, x, ɛ) .$

The degree to which {x_n} is a convergent sequence is defined by $Conv ({x_{n}}) = ⋁_{x \in X} Conv ({x_{n}}, x) .$

Definition 3.2. Let (X, M, ∗) be a fuzzy metric space and {x_n} be a sequence in X. The degree to which {x_n} is a Cauchy sequence is defined by $Cauchy ({x_{n}}) = ⋀_{ɛ > 0} ⋀_{p \in ℕ} ⋁_{N \in ℕ} ⋀_{n > N} M (x_{n}, x_{n + p}, ɛ) .$

Remark 3.3. The above notions allow us to talk on the convergence and the Cauchyness of a sequence in some degree even if the sequence is not convergent or not a Cauchy sequence. Conv ({x_n}) and Cauchy ({x_n}) are natural measures for which {x_n} is a convergent sequence or a Cauchy sequence. We will show that the degrees Conv ({x_n}) and Cauchy ({x_n}) naturally suggest fuzzy extensions of these two concepts in metric spaces to fuzzy metric spaces and naturally conform to the requirements of fuzzy mathematics.

As we all know, a convergence sequence must be a Cauchy sequence in metric spaces. The next result gives the fuzzy counterpart, which is represented by an inequality.

Lemma 3.4. Let (X, M, ∗) be a fuzzy metric space and {x_n} be a sequence in X. Then $Conv ({x_{n}}) ⩽ Cauchy ({x_{n}}) .$

Proof. It suffices to verify that $\begin{matrix} ⋁_{x \in X} ⋀_{ɛ > 0} ⋁_{N \in ℕ} ⋀_{n > N} M (x_{n}, x, ɛ) \\ ⩽ ⋀_{ɛ > 0} ⋀_{p \in ℕ} ⋁_{N \in ℕ} ⋀_{n > N} M (x_{n}, x_{n + p}, ɛ) . \end{matrix}$

Take any t ∈ (0, 1) such that $t < ⋁_{x \in X} ⋀_{ɛ > 0} ⋁_{N \in ℕ} ⋀_{n > N} M (x_{n}, x, ɛ) .$

Then there exists x ∈ X such that for each ɛ > 0, there exists $N_{ɛ} \in ℕ$ such that t ⩽ M (x_n, x, ɛ) for each n > N_ɛ. Now for each ɛ > 0 and each $p \in ℕ$ , let N₀ = N_ɛ. Then for each n > N₀, it follows that n + p > N₀. This implies that $t ⩽ M (x_{n}, x, ɛ / 2) and t ⩽ M (x_{n + p}, x, ɛ / 2) .$

By (KM3) and (KM4), we obtain that $\begin{matrix} t & ⩽ & M (x_{n}, x, ɛ / 2) * M (x_{n + p}, x, ɛ / 2) \\ ⩽ & M (x_{n}, x_{n + p}, ɛ) . \end{matrix}$

This means that $t ⩽ ⋀_{ɛ > 0} ⋀_{p \in ℕ} ⋁_{N \in ℕ} ⋀_{n > N} M (x_{n}, x_{n + p}, ɛ) .$

By the arbitrariness of t, we get $\begin{matrix} ⋁_{x \in X} ⋀_{ɛ > 0} ⋁_{N \in ℕ} ⋀_{n > N} M (x_{n}, x, ɛ) \\ ⩽ ⋀_{ɛ > 0} ⋀_{p \in ℕ} ⋁_{N \in ℕ} ⋀_{n > N} M (x_{n}, x_{n + p}, ɛ), \end{matrix}$ as desired. □

For a metric space, it is complete if and only if every Cauchy sequence is convergent. This result was generalized to fuzzy metric spaces in different ways. Now we use the implication operation “→", which is compatible with “∗", to define the degree of completeness of a fuzzy metric space.

Definition 3.5. Let (X, M, ∗) be a fuzzy metric space. The degree to which (X, M, ∗) is complete is defined by $\begin{matrix} Complete ((X, M, *)) \\ = ⋀_{{x_{n}}} (Cauchy ({x_{n}}) \to Conv ({x_{n}})) . \end{matrix}$

Remark 3.6. The above concept also allows us to talk on the completeness of a metric space in some degree even if the space is not complete. Complete ((X, M, ∗)) provides a natural measure for which (X, M, ∗) is a complete metric space.

Adopting the same way, we will propose the following two notions, which describe the degree to which a map is a contractive map and to which a map has fixed points, respectively.

Definition 3.7. Let f be a map of (X, M, ∗) into itself. The degree to which f is contractive is defined by $\begin{matrix} Contr (f) & = & ⋁_{k \in (0, 1)} ⋀_{x, y \in X} ⋀_{ɛ > 0} (M (x, y, ɛ) \\ \to M (f (x), f (y), k ɛ)) . \end{matrix}$

Definition 3.8. Let f be a map of (X, M, ∗) into itself and x ∈ X. The degree to which x is a fixed point of f is defined by $Fix (f, x) = ⋀_{ɛ > 0} M (f (x), x, ɛ) .$

The degree to which f has fixed points is defined by $Fix (f) = ⋁_{x \in X} Fix (f, x) = ⋁_{x \in X} ⋀_{ɛ > 0} M (f (x), x, ɛ) .$

Now we give the main result in this paper. Namely, we gave a degree representation of the Banach contraction principle in fuzzy metric spaces.

Theorem 3.9. Let (X, M, ∗) be a fuzzy metric space and f be a map of (X, M, ∗) into itself. Then $Complete ((X, M, *)) * Contr (f) ⩽ Fix (f) .$

Proof. It suffices to prove the following inequality: $\begin{matrix} ⋀_{{x_{n}}} (Cauchy ({x_{n}}) \to Conv ({x_{n}})) * Contr (f) \\ ⩽ Fix (f) . \end{matrix}$

That is, $\begin{matrix} ⋀_{{x_{n}}} (⋀_{ɛ > 0} ⋀_{p \in ℕ} ⋁_{n_{0} \in ℕ} ⋀_{n > n_{0}} M (x_{n}, x_{n + p}, ɛ) \\ \to ⋁_{x \in X} ⋀_{ɛ > 0} ⋁_{m_{0} \in ℕ} ⋀_{m > m_{0}} M (x_{m}, x, ɛ)) \\ * ⋁_{k \in (0, 1)} ⋀_{x, y \in X} ⋀_{ɛ > 0} (M (x, y, ɛ) \\ \to M (f (x), f (y), k ɛ)) \\ ⩽ ⋁_{x \in X} ⋀_{s > 0} M (f (x), x, s) . \end{matrix}$

For convenience, denote $\begin{matrix} (1) & ≜ & ⋀_{{x_{n}}} (⋀_{ɛ > 0} ⋀_{p \in ℕ} ⋁_{n_{0} \in ℕ} ⋀_{n > n_{0}} M (x_{n}, x_{n + p}, ɛ) \\ \to ⋁_{x \in X} ⋀_{ɛ > 0} ⋁_{m_{0} \in ℕ} ⋀_{m > m_{0}} M (x_{m}, x, ɛ)), \\ (2) & ≜ & ⋁_{k \in (0, 1)} ⋀_{x, y \in X} ⋀_{ɛ > 0} (M (x, y, ɛ) \\ \to M (f (x), f (y), k ɛ)) \end{matrix}$ and $(3) ≜ ⋁_{x \in X} ⋀_{s > 0} M (f (x), x, s) .$

Take any t ∈ (0, 1) such that t<(1)∗(2). Then there exists r ∈ (0, 1) such that t< r <(1)∗(2). Since ∗ is order-preserving, we know that r<(2). Then there exists k ∈ (0, 1) such that r < M (x, y, ɛ) → M (f (x) , f (y) , kɛ) for all x, y ∈ X and ɛ > 0. By (ADJ), it follows that $\begin{matrix} (4) r * M (x, y, ɛ) ⩽ M (f (x), f (y), k ɛ) . \end{matrix}$

Take any x₀ ∈ X and define x_n = f (x_n-1) for n ⩾ 1. Then by r<(1), we have $\begin{matrix} (5) r & < & ⋀_{ɛ > 0} ⋀_{p \in ℕ} ⋁_{n_{0} \in ℕ} ⋀_{n > n_{0}} M (x_{n}, x_{n + p}, ɛ) \\ \to ⋁_{x \in X} ⋀_{ɛ > 0} ⋁_{m_{0} \in ℕ} ⋀_{m > m_{0}} M (x_{m}, x, ɛ) . \end{matrix}$

For each $n \in ℕ$ , by (4), we obtain $\begin{matrix} M (x_{n}, x_{n + 1}, ɛ / p) \\ = M (f (x_{n - 1}, f (x_{n}), ɛ / p) \\ ⩾ r * M (x_{n - 1}, x_{n}, ɛ / pk) \\ = r * M (f (x_{n - 2}, f (x_{n - 1}), ɛ / pk) \\ ⩾ r^{2} * M (x_{n - 2}, x_{n - 1}, ɛ / {pk}^{2}) \\ \dots \dots \\ ⩾ r^{n} * M (x_{0}, x_{1}, ɛ / {pk}^{n}) \\ = r * M (x_{0}, x_{1}, ɛ / {pk}^{n}) . (* is idempotent) \end{matrix}$

Then for fixed $n_{0} \in ℕ$ and n ⩾ n₀, by (KM3), it follows that $\begin{matrix} M (x_{n}, x_{n + p}, ɛ) \\ ⩾ M (x_{n}, x_{n + 1}, ɛ / p) * M (x_{n + 1}, x_{n + 2}, ɛ / p) * \\ \dots * M (x_{n + p - 1}, x_{n + p}, ɛ / p) \\ ⩾ r * M (x_{0}, x_{1}, ɛ / {pk}^{n}) * r * M (x_{0}, x_{1}, ɛ / {pk}^{n + 1}) * \\ \dots * r * M (x_{0}, x_{1}, ɛ / {pk}^{n + p - 1}) \\ ⩾ r^{p} * M (x_{0}, x_{1}, ɛ / {pk}^{n_{0}}) (by Lemma 2.7) \\ = r * M (x_{0}, x_{1}, ɛ / {pk}^{n_{0}}) \end{matrix}$ for each ɛ > 0 and $p \in ℕ$ . Hence, by (KM6), we obtain $\begin{matrix} ⋀_{ɛ > 0} ⋀_{p \in ℕ} ⋁_{n_{0} \in ℕ} ⋀_{n > n_{0}} M (x_{n}, x_{n + p}, ɛ) \\ ⩾ ⋀_{ɛ > 0} ⋀_{p \in ℕ} ⋁_{n_{0} \in ℕ} r * M (x_{0}, x_{1}, ɛ / {pk}^{n_{0}}) \\ = r . \end{matrix}$

The equality holds from (KM5) since $⋁_{n_{0} \in ℕ} M (x_{0}, x_{1}, ɛ / {pk}^{n_{0}}) = lim_{t \to \infty} M (x_{0}, x_{1}, t) = 1 .$

By (5), it follows that $r < r \to ⋁_{x \in X} ⋀_{ɛ > 0} ⋁_{m_{0} \in ℕ} ⋀_{m > m_{0}} M (x_{m}, x, ɛ) .$

This implies that $t < r ⩽ ⋁_{x \in X} ⋀_{ɛ > 0} ⋁_{m_{0} \in ℕ} ⋀_{m > m_{0}} M (x_{m}, x, ɛ) .$

Then there exists x₀ ∈ X such that for each s > 0, we can choose $m_{0} \in ℕ$ such that t < M (x_m, x₀, s/2) for all m > m₀. Hence, it follows that $\begin{matrix} M (f (x_{0}), x_{0}, s) \\ ⩾ M (f (x_{0}), f (x_{m_{0} + 1}), s / 2) \\ * M (f (x_{m_{0} + 1}), x_{0}, s / 2) \\ ⩾ r * M (x_{0}, x_{m_{0} + 1}, s / 2 k) * M (x_{m_{0} + 2}, x_{0}, s / 2) \\ ⩾ t * M (x_{0}, x_{m_{0} + 1}, s / 2) * M (x_{m_{0} + 2}, x_{0}, s / 2) \\ (by Lemma 2.7) \\ ⩾ t * t * t \\ = t . \end{matrix}$

Therefore, t ⩽ ⋁ _x∈X ⋀ _s>0M (f (x) , x, s) . By the arbitrariness of t, we obtain (1)∗(2)⩽(3). □

4 Examples

In this section, we mainly give two examples to show the rationality of the Banach contraction principle in terms of the degree representation.

Example 4.1. Let X = {x, y, z} and define M : X × X × [0, + ∞) ⟶ [0, 1] by

If t = 0, then M (a, b, t) =0 for each a, b ∈ X;

If t > 0, then $\begin{matrix} M (x, x, t) = M (y, y, t) = M (z, z, t) = 1; \\ M (x, y, t) = M (y, x, t) = 0.25; \\ M (x, z, t) = M (z, x, t) = 0.5; \\ M (y, z, t) = M (z, y, t) = 0.75 . \end{matrix}$

It is easy to check that M is a fuzzy metric in the sense of Definition 2.6 whenever ∗ =⋄. Next we compute Complete ((X, M, ∗)). For this, take each {x_n} ⊆ X. Then there are the following fourcases:

Case 1: If Cauchy ({x_n}) =1, then it follows that $\begin{matrix} {Conv ({x_{n}}, a) ∣ a \in X} = {1, 0.25, 0.5}, \\ or & {Conv ({x_{n}}, a) ∣ a \in X} = {1, 0.25, 0.75}, \\ or & {Conv ({x_{n}}, a) ∣ a \in X} = {1, 0.5, 0.75} . \end{matrix}$

This implies that $\begin{matrix} Cauchy ({x_{n}}) \to Conv ({x_{n}}) \\ = 1 \to 1 \\ = 1 . \end{matrix}$

Case 2: If Cauchy ({x_n}) =0.25, then it follows that {Conv ({x_n} , a) ∣ a ∈ X} = {0.25, 0.5}. This implies that $\begin{matrix} Cauchy ({x_{n}}) \to Conv ({x_{n}}) \\ = 0.25 \to (0.5 \lor 0.25) \\ = 1 . \end{matrix}$

Case 3: If Cauchy ({x_n}) =0.5, then it follows that {Conv ({x_n} , a) ∣ a ∈ X} = {0.25, 0.5}. This implies that $\begin{matrix} Cauchy ({x_{n}}) \to Conv ({x_{n}}) \\ = 0.5 \to (0.5 \lor 0.25) \\ = 1 . \end{matrix}$

Case 4: If Cauchy ({x_n}) =0.75, then it follows that {Conv ({x_n} , a) ∣ a ∈ X} = {0.25, 0.75}. This implies that $\begin{matrix} Cauchy ({x_{n}}) \to Conv ({x_{n}}) \\ = 0.75 \to (0.25 \lor 0.75) \\ = 1 . \end{matrix}$

Considering these four cases, we obtain $Complete ((X, M, *)) = 1 .$

Further, define f : X ⟶ X by $f (x) = f (y) = f (z) = x .$ Then we have $\begin{matrix} Contr (f) \\ = ⋁_{k \in (0, 1)} ⋀_{a, b \in X} ⋀_{ɛ > 0} (M (a, b, ɛ) \\ \to M (f (a), f (b), k ɛ)) \\ = ⋁_{k \in (0, 1)} ⋀_{a, b \in X} ⋀_{ɛ > 0} (M (a, b, ɛ) \to M (x, x, k ɛ)) \\ = ⋁_{k \in (0, 1)} ⋀_{a, b \in X} ⋀_{ɛ > 0} (M (a, b, ɛ) \to 1) \\ = 1 \end{matrix}$ and $\begin{matrix} Fix (f) & = & ⋁_{a \in X} ⋀_{ɛ > 0} M (f (a), a, ɛ) \\ = & ⋁_{a \in X} ⋀_{ɛ > 0} M (x, a, ɛ) \\ \geq & ⋀_{ɛ > 0} M (x, x, ɛ) \\ = & 1 . \end{matrix}$

This shows $Complete ((X, M, *)) * Contr (f) ⩽ Fix (f),$ as desired.

Example 4.2. Let d be an ordinary metric and define M^d : X × X × [0, + ∞) ⟶ [0, 1] as follows: $M^{d} (x, y, t) = \frac{t}{t + d (x, y)} .$

Then M^d is a fuzzy metric, which is called the standard fuzzy metric. Take each {x_n} ⊆ X. Then

$\begin{matrix} Cauchy ({x_{n}}) \\ = ⋀_{ɛ > 0} ⋀_{p \in ℕ} ⋁_{N \in ℕ} ⋀_{n > N} M^{d} (x_{n}, x_{n + p}, ɛ) \\ = ⋀_{ɛ > 0} ⋀_{p \in ℕ} ⋁_{N \in ℕ} ⋀_{n > N} \frac{ɛ}{ɛ + d (x_{n}, x_{n + p})} \\ = ⋀_{ɛ > 0} \frac{ɛ}{ɛ + ⋁_{p \in ℕ} ⋀_{N \in ℕ} ⋁_{n > N} d (x_{n}, x_{n + p})} \\ = {\begin{matrix} 1, & if {x_{n}} is a Cauchy sequence in (X, d), \\ 0, & otherwise . \end{matrix} \end{matrix}$

Take each a ∈ X. Then $\begin{matrix} Conv ({x_{n}}, a) \\ = ⋀_{ɛ > 0} ⋁_{N \in ℕ} ⋀_{n > N} M^{d} (x_{n}, a, ɛ) \\ = ⋀_{ɛ > 0} ⋁_{N \in ℕ} ⋀_{n > N} \frac{ɛ}{ɛ + d (x_{n}, a)} \\ = ⋀_{ɛ > 0} \frac{ɛ}{ɛ + ⋀_{N \in ℕ} ⋁_{n > N} d (x_{n}, a)} \\ = {\begin{matrix} 1, & if {x_{n}} converges to a in (X, d), \\ 0, & otherwise . \end{matrix} \end{matrix}$

This implies that $\begin{matrix} Conv ({x_{n}}) \\ = ⋁_{a \in X} Conv ({x_{n}}, a) \\ = {\begin{matrix} 1, & if {x_{n}} is convergent in (X, d), \\ 0, & otherwise . \end{matrix} \end{matrix}$

Further, if f is a contractive map from (X, d) to itself, then there exists k₀ ∈ (0, 1) such that d (f (x) , f (y)) ≤ k₀ · d (x, y). This implies that $\begin{matrix} Contr (f) & = & ⋁_{k \in (0, 1)} ⋀_{x, y \in X} ⋀_{ɛ > 0} (M^{d} (x, y, ɛ) \to M^{d} (f (x), f (y), k ɛ)) \\ = & ⋁_{k \in (0, 1)} ⋀_{x, y \in X} ⋀_{ɛ > 0} (\frac{ɛ}{ɛ + d (x, y)} \to \frac{k ɛ}{k ɛ + d (f (x), f (y))}) \\ \geq & ⋀_{x, y \in X} ⋀_{ɛ > 0} (\frac{ɛ}{ɛ + d (x, y)} \to \frac{k_{0} ɛ}{k_{0} ɛ + d (f (x), f (y))}) \\ \geq & ⋀_{x, y \in X} ⋀_{ɛ > 0} (\frac{ɛ}{ɛ + d (x, y)} \to \frac{k_{0} ɛ}{k_{0} ɛ + k_{0} d (x, y)}) \\ = & 1 . \end{matrix}$

Then we have $\begin{matrix} Complete ((X, M, *)) \\ = ⋀_{{x_{n}}} (Cauchy ({x_{n}}) \to Conv ({x_{n}})) \\ = {\begin{matrix} 1, & if every Cauchy sequence is \\ convergent in (X, d), \\ 0, & otherwise . \end{matrix} \end{matrix}$

By Theorem 3.9, we obtain $\begin{matrix} Fix (f) & \geq & Complete ((X, M, *)) * Contr (f) \\ = & 1 * 1 = 1 \end{matrix}$ provided that each Cauchy sequence in (X, d) is convergent and f is a contractive map from (X, d) to itself. Moreover, it follows from the definition of Fix (f), we have $\begin{matrix} Fix (f, x) & = & ⋀_{ɛ > 0} M^{d} (f (x), x, ɛ) \\ = & ⋀_{ɛ > 0} \frac{ɛ}{ɛ + d (f (x), x)} \\ = & 1 . \end{matrix}$

This implies that there exists x₀ ∈ X such that d (f (x₀) , x₀) =0, which means that x₀ is a fixed point of f. In this case, we can see that the Banach contraction principle in classical metric spaces can be included as a special case of Theorem 3.9.

5 Conclusion

In this paper, we provided a new approach to the Banach contraction principle in fuzzy metric spaces. The main advantage is that the present form gave a degree description by using an inequality. To some extent, this approach is more compatible with the essential requirement of fuzzy set theory. By Example 4.2, we can observe that Theorem 3.9 contains the classical Banach contraction principle in the framework of metric spaces as a special case. Also, by a concrete example in Example 4.1, we showed the rationality of Theorem 3.9. Therefore, the Banach contraction principle with the degree representation in Theorem 3.9 is more appropriate in the framework of fuzzy metric spaces.

As we all know, the Banach contraction principle is a powerful tool in differential equations, especially for the existence of solutions of differential equations. In this paper, we gave a degree representation of the Banach contraction principle in the framework of fuzzy metric spaces, which generalized the Banach contraction principle in classical metric spaces as shown in Example 4.2. Certainly, this degree representation can be used to examine the existence of solutions of differential equations. Moreover, a natural question has risen: Could this degree representation be used to examine the existence of solutions of fuzzy differential equations or prove an existence result for some special kinds of fixed point theorems, such as [5, 10]? In the future, we will consider how to apply Theorem 3.9 to differential equations and fuzzy differential equations.

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