Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces

Abstract

In this paper, we first present a fixed point theorem for set-valued fuzzy contraction type maps in complete fuzzy metric spaces which extends and improves some well-know results in literature. Then by presenting an endpoint result we initiate endpoint theory for fuzzy contraction maps in fuzzy metric spaces.

⁰2000 Mathematics Subject Classification: 47H10, 54H25.

1. Introduction and preliminaries

Many authors have introduced the concept of fuzzy metric spaces in different ways [1–4]. Kramosil and Michalek [5] introduced the fuzzy metric space by generalizing the concept of the probabilistic metric space to fuzzy situation. George and Veeramani [6, 7] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [5] and obtained a Hausdorff topology for this kind of fuzzy metric spaces. Recently, the fixed point theory in fuzzy metric spaces has been studied by many authors [8–18]. In [11], the following definition is given.

Definition 1.1. A sequence (t _n ) of positive real numbers is said to be an s-increasing sequence if there exists m₀ ∈ ℕ such that t _m + 1 ≤ t_m+1, for all m ≥ m₀.

Gregori and Sapena [11] proved the following fixed point theorem.

Theorem 1.2. Let (X, M, *) be a complete fuzzy metric space such that for every s-increasing sequence (t _n ) and every x, y ∈ X

$$\underset{n\to \infty }{lim}{*}_{i=n}^{\infty }M\left(x,y,t_n\right)=1.$$

Suppose f : X → X is a map such that for each x, y ∈ X and t > 0, we have

$$M\left(fx,fy,kt\right)\ge M\left(x,y,t\right),$$

where 0 < k < 1. Then, f has a unique fixed point.

In this article, we first give a fixed point theorem for set-valued contraction maps which improve and generalize the above-mentioned result of Gregori and Sapena. Then, in Section 2, we initiate endpoint theory in fuzzy metric spaces by presenting an endpoint result for set-valued maps.

To set up our results in the next section we recall some definitions and facts.

Definition 1.3 (3). A binary operation * : [0, 1] × [0, 1] → [0, 1] is called a continuous t-norm if ([0,1], *) is an abelian topological monoid with unit 1 such that a * b ≤ c * d whenever a ≤ c and b ≤ d for all a, b, c, ∈ [0, 1]. Examples of t-norm are a * b = ab and a * b = min{a, b}.

Definition 1.4 (6). The 3-tuple (X, M, *) is called a fuzzy metric space if X is an arbitrary non-empty set, * is a continuous t-norm, and M is a fuzzy set on X² × [0, ∞) satisfying the following conditions, for each x, y, z ∈ X and t, s > 0,

• (1) M(x, y, t) > 0,

• (2) M(x, y, t) = 1 if and only if x = y,

• (3) M(x, y, t) = M(y, x, t),

• (4) M(x, y, t) * M(y, z, s) ≤ M(x, z, t + s),

• (5) M(x, y, t) : (0, ∞) → [0,1] is continuous.

Example 1.5. [6] Let (X, d) be a metric space. Define a * b = ab (or a * b = min{a, b}) and for all x, y ∈ X and t > 0,

$$M\left(x,y,t\right)=\frac{t}{t+d\left(x,y\right)}.$$

Then (X, M, *) is a fuzzy metric space. We call this fuzzy metric M induced by the metric d the standard fuzzy metric.

Definition 1.6. Let (X, M, *) be a fuzzy metric space.

• (1) A sequence {x _n } is said to be convergent to a point x ∈ X if lim_n→∞ M(x _n , x, t) = 1 for all t > 0.

• (2) A sequence {x _n } is called a Cauchy sequence if

  $$\underset{m,n\to \infty }{lim}M\left(x_m,x_n,t\right)=1,$$

     for all t > 0.

• (3) A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.

• (4) A subset A ⊆ X is said to be closed if for each convergent sequence {x _n } with x _n ∈ A and x _n → x, we have x ∈ A.

• (5) A subset A ⊆ X is said to be compact if each sequence in A has a convergent subsequence.

Throughout the article, let $\mathcal{K}\left(X\right)$ denote the class of all compact subsets of X.

Lemma 1.7. [10]For all x, y ∈ X, M(x, y,.) is non-decreasing.

Definition 1.8. Let (X, M, *) be a fuzzy metric space. M is said to be continuous on X² × (0, ∞) if

$$\underset{n\to \infty }{lim}M\left(x_n,y_n,t_n\right)=M\left(x,y,t\right),$$

whenever {(x _n , y _n , t _n )} is a sequence in X² × (0, ∞) which converges to a point (x, y, t) ∈ X² × (0, ∞); i.e.,

$$\underset{n\to \infty }{lim}M\left(x_n,x,t\right)=\underset{n\to \infty }{lim}M\left(y_n,y,t\right)=1\mathsf{\text{and}}\underset{n\to \infty }{lim}M\left(x,y,t_n\right)=M\left(x,y,t\right).$$

Lemma 1.9. [10]M is a continuous function on X² × (0, ∞).

2. Fixed point theory

The following lemma is essential in proving our main result.

Lemma 2.1. Let (X, M, *) be a fuzzy metric space such that for every x, y ∈X, t > 0 and h > 1

$$\underset{n\to \infty }{lim}{*}_{i=n}^{\infty }M\left(x,y,t{h}^i\right)=1.$$

(2.1)

Suppose {x _n } is a sequence in X such that for all n ∈ ℕ,

$$M\left(x_n,x_{n+1},\alpha t\right)\ge M\left(x_{n-1},x_n,t\right),$$

where 0 < α < 1. Then {x _n } is a Cauchy sequence.

Proof. For each n ∈ ℕ and t > 0, we have

$$M\left(x_n,x_{n+1},t\right)\ge M\left(x_{n-1},x_n,\frac{1}{\alpha }t\right)\ge M\left(x_{n-2},x_{n-1},\frac{1}{{\alpha }^2}t\right)\ge \cdots \ge M\left(x_0,x_1,\frac{1}{{\alpha }^{n-1}}t\right).$$

Thus for each n ∈ ℕ, we get

$$M\left(x_n,x_{n+1},t\right)\ge M\left(x_0,x_1,\frac{1}{{\alpha }^{n-1}}t\right).$$

Pick the constants h > 1 and l ∈ ℕ such that

$$h\alpha <1\mathsf{\text{and}}\sum _{i=l}^{\infty }\frac{1}{h^i}=\frac{\frac{1}{h^l}}{1-\frac{1}{h}}<1.$$

Hence, for m ≥ n, we get

$$\begin{array}{ll}\hfill M\left(x_n,x_m,t\right)& \ge M\left(x_n,x_m,\left(\frac{1}{h^l}+\frac{1}{h^{l+1}}+\cdots +\frac{1}{h^{l+m}}\right)t\right)\\ \ge M\left(x_n,x_{n+1},\frac{1}{h^l}t\right)*M\left(x_{n+1},x_{n+2},\frac{1}{h^{l+1}}t\right)*\cdots *M\left(x_{m-1},x_m,\frac{1}{h^{l+m}}t\right)\\ \ge M\left(x_0,x_1,\frac{1}{{\alpha }^{n-1}{h}^l}t\right)*M\left(x_0,x_1,\frac{1}{{\alpha }^n{h}^{l+1}}t\right)*\cdots *M\left(x_0,x_1,\frac{1}{{\alpha }^{m-2}{h}^{l+m-n-2}}t\right)\\ \ge M\left(x_0,x_1,\frac{1}{{\left(\alpha h\right)}^{n-1}}t\right)*M\left(x_0,x_1,\frac{1}{{\left(\alpha h\right)}^n}t\right)*\cdots *M\left(x_0,x_1,\frac{1}{{\left(\alpha h\right)}^{m-2}}t\right)\\ \ge {*}_{i=n}^{\infty }M\left(x_0,x_1,\frac{1}{{\left(\alpha h\right)}^{i-1}}t\right)\end{array}$$

Then, from the above, we have

$$\underset{m,n\to \infty }{lim}M\left(x_n,x_m,t\right)\ge \underset{n\to \infty }{lim}{*}_{i=n}^{\infty }M\left(x_0,x_1,\frac{1}{{\left(\alpha h\right)}^{i-1}}t\right)=1,$$

for each t > 0. Therefore, we get

$$\underset{m,n\to \infty }{lim}M\left(x_n,x_m,t\right)=1,$$

for each t > 0 and so {x _n } is a Cauchy sequence.

In 2004, Rodríguez-López and Romaguera [19] introduced Hausdorff fuzzy metric on the set of the non-empty compact subsets of a given fuzzy metric space.

Definition 2.2. ([19]) Let (X, M, *) be a fuzzy metric space. For each $A,B\in \mathcal{K}\left(X\right)$ and t > 0, set

$$H_M\left(A,B,t\right)=min\left\{\underset{x\in A}{inf}\underset{y\in B}{sup}M\left(x,y,t\right),\underset{y\in B}{inf}\underset{x\in A}{sup}M\left(x,y,t\right)\right\}.$$

Lemma 2.3. [19]Let (X, M, *) be a fuzzy metric space. Then, the 3-tuple$\left(\mathcal{K}\left(X\right),H_M,*\right)$is a fuzzy metric space.

Now we are ready to prove our first main result.

Theorem 2.4. Let (X, M, *) be a complete fuzzy metric. Suppose F : X → X is a set-valued map with non-empty compact values such that for each x, y ∈ X and t > 0, we have

$$H_M\left(Fx,Fy,\alpha \left(d\left(x,y,t\right)\right)t\right)\ge M\left(x,y,t\right),$$

(2.2)

where α : [0, ∞) → [0,1) satisfying

$$\underset{r\to t^{+}}{limsup}\alpha \left(r\right)<1,\forall t\in \left[0,\infty \right),$$

and$d\left(x,y,t\right)=\frac{t}{M\left(x,y,t\right)}-t$. Furthermore, assume that (X, M, *) satisfies (2.1) for some x₀ ∈ X and x₁ ∈ Fx₀. Then F has a fixed point.

Proof. Let t > 0 be fixed. Notice first that if A and B are non-empty compact subsets of X and x ∈ A then by [19, Lemma 1], there exists a y ∈ B such that

$$H_M\left(A,B,t\right)\le \underset{b\in B}{sup}M\left(x,b,t\right)=M\left(x,B,t\right)=M\left(x,y,t\right).$$

Thus given α ≤ H _M (A, B, t) there exists a point y ∈ B such that

$$M\left(x,y,t\right)\ge \alpha .$$

Let x₀ ∈ X and x₁ ∈ Fx₀. If Fx₀ = Fx₁ then x₁ ∈ Fx₁ and x₁ is a fixed point of F and we are finished. So, we may assume that Fx₀ ≠ Fx₁. From (2.2), we get

$$H_M\left(F{x}_0,F{x}_1,\alpha \left(d\left(x_0,x_1,t\right)\right)t\right)\ge M\left(x_0,x_1,t\right).$$

Since x₁ ∈ Fx₀ and F is compact valued then by Rodríguez-López and Romaguera [19, Lemma 1] there exists a x₂ ∈ Fx₁ satisfying

$$\begin{array}{ll}\hfill M\left(x_1,x_2,t\right)\ge M\left(x_1,x_2,\alpha \left(d\left(x_0,x_1,t\right)\right)t\right)& =\underset{y\in F{x}_1}{sup}M\left(x_1,y,\alpha \left(d\left(x_0,x_1,t\right)\right)t\right)\\ \ge H_M\left(F{x}_0,F{x}_1,\alpha \left(d\left(x_0,x_1,t\right)\right)t\right)\\ \ge M\left(x_0,x_1,t\right).\end{array}$$

Continuing this process, we can choose a sequence {x _n }_{n ≥ 0}in X such that x_n+1∈ Fx _n satisfying

$$\begin{array}{ll}\hfill M\left(x_{n+1},x_{n+2},t\right)\ge M\left(x_{n+1},x_{n+2},\alpha \left(d\left(x_n,x_{n+1},t\right)\right)t\right)& =\underset{y\in F{x}_{n+1}}{sup}M\left(x_{n+1},y,\alpha \left(d\left(x_n,x_{n+1},t\right)\right)t\right)\\ \ge H_M\left(F{x}_n,F{x}_{n+1},\alpha \left(d\left(x_n,x_{n+1},t\right)\right)t\right)\\ \ge M\left(x_n,x_{n+1},t\right).\end{array}$$

Then, the sequence {M(x_n+1, x_n+2, t)} _n is non-decreasing.

Thus {d(x_n+1, x_n+2, t)} _n is a non-negative non-increasing sequence and so is convergent, say to, l ≥ 0. Since by the assumption

$$\underset{n\to \infty }{limsup}\alpha \left(d\left(x_{n+1},x_{n+2},t\right)\right)\le \underset{r\to t^{+}}{limsup}\alpha \left(r\right)<1,$$

then there exists k < 1 and N ∈ ℕ such that

$$\alpha \left(d\left(x_{n+1},x_{n+2},t\right)\right)<k,\forall n>N.$$

(2.4)

Since M(x, y,.) is non-decreasing then (2.3) together with (2.4) yield

$$M\left(x_{n+1},x_{n+2},kt\right)\ge M\left(x_{n+1},x_{n+2},\alpha \left(d\left(x_n,x_{n+1},t\right)\right)t\right)\ge M\left(x_n,x_{n+1},t\right).$$

Then from the above, we get

$$M\left(x_{n+1},x_{n+2},kt\right)\ge M\left(x_n,x_{n+1},t\right).$$

Hence by Lemma 2.1, we get {x _n }, which is a Cauchy sequence. Since (X, M, *) is a complete fuzzy metric space, then there exists $\bar{x}\in X$ such that ${lim}_{n\to \infty }{x}_n=\bar{x}$, that means ${lim}_{n\to \infty }M\left(x_n,\bar{x},t\right)=1$, for each t > 0. Thus, ${lim}_{n\to \infty }d\left(x_n,\bar{x},t\right)=0$, for each t > 0. Since

$$\underset{n\to \infty }{limsup}\alpha \left(d\left(x_n,\bar{x},t\right)\right)\le \underset{r\to 0^{+}}{limsup}\alpha \left(r\right)<1,$$

then there exists k < l < 1 such that

$$\underset{n\to \infty }{limsup}\alpha \left(d\left(x_n,\bar{x},t\right)\right)<l.$$

Now we claim that $\bar{x}\in F\bar{x}$. To prove the claim notice first that since $H_M\left(F{x}_n,F\bar{x},lt\right)\ge H_M\left(F{x}_n,F\bar{x},kt\right)\ge H_M\left(F{x}_n,F\bar{x},\alpha \left(d\left(x_n,\bar{x},t\right)\right)t\right)\ge M\left(x_n,\bar{x},t\right),$ and ${lim}_{n\to \infty }M\left(x_n,\bar{x},t\right)=1$ then for each t > 0, we get

$$\underset{n\to \infty }{lim}{H}_M\left(F{x}_n,F\bar{x},t\right)=1.$$

(2.5)

Since x_n+1∈ Fx _n then from (2.5), we have

$$\underset{n\to \infty }{lim}\underset{y\in F\bar{x}}{sup}M\left(x_{n+1},y,t\right)=1.$$

Thus there exists a sequence $y_n\in F\bar{x}$ such that

$$\underset{n\to \infty }{lim}M\left(x_n,y_n,t\right)=1,$$

for each t > 0. For each n ∈ ℕ, we have

$$M\left(y_n,\bar{x},s+t\right)\ge M\left(y_n,x_n,s\right)*M\left(x_n,\bar{x},t\right).$$

Hence, from the above, we get

$$\underset{n\to \infty }{lim}M\left(y_n,\bar{x},t\right)=1,$$

which means ${lim}_{n\to \infty }{y}_n=\bar{x}$. Since $F\bar{x}$ is closed (note that $F\bar{x}$ is compact), $y_n\to \bar{x}$ and $y_n\in F\bar{x}$ then, we get $\bar{x}\in F\bar{x}$.

Corollary 2.5. Let (X, M, *) be a complete fuzzy metric. Suppose F : X → X is a set-valued map with non-empty compact values such that for each x, y ∈ X and t > 0, we have

$$H_M\left(Fx,Fy,kt\right)\ge M\left(x,y,t\right),$$

where 0 < k < 1. Furthermore, assume that (X, M, *) satisfies (2.1) for some x₀ ∈ X and x₁ ∈ Fx₀. Then F has a fixed point.

From Corollary 2.5, we get the following improvement of the above mentioned result of Gregori and Sapena [11] (note that for each t > 0 and h > 1, the sequence t _n = thⁿis s-increasing).

Theorem 2.6. Let (X, M, *) be a complete fuzzy metric space. Suppose f : X → X is a map such that for each x, y ∈ X and t > 0, we have

$$M\left(fx,fy,kt\right)\ge M\left(x,y,t\right),$$

where 0 < k < 1. Furthermore, assume that (X, M, *) satisfies (2.1) for some x₀ ∈ X, each t > 0 and h > 1. Then f has a fixed point.

Let (X, d) be a metric space and A and B are non-empty closed bounded subsets of X. Now set

$$H\left(A,B\right)=max\left\{\underset{x\in A}{sup}\underset{y\in B}{inf}d\left(x,y\right),\underset{y\in B}{sup}\underset{x\in A}{inf}d\left(x,y\right)\right\}.$$

Then H is called the Hausdorff metric. Now, we are ready to derive the following version of Mizoguchi-Takahashi fixed point theorem [20].

Corollary 2.7. Let (X, d) be a complete metric space. Suppose F : M → M is a set-valued map with non-empty compact values such that for some k < 1

$$H\left(Fx,Fy\right)\le \alpha \left(d\left(x,y\right)\right)d\left(x,y\right),$$

where α : [0, ∞) → [0,1) satisfying

$$\underset{r\to t^{+}}{limsup}\alpha \left(r\right)<1,\forall t\in \left[0,\infty \right).$$

Then F has a fixed point.

Proof. Let (X, M, *) be standard fuzzy metric space induced by the metric d with a * b = ab. Now we show that the conditions of Theorem 2.4 are satisfied. Since (X, d) is a complete metric space then (X, M, *) is complete. It is easy to see that (X, M, *) satisfies (2.1). For each non-empty closed bounded subsets of X, we have

$$\begin{array}{ll}\hfill H_M\left(A,B,t\right)& =min\left\{\underset{x\in A}{inf}\underset{y\in B}{sup}M\left(x,y,t\right),\underset{y\in B}{inf}\underset{x\in A}{sup}M\left(x,y,t\right)\right\}\\ =min\left\{\underset{x\in A}{inf}\underset{y\in B}{sup}\frac{t}{t+d\left(x,y\right)},\underset{y\in B}{inf}\underset{x\in A}{sup}\frac{t}{t+d\left(x,y\right)}\right\}\\ =min\left\{\frac{t}{t+{sup}_{x\in A}{inf}_{y\in B}d\left(x,y\right)},\frac{t}{t+{sup}_{y\in B}{inf}_{x\in A}d\left(x,y\right)}\right\}\\ =\frac{t}{t+max\left\{{sup}_{x\in A}{inf}_{y\in B}d\left(x,y\right),{sup}_{y\in B}{inf}_{x\in A}d\left(x,y\right)\right\}}\\ =\frac{t}{t+H\left(A,B\right)}.\end{array}$$

By the above and our assumption, we have

$$\begin{array}{ll}\hfill H_M\left(Fx,Fy,\alpha \left(d\left(x,y,t\right)\right)t\right)& =\frac{\alpha \left(d\left(x,y\right)\right)t}{\alpha \left(d\left(x,y\right)\right)t+H\left(Fx,Fy\right)}\\ \ge \frac{\alpha \left(d\left(x,y\right)\right)t}{\alpha \left(d\left(x,y\right)\right)\left(t+d\left(x,y\right)\right)}\\ =\frac{t}{t+d\left(x,y\right)}\\ =M\left(x,y,t\right),\end{array}$$

for each t > 0 and each x, y ∈ X. Therefore, the conclusion follows from Theorem 2.4.

3. Endpoint theory

Let X be a non-empty set and let F : X → 2^Xbe a set-valued map. An element x ∈ X is said to be an endpoint (invariant or stationary point) of F, if Fx = {x}. The investigation of the existence and uniqueness of endpoints of set-valued contraction maps in metric spaces have received much attention in recent years [21–26].

Definition 3.1. Let (X, M, *) be a fuzzy metric space and let F : X → X be a multi-valued mapping. We say that F is continuous if for any convergent sequence x _n → x₀ we have H _M (Fx _n , Fx₀, t) → 1 as n → ∞, for each t > 0.

As far as we know the following is the first endpoint result for set-valued contraction type maps in fuzzy metric spaces.

Theorem 3.2. Let (X, M, *) be a complete fuzzy metric space and let$F:X\to \mathcal{K}\left(X\right)$be a continuous set-valued mapping. Suppose that for each x ∈ X there exists y ∈ Fx satisfying

$$H_M\left(y,Fy,kt\right)\ge M\left(x,y,t\right),\forall t>0,$$

(3.1)

where k ∈ [0,1). Then, F has an endpoint.

Proof. For each x ∈ X, define the function f : X → [0, ∞) by f(x, t) = H _M (x, Fx, t) = inf_y∈FxM(x, y, t), x ∈ X. Suppose that {x _n } converges to x; then for any y ∈ Fx and z ∈ Fx _n , we have

$$\begin{array}{ll}\hfill M\left(x,y,t\right)& \ge M\left(x,x_n,\frac{t}{3}\right)*M\left(x_n,z,\frac{t}{3}\right)*M\left(z,y,\frac{t}{3}\right)\\ \ge M\left(x,x_n,\frac{t}{3}\right)*H_M\left(x_n,F{x}_n,\frac{t}{3}\right)*H_M\left(z,Fx,\frac{t}{3}\right)\\ \ge M\left(x,x_n,\frac{t}{3}\right)*f\left(x_n,\frac{t}{3}\right)*H_M\left(F{x}_n,Fx,\frac{t}{3}\right).\end{array}$$

Since y ∈ Fx is arbitrary then from the above, we get

$$f\left(x,t\right)=H_M\left(x,Fx,t\right)\ge M\left(x,x_n,\frac{t}{3}\right)*f\left(x_n,\frac{t}{3}\right)*H_M\left(F{x}_n,Fx,\frac{t}{3}\right).$$

It follows from the continuity of F that

$$f\left(x,t\right)\ge \underset{n\to \infty }{limsup}\left(M\left(x,x_n,\frac{t}{3}\right)*f\left(x_n\right)*H_M\left(F{x}_n,Fx,\frac{t}{3}\right)\right)=\underset{n\to \infty }{limsup}f\left(x_n,\frac{t}{3}\right).$$

Hence,

$$f\left(x,t\right)\ge \underset{n\to \infty }{limsup}f\left(x_n,\frac{t}{3}\right),$$

whenever x _n → x. Let x₀ ∈ X. Then by (3.1) there exists a x₁ ∈ Fx₀ such that

$$H_M\left(x_1,F{x}_1,kt\right)\ge M\left(x_0,x_1,t\right).$$

Continuing this process, we can choose a sequence {x _n }_n≥0in X such that x_n+1∈ Fx _n satisfying

$$H_M\left(x_{n+1},F{x}_{n+1},kt\right)\ge M\left(x_n,x_{n+1},t\right).$$

(3.2)

From the definition of H _M (x _n , Tx _n ), we have

$$M\left(x_n,x_{n+1},t\right)\ge H_M\left(x_n,F{x}_n,t\right).$$

(3.3)

From (3.2) and (3.3), we get

$$\begin{array}{ll}\hfill H_M\left(x_{n+1},F{x}_{n+1},kt\right)& \ge M\left(x_n,x_{n+1},t\right)\\ \ge H_M\left(x_n,F{x}_n,t\right)\\ \ge H_M\left(x_n,F{x}_n,kt\right)\\ \ge M\left(x_{n-1},x_n,\frac{1}{k}t\right),\end{array}$$

(3.4)

which implies that {H _M (x _n , Fx _n , kt)} _n is a non-negative non-decreasing sequence of real numbers and so is convergent. To find the limit of {H(x _n , Fx _n , kt)} _n notice that

$$\begin{array}{ll}\hfill H_M\left(x_{n+1},F{x}_{n+1},kt\right)& \ge H_M\left(x_n,F{x}_n,t\right)\\ \ge H_M\left(x_{n-1},F{x}_{n-1},\frac{1}{k}t\right)\ge \cdots \ge H_M\left(x_0,F{x}_0,\frac{1}{k^n}t\right).\end{array}$$

(3.5)

Since Fx₀ is compact then there exists a y₀ ∈ Fx₀ such that

$$H_M\left(x_0,F{x}_0,\frac{1}{k^n}t\right)=M\left(x_0,y_0,\frac{1}{k^n}t\right).$$

(3.6)

(3.5) together with (3.6) imply that for each n ∈ ℕ

$$H_M\left(x_{n+1},F{x}_{n+1},kt\right)\ge M\left(x_0,y_0,\frac{1}{k^n}t\right).$$

From (2.1) we have ${lim}_{n\to \infty }M\left(x_0,y_0,\frac{1}{k^n}t\right)=1$ and so

$$\underset{n\to \infty }{lim}{H}_M\left(x_n,F{x}_n,t\right)=1,\forall t>0.$$

From (3.2), we get

$$M\left(x_n,x_{n+1},t\right)\ge M\left(x_{n-1},x_n,\frac{1}{k}t\right),$$

from which and Lemma (2.1), we get {x _n } is a Cauchy sequence. Since (X, M, *) is a complete fuzzy metric space then there exists a $\bar{x}\in X$ such that ${lim}_{n\to \infty }{x}_n=\bar{x}$. By assumption the function f(x) = H _M (x, Fx, t) is upper semicontinuous, then

$$H_M\left(\bar{x},F\bar{x},t\right)\ge \underset{n\to \infty }{lim}{H}_M\left(x_n,F{x}_n,t\right)=1.$$

Thus

$$H_M\left(\bar{x},F\bar{x},t\right)=1,$$

and so $F\bar{x}=\left\{\bar{x}\right\}$.