# 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 . Kramosil and Michalek  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  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 . In , 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 such that t m + 1 ≤ tm+1, for all mm0.

Gregori and Sapena  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

$\underset{n\to \infty }{lim}\phantom{\rule{2.77695pt}{0ex}}{*}_{i=n}^{\infty }M\left(x,y,{t}_{n}\right)=1.$

Suppose f : XX 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 * bc * d whenever ac and bd 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. (1)

M(x, y, t) > 0,

2. (2)

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

3. (3)

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

4. (4)

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

5. (5)

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

Example 1.5.  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. (1)

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

2. (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.

1. (3)

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

2. (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.

3. (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. 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.,

$\underset{n\to \infty }{lim}M\left({x}_{n},x,t\right)=\underset{n\to \infty }{lim}M\left({y}_{n},y,t\right)=1\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}\underset{n\to \infty }{lim}M\left(x,y,{t}_{n}\right)=M\left(x,y,t\right).$

Lemma 1.9. 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

$\underset{n\to \infty }{lim}\phantom{\rule{2.77695pt}{0ex}}{*}_{i=n}^{\infty }M\left(x,y,t{h}^{i}\right)=1.$
(2.1)

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

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

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

Proof. For each n 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 , 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 such that

$h\alpha <1\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}\sum _{i=l}^{\infty }\frac{1}{{h}^{i}}=\frac{\frac{1}{{h}^{l}}}{1-\frac{1}{h}}<1.$

Hence, for mn, we get

$\begin{array}{ll}\hfill M\left({x}_{n},{x}_{m},t\right)& \ge M\left({x}_{n},{x}_{m},\left(\frac{1}{{h}^{l}}+\frac{1}{{h}^{l+1}}+\cdots +\frac{1}{{h}^{l+m}}\right)t\right)\phantom{\rule{2em}{0ex}}\\ \ge M\left({x}_{n},{x}_{n+1},\frac{1}{{h}^{l}}t\right)*M\left({x}_{n+1},{x}_{n+2},\frac{1}{{h}^{l+1}}t\right)*\cdots *M\left({x}_{m-1},{x}_{m},\frac{1}{{h}^{l+m}}t\right)\phantom{\rule{2em}{0ex}}\\ \ge M\left({x}_{0},{x}_{1},\frac{1}{{\alpha }^{n-1}{h}^{l}}t\right)*M\left({x}_{0},{x}_{1},\frac{1}{{\alpha }^{n}{h}^{l+1}}t\right)*\cdots *M\left({x}_{0},{x}_{1},\frac{1}{{\alpha }^{m-2}{h}^{l+m-n-2}}t\right)\phantom{\rule{2em}{0ex}}\\ \ge M\left({x}_{0},{x}_{1},\frac{1}{{\left(\alpha h\right)}^{n-1}}t\right)*M\left({x}_{0},{x}_{1},\frac{1}{{\left(\alpha h\right)}^{n}}t\right)*\cdots *M\left({x}_{0},{x}_{1},\frac{1}{{\left(\alpha h\right)}^{m-2}}t\right)\phantom{\rule{2em}{0ex}}\\ \ge {*}_{i=n}^{\infty }M\left({x}_{0},{x}_{1},\frac{1}{{\left(\alpha h\right)}^{i-1}}t\right)\phantom{\rule{2em}{0ex}}\end{array}$

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  introduced Hausdorff fuzzy metric on the set of the non-empty compact subsets of a given fuzzy metric space.

Definition 2.2. () 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. 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 : XX 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

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

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 αH 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 Fx0Fx1. 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

$\begin{array}{ll}\hfill M\left({x}_{1},{x}_{2},t\right)\ge M\left({x}_{1},{x}_{2},\alpha \left(d\left({x}_{0},{x}_{1},t\right)\right)t\right)& =\underset{y\in F{x}_{1}}{sup}M\left({x}_{1},y,\alpha \left(d\left({x}_{0},{x}_{1},t\right)\right)t\right)\phantom{\rule{2em}{0ex}}\\ \ge {H}_{M}\left(F{x}_{0},F{x}_{1},\alpha \left(d\left({x}_{0},{x}_{1},t\right)\right)t\right)\phantom{\rule{2em}{0ex}}\\ \ge M\left({x}_{0},{x}_{1},t\right).\phantom{\rule{2em}{0ex}}\end{array}$

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

$\begin{array}{ll}\hfill M\left({x}_{n+1},{x}_{n+2},t\right)\ge M\left({x}_{n+1},{x}_{n+2},\alpha \left(d\left({x}_{n},{x}_{n+1},t\right)\right)t\right)& =\underset{y\in F{x}_{n+1}}{sup}M\left({x}_{n+1},y,\alpha \left(d\left({x}_{n},{x}_{n+1},t\right)\right)t\right)\phantom{\rule{2em}{0ex}}\\ \ge {H}_{M}\left(F{x}_{n},F{x}_{n+1},\alpha \left(d\left({x}_{n},{x}_{n+1},t\right)\right)t\right)\phantom{\rule{2em}{0ex}}\\ \ge M\left({x}_{n},{x}_{n+1},t\right).\phantom{\rule{2em}{0ex}}\end{array}$

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

Thus {d(xn+1, xn+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 such that

$\alpha \left(d\left({x}_{n+1},{x}_{n+2},t\right)\right)N.$
(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 $\stackrel{̄}{x}\in X$ such that ${lim}_{n\to \infty }{x}_{n}=\stackrel{̄}{x}$, that means ${lim}_{n\to \infty }M\left({x}_{n},\stackrel{̄}{x},t\right)=1$, for each t > 0. Thus, ${lim}_{n\to \infty }d\left({x}_{n},\stackrel{̄}{x},t\right)=0$, for each t > 0. Since

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

then there exists k < l < 1 such that

$\underset{n\to \infty }{limsup}\alpha \left(d\left({x}_{n},\stackrel{̄}{x},t\right)\right)

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

$\underset{n\to \infty }{lim}{H}_{M}\left(F{x}_{n},F\stackrel{̄}{x},t\right)=1.$
(2.5)

Since xn+1 Fx n then from (2.5), we have

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

Thus there exists a sequence ${y}_{n}\in F\stackrel{̄}{x}$ such that

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

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

$M\left({y}_{n},\stackrel{̄}{x},s+t\right)\ge M\left({y}_{n},{x}_{n},s\right)*M\left({x}_{n},\stackrel{̄}{x},t\right).$

Hence, from the above, we get

$\underset{n\to \infty }{lim}M\left({y}_{n},\stackrel{̄}{x},t\right)=1,$

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

Corollary 2.5. Let (X, M, *) be a complete fuzzy metric. Suppose F : XX 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  (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 : XX 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 .

Corollary 2.7. Let (X, d) be a complete metric space. Suppose F : MM 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

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

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

$\begin{array}{ll}\hfill {H}_{M}\left(A,B,t\right)& =min\left\{\underset{x\in A}{inf}\underset{y\in B}{sup}M\left(x,y,t\right),\underset{y\in B}{inf}\underset{x\in A}{sup}M\left(x,y,t\right)\right\}\phantom{\rule{2em}{0ex}}\\ =min\left\{\underset{x\in A}{inf}\underset{y\in B}{sup}\frac{t}{t+d\left(x,y\right)},\underset{y\in B}{inf}\underset{x\in A}{sup}\frac{t}{t+d\left(x,y\right)}\right\}\phantom{\rule{2em}{0ex}}\\ =min\left\{\frac{t}{t+{sup}_{x\in A}{inf}_{y\in B}d\left(x,y\right)},\frac{t}{t+{sup}_{y\in B}{inf}_{x\in A}d\left(x,y\right)}\right\}\phantom{\rule{2em}{0ex}}\\ =\frac{t}{t+max\left\{{sup}_{x\in A}{inf}_{y\in B}d\left(x,y\right),{sup}_{y\in B}{inf}_{x\in A}d\left(x,y\right)\right\}}\phantom{\rule{2em}{0ex}}\\ =\frac{t}{t+H\left(A,B\right)}.\phantom{\rule{2em}{0ex}}\end{array}$

By the above and our assumption, we have

$\begin{array}{ll}\hfill {H}_{M}\left(Fx,Fy,\alpha \left(d\left(x,y,t\right)\right)t\right)& =\frac{\alpha \left(d\left(x,y\right)\right)t}{\alpha \left(d\left(x,y\right)\right)t+H\left(Fx,Fy\right)}\phantom{\rule{2em}{0ex}}\\ \ge \frac{\alpha \left(d\left(x,y\right)\right)t}{\alpha \left(d\left(x,y\right)\right)\left(t+d\left(x,y\right)\right)}\phantom{\rule{2em}{0ex}}\\ =\frac{t}{t+d\left(x,y\right)}\phantom{\rule{2em}{0ex}}\\ =M\left(x,y,t\right),\phantom{\rule{2em}{0ex}}\end{array}$

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 → 2Xbe 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 .

Definition 3.1. Let (X, M, *) be a fuzzy metric space and let F : XX 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

${H}_{M}\left(y,Fy,kt\right)\ge M\left(x,y,t\right),\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{1em}{0ex}}t>0,$
(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) = infyFxM(x, y, t), x X. Suppose that {x n } converges to x; then for any y Fx and z Fx n , we have

$\begin{array}{ll}\hfill M\left(x,y,t\right)& \ge M\left(x,{x}_{n},\frac{t}{3}\right)*M\left({x}_{n},z,\frac{t}{3}\right)*M\left(z,y,\frac{t}{3}\right)\phantom{\rule{2em}{0ex}}\\ \ge M\left(x,{x}_{n},\frac{t}{3}\right)*{H}_{M}\left({x}_{n},F{x}_{n},\frac{t}{3}\right)*{H}_{M}\left(z,Fx,\frac{t}{3}\right)\phantom{\rule{2em}{0ex}}\\ \ge M\left(x,{x}_{n},\frac{t}{3}\right)*f\left({x}_{n},\frac{t}{3}\right)*{H}_{M}\left(F{x}_{n},Fx,\frac{t}{3}\right).\phantom{\rule{2em}{0ex}}\end{array}$

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 xn+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

$\begin{array}{ll}\hfill {H}_{M}\left({x}_{n+1},F{x}_{n+1},kt\right)& \ge M\left({x}_{n},{x}_{n+1},t\right)\phantom{\rule{2em}{0ex}}\\ \ge {H}_{M}\left({x}_{n},F{x}_{n},t\right)\phantom{\rule{2em}{0ex}}\\ \ge {H}_{M}\left({x}_{n},F{x}_{n},kt\right)\phantom{\rule{2em}{0ex}}\\ \ge M\left({x}_{n-1},{x}_{n},\frac{1}{k}t\right),\phantom{\rule{2em}{0ex}}\end{array}$
(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

$\begin{array}{ll}\hfill {H}_{M}\left({x}_{n+1},F{x}_{n+1},kt\right)& \ge {H}_{M}\left({x}_{n},F{x}_{n},t\right)\phantom{\rule{2em}{0ex}}\\ \ge {H}_{M}\left({x}_{n-1},F{x}_{n-1},\frac{1}{k}t\right)\ge \cdots \ge {H}_{M}\left({x}_{0},F{x}_{0},\frac{1}{{k}^{n}}t\right).\phantom{\rule{2em}{0ex}}\end{array}$
(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

${H}_{M}\left({x}_{n+1},F{x}_{n+1},kt\right)\ge M\left({x}_{0},{y}_{0},\frac{1}{{k}^{n}}t\right).$

From (2.1) we have ${lim}_{n\to \infty }M\left({x}_{0},{y}_{0},\frac{1}{{k}^{n}}t\right)=1$ and so

$\underset{n\to \infty }{lim}{H}_{M}\left({x}_{n},F{x}_{n},t\right)=1,\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{1em}{0ex}}t>0.$

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 $\stackrel{̄}{x}\in X$ such that ${lim}_{n\to \infty }{x}_{n}=\stackrel{̄}{x}$. By assumption the function f(x) = H M (x, Fx, t) is upper semicontinuous, then

${H}_{M}\left(\stackrel{̄}{x},F\stackrel{̄}{x},t\right)\ge \underset{n\to \infty }{lim}{H}_{M}\left({x}_{n},F{x}_{n},t\right)=1.$

Thus

${H}_{M}\left(\stackrel{̄}{x},F\stackrel{̄}{x},t\right)=1,$

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

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## Acknowledgements

This research was in part supported by the grant from IPM (90470017). The second author was also partially supported by the Center of Excellence for Mathematics, University of Shahrekord.

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Correspondence to Fatemeh Kiany.

### 4. Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors read and approved the final manuscript.

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Kiany, F., Amini-Harandi, A. Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces. Fixed Point Theory Appl 2011, 94 (2011). https://doi.org/10.1186/1687-1812-2011-94

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• DOI: https://doi.org/10.1186/1687-1812-2011-94

### Keywords

• Fixed point
• Endpoint
• Set-valued fuzzy contraction map
• Fuzzy metric space
• Topology 