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# On locally contractive fuzzy set-valued mappings

*Journal of Inequalities and Applications*
**volume 2014**, Article number: 74 (2014)

## Abstract

We prove the existence of common fuzzy fixed points for a sequence of locally contractive fuzzy mappings satisfying generalized Banach type contraction conditions in a complete metric space by using iterations. Our main result generalizes and unifies several well-known fixed-point theorems for multivalued maps. Illustrative examples are also given.

**MSC:**46S40, 47H10, 54H25.

## 1 Introduction

The Banach contraction theorem and its subsequent generalizations play a fundamental role in the field of fixed point theory. In particular, Heilpern introduced in [1] the notion of a fuzzy mapping in a metric linear space and proved a Banach type contraction theorem in this framework. Subsequently several other authors [2–10] have studied and established the existence of fixed points of fuzzy mappings. The aim of this paper is to prove a common fixed-point theorem for a sequence of fuzzy mappings in the context of metric spaces without the assumption of linearity. Our results generalize and unify several typical theorems of the literature.

## 2 Preliminaries

Given a metric space (X,d), denote by CB(X) the family of all nonempty closed bounded subsets of (X,d). As usual, for \zeta \in X and A\in CB(X), we define

Then the Hausdorff metric *H* on CB(X) induced by *d* is defined as

for all A,B\in CB(X).

A fuzzy set in (X,d) is a function with domain *X* and values in I=[0,1]. {I}^{X} denotes the collection of all fuzzy sets in *X*. If *A* is a fuzzy set and \zeta \in X, then the function value A(\zeta ) is called the grade of membership of *ζ* in *A*. The *α*-level set of a fuzzy set *A* is denoted by {A}_{\alpha}, and it is defined as follows:

According to Heilpern [1], a fuzzy set *A* in a metric linear space (X,d) is said to be an approximate quantity if {A}_{\alpha} is compact and convex in *X*, for each \alpha \in (0,1], and {sup}_{\zeta \in X}A(\zeta )=1. The family of all approximate quantities of the metric linear space (X,d) is denoted by W(X).

Now, for A,B\in W(X) and \alpha \in [0,1], define

and

It is well known that {d}_{\mathrm{\infty}} is a metric on W(X).

In case that (X,d) is a (non-necessarily linear) metric space, we also define

whenever A,B\in {I}^{X} and {A}_{\alpha},{B}_{\alpha}\in CB(X), \alpha \in [0,1].

In the sequel the letter ℕ will denote the set of positive integer numbers.

The following well-known properties on the Hausdorff metric (see *e.g.* [11]) will be useful in the next section.

**Lemma 2.1** *Let* (X,d) *be a metric space and let* A,B\in CB(X) *with* H(A,B)<r, r>0. *If* a\in A, *then there exists* b\in B *such that* d(a,b)<r.

**Lemma 2.2** *Let* (X,d) *be a metric space and let* {\{{A}_{n}\}}_{n=1}^{\mathrm{\infty}} *be a sequence in* CB(X) *such that* {lim}_{n\to \mathrm{\infty}}H({A}_{n},A)=0, *for some* A\in CB(X). *If* {\xi}_{n}\in {A}_{n}, *for all* n\in \mathbb{N}, *and* d({\xi}_{n},\xi )\to 0, *then* \xi \in A.

Now, let *X* be an arbitrary set and let *Y* be a metric space. A mapping *T* is called fuzzy mapping if *T* is a mapping from *X* into {I}^{Y}. In fact, a fuzzy mapping *T* is a fuzzy subset on X\times Y with membership function T(\zeta ). The value T(\zeta )(\xi ) is the grade of membership of *ξ* in T(\zeta ).

If (X,d) is a metric space and *T* is a (fuzzy) mapping from *X* into {I}^{X}, we say that \xi \in X is a fixed point of *T* if \xi \in T{(\xi )}_{1}.

We conclude this section with the notion of contractiveness that will be used in our main result.

**Definition 2.3** (compare [12])

Let \epsilon \in (0,\mathrm{\infty}]. A function \psi :[0,\epsilon )\to [0,1) is said to be a *MT*-function if it satisfies Mizoguchi-Takahashi’s condition (*i.e.*, lim{sup}_{r\to {t}^{+}}\psi (r)<1, for all t\in [0,\epsilon )).

Clearly, if \psi :[0,\epsilon )\to [0,1) is a nondecreasing function or a nonincreasing function, then it is a *MT*-function. So the set of *MT*-functions is a rich class.

## 3 Fixed points of fuzzy mappings

Fixed-point theorems for locally contractive mappings were studied, among others, by Edelstein [13], Beg and Azam [14], Holmes [15], Hu [11], Hu and Rosen [16], Ko and Tasi [17], Kuhfitting [18] and Nadler [19].

Heilpern [1] established a fixed-point theorem for fuzzy contraction mappings in metric linear spaces, which is a fuzzy extension of Banach’s contraction principle. Afterwards Azam *et al.* [4, 5], and Lee and Cho [10] further extended Banach’s contraction principle to fuzzy contractive mappings in Heilpern’s sense. In our main result (Theorem 3.1 below) we establish a common fixed-point theorem for a sequence of generalized fuzzy uniformly locally contraction mappings on a complete metric space without the requirement of linearity. This is a generalization of many conventional results of the literature.

Let \epsilon \in (0,\mathrm{\infty}], and \lambda \in (0,1). A metric space (X,d) is said to be *ε*-chainable if given \zeta ,\xi \in X, there exists an *ε*-chain from *ζ* to *ξ* (*i.e.*, a finite set of points \zeta ={\zeta}_{0}, {\zeta}_{1},{\zeta}_{2},\dots ,{\zeta}_{m}=\xi such that d({\zeta}_{j-1},{\zeta}_{j})<\epsilon, for all j=1,2,\dots ,m). A mapping T:X\to X is called an (\epsilon ,\lambda ) uniformly locally contractive mapping if \zeta ,\zeta \in X and 0<d(\zeta ,\zeta )<\epsilon, implies d(T\zeta ,T\xi )\le \lambda d(\zeta ,\xi ). A mapping T:X\to W(X) is called an (\epsilon ,\lambda ) uniformly locally contractive fuzzy mapping if \zeta ,\xi \in X and 0<d(\zeta ,\xi )<\epsilon, imply {d}_{\mathrm{\infty}}(T(\zeta ),T(\xi ))\le \lambda d(\zeta ,\xi ). We remark that a globally contractive mapping can be regarded as an (\mathrm{\infty},\lambda ) uniformly locally contractive mapping and for some special spaces every locally contractive mapping is globally contractive.

**Theorem 3.1** *Let* \epsilon \in (0,\mathrm{\infty}], (X,d) *a complete* *ε*-*chainable metric space and* {\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}} *a sequence of fuzzy mappings from* *X* *into* {I}^{X} *such that*, *for each* \zeta \in X *and* i\in \mathbb{N}, {T}_{i}{(\zeta )}_{1}\in CB(X). *If*

*for all* i,j\in \mathbb{N}, *where* \psi :[0,\epsilon )\to [0,1) *is a* *MT*-*function*, *then the sequence* {\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}} *has a common fixed point*, *i*.*e*., *there is* {\xi}^{\ast}\in X *such that* {\xi}^{\ast}\in {T}_{i}{({\xi}^{\ast})}_{1}, *for all* i\in \mathbb{N}.

*Proof* Let {\xi}_{0} be an arbitrary, but fixed element of *X*. Find {\xi}_{1}\in X such that {\xi}_{1}\in {T}_{1}{({\xi}_{0})}_{1}. Let

be an arbitrary *ε*-chain from {\xi}_{0} to {\xi}_{1}. (We suppose, without loss of generality, that {\zeta}_{(1,i)}\ne {\zeta}_{(1,j)}, for each i,j\in \{0,1,2,\dots ,m\} with i\ne j.)

Since 0<d({\zeta}_{(1,0)},{\zeta}_{(1,1)})<\epsilon, we deduce that

Rename {\xi}_{1} as {\zeta}_{(2,0)}. Since {\zeta}_{(2,0)}\in {T}_{1}{({\zeta}_{(1,0)})}_{1}, using Lemma 2.1 we find {\zeta}_{(2,1)}\in {T}_{2}{({\zeta}_{(1,1)})}_{1} such that

Similarly we may choose an element {\zeta}_{(2,2)}\in {T}_{2}{({\zeta}_{(1,2)})}_{1} such that

Thus we obtain a set \{{\zeta}_{(2,0)},{\zeta}_{(2,1)},{\zeta}_{(2,2)},\dots ,{\zeta}_{(2,m)}\} of m+1 points of *X* such that {\zeta}_{(2,0)}\in {T}_{1}{({\zeta}_{(1,0)})}_{1} and {\zeta}_{(2,j)}\in {T}_{2}{({\zeta}_{(1,j)})}_{1}, for j=1,2,\dots ,m, with

for j=0,1,2,\dots ,m-1.

Let {\zeta}_{(2,m)}={\xi}_{2}. Thus the set of points {\xi}_{1}={\zeta}_{(2,0)},{\zeta}_{(2,1)},{\zeta}_{(2,2)},\dots ,{\zeta}_{(2,m)}={\xi}_{2}\in {T}_{2}{({\xi}_{1})}_{1} is an *ε*-chain from {\xi}_{0} to {\xi}_{1}. Rename {\xi}_{2} as {\zeta}_{(3,0)}. Then by the same procedure we obtain an *ε*-chain

from {\xi}_{2} to {\xi}_{3}. Inductively, we obtain

with

for j=0,1,2,\dots ,m-1.

Consequently, we construct a sequence {\{{\xi}_{n}\}}_{n=1}^{\mathrm{\infty}} of points of *X* with

for all n\in \mathbb{N}.

For each j\in \{0,1,2,\dots ,m-1\}, we deduce from (2) that {\{d({\zeta}_{(n,j)},{\zeta}_{(n,j+1)})\}}_{n=1}^{\mathrm{\infty}} is a decreasing sequence of non-negative real numbers and therefore there exists {l}_{j}\ge 0 such that

By assumption, lim{sup}_{t\to {l}_{j}^{+}}\psi (t)<1, so there exists {n}_{j}\in \mathbb{N} such that \psi (d({\zeta}_{(n,j)},{\zeta}_{(n,j+1)}))<s({l}_{j}), for all n\ge {n}_{j} where lim{sup}_{t\to {l}_{j}^{+}}\psi (t)<s({l}_{j})<1.

Now put

Then, for every n>{n}_{j}, we obtain

Putting N=max\{{n}_{j}:j=0,1,2,\dots ,m-1\}, we have

for all n>N+1. Hence

whenever p>n>N+1.

Since {M}_{j}<1, for all j\in \{0,1,2,\dots ,m-1\}, it follows that {\{{\xi}_{n}\}}_{n=1}^{\mathrm{\infty}} is a Cauchy sequence. Since (X,d) is complete, there is {\xi}^{\ast}\in X such that {\xi}_{n}\to {\xi}^{\ast}. So for each \delta \in (0,\epsilon ] there is {M}_{\delta}\in \mathbb{N} such that n>{M}_{\delta} implies d({\xi}_{n},{\xi}^{\ast})<\delta. This in view of inequality (1) implies {D}_{1}({T}_{n+1}({\xi}_{n}),{T}_{i}({\xi}^{\ast}))<\delta, for all i\in \mathbb{N}. Consequently, H({T}_{n+1}{({\xi}_{n})}_{1},{T}_{i}{({\xi}^{\ast})}_{1})\to 0. Since {\xi}_{n+1}\in {T}_{n+1}{({\xi}_{n})}_{1} with d({\xi}_{n+1},{\xi}^{\ast})\to 0, we deduce from Lemma 2.2 that {\xi}^{\ast}\in {T}_{i}{({\xi}^{\ast})}_{1}, for all i\in \mathbb{N}. This completes the proof. □

**Corollary 3.2** *Let* \epsilon \in (0,\mathrm{\infty}], (X,d) *a complete* *ε*-*chainable metric space and* {\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}} *a sequence of fuzzy mappings from* *X* *into* {I}^{X} *such that*, *for each* \zeta \in X *and* i\in \mathbb{N}, {T}_{i}{(\zeta )}_{1}\in CB(X). *If*

*for all* i,j\in \mathbb{N}, *where* \lambda \in (0,1), *then the sequence* {\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}} *has a common fixed point*.

*Proof* Apply Theorem 3.1 when *ψ* is the *MT*-function defined as \psi (t)=\lambda, for all t\in [0,\epsilon ). □

**Corollary 3.3** *Let* \epsilon \in (0,\mathrm{\infty}], (X,d) *a complete* *ε*-*chainable metric linear space and* {\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}} *a sequence of fuzzy mappings from* *X* *into* W(X) *satisfying the following condition*:

*for all* i,j\in \mathbb{N}, *where* \psi :[0,\epsilon )\to [0,1) *is a* *MT*-*function*. *Then the sequence* {\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}} *has a common fixed point*.

*Proof* Since W(X)\subseteq CB(X) and {D}_{1}({T}_{i}(\zeta ),{T}_{j}(\xi ))\le {d}_{\mathrm{\infty}}({T}_{i}(\zeta ),{T}_{j}(\xi )), for all i,j\in \mathbb{N}, the result follows immediately from Theorem 3.1. □

**Corollary 3.4** *Let* \epsilon \in (0,\mathrm{\infty}], (X,d) *a complete* *ε*-*chainable metric linear space and* {\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}} *a sequence of fuzzy mappings from* *X* *into* W(X) *satisfying the following condition*:

*for all* i,j\in \mathbb{N}, *where* \lambda \in (0,1). *Then the sequence* {\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}} *has a common fixed point*.

**Corollary 3.5** [4]

*Let* \epsilon \in (0,\mathrm{\infty}], (X,d) *a complete* *ε*-*chainable metric linear space and* {T}_{1}, {T}_{2}, *two fuzzy mappings from* *X* *into* W(X) *satisfying the following condition*:

*for* i,j=1,2, *where* \psi :[0,\epsilon )\to [0,1) *is a* *MT*-*function*. *Then* {T}_{1} *and* {T}_{2} *have a common fixed point*.

*Let* \epsilon \in (0,\mathrm{\infty}], (X,d) *a complete* *ε*-*chainable metric linear space and* *T*: X\to W(X) *an* (\epsilon ,\lambda ) *uniformly locally contractive fuzzy mapping*. *Then* *T* *has a fixed point*.

**Corollary 3.7** *Let* \epsilon \in (0,\mathrm{\infty}], (X,d) *a complete* *ε*-*chainable metric space and* *S* *be a multivalued mapping from* *X* *into* CB(X) *satisfying the following condition*:

*where* \psi :[0,\epsilon )\to [0,1) *is a* *MT*-*function*. *Then* *S* *has a fixed point*.

*Proof* Define a fuzzy mapping *T* from *X* into {I}^{X} as T(\xi )(t)=1 if t\in S(\xi ) and T(\xi )(t)=0, otherwise. Then T{(\xi )}_{1}=S(\xi ), for all \xi \in X, so T{(\xi )}_{1}\in CB(X), for all \xi \in X. Since

for all \zeta ,\xi \in X, we deduce that condition (1) of Theorem 3.1 is satisfied for *T*. Hence *T* has a fixed point {\xi}^{\ast}, *i.e.*, {\xi}^{\ast}\in T{({\xi}^{\ast})}_{1}. We conclude that {\xi}^{\ast}\in S({\xi}^{\ast}). The proof is complete. □

**Corollary 3.8** [13]

*Let* \epsilon \in (0,\mathrm{\infty}], (X,d) *a complete* *ε*-*chainable metric space and* *S* *be a multivalued mapping from* *X* *into* CB(X) *satisfying the following condition*:

*where* \lambda \in (0,1). *Then* *S* *has a fixed point*.

**Corollary 3.9** ([20, 21], see also [9, 13])

*Let* (X,d) *be a complete metric space*, *S* *a multivalued mapping from* *X* *into* CB(X) *and* \psi :[0,\mathrm{\infty})\to [0,1) *a* *MT*-*function such that*

*for all* \zeta ,\xi \in X. *Then* *S* *has a fixed point in* *X*.

*Proof* Apply Corollary 3.8 with \epsilon =\mathrm{\infty}. □

We conclude the paper with two examples to support Theorem 3.1 and Corollary 3.2.

**Example 3.10** Let (X,d) be the compact, and thus complete, metric space such that X=[0,1], and d(x,y)=|x-y|, for all x,y\in X. Let *λ* be a constant such that \lambda \in [1/14,1) and let {\{{T}_{k}\}}_{k=1}^{\mathrm{\infty}} be the sequence of fuzzy mappings defined from *X* into {I}^{X} as follows:

For each x,y\in X with x\ne y, and i,j\in \mathbb{N} we have

Hence, for \psi (t)=\lambda, the conditions of Corollary 3.2, and hence of Theorem 3.1, are satisfied for any \epsilon \in (0,\mathrm{\infty}], whereas *X* is not linear. Therefore all previous relevant fixed point results Corollaries 3.3-3.6 on metric linear spaces are not applicable.

**Example 3.11** Let (X,d) be the complete metric space such that X=[0,\mathrm{\infty}), d(x,x)=0, for all x\in X, and d(x,y)=max\{x,y\} whenever x\ne y (in the sequel we shall write x\vee y instead of max\{x,y\}).

Note that a sequence {\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}} is a Cauchy sequence in (X,d) if and only if d({x}_{n},0)\to 0. Moreover, x=0 is the only non-isolated point of *X* for the topology induced by *d*.

Let \psi :[0,\mathrm{\infty})\to [0,1) be the *MT*-function defined as

and let {\{{T}_{k}\}}_{k=1}^{\mathrm{\infty}} be the sequence of fuzzy mappings defined from *X* into {I}^{X} as follows:

Observe that, for 0\le x\le 1,

and, for x>1,

Therefore {T}_{k}{(x)}_{1}\in CB(X), for all x\in X and k\in \mathbb{N} (recall that each x\ne 0 is an isolated point for the induced topology, so every bounded interval belongs to CB(X)).

We show that condition (1) of Theorem 3.1 is satisfied for \epsilon =\mathrm{\infty} and *ψ* as defined above. Indeed, let x,y\in X with x\ne y and j,k\in \mathbb{N}. Assume without loss of generality that x>y.

If x,y>1, for each b\in {T}_{j}{(y)}_{1}, we obtain

Similarly, for each a\in {T}_{k}{(x)}_{1}, we obtain

Consequently

If x>1 and y\le 1, we deduce, in a similar way, that

Finally, if x,y\le 1, we deduce

We have shown that all conditions of Theorem 3.1 are satisfied (in fact x=0 is the only fixed point of *T*).

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## Acknowledgements

The third author thanks the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.

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Ahmad, J., Azam, A. & Romaguera, S. On locally contractive fuzzy set-valued mappings.
*J Inequal Appl* **2014, **74 (2014). https://doi.org/10.1186/1029-242X-2014-74

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DOI: https://doi.org/10.1186/1029-242X-2014-74

### Keywords

- fixed point
- fuzzy mapping
- contractive mapping
- locally contractive