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Common fixed point theorem for a hybrid pair of mappings in Hausdorff fuzzy metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 225 (2012)
Abstract
In this paper, we prove a coupled fixed point theorem for a multivalued fuzzy contraction mapping in complete Hausdorff fuzzy metric spaces. As an application of the first theorem, a coupled coincidence and coupled common fixed point theorem has been proved for a hybrid pair of multivalued and singlevalued mappings. It is worth mentioning that to find coupled coincidence points, we do not employ the condition of continuity of any mapping involved therein. Also, coupled coincidence points are obtained without exploiting any type of commutativity condition. Our results extend, improve, and unify some wellknown results in the literature.
MSC:47H10, 47H04, 47H07.
1 Introduction and preliminaries
Bhaskar and Lakshmikantham [1] introduced the concept of a coupled fixed point of a mapping F from X\times X to X and established some coupled fixed point theorems in partially ordered sets. Later on some authors gave improved and generalized results in this context. For details, we refer to [2, 3].
The concept of fuzzy sets was initiated by Zadeh [4] in 1965. Fuzzy metric spaces were introduced by Kramosil and Michalek [5]. George and Veeramani [6, 7] modified the notion of fuzzy metric spaces by using continuous tnorm and generalized the concept of a probabilistic metric space to a fuzzy situation. Then a number of authors started the study of fixed point theory in fuzzy metric spaces; for a detailed survey, we refer to [8–18] and the references therein. Recently López and Romaguera [19] introduced a Hausdorff fuzzy metric on a set of nonempty compact subsets of a given fuzzy metric space. In 2011, Kiany et al. [20] proved fixed point and endpoint theorems for setvalued fuzzy contraction maps in fuzzy metric spaces.
Recently Abbas [21] introduced the concept of coupled fixed points of a mapping F:X\times X\to {2}^{X} (a collection of all nonempty subsets of X) and coupled coincidence points of a hybrid pair F and g:X\to X. The aim of this paper is to obtain a coupled fixed point theorem for F and a coupled coincidence and coupled common fixed point theorem for a hybrid pair \{F,g\} which satisfies a contractive condition in complete Hausdorff fuzzy metric spaces. It is to be noted that to find coupled coincidence points, we do not employ the condition of commutativity and continuity of any mapping involved therein. Our results unify, extend, and generalize various known comparable results given in existing literature (see, for example, [20] and some references therein).
Definition 1 [22]
A binary operation \ast :{[0,1]}^{2}\to [0,1] is called a continuous tnorm if

(1)
∗ is associative and commutative;

(2)
∗ is continuous;

(3)
a\ast 1=a for all a\in [0,1];

(4)
a\ast b\le c\ast d whenever a\le c and b\le d.
Definition 2 [6]
Let X be a nonempty set and ∗ be a continuous tnorm. If a mapping M:{X}^{2}\times [0,\mathrm{\infty})\to [0,1] satisfies the following conditions:
(F1) M(x,y,t)>0;
(F2) M(x,y,t)=1 if and only if x=y;
(F3) M(x,y,t)=M(y,x,t);
(F4) M(x,y,t)\ast M(y,z,s)\le M(x,z,t+s);
(F5) M(x,y,t):(0,\mathrm{\infty})\to [0,1] is continuous;
for each x,y,z\in X and s,t>0, then 3tuple (X,M,\ast ) is called a fuzzy metric space.
Example 3 [6]
Let (X,d) be a metric space. Define a\ast b=min\{a,b\} and
for all x,y\in X and t>0. Then (X,M,\ast ) is a fuzzy metric space. We call this a fuzzy metric M, the standard fuzzy metric induced by d.
Definition 4 [6]
Let (X,M,\ast ) be a fuzzy metric space.

(i)
A sequence \{{x}_{n}\} is said to be convergent to a point x\in X if {lim}_{n\to \mathrm{\infty}}M({x}_{n},x,t)=1 for all t>0.

(ii)
A sequence \{{x}_{n}\} is said to be a Cauchy sequence if {lim}_{n\to \mathrm{\infty}}M({x}_{m},{x}_{n},t)=1 for all t>0.

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

(iv)
A subset A\subseteq X is said to be closed if for each convergent sequence \{{x}_{n}\} with {x}_{n}\in A and {x}_{n}\to x, we have x\in A.

(v)
A subset A\subseteq X is said to be compact if each sequence in A has a convergent subsequence. The set of all compact subsets of X will be denoted by K(X).
Lemma 5 [10]
For all x,y\in X, M(x,y,\cdot ) is nondecreasing.
Definition 6 Let (M,X,\ast ) be a fuzzy metric space, M is said to be continuous on {X}^{2}\times (0,\mathrm{\infty}) if
whenever \{({x}_{n},{y}_{n},{t}_{n})\} is a sequence in {X}^{2}\times (0,\mathrm{\infty}) which converges to a point (x,y,t)\in {X}^{2}\times (0,\mathrm{\infty}); that is,
Lemma 7 [10]
M is a continuous function on {X}^{2}\times (0,\mathrm{\infty}).
Kiany et al. [20] introduced the following lemma in fuzzy metric spaces.
Lemma 8 [20]
Let (X,M,\ast ) be a fuzzy metric space satisfying
for every x,y\in X, t>0, and h>1. Suppose \{{x}_{n}\} is a sequence in X satisfying
for all n\in \mathbb{N} and 0<\alpha <1. Then \{{x}_{n}\} is a Cauchy sequence.
Lemma 9 [19]
Let (X,M,\ast ) be a fuzzy metric space. Then, for each a\in X, B\in K(X), and t>0, there is a {b}_{0}\in B such that M(a,B,t)=M(a,{b}_{0},t), where
Definition 10 [19]
Let (X,M,\ast ) be a fuzzy metric space. For each A,B\in K(X) and t>0, set
The 3tuple (K(X),{H}_{M},\ast ) is called a Hausdorff fuzzy metric space.
Lemma 11 [23]
Let X be a nonempty set and g:X\to X be a mapping. Then there exists a subset E\subseteq X such that g(E)=g(X) and g:E\to X is onetoone.
Theorem 12 [20]
Let (X,M,\ast ) be a complete fuzzy metric. Suppose F:X\times X\to K(X) is a multivalued mapping such that
for each x,y\in X and t>0, where \varphi :[0,\mathrm{\infty})\to [0,1] satisfying {lim\hspace{0.17em}sup}_{r\to {t}^{+}}\varphi (r)<1, for all t\in [0,\mathrm{\infty}), and d(x,y,t)=\frac{t}{M(x,y,t)}t. Furthermore, assume that (X,M,\ast ) satisfies (1) for some {x}_{0} and {x}_{1}\in F({x}_{0}). Then F has a fixed point.
We also need the following definitions given in [21].
Definition 13 [21]
Let X be a nonempty set, F:X\times X\to {2}^{X} (a collection of all nonempty subsets of X) and g:X\to X. An element (x,y)\in X\times X is called
(C1) a coupled fixed point of F if x\in F(x,y) and y\in F(y,x);
(C2) a coupled coincidence point of a hybrid pair \{F,g\} if g(x)\in F(x,y) and g(y)\in F(y,x);
(C3) a coupled common fixed point of a hybrid pair \{F,g\} if x=g(x)\in F(x,y) and y=g(y)\in F(y,x).
We denote the set of coupled coincidence points of mappings F and g by C(F,g). Note that if (x,y)\in C(F,g), then (y,x) is also in C(F,g).
Definition 14 [21]
Let F:X\times X\to {2}^{X} be a multivalued mapping and g be a selfmap on X. The hybrid pair \{F,g\} is called wcompatible if g(F(x,y))\subseteq F(gx,gy) whenever (x,y)\in C(F,g).
Definition 15 [21]
Let F:X\times X\to {2}^{X} be a multivalued mapping and g be a selfmapping on X. The mapping g is called Fweakly commuting at some point (x,y)\in X\times X if {g}^{2}(x)\in F(gx,gy) and {g}^{2}(y)\in F(gy,gx).
2 Coupled fixed and coincidence point theorems
In the following theorem, we obtain a coupled fixed point for a multivalued mapping satisfying a contractive condition.
Theorem 16 Let (X,M,\ast ) be a complete fuzzy metric space and let F:X\times X\to K(X) be a setvalued mapping satisfying
for each x,y,u,v\in X, t>0. Suppose that d(x,u,t)=\frac{t}{M(x,u,t)}t and \varphi :[0,\mathrm{\infty})\to [0,1) is a mapping satisfying
for all t\in [0,\mathrm{\infty}). Furthermore, assume that (X,M,\ast ) satisfies (1) for some {x}_{0},{x}_{1}\in F({x}_{0},{y}_{0}) and {y}_{0},{y}_{1}\in F({y}_{0},{x}_{0}). Then F has a coupled fixed point.
Proof Let {x}_{0},{y}_{0}\in X be arbitrary. Choose {x}_{1}\in F({x}_{0},{y}_{0}) and {y}_{1}\in F({y}_{0},{x}_{0}). Since F is compact valued, then by Lemma 9 there exists {x}_{2}\in F({x}_{1},{y}_{1}) such that
Since F is compact valued, there exists {y}_{2}\in F({y}_{1},{x}_{1}) such that
Continuing this process, we obtain a sequence {\{{x}_{n}\}}_{n\ge 0} and {\{{y}_{n}\}}_{n\ge 0} in X such that {x}_{n+1}\in F({x}_{n},{y}_{n}) and {y}_{n+1}\in F({y}_{n},{x}_{n}) satisfying
That is, we obtain
Similarly,
Inequalities (3) and (4) show that the sequences {\{M({x}_{n},{x}_{n+1},t)\}}_{n} and {\{M({y}_{n},{y}_{n+1},t)\}}_{n} are nondecreasing. Thus, d({x}_{n},{x}_{n+1},t) and d({y}_{n},{y}_{n+1},t) are nonnegative nonincreasing and so they are convergent, say, to {l}_{1}\ge 0 and {l}_{2}\ge 0. Since, by the given assumption,
then there exist {k}_{1}<1, {k}_{2}<1, and {N}_{1},{N}_{2}\in \mathbb{N} such that
Since M(x,y,\cdot ) is nondecreasing, then (3), (4), and (6) yield
Then we obtain
Hence, by Lemma 8, \{{x}_{n}\} and \{{y}_{n}\} are Cauchy sequences. Since (X,M,\ast ) is a complete fuzzy metric space, then there exist x and y in X such that {lim}_{n\to \mathrm{\infty}}{x}_{n}=x and {lim}_{n\to \mathrm{\infty}}{y}_{n}=y, we have
Thus
Since
then there exist {\lambda}_{1}, {\lambda}_{2} such that {\lambda}_{1}<1, {\lambda}_{2}<1, and {n}_{1},{n}_{2}\in \mathbb{N} such that
Now, we show that x\in F(x,y) and y\in F(y,x). Consider
On taking limit as n\to \mathrm{\infty}, we obtain
So that
Similarly, we can obtain
Since {x}_{n+1}\in F({x}_{n},{y}_{n}) and {y}_{n+1}\in F({y}_{n},{x}_{n}), from (7) and (8) we obtain
There exist sequences {w}_{n}\in F(x,y) and {z}_{n}\in F(y,x) such that
for each t>0. Now, for each n\in \mathbb{N}, we have
On taking limit as n\to \mathrm{\infty}, we get
Similarly, for each n\in \mathbb{N}, we have
On taking limit as n\to \mathrm{\infty}, we get

(9)
and (10) imply
\underset{n\to \mathrm{\infty}}{lim}{w}_{n}=x\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{n\to \mathrm{\infty}}{lim}{z}_{n}=y.
Since F(x,y) and F(y,x) are compact, we get x\in F(x,y) and y\in F(y,x). □
Corollary 17 Let (X,M,\ast ) be a complete fuzzy metric space and let F:X\times X\to K(X) be a mapping satisfying
for each x,y,u,v\in X, t>0, and 0<k<1. Suppose that (X,M,\ast ) satisfies (1) for some {x}_{0},{x}_{1}\in F({x}_{0},{y}_{0}) and {y}_{0},{y}_{1}\in F({y}_{0},{x}_{0}). Then F has a coupled fixed point.
Corollary 18 Let (X,M,\ast ) be a complete fuzzy metric space and let F:X\times X\to K(X) be a setvalued mapping satisfying
for each x,y,u,v\in X, t>0. Suppose that d(x,u,t)=\frac{t}{M(x,u,t)}t and \varphi :[0,\mathrm{\infty})\to [0,1) is a mapping satisfying
for all t\in [0,\mathrm{\infty}). Furthermore, assume that (X,M,\ast ) satisfies (1) for some {x}_{0},{x}_{1}\in F({x}_{0},{y}_{0}) and {y}_{0},{y}_{1}\in F({y}_{0},{x}_{0}). Then F has a coupled fixed point.
Corollary 19 Let (X,M,\ast ) be a complete fuzzy metric space and let F:X\times X\to K(X) be a mapping satisfying
for each x,y,u,v\in X, t>0, and 0<k<1. Suppose that (X,M,\ast ) satisfies (1) for some {x}_{0},{x}_{1}\in F({x}_{0},{y}_{0}) and {y}_{0},{y}_{1}\in F({y}_{0},{x}_{0}). Then F has a coupled fixed point.
Example 20 Let X=\{0,1,2\} and d:X\times X\to R be defined as
Hence, (X,d) is a metric space. Consider
for t>0 and (X,M,\ast ) satisfies (1). Define F:X\times X\to CB(X) as follows:
For x\in \{0,2\} and u\in \{0,2\} and for x=1 and u=1, we have
for any k\in (0,1). Hence, (11) is satisfied. For the rest of cases,
and
Hence, for all x,y,u,v\in X, t>0, and k=\frac{2}{3}, (11) holds. All the conditions of Corollary 17 and Theorem 16 with \varphi (t)=k are satisfied. Moreover, (0,0) and (2,2) are coupled fixed points of F.
Example 21 Let X=[0,1] be endowed with the usual metric d(x,y)=xy and F(x,y)=[ln(1+x),1] for each x,y\in X. Let M(x,y,t)=\frac{t}{t+d(x,y)} and let \varphi (t)=\frac{ln(1+t)}{t} for each t>0. Then we have
for each x,y\in X and t>0. Then by Corollary 18, F has a coupled fixed point ((0,0) is a coupled fixed point of F).
Example 22 Let X=R be endowed with the usual metric d(x,y)=xy and let f(x,y)=\frac{1}{2}x for each x,y\in X. Let M(x,y,t)=\frac{t}{t+d(x,y)} and let \varphi (t)=\frac{1}{2} for each t>0. Then we have
for each x,y\in X and t>0. Then by Corollary 19, f has a coupled fixed point ((0,0) is a coupled fixed point of f).
Now, as an application of the above theorem, we obtain a coupled coincidence and common fixed point theorem for a hybrid pair of multivalued and singlevalued mappings.
Theorem 23 Let (X,M,\ast ) be a complete fuzzy metric space and let F:X\times X\to K(X) and g:X\to X be mappings satisfying
for each x,y,u,v\in X, t>0. Suppose that d(x,u,t)=\frac{t}{M(x,u,t)}t and \varphi :[0,\mathrm{\infty})\to [0,1) is a mapping satisfying
for all t\in [0,\mathrm{\infty}). Furthermore, assume that F(X\times X)\subseteq g(X) and (X,M,\ast ) satisfies (1) for some g{x}_{0},g{x}_{1}\in F({x}_{0},{y}_{0}) and g{y}_{0},g{y}_{1}\in F({y}_{0},{x}_{0}). Then F and g have a coupled coincidence point. Moreover, F and g have a coupled common fixed point if one of the following conditions holds:

(a)
F and g are wcompatible, {lim}_{n\to \mathrm{\infty}}{g}^{n}x=u and {lim}_{n\to \mathrm{\infty}}{g}^{n}y=v for some (x,y)\in C(F,g), u,v\in X, and g is continuous at u and v.

(b)
g is Fweakly commuting for some (x,y)\in C(g,F), and gx and gy are fixed points of g, that is, {g}^{2}x=gx and {g}^{2}y=gy.

(c)
g is continuous at x, y for some (x,y)\in C(g,F) and for some u,v\in X, {lim}_{n\to \mathrm{\infty}}{g}^{n}u=x and {lim}_{n\to \mathrm{\infty}}{g}^{n}v=y.
Proof By Lemma 11 there exists E\subseteq X such that g:E\to X is onetoone and g(E)=g(X). Now, define a mapping \mathcal{A}:g(E)\times g(E)\to K(X) by
and since g is oneone on E, so \mathcal{A} is well defined. Further,
Hence, \mathcal{A} satisfies (2) and all the conditions of Theorem 16. By using Theorem 16 with a mapping \mathcal{A}, it follows that \mathcal{A} has a coupled fixed point (u,v)\in g(E)\times g(E). Finally, it is left to prove that F and g have a coupled coincidence point. Since \mathcal{A} has a coupled fixed point (u,v)\in g(E)\times g(E), we get
Since F(X\times X)\subseteq g(X), so there exist {u}_{1},{v}_{1}\in X\times X such that g{u}_{1}=u and g{v}_{1}=v. Thus, it follows from (15),
This implies that ({u}_{1},{v}_{1})\in X\times X is a coupled coincidence point of F and g. Hence, C(F,g) is nonempty. Suppose now that (a) holds. Then, for some (x,y)\in C(F,g),
where u,v\in X. Since g is continuous at u and v, we have that u and v are fixed points of g. As F and g are wcompatible, so
That is, for all n\ge 1,
Using (12), we obtain
On taking limit as n\to \mathrm{\infty}, we get
This implies gu\in F(u,v). Similarly, gv\in F(v,u). Consequently, u=gu\in F(u,v) and v=gv\in F(v,u). Hence, (u,v) is a coupled common fixed point of F and g. Suppose now that (b) holds. If for some (x,y)\in C(F,g), g is Fcommuting and {g}^{2}x=gx and {g}^{2}y=gy, then
Hence, (gx,gy) is a coupled common fixed point of F and g. Suppose now that (c) holds and assume that for some (x,y)\in C(g,F) and for some u,v\in X, {lim}_{n\to \mathrm{\infty}}{g}^{n}u=x and {lim}_{n\to \mathrm{\infty}}{g}^{n}v=y. By the continuity of g at x and y, we get
Hence, (x,y) is a coupled common fixed point of F and g. □
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Acknowledgements
The authors would like to thank the editor and anonymous reviewers for their helpful comments that helped to improve the presentation of this paper. The third author was partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Iran and by a grant from IPM (No. 91470412).
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Abbas, M., Ali, B. & AminiHarandi, A. Common fixed point theorem for a hybrid pair of mappings in Hausdorff fuzzy metric spaces. Fixed Point Theory Appl 2012, 225 (2012). https://doi.org/10.1186/168718122012225
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DOI: https://doi.org/10.1186/168718122012225
Keywords
 coupled fixed point
 coupled coincidence point
 coupled common fixed point
 tnorm