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Setvalued mappings in partially ordered fuzzy metric spaces
Journal of Inequalities and Applications volume 2014, Article number: 157 (2014)
Abstract
In this paper, we provide coincidence point and fixed point theorems satisfying an implicit relation, which extends and generalizes the result of Gregori and Sapena, for setvalued mappings in complete partially ordered fuzzy metric spaces. Also we prove a fixed point theorem for setvalued mappings on complete partially ordered fuzzy metric spaces which generalizes results of Mihet and Tirado.
MSC:54E40, 54E35, 54H25.
1 Preliminaries
The concept of fuzzy metric space was introduced by Kramosil and Michalek [1] and the modified concept by George and Veeramani [2] (for other modifications see [3, 4]). Furthermore, the fixed point theory in this kind of spaces has been intensively studied (see [5–14]).
The applications of fixed point theorems are remarkable in different disciplines of mathematics, engineering, and economics in dealing with problems in approximation theory, game theory, and many others (see [15] and references therein).
In 2004 RodríguezLópez and Romaguera [16] introduced the Hausdorff fuzzy metric of a given fuzzy metric space in the sense of George and Veeramani on the set of nonempty compact subsets.
Some fixed point results for setvalued mappings on fuzzy metric space can be found in [17, 18] and references therein.
The aim of this paper is to prove a coincidence point and fixed point theorem on a partially ordered fuzzy metric space satisfying an implicit relation and another fixed point theorem. Our result substantially generalizes and extends the result of Gregori and Sapena [8] and results of Miheţ [19] and Tirado [20] and also the result of Latif and Beg [21] for setvalued mappings in complete partially ordered fuzzy metric spaces. Implicit relations have been considered by several authors in connection with solving nonlinear functional equations (see [22–25]).
For the sake of completeness, we briefly recall some basic concepts used in the following.
Definition 1.1 [26]
A binary operation \ast :[0,1]\times [0,1]\to [0,1] is called a continuous tnorm if it satisfies the following conditions:

(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, for each a,b,c,d\in [0,1].
The three basic continuous tnorms are: (i) The minimum tnorm is defined by a\ast b=min\{a,b\}. (ii) The product tnorm is defined by a\ast b=ab. (iii) The Łukasiewicz tnorm is defined by a\ast b=max\{a+b1,0\}.

(i)
A tnorm ∗ is said to be Hadžićtype tnorm, if the family {\{{\ast}^{n}\}}_{n\ge 0} of its iterates defined for each s\in [0,1] by {\ast}^{0}(s)=1, {\ast}^{n}(s)=({\ast}^{n1}(s))\ast s, for all n\ge 0, are equicontinuous at s=1, that is, given \lambda >0, there exists \eta (\lambda )\in (0,1) such that for all n\ge 0
1\ge s>\eta (\lambda )\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{\ast}^{n}(s)>1\lambda .
The tnorm ∗, defined by a\ast b=min\{a,b\} is a trivial example of the tnorm of Hadžićtype, but there are other tnorms of Hadžićtype (see [27]).

(ii)
If ∗ be a tnorm and {\{{x}_{n}\}}_{n\ge 1} is a sequence of numbers in [0,1], one defines recurrently {\ast}_{i=1}^{n}{x}_{i} by {\ast}_{i=1}^{1}{x}_{i}={x}_{1} and {\ast}_{i=1}^{n}{x}_{i}=\ast ({\ast}_{i=1}^{n1}{x}_{i},{x}_{n}), \mathrm{\forall}n\ge 2. {\ast}_{i=1}^{\mathrm{\infty}}{x}_{i} is defined as {lim}_{n\to \mathrm{\infty}}{\ast}_{i=1}^{n}{x}_{i} and {\ast}_{i=n}^{\mathrm{\infty}}{x}_{i} as {\ast}_{i=1}^{\mathrm{\infty}}{x}_{n+i}.
If q\in (0,1) is given, we say that the tnorm is geometrically convergent (gconvergent) if {lim}_{n\to \mathrm{\infty}}{\ast}_{i=n}^{\mathrm{\infty}}(1{q}^{i})=1.
The Łukasiewicz tnorm and tnorms of Hadžićtype are examples of gconvergent tnorms. Other examples be found in [28]. Also note that if the tnorm ∗ is gconvergent, then {sup}_{t<1}t\ast t=1.
Proposition 1.3 [28]

(i)
For a\ast b\ge max\{a+b1,0\} the following implication holds:
\underset{n\to \mathrm{\infty}}{lim}\underset{i=1}{\overset{\mathrm{\infty}}{\ast}}{x}_{n+i}=1\phantom{\rule{1em}{0ex}}\iff \phantom{\rule{1em}{0ex}}\sum _{n=1}^{\mathrm{\infty}}(1{x}_{n})<\mathrm{\infty}. 
(ii)
If ∗ is of Hadžićtype, then {lim}_{n\to \mathrm{\infty}}{\ast}_{i=1}^{\mathrm{\infty}}{x}_{n+i}=1, for every sequence {\{{x}_{n}\}}_{n\in \mathbb{N}} in [0,1] such that {lim}_{n\to \mathrm{\infty}}{x}_{n}=1.
Definition 1.4 [2]
A 3tuple (X,M,\ast ) is called a fuzzy metric space if X is an arbitrary nonempty set, ∗ is a continuous tnorm and M is a fuzzy set on {X}^{2}\times (0,\mathrm{\infty}), satisfying the following conditions for each x,y,z\in X and t,s>0:
(FM1) M(x,y,t)>0,
(FM2) M(x,y,t)=1 for all t>0 if and only if x=y,
(FM3) M(x,y,t)=M(y,x,t),
(FM4) M(x,z,t+s)\ge M(x,y,t)\ast M(y,z,s),
(FM5) M(x,y,\cdot ):(0,\mathrm{\infty})\to [0,1] is continuous.
Definition 1.5 [2]
Let (X,M,\ast ) be a fuzzy metric space. A sequence \{{x}_{n}\} in X is called a Cauchy sequence, if, for each \u03f5\in (0,1) and t>0, there exists {n}_{0}\in \mathbb{N} such that M({x}_{n},{x}_{m},t)>1\u03f5 for all n,m\ge {n}_{0}. A sequence \{{x}_{n}\} in a fuzzy metric space (X,M,\ast ) is said to be convergent to x\in X if {lim}_{n\to \mathrm{\infty}}M({x}_{n},x,t)=1 for all t>0. A 3tuple (X,M,\ast ) is complete if every Cauchy sequence is convergent in X.
Lemma 1.6 [7]
Let (X,M,\ast ) be a fuzzy metric space. Then M(x,y,t) is nondecreasing with respect to t for all x,y\in X.
Definition 1.7 [16]
Let (X,M,\ast ) be a fuzzy metric space. M is said to be continuous on {X}^{2}\times (0,\mathrm{\infty}), if
whenever a sequence \{({x}_{n},{y}_{n},{t}_{n})\} in {X}^{2}\times (0,\mathrm{\infty}) converges to a point (x,y,t)\in {X}^{2}\times (0,\mathrm{\infty}), that is,
Lemma 1.8 [16]
Let (X,M,\ast ) be a fuzzy metric space. Then M is continuous function on {X}^{2}\times (0,\mathrm{\infty}).
Definition 1.9 [6]
Let (X,M,\ast ) be a fuzzy metric space. The fuzzy metric M is triangular if it satisfies the condition
for every x,y,z\in X and every t>0.
Example 1.10 [2]
Let (X,d) be a metric space. Define a\ast b=ab (or a\ast b=min\{a,b\}) and for all x,y\in X and t>0,
Then (X,{M}_{d},\ast ) is a fuzzy metric space. We call the fuzzy metric {M}_{d} induced by the metric d the standard fuzzy metric. Note that every standard fuzzy metric is triangular.
Definition 1.11 Let (X,M,\ast ) is a fuzzy metric space and t>0. (i): 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 as n\to \mathrm{\infty}, we have x\in A.
(ii): A\subseteq X is said to be compact if each sequence in A has a convergent subsequence.
Throughout the article, let \mathcal{P}(X), \mathcal{C}(X), and \mathcal{K}(X) denote the set of all nonempty subsets, the set of all nonempty closed subsets, and the set of all nonempty compact subsets of X, respectively.
Definition 1.12 Let X be a nonempty set. A point x\in X is called a coincidence point of the mappings F:X\to \mathcal{P}(X) and f:X\to X if fx\in Fx. Point x\in X is called a fixed point of the mappings F:X\to \mathcal{P}(X) if fx\in Fx.
Theorem 1.13 [16]
Let (X,M,\ast ) be a fuzzy metric space. For each A,B\in \mathcal{K}(X) and t>0 define
where M(a,B,t):=sup\{M(a,b,t):b\in B\}. Then the 3tuple (\mathcal{K}(X),{H}_{M},\ast ) is a fuzzy metric space.
The fuzzy metric ({H}_{M},\ast ) will be called the Hausdorff fuzzy metric of (M,\ast ) on \mathcal{K}(X).
Lemma 1.14 [16]
Let (X,M,\ast ) be a fuzzy metric space. Then, for each a\in X, B\in \mathcal{K}(X) and t>0, there is {b}_{0}\in B such that
2 Main results
Throughout this section, ∗ denotes a continuous tnorm and the set of all continuous realvalued mappings T:{[0,1]}^{6}\to R satisfying the following properties:
{\mathcal{T}}_{1}: T({t}_{1},{t}_{2},\dots ,{t}_{6}) is nonincreasing in {t}_{2},\dots ,{t}_{6}.
{\mathcal{T}}_{2}: If there exists k\in (0,1) such that for each t>0, we have
where u,v,w:(0,\mathrm{\infty})\to [0,1] are nondecreasing functions with u(t),v(t),w(t)\in (0,1], then w(kt)\ge v(t).
{\mathcal{T}}_{3}: For each t>0 and some k\in (0,1), the condition
implies w(kt)\ge v(t).
Now we give our main result.
Theorem 2.1 Let (X,M,\ast ) be a complete fuzzy metric space with Hadžićtype tnorm ∗ such that M(x,y,t)\to 1 as t\to \mathrm{\infty}, for all x,y\in X. Let ⪯ be a partial order defined on X. Let F:X\to \mathcal{K}(X) be a setvalued mapping with nonempty compact values and f:X\to X a mapping such that f(X) is closed and for some T\in \mathcal{T} and all comparable elements x,y\in X, and t>0, we have
Also suppose that the following conditions are satisfied:

(i)
F(X)\subseteq f(X),

(ii)
fy\in F(x) implies x\u2aafy,

(iii)
if {y}_{n}\in F({x}_{n}) is a sequence such that {y}_{n}\to y=fx, then {x}_{n}\u2aafx for all n.
Then F and f have a coincidence point, that is, there exists x\in X such that fx\in F(x).
Proof Let t>0 be fixed and {x}_{0}\in X. By using (i) and (ii), there exists {x}_{1}\in X such that {x}_{0}\u2aaf{x}_{1} and {y}_{0}=f{x}_{1}\in F{x}_{0}. Now from (i), (ii), and by Lemma 1.14, for {x}_{1}\in X there is {x}_{2}\in X such that {x}_{1}\u2aaf{x}_{2} and {y}_{1}=f{x}_{2}\in F{x}_{1} with
thus
On the other hand by x={x}_{0} and y={x}_{1} in (2.1), we have
Now since M(f{x}_{0},F{x}_{0},t)\ge M(f{x}_{0},{y}_{0},t), M(f{x}_{1},F{x}_{1},t)\ge M({y}_{0},{y}_{1},t), also
and M(f{x}_{1},F{x}_{0},t)\ge M({y}_{0},{y}_{0},t)=1, and by using {\mathcal{T}}_{1}, we get
This means that
where w(t)={H}_{M}(F{x}_{0},F{x}_{1},t), v(t)=M(f{x}_{0},f{x}_{1},t), u(t)=M(f{x}_{1},{y}_{1},t), then from {\mathcal{T}}_{2}, we have (w(kt)\ge v(t))
hence by (2.2), we obtain
Again by (i), (ii), and by Lemma 1.14, there exists {x}_{3}\in X such that {x}_{2}\u2aaf{x}_{3} with {y}_{2}=f{x}_{3}\in F{x}_{2} that satisfying in
Since {x}_{1}\u2aaf{x}_{2} thus by replacing x={x}_{1} and y={x}_{2} in (2.1) and from {\mathcal{T}}_{1}, we obtain
Now by w(t)={H}_{M}(F{x}_{1},F{x}_{2},t), v(t)=M({y}_{0},{y}_{1},t), u(t)=M({y}_{1},{y}_{2},t), the property {\mathcal{T}}_{2} implies
thus from (2.3), we get
Repeatedly, there exists {x}_{4}\in X with {x}_{3}\u2aaf{x}_{4} such that {y}_{3}=f{x}_{4}\in F{x}_{3} and {H}_{M}(F{x}_{2},F{x}_{3},t)\le M({y}_{2},{y}_{3},t), and
therefore
Continuing the process, we can have a sequence \{{x}_{n}\} in X with {x}_{n}\u2aaf{x}_{n+1} such that, for n\ge 0, {y}_{n}=f{x}_{n+1}\in F{x}_{n}, and
and
From (2.4), we conclude that, for each i\ge 1,
Next, we prove that the sequence {y}_{n} is Cauchy. Suppose that \delta >0 and \u03f5\in (0,1) are given. Then, by Lemma 1.6 and (FM4), for all m>n,
On the other hand, putting t=\delta {k}^{i}(1k) in (2.6), for all n\ge 0, i\ge 1, we get
Then by replacing the above inequality in (2.7), we obtain, for all m>n,
By hypothesis, ∗ is a tnorm of Hadžićtype, and there exists \eta \in (0,1) such that for all m>n,
By M(f{x}_{0},{y}_{0},t)\to 1 as t\to \mathrm{\infty}, there exists {n}_{0} such that, for all n\ge {n}_{0},
From (2.5) and the above inequality, we have
therefore, (2.8) and (2.9) imply that, for all n\ge {n}_{0} and each m>n,
This shows that \{{y}_{n}\} is a Cauchy sequence. Since X is complete, there exists some y\in X such that
Now, since f(X) is closed, there exists \overline{x}\in X such that y=f\overline{x}\in f(X). Also (ii) implies that {x}_{n}\u2aaf\overline{x} for any n. Thus from (2.1), we have
By taking the limit as n\to \mathrm{\infty}, by the continuity of T, and from Lemma 1.8, we get
Now by using {\mathcal{T}}_{3}, we have
on the other hand {H}_{M}(F{x}_{n},F\overline{x},kt)\le M({y}_{n},F\overline{x},kt), so
It follows that M(y,F\overline{x},t)=1 for each t>0. Now since F\overline{x} is closed (note that F\overline{x} is compact), we get f\overline{x}=y\in F\overline{x}, thus \overline{x} is a coincidence point of F and f. The proof is complete. □
Remark 2.2 In Theorem 2.1, we proved that the sequence {y}_{n} is Cauchy; one can replace the condition ‘∗ is Hadžićtype tnorm and M(x,y,t)\to 1 as t\to \mathrm{\infty}, for all x,y\in X’ with the following: ‘{lim}_{n\to \mathrm{\infty}}{\ast}_{i=n}^{\mathrm{\infty}}M(x,y,t{h}^{i})=1 for each h>1’. To see this, choose some q>1 and n\in \mathbb{N} such that kq<1 and {\sum}_{i={n}_{1}}^{\mathrm{\infty}}\frac{1}{{q}^{i}}\le 1. Then from (FM4) and (2.5), for every m>n\ge {n}_{1}, we have
Thus, \{{y}_{n}\} is a Cauchy sequence. Then we have the following theorem.
Theorem 2.3 Let (X,M,\ast ) be a complete fuzzy metric space and suppose for each h>1, {lim}_{n\to \mathrm{\infty}}{\ast}_{i=n}^{\mathrm{\infty}}M(x,y,t{h}^{i})=1. Let ⪯ be a partial order defined on X. Let F:X\to \mathcal{K}(X) be a setvalued mapping with nonempty compact values and f:X\to X a mapping such that f(X) is closed and for some T\in \mathcal{T} and all comparable elements x,y\in X, and t>0, we have
Also suppose that the following conditions are satisfied:

(i)
F(X)\subseteq f(X),

(ii)
fy\in F(x) implies x\u2aafy,

(iii)
if {y}_{n}\in F({x}_{n}) is a sequence such that {y}_{n}\to y=fx, then {x}_{n}\u2aafx for all n.
Then F and f have a coincidence point, that is, there exists x\in X such that fx\in F(x).
If in Theorem 2.1 and 2.3 we put T({u}_{1},\dots ,{u}_{6}):=\frac{{u}_{1}(kt)}{{u}_{2}(t)}, where k\in (0,1), then we have the following corollaries.
Corollary 2.4 Let (X,M,\ast ) be a complete fuzzy metric space with Hadžićtype tnorm ∗ such that M(x,y,t)\to 1 as t\to \mathrm{\infty}, for all x,y\in X. Let ⪯ be a partial order defined on X. Let F:X\to \mathcal{K}(X) be a setvalued mapping with nonempty compact values and f:X\to X a mapping such that f(X) be closed and for all comparable elements x,y\in X, and t>0, we have
Also suppose that the following conditions are satisfied:

(i)
F(X)\subseteq f(X),

(ii)
fy\in F(x) implies x\u2aafy,

(iii)
if {y}_{n}\in F({x}_{n}) is a sequence such that {y}_{n}\to y=fx, then {x}_{n}\u2aafx for all n.
Then there exists x\in X such that fx\in F(x).
Corollary 2.5 Let (X,M,\ast ) be a complete fuzzy metric space and suppose for each h>1, {lim}_{n\to \mathrm{\infty}}{\ast}_{i=n}^{\mathrm{\infty}}M(x,y,t{h}^{i})=1. Let ⪯ be a partial order defined on X. Let F:X\to \mathcal{K}(X) be a setvalued mapping with nonempty compact values and f:X\to X a mapping such that f(X) be closed and for all comparable elements x,y\in X, and t>0, we have
Also suppose that the following conditions are satisfied:

(i)
F(X)\subseteq f(X),

(ii)
fy\in F(x) implies x\u2aafy,

(iii)
if {y}_{n}\in F({x}_{n}) is a sequence such that {y}_{n}\to y=fx, then {x}_{n}\u2aafx for all n.
Then there exists x\in X such that fx\in F(x).
Putting f=I (the identity mapping) in Corollary 2.4 and 2.5, we get the following corollaries.
Corollary 2.6 Let (X,M,\ast ) be a complete fuzzy metric space with Hadžićtype tnorm ∗ such that M(x,y,t)\to 1 as t\to \mathrm{\infty}, for some {x}_{0}\in X and {x}_{1}\in F{x}_{0}. Let ⪯ be a partial order defined on X. Let F:X\to \mathcal{K}(X) be a setvalued mapping with nonempty compact values for all comparable elements x,y\in X, and t>0, we have
Also suppose that the following conditions are satisfied:

(i)
y\in F(x) implies x\u2aafy,

(ii)
if {y}_{n}\in F({x}_{n}) is a sequence such that {y}_{n}\to x, then {x}_{n}\u2aafx for all n.
Then F has a fixed point.
Corollary 2.7 Let (X,M,\ast ) be a complete fuzzy metric space and suppose for each h>1, {lim}_{n\to \mathrm{\infty}}{\ast}_{i=n}^{\mathrm{\infty}}M(x,y,t{h}^{i})=1 for some {x}_{0}\in X and {x}_{1}\in F{x}_{0}. Let ⪯ be a partial order defined on X. Let F:X\to \mathcal{K}(X) be a setvalued mapping with nonempty compact values for all comparable elements x,y\in X, and t>0, we have
Also suppose that the following conditions are satisfied:

(i)
y\in F(x) implies x\u2aafy,

(ii)
if {y}_{n}\in F({x}_{n}) is a sequence such that {y}_{n}\to x, then {x}_{n}\u2aafx for all n.
Then F has a fixed point.
Remark 2.8 Note that we assumed the implicit relation (2.1) only for the comparable elements of the partially ordered fuzzy metric space.
Remark 2.9 Corollary 2.7 improves and generalizes the mentioned result of Gregori and Sapena (see Theorem 4.8 of [8]) for setvalued mappings in complete partially ordered fuzzy metric spaces.
In continuation, in the spirit of Miheţ [19], we introduce the notion of a setvalued fuzzy order ψcontraction of (\u03f5,\lambda )type mappings and give a fixed point theorem in partially ordered fuzzy metric spaces.
Definition 2.10 Let (X,M,\ast ) be a fuzzy metric space and \psi :(0,1)\to (0,1). A mapping F:X\to \mathcal{C}(X) is a setvalued fuzzy order ψcontraction of (\u03f5,\lambda )type if the following implication holds:
for every \u03f5>0, \lambda \in (0,1) and all comparable elements x,y\in X.
If \psi (t)=\alpha t (t\in (0,1)) for some \alpha \in (0,1), then F will be called a setvalued fuzzy order αcontraction of (\u03f5,\lambda )type.
Also note that if \psi (t)<t for all t\in (0,1), then every setvalued fuzzy order ψcontraction of (\u03f5,\lambda )type satisfies the relation
for all comparable elements x,y\in X and t>0. Indeed, if for some comparable x,y\in X and t>0 there exists p\in Fx such that for all q\in Fy, we have M(p,q,t)<M(x,y,t); then there is \lambda \in (0,1) such that M(p,q,t)<1\lambda <M(x,y,t), that is, M(x,y,t)>1\lambda and M(p,q,t)<1\lambda <1\psi (\lambda ), which is a contradiction.
Example 2.11 Let (X,M,\ast ) be a fuzzy metric space. Let F:Y\to \mathcal{C}(Y) be a setvalued mapping, where Y\in \mathcal{C}(X). If there is \alpha \in (0,1) such that
for all comparable elements x,y\in X and t>0, then F is a setvalued fuzzy order αcontraction of (\u03f5,\lambda )type. Indeed, if M(x,y,\u03f5)>1\lambda, then for every comparable elements x,y\in X and some \alpha \in (0,1), we have
thus M(p,q,\u03f5)>1\alpha \lambda.
Now we state our main theorem.
Theorem 2.12 Let (X,M,\ast ) be a complete fuzzy metric space with {sup}_{t<1}t\ast t=1. Y\in \mathcal{C}(X) and F:Y\to \mathcal{C}(Y) be a setvalued fuzzy order ψcontraction of (\u03f5,\lambda )type, where {lim}_{n\to \mathrm{\infty}}{\psi}^{(n)}(t)=0 for all t\in (0,1). Let ‘⪯’ be a partial order defined on X, and {lim}_{n\to \mathrm{\infty}}{\ast}_{i=n}^{\mathrm{\infty}}(1{\psi}^{(i)}(\xi ))=1 for all \xi \in (0,1). Suppose that there exist {x}_{0}\in Y and {x}_{1}\in F{x}_{0} such that M({x}_{0},{x}_{1},{0}_{+})>0 and the following two conditions hold:

(i)
y\in F(x) implies x\u2aafy,

(ii)
if {x}_{n} is a sequence with {x}_{n+1}\in F{x}_{n} and {x}_{n}\to x, then {x}_{n}\u2aafx for all n.
Then F has a fixed point.
Proof Since there exist {x}_{0}\in Y and {x}_{1}\in F{x}_{0} such that M({x}_{0},{x}_{1},{0}_{+})>0, we have {x}_{0}\u2aaf{x}_{1} with M({x}_{0},{x}_{1},{0}_{+})>0. We may suppose that M({x}_{0},{x}_{1},{0}_{+})<1. For, if we assume the contrary, then M({x}_{0},{x}_{1},t)=1 for all t>0, that is, {x}_{0}={x}_{1}\in F{x}_{0} and we have finished the proof. Therefore, for some {\delta}_{1}\in (0,1) and every t>0, \delta \in ({\delta}_{1},1), we have
Since F is a setvalued fuzzy order ψcontraction of (\u03f5,\lambda )type mapping, there exists {x}_{2}\in F{x}_{1} with {x}_{1}\u2aaf{x}_{2} such that M({x}_{1},{x}_{2},t)>1\psi (\delta ). Repeating this argument, we get a sequence \{{x}_{n}\} in Y such that {x}_{n+1}\in F{x}_{n} with {x}_{n}\u2aaf{x}_{n+1} and such that
Suppose that \u03f5>0 and \lambda \in (0,1) are given. Since {lim}_{n\to \mathrm{\infty}}{\ast}_{i=n}^{\mathrm{\infty}}(1{\psi}^{(i)}(\xi ))=1 for all \xi \in (0,1), there exists {n}_{0}\in \mathbb{N} such that for all n\ge {n}_{0} and all \xi \in (0,1), and we have
Now by using (FM4) and from (2.11)(2.12), for all m>n\ge {n}_{0}, we get
This shows that \{{x}_{n}\} is a Cauchy sequence. Since X is complete, \{{x}_{n}\} converges to some \overline{x}\in X, that is, {lim}_{n\to \mathrm{\infty}}M({x}_{n},\overline{x},t)=1. Now we prove that \overline{x}\in F\overline{x}. But F\overline{x}=\overline{F\overline{x}}; then it is enough to show that for every {\u03f5}^{\prime}>0 and {\lambda}^{\prime}\in (0,1) there exists z\in F\overline{x} such that M(\overline{x},z,{\u03f5}^{\prime})>1{\lambda}^{\prime}.
Let {\u03f5}^{\prime}>0 and {\lambda}^{\prime}\in (0,1) be arbitrary. From {sup}_{t<1}t\ast t=1, it follows that there exists {\lambda}_{1}({\lambda}^{\prime})\in (0,1) such that
Also for {\lambda}_{1} there are {\lambda}_{2}\in (0,1) such that
Now put {\lambda}_{3}=min\{{\lambda}_{1},{\lambda}_{2}\}. We prove that there exists \mu \in (0,1) such that \psi (\mu )<{\lambda}_{3}. For, if \psi (t)\ge {\lambda}_{3} for every t\in (0,1), then {\psi}^{n}(t)\ge {\lambda}_{3} for every n\in \mathbb{N} and every t\in (0,1), therefore {\ast}_{i=n}^{\mathrm{\infty}}(1{\psi}^{(i)}(\xi ))\le {\ast}_{i=n}^{\mathrm{\infty}}(1{\lambda}_{3})\le 1{\lambda}_{3} for all n\in \mathbb{N}, which means that 1={lim}_{n\to \mathrm{\infty}}{\ast}_{i=n}^{\mathrm{\infty}}(1{\psi}^{(i)}(\xi ))\le 1{\lambda}_{3}<1, and this is a contradiction.
Since {lim}_{n\to \mathrm{\infty}}M({x}_{n},\overline{x},t)=1 for all t>0, there exists {n}_{1}\in \mathbb{N} such that for all n\ge {n}_{1}, and we have M({x}_{n},\overline{x},\frac{{\u03f5}^{\prime}}{3})>1\mu; thus, since {x}_{n}\u2aaf\overline{x} and by using (2.10), there exists z\in F\overline{x} such that
On the other hand {lim}_{n\to \mathrm{\infty}}{\psi}^{(n)}(t)=0 for every t\in (0,1). Therefore (2.11) implies the existence of the element {n}_{2}\in \mathbb{N} such that for all n\ge {n}_{2}, we have
Also since {lim}_{n\to \mathrm{\infty}}{x}_{n}=\overline{x}, there exists {n}_{3}\in \mathbb{N} such that for all n\ge {n}_{3},
Now if n\ge max\{{n}_{1},{n}_{2},{n}_{3}\}, then by (2.13)(2.17), we get
Hence \overline{x}\in F\overline{x}=\overline{F\overline{x}}, consequently \overline{x} is a fixed point of F. The theorem is proved. □
Corollary 2.13 Let (X,M,\ast ) be a complete fuzzy metric space with Lukasiewicz tnorm and ‘⪯’ be a partial order defined on X. Let Y\in \mathcal{C}(X) and F:Y\to \mathcal{C}(Y) be a setvalued mapping with the property that there is \alpha \in (0,1) such that
for all comparable elements x,y\in X and t>0, and the following conditions hold:

(i)
y\in F(x) implies x\u2aafy,

(ii)
if {x}_{n} is a sequence with {x}_{n+1}\in F{x}_{n} and {x}_{n}\to x, then {x}_{n}\u2aafx for all n.
Then F has a fixed point.
Proof By using Definition 1.2, {sup}_{t<1}t\ast t=1. Also, from Example 2.11 it follows that F is a setvalued fuzzy order ψcontraction of (\u03f5,\lambda )type with \psi (t)=\alpha t. Since, for all \lambda \in (0,1), {\sum}_{i=1}^{\mathrm{\infty}}{\psi}^{(i)}(\lambda )={\sum}_{i=1}^{\mathrm{\infty}}{\alpha}^{i}\lambda <\mathrm{\infty}, from Proposition 1.3, we have {lim}_{n\to \mathrm{\infty}}{\ast}_{i=n}^{\mathrm{\infty}}(1{\psi}^{(i)}(\lambda ))=1. Next, since
for all comparable elements x,y\in X and t>0, there exist {x}_{0}\in Y and {x}_{1}\in F{x}_{0} such that M({x}_{0},{x}_{1},{0}_{+})>0. Consequently, by the preceding theorem, F has a fixed point. □
Corollary 2.14 Let (X,M,\ast ) be a complete fuzzy metric space with a continuous gconvergent tnorm and ‘⪯’ be a partial order defined on X. Let Y\in \mathcal{C}(X) and F:Y\to \mathcal{C}(Y) be a setvalued fuzzy order αcontraction of (\u03f5,\lambda )type. If there exist {x}_{0}\in Y and {x}_{1}\in F{x}_{0} such that M({x}_{0},{x}_{1},{0}_{+})>0 and the following two conditions hold:

(i)
y\in F(x) implies x\u2aafy,

(ii)
if {x}_{n} is a sequence with {x}_{n+1}\in F{x}_{n} and {x}_{n}\to x, then {x}_{n}\u2aafx for all n.
Then F has a fixed point.
Theorem 2.12 and Corollary 2.13 are, respectively, generalizations of the theorems of Mihet [19] and Tirado [20] to the setvalued case in partial ordered fuzzy metric spaces.
Now we introduce a definition and, by using it, we shall state fixed and common fixed point theorems in the partially ordered fuzzy metric space. Our results generalize and extend Theorems 4.1 and 4.2 of [21] to setvalued mappings in complete partially ordered fuzzy metric spaces.
Definition 2.15 Let Y be a nonempty subset of fuzzy metric space (X,M,\ast ). Mapping F:Y\to \mathcal{P}(X) is called fuzzy order Ksetvalued mapping, if for all x\in Y, {u}_{x}\in Fx, there exists {u}_{y}\in Fy with {u}_{x}\u2aaf{u}_{y} such that
for every t>0 and y\in Y with x\u2aafy and some k\in (0,\frac{1}{2}).
Theorem 2.16 Let (X,M,\ast ) be a complete fuzzy metric space, with M triangular, and ‘⪯’ a partial order on X. Let Y\in \mathcal{C}(X) and F:Y\to \mathcal{C}(Y) be a fuzzy order Ksetvalued mapping. Also let there for some {x}_{0}\in Y exist {x}_{1}\in F{x}_{0} with {x}_{0}\u2aaf{x}_{1}, and the following condition is satisfied:
If {x}_{n}\to x is a sequence in Y whose consecutive terms are comparable, then {x}_{n}\u2aafx, for all n.
Then F has a fixed point in X.
Proof By the hypothesis, for {x}_{0}\in Y there exists {x}_{1}\in F{x}_{0} such that {x}_{0}\u2aaf{x}_{1}. Now because F is a fuzzy order Ksetvalued mapping, there exists {x}_{2}\in F{x}_{1} such that {x}_{1}\u2aaf{x}_{2} and
thus
Then it follows by induction that
where \{{x}_{n}\} is a sequence whose consecutive terms are comparable, that is, {x}_{n+1}\in F{x}_{n}. Now we prove that \{{x}_{n}\} is a Cauchy sequence. By putting \lambda =\frac{k}{1k}, and by (2.19), and since M is triangular, we have for all m>n
For each t>0 and each \u03f5\in (0,1), we can choose a sufficiently large {n}_{0}\in \mathbb{N} such that
Thus from (2.20) and (2.21), M({x}_{n},{x}_{m},t)>1\u03f5, for all m,n>{n}_{0} and t>0. This shows that the sequence \{{x}_{n}\} is Cauchy, and, since X is complete, it converges to a point x\in X. But Y is closed, thus x\in Y and also by using the hypothesis {x}_{n}\u2aafx. Now we show that x\in Fx. From {x}_{n}\in F{x}_{n1}, and {x}_{n1}\u2aafx for all n, since F is a fuzzy order Ksetvalued mapping, there exists {u}_{n}\in Fx such that {x}_{n}\u2aaf{u}_{n}, and
Now since M is triangular, by using (2.22), we get
and so, letting n\to \mathrm{\infty}, {u}_{n}\to x. Consequently, since Fx is closed, we have x\in Fx. Then F has a fixed point. □
From the above theorem we can immediately obtain the following generalization for getting a common fixed point.
Theorem 2.17 Let (X,M,\ast ) be a complete fuzzy metric space, with M triangular, and ‘⪯’ a partial order on X. Let Y\in \mathcal{C}(X) and, for every n\in \mathbb{N}, {F}_{n}:Y\to \mathcal{C}(Y) be a sequence of mappings such that, for every two mappings {F}_{i}, {F}_{j} and for all x\in Y, {u}_{x}\in {F}_{i}(x), there exists {u}_{y}\in {F}_{j}(y) with {u}_{x}\u2aaf{u}_{y} such that
for every t>0 and y\in Y with x\u2aafy and some k\in (0,\frac{1}{2}). Also let there exist, for some {x}_{0}\in Y, {x}_{1}\in {F}_{1}{x}_{0} with {x}_{0}\u2aaf{x}_{1}, and the following condition be satisfied:
If {x}_{n}\to x is a sequence in Y whose consecutive terms are comparable, then {x}_{n}\u2aafx, for all n.
Then there exists x\in Y such that x\in \bigcap {F}_{n}x, that is, \{{F}_{n}\} has a common fixed point.
Proof We can find {x}_{2}\in {F}_{2}{x}_{1} such that {x}_{1}\u2aaf{x}_{2} and that
Also for {x}_{2} there exists {x}_{3}\in {F}_{3}{x}_{2} with {x}_{2}\u2aaf{x}_{3} and such that
By continuing this process, we get
where \{{x}_{n}\} is a sequence with {x}_{n+1}\in {F}_{n+1}{x}_{n}. Now similar to the proof of the preceding theorem, we can prove that \{{x}_{n}\} is a Cauchy sequence and by the completeness of X it follows that \{{x}_{n}\} converges to some x\in X. Furthermore, x\in Y and {x}_{n}\u2aafx. Now suppose that {F}_{N} is any arbitrary member of {F}_{n}. Since {x}_{n}\in {F}_{n}{x}_{n1}, {x}_{n1}\u2aafx for all n, and by the hypothesis, there exists {u}_{n}\in {F}_{N}x such that {x}_{n}\u2aaf{u}_{n}, and
thus
Next by the letting n\to \mathrm{\infty}, we get {u}_{n}\to x, and then x\in {F}_{N}x. As {F}_{N} is an arbitrary member of {F}_{n}, x\in \bigcap {F}_{n}x, and x is a common fixed point of \{{F}_{n}\}. The theorem is proved. □
Example 2.18 Let X=[0,\mathrm{\infty}) with tnorm defined a\ast b=min\{a,b\} for all a,b\in [0,1] and M(x,y,t)=\frac{t}{t+\mid xy\mid}, for all x,y\in X and t>0. Then (X,M,\ast ) is a complete fuzzy metric space. Let the natural ordering ≤ of the numbers as the partial ordering ⪯. Define Y=[0,1] and F:Y\to \mathcal{C}(Y) as Fx=\{z,\frac{x}{5}\} for each 0\le x<\frac{1}{2}, and \{z,\frac{x}{4}\} for each \frac{1}{2}\le x\le 1, where z\in Y is an arbitrary. If x,y\in Y such that x\u2aafy and {u}_{x}=z\in Fx, then there exists {u}_{y}=z\in Fy such that {u}_{x}\u2aaf{u}_{y} and (2.18) is satisfied. Thus F is a fuzzy order Ksetvalued mapping. But if {u}_{x}\ne z\in Fx, then three cases arise.
Case (i). If 0\le x\le y<\frac{1}{2}, then for every t>0
Case (ii). If \frac{1}{2}\le x\le y\le 1, then for every t>0
Case (iii). If 0\le x<\frac{1}{2}\le y\le 1, then for every t>0
Hence F is a fuzzy order Ksetvalued mapping with k=\frac{4}{9}<\frac{1}{2}. Moreover, there exists {x}_{0}=0 (or {x}_{0}=z) with {x}_{1}=0 ({x}_{1}=z) such that {x}_{0}\u2aaf{x}_{1}. Thus all the hypotheses of Theorem 2.16 are satisfied and x=0 (or x=z) is the fixed point of F.
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Sadeghi, Z., Vaezpour, S.M., Park, C. et al. Setvalued mappings in partially ordered fuzzy metric spaces. J Inequal Appl 2014, 157 (2014). https://doi.org/10.1186/1029242X2014157
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DOI: https://doi.org/10.1186/1029242X2014157