# Common Fixed Point Theorems of Multivalued Maps in Fuzzy

Authors: A. F. Sayed
Journal: Journal of Mathematics
 (2013)

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## Abstract

In the setting of fuzzy ultrametric spaces, we study common fixed point theorems of multivalued maps. Our results unify, extend, and generalize some related common fixed point theorems of the literature for both ultrametric spaces (Wang and Song (2013), Gajić (2002) and (2001)) and fuzzy metric spaces (Vijayaraju and Sajath (2011)).


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## Body

## 1. Introduction

In 1965, Zadeh [1] introduced the theory of fuzzy sets. Many authors introduced the notion of fuzzy metric space in different ways. George and Veeramani [2] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [3] and defined Hausdorff topology in fuzzy metric space. Several authors [4–10] studied and developed the concept in different directions and proved fixed point theorems in fuzzy metric spaces. Vijayaraju and Sajath [11] extended some previous results and proved some common fixed points theorems for hybrid pair of single and multivalued maps under hybrid contractive conditions. Wang and Song [12] established some results on coincidence and common fixed point for two pairs of multivalued and single-valued maps in ultrametric spaces. In 2009, Savchenko and Zarichnyi [13] introduced the concept of fuzzy ultrametric space. Sedghi and Shobe [14] proved common fixed point theorems for self-maps satisfying contractive conditions on spherically complete fuzzy ultrametric spaces. In this paper, in the setting of fuzzy ultrametric spaces, we study common fixed point theorems of multivalued maps. Our results unify, extend, and generalize some related common fixed point theorems of the literature for both ultrametric spaces [12, 15, 16] and fuzzy metric spaces [11].


## 2. Preliminaries and Notations

Definition 1 (see [1]).
LetXbe any nonempty set. A fuzzy setAinXis a function with domainXand values in [0,1].Definition 2 (see [17]).
A binary operation*:[0,1]2→[0,1] is called a continuous triangular norm (shortly t-norm) if it satisfies the following conditions:(1)
* is associative and commutative,(2)
* is continuous,(3)
a*1=a for all a,b,c,d∈[0,1],(4)
a*b≤c*d whenever a≤c and b≤d for all a,b,c,d∈[0,1].Definition 3 (see [2]).
The 3-tuple (X, M, *) is called a fuzzy metric space ifXis an arbitrary (nonempty) set, * is a continuous t-norm, and M is a fuzzy set on X2×[0,∞) satisfying the following conditions, for all x,y,z∈X and each t and 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,t)≤M(x,z,t+s),(5)
M(x,y,·):(0,∞)→[0,1] is continuous.
Mis called a fuzzy metric onX. The functions M(x,y,t) denote the degree of nearness between x and y with respect tot, respectively.
Let (X,M, *) be a fuzzy metric space. For t>0, the open ball B(x,r,t) with center x∈X and radius 0<r<1 is defined by
(1)B(x,r,t)={y∈X:M(x,y,t)>1-r}.
A subsetA⊂X is called open if, for each x∈A, there existt>0and 0<r<1 such that B(x,r,t)⊂A. Let τ denote the family of all open subsets ofX. Then τ is a topology onX induced by the fuzzy metricM. This topology is Hausdorff and first countable.Definition 4 (see [18]).
Let (X,d) be a metric space. If the metricdsatisfies strong triangle inequality
(2)d(x,y)≤max{d(x,z),d(z,y)},∀x,y,z∈X,
thendis called an ultrametric onXand the pair (X,d) is called an ultrametric space.Definition 5 (see [18]).
An ultrametric space (X,d) is said to be spherically complete if every shrinking collection of balls inXhas a nonempty intersection.Definition 6 (see [14]).
Let (X,M, *) be a fuzzy metric space. If the fuzzy metricMsatisfies strong triangle inequality
(3)M(x,y,t)≥min{M(x,z,t),M(z,y,t)},∀x,y,z∈X,t>0,
thenMis called a fuzzy ultrametric onXand the 3-tuple (X,M, *) is called a fuzzy ultrametric space.Definition 7 (see [14]).
A fuzzy ultrametric space (X,M, *) is said to be spherically complete if every shrinking collection of balls inXhas a nonempty intersection.Remark 8 (see [14]).
(i) Letdbe an ultrametric onXand a*b=a·b for all a,b∈[0,1]. For each t∈(0,∞), define
(4)M(x,y,t)=(11+d(x,y))ψ(t),for all x,y,z∈X,t>0, and ψ∈Ψ. Then fuzzy metricMis also a fuzzy ultrametric.
(ii) Let an ultrametric space (X,d) be spherically complete and a*b=a·b for all a,b∈[0,1]. For each t∈(0,∞), define
(5)M(x,y,t)=(11+d(x,y))ψ(t),
for all x,y,z∈X,t>0, and ψ∈Ψ. Then fuzzy metric space (X,M, *) is also spherically complete.


## 3. Main Results

Let (X,M, *) be a fuzzy ultrametric space. Let CB(X) denote the set of all nonempty closed bounded subsets ofX. For every A,B,C∈CB(X) and t>0, we define
(6)M∇(A,B,t)=min{⋀a∈AM∇(a,B,t),⋀b∈BM∇(b,A,t)},
where M∇(x,C,t)=⋁c∈CM(x,c,t).Remark 9 (see [8]).
It is obvious that, for allA,B∈CB(X) and t>0, we have(i)
M∇(A,B,t)≤M∇(a,B,t), whenever a∈A,(ii)
M∇(A,B,t)=1 if and only if A=B.
It is easy to prove that(CB(X),M∇,*) is a Hausdorff ultrametric space.Definition 10.
Let (X,M, *) be a fuzzy ultrametric space. An element x∈X is said to be a coincidence point of a multivalued map T:X→CB(X) and a single-valued map f:X→X if fx∈TX. We denote
(7)C(f,T)={x∈X∣fx∈Tx}
the set of coincidence points of a multivalued mapTand single-valued mapf.Definition 11.
Let (X,M, *) be a fuzzy ultrametric space, T:X→CB(X) a multivalued map, and f:X→X if fx∈TX is a single-valued map. Tandfare said to be coincidentally computing at z∈X if fz∈Tz⇒fTz⊆Tfz.Definition 12.
Let (X,M, *) be a fuzzy ultrametric space. An element x∈X is said to be a common fixed point of two multivalued maps S,T:X→CB(X) and a single-valued map f:X→X if
(8)x=fx∈Sx∩Tx.Theorem 13.
Let (X,M, *) be a fuzzy ultrametric space. Let S,T:X→CB(X) be a pair of multivalued maps and f,g:X→X a pair of single-valued maps satisfying the following conditions:(a)
fg(x) is spherically complete;(b)
M∇(Sx,Ty,t)>min{M(fx,gy,t),M∇(fx,Sx,t),M∇(gy,Ty,t)},  for all x,y∈X, with fx≠gy, and for t>0;(c)
fS=Sg,fg=gf,fT=Tf,gS=Sg,gT=Tg,ST=TS;(d)
IfS(X)⊆f(X),T(X)⊆g(X), then there exist pointsuandvinX, such that
(9)fu∈Su,gv∈Tv,fu=gv,Su=Tv.Proof.
LetBa=(fga;min{M∇(fga,Sga,t),M∇(fga,Tfa,t)}) denote the closed ball with centered fga and radius
(10)min{M∇(fga,Sga,t),M∇(fga,Tfa,t)}.
Let A be the collection of all the spheres for all a∈fg(X). Then the relation
(11)Ba≤BbifBb⊆Ba
is a partial order on A.
Consider a totally ordered subfamilyA1 of A. Since fg(X) is spherically complete, we have
(12)⋂Ba∈A1Ba=B≠ϕ.
Letfgb∈B, where b∈fg(X) and Ba∈A1. Then fgb∈Ba. Hence
(13)M(fgb,fga,t)≥min{M∇(fga,Sga,t),M∇(fga,Tfa,t)}.
If a=b, then Ba=Bb. Assume that a≠b. Let, for all x∈Bb,
(14)M(x,fgb,t)≥min{M∇(fgb,Sgb,t),M∇(fgb,Tfb,t)}.
SinceSgais a nonempty closed bounded set, then there exists an element p∈Sga, such that
(15)M∇(fga,Sga,t)=M(fga,p,t).Tfa is a nonempty closed bounded set; then there exists an element q∈Tfa such that
(16)M∇(fga,Tfa,t)=M(fga,q,t),min{M∇(fgb,Sgb,t),M∇(fgb,Tfb,t)}=min{⋁c∈SgbM(fgb,c,t),⋁d∈TgbM(fgb,d,t)}≥min{M(fga,p,t),⋁d∈TfbM(p,d,t)}M(fgb,fga,t),M(fga,q,t),⋁c∈SgbM(q,c,t),M(fgb,fga,t),M(fga,p,t),⋁d∈TfbM(p,d,t)}≥min{M(fgb,fga,t),M∇(fga,Tfa,t),M∇(Tfa,Sgb,t),M(fgb,fga,t),M∇(fga,Sga,t),M∇(Sga,Tfb,t)}>min{M(fgb,fga,t),M∇(fga,Tfa,t),M∇(fga,Sga,t),min{M∇(fgb,Sgb,t)}M(fgb,fga,t),M∇(fga,Tfa,t),M∇(fgb,Sgb,t)}},min{M(fga,fgb,t),M∇(fga,Sga,t),M∇(gfb,Tfb,t)}≥min{M∇(fga,Tfa,t),M∇(fga,Sga,t)}.
From the two conditions(a),(b) and (13), we now have
(17)M(x,fga,t)≥min{M(x,fgb,t),M(fgb,fga,t)}≥min{M∇(fga,Sga,t),M∇(fga,Tfa,t)}.
Sox∈Ba; we have just proved that Bb⊆Ba, for every Ba∈A1. Thus Bb is an upper bound in A for the family A1, and hence by Zorn’s lemma, there is a maximal element in A, say BZ,z∈fg(X). There exists an element w∈X such that z=fgw.
Suppose(18)f(gfgw)∉S(gfgw),g(ffgw)∉T(ffgw).
SinceSgfgw and Tffgw are nonempty closed bounded sets, then there exist k∈Sgfgw and l∈Tffgw such that
(19)M∇(fgfgw,Sgfgw,t)=M(fgfgw,k,t),(20)M∇(fgfgw,Tffgw,t)=M(fgfgw,l,t).
From the two conditions (b), (c) and (19), we have
(21)M∇(Sgfgw,TSfgw,t)=⋁e∈TSfgwM(gSfgw,e,t)≥min{⋁e∈TSfgwM(k,e,t)}M∇(Sgfgw,fgfgw,t),M(fgfgw,k,t),⋁e∈TSfgwM(k,e,t)}≥min{M∇(Sgfgw,fgfgw,t),M∇(Sgfgw,TSfgw,t)}>min{M∇(Sgfgw,fgfgw,t),min{M∇(gSfgw,TSfgw,t)}M(fgfgw,fSfgw,t),M∇(fgfgw,Sgfgw,t),M∇(gSfgw,TSfgw,t)}}=M∇(fgfgw,Sgfgw,t),(22)M∇(Tffgw,STfgw,t)=⋁h∈STfgwM(fTfgw,h,t)≥min{M∇(Tffgw,fgfgw,t),M∇(Tffgw,STfgw,t){M∇(Tffgw,fgfgw,t),}>min{M∇(Tffgw,gffgw,t),min{M∇(gffgw,Tffgw,t)}(fTfgw,gffgw,t),M∇(fTfgw,STfgw,t),M∇(gffgw,Tffgw,t)}}=M∇(Tffgw,fgfgw,t).
From the two conditions (b), (c) and (19)–(22), we have
(23)M∇(fggSw,SggSw,t)=⋁m∈SggSwM(fggSw,m,t)≥min{,⋁m∈SggSwM(l,m,t)}M∇(fggSw,TSfgw,t),M(TSfgw,Tffgw,t),M∇(Tffgw,fgfgw,t),M(gffgw,l,t),⋁m∈SggSwM(l,m,t)}≥min{M∇(fgfgw,Sgfgw,t),M∇(fgfgw,Tffgw,t),min{M∇(gffgSw,Tffgw,t)}M(fggsw,gffgw,t),M∇(fggSw,SggSw,t),M∇(gffgSw,Tffgw,t)}}≥min{M∇(fgfgw,Sgfgw,t),M∇(fgfgw,Tffgw,t)},M∇(fgfTw,TffTw,t)=⋁n∈TffTwM(fgfTw,n,t)≥min{⋁n∈TffTwM(k,n,t)}M∇(fgfTw,STfgw,t),M(STfgw,Sgfgw,t),M∇(Sgfgw,fgfgw,t),M(fgfgw,k,t),⋁n∈TffTwM(k,n,t)}≥min{M∇(fgfgw,Tffgw,t),M∇(fgfgw,Sgfgw,t),M∇(Sgfgw,TffTw,t){M∇(fgfgw,Tffgw,t),}≥min{M∇(fgfgw,Tffgw,t),M∇(fgfgw,Sgfgw,t),min{M∇(gffTw,TffTw,t)}M(fgfgw,gffTw,t),M∇(fgfgw,Sgfgw,t),M∇(gffTw,TffTw,t)}}≥min{M∇(fgfgw,Tffgw,t),M∇(fgfgw,Sgfgw,t)}.
From (21) and (23), we have
(24)min{M(Sgfgw,TSfgw,t),M∇(fggSw,SggSw,t)}>min{M∇(fgfgw,Sgfgw,t),M∇(fgfgw,Tffgw,t)}.
From (22) and (3.7), we have
(25)min{M(STfgw,Tffgw,t),M∇(fgfTw,TffTw,t)}>min{M∇(fgfgw,Tffgw,t),M∇(fgfgw,Sgfgw,t){M∇(fgfgw,Tffgw,t),}.
If(26)min{M∇(fgfgw,Sgfgw,t),M∇(fgfgw,Tffgw,t)}=M∇(fgfgw,Sgfgw,t),
then from (24), we have fgfgw∉BgSw⇒fgz∉BgSw.
HenceBz⊄BgSw. It is a contradiction to the maximality of Bz in A, since gSw⊆gf(X)=fg(X).
If(27)min{M∇(fgfgw,Sgfgw,t),M∇(fgfgw,Tffgw,t)}=M∇(fgfgw,Tffgw,t),
then from (25), we have fgfgw∉BfTw⇒fgz∉BfTw.
HenceBz⊄BfTw. It is a contradiction to the maximality of Bz in A, since fTw⊆fg(X),
So(28)f(gfgw)∈S(ffgw),g(gfgw)∈T(ffgw).
In addition, f(gfgw)=g(ffgw).
Using the two conditions (b), (c) and (28), we obtain
(29)M∇(Sgfgw,Tffgw,t)>min{M(fgfgw,gffgw,t),M∇(fgfgw,Sgfgw,t),M∇(gffgw,Tffgw,t)}=1.
HenceS(gfgw)=T(ffgw). Then the proof is completed.Theorem 14.
Let (X,M,*) be a fuzzy ultrametric space. Let S,T:X→CB(X) be a pair of multivalued maps and let f:X→X be a single-valued map satisfying the following conditions:(a)
f(x) is spherically complete;(b)
M∇(Sx,Ty,t)>min{M(fx,fy,t),M∇(fx,Sx,t),M∇(fy,Ty,t)},  for all x,y∈X, with x≠y, and for t>0;(c)
fS=Sf,fT=Tf,ST=TS;(d)
IfS(X)⊆f(X),T(X)⊆f(X).
Thenf,S, and T have a coincidence point in X. Moreover, if f and S,f and T are coincidentally commuting at z∈C(f,T) and ffz=fz, then f,S, and T have a common fixed point inX.Proof.
Iff=g in Theorem 13, we obtain that there exist points u andvinXsuch that
(30)fu∈Su,fv∈Tv,fu=fv,Su=Tv.
Asu∈C(f,S),f and S are coincidentally commuting at u and ffu=fu.
Writingw=fu, then w∈Su,w∈Tv. Then we have
(31)fw=w,w=fw∈f(Su)⊆S(fu)=Sw.
Now, since alsou∈C(f,T),f and T are coincidentally commuting at u and ffu=fu, so we obtain
(32)w=fw∈f(Tv)⊆T(fv)=Tw.
Thus, we have proved thatw=fw∈Sw∩Tw; that is, w is a common fixed point of f,S, and T.The proof of the following corollary is easy, so it is omitted.Corollary 15.
Let (X,M,*) be a spherically complete fuzzy ultrametric space. Let S,T:X→CB(X) be a pair of multivalued maps satisfying the following conditions:(a)
M∇(Sx,Ty,t)>min{M(x,y,t),M∇(x,Sx,t),M∇(y,Ty,t)}, for all x,y∈X, with x≠y, and for t>0;(b)
ST=TS.
Then, there exists a pointz in X such that z∈Sz∩Tz and Sz=Tz.Now we give an example to illustrate Theorem13.Example 16.
Let (X,M,*) be a fuzzy ultrametric space in which X=[0,1],a*b=min{a,b} for all a,b∈[0,1], and
(33)M(x,y,t)=tt+|x-y|
for all t>0.
Define the mapsS,T,f,g on X as follows:
(34)Sx=Tx=1,fx=x+12,gx=2x+13,∀x∈X.
Define(35)W(t1,t2,t3,t4,t5)=t1-(min{t2,t3,t4,t5})h,
where W is a function from [0,1]5→[-1,1] satisfying the following conditions:(W1)
W(t1,…,t5) is increasing in t1 and decreasing in t2,…,t5(W2)
W(u,v,v,v,v,)≥0 implies that u>v, for all u∈[0,1] and v∈[0,1).
Then for anyh∈(0,1), the following inequality
(36)M∇(Sx,Ty,t)>(min{M(fx,gy,t),M∇(fx,Sx,t),M∇(gy,Ty,t)})h
is satisfied for all x,y∈X, with x≠y, and for all t>0 since the LHS of the inequality is 1.
Clearly all conditions in Theorem13 are satisfied, so 1 is the unique common fixed point of S,T,f, and g.


## 4. Conclusion

In this paper, we get coincidence point theorems and common fixed point theorems for two pairs of multivalued and single-valued maps satisfying different contractive conditions on spherically complete fuzzy ultrametric space, which are generalized results for both ultrametric spaces [12, 15, 16] and fuzzy metric spaces [11].



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