# Related Fixed Point of Set-Valued Mappings in Three Menger Spaces

Authors: Ismat Beg; Sunny Chauhan
Journal: International Journal of Analysis
 (2013)

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## Abstract

We prove a related fixed point theorem for set-valued mappings in three Menger spaces. Some examples are also furnished to support the results. Our main result generalizes and extends several known results in the literature.


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## Body

## 1. Introduction

Fisher [1] initiated the study of conditions for the existence of a relation connecting the fixed points of two mappings in two different metric spaces (also see [2]). Afterward, Fisher and Tűrkoğlu [3] proved a related fixed point theorem for set-valued mappings in two metric spaces. In 2003, Chourasia and Fisher [4] established a related fixed point theorem for two pairs of set valued mappings in two metric spaces. Several mathematicians have extensively developed metrical related fixed point theorems (see, e.g., [5–8]). Later on, Pant [9] firstly studied related fixed point theorems for single valued mappings in two complete Menger spaces and generalized the results of Fisher [1, 2]. Pant and Kumar [10] extended the results of Namdeo et al. [7] to two pairs of single-valued mappings.In this paper, we prove a related fixed point theorem for set-valued mappings in three complete Menger spaces. Our result generalizes the result of Jain et al. [6] and extends the result of Fisher and Tűrkoğlu [3].


## 2. Preliminaries

Definition 1 (see [11]).
A mappingΔ:[0,1]×[0,1]→[0,1] is called a t-norm if(i)
Δ(a,1)=a, Δ(0,0)=0,(ii)
Δ(a,b)=Δ(b,a),(iii)
Δ(a,b)≤Δ(c,d), whenever a≤c and b≤d,(iv)
Δ(a,Δ(b,c))=Δ(Δ(a,b),c), for all a,b,c,d∈[0,1].Examples oft-norms are ΔM(a,b)=min{a,b}, ΔP(a,b)=ab, and ΔL(a,b)=max{a+b-1,0}.Definition 2 (see [11]).
A mappingF∶ℝ→ℝ+ is called distribution function if it is nondecreasing, left continuous with inf{F(t)∶t∈ℝ}=0 and sup{F(t)∶t∈ℝ}=1.Letℑ be the set of all distribution functions whereas H stands for the specific distribution function (also known as Heaviside function) defined by
(1)H(t)={0,ift≤0;1,ift>0.Definition 3 (see [11]).
LetX be a nonempty set. An ordered pair (X,ℱ) is called a PM-space where ℱ is a mapping from ℱ∶X×X→ℑ satisfying the following conditions:(i)
Fx,y(t)=H(t) if and only if x=y,(ii)
Fx,y(t)=Fy,x(t),(iii)
Fx,z(t)=1 and Fz,y(s)=1; then Fx,y(t+s)=1 for all x,y,z∈X and t,s>0.Every metric space(X,d) can be realized as a PM-space by considering        ℱ∶X×X→ℑ defined by Fx,y(t)=H(t-d(x,y)) for all x,y∈X. So PM-spaces offer a wider framework (than that of the metric spaces) and are general enough to cover even wider statistical situations.Definition 4 (see [12]).
A Menger space(X,ℱ,Δ) is a triplet where (X,ℱ) is a PM-space and Δ is a t-norm satisfying the following condition:
(2)Fx,y(t+s)≥Δ(Fx,z(t),Fz,y(s)),
for all x,y,z∈X and t,s>0.Definition 5 (see [11]).
Let(X,ℱ,Δ) be a Menger space and Δ a continuous t-norm. A sequence {xn} in X is said to be(i)
convergent to a pointx in X if and only if for every ε>0 and λ>0, there exists a positive integer N(ε,λ) such that Fxn,x(ε)>1-λ for all n≥N(ε,λ).(ii)
Cauchy if for everyε>0 and λ∈(0,1), there exists a positive integer N(ε,λ) such that Fxn,xm(ε)>1-λ for all n,m≥N(ε,λ).A Menger space in which every Cauchy sequence is convergent is said to be complete.Lemma 6 (see [13]).
Let(X,ℱ,Δ) be a Menger space. If there exists a constant k∈(0,1) such that
(3)Fx,y(kt)≥Fx,y(t),
for all t>0 with fixed x,y∈X, then x=y.We need the following definition due to Chang et al. [14] for subsequent use in next section.Definition 7 (see [14]).
Let(X,ℱ,Δ) be a Menger space and 𝒰 a nonempty subset of X. Then 𝒰 is said to be probabilistically bounded if supt>0infx,y∈𝒰Fx,y(t)=1.IfX is itself probabilistically bounded, then X is said to be a probabilistically bounded space.Let𝒞ℬ(X) be a family of nonempty closed and bounded subsets of a Menger space (X,ℱ,Δ). For all 𝒰,𝒱∈𝒞ℬ(X) and for every t>0, we define
(4)DF𝒰,𝒱(t)=sup{Fa,b(t);a∈𝒰,b∈𝒱},δF𝒰,𝒱(t)=inf{Fa,b(t);a∈𝒰,b∈𝒱}.
If the set 𝒰 consists of a single point a, we write Fδ𝒰,𝒱(t)=Fa,𝒱δ(t). Also if the set 𝒱 consists of a single point b, we write Fδ𝒰,𝒱(t)=Fa,bδ(t). It follows immediately from the definition that(i)
δF𝒰,𝒱(t)=δF𝒱,𝒰(t)≥0,(ii)
δF𝒰,𝒱(t)=1 if and only if 𝒰=𝒱={a}, for all 𝒰,𝒱∈𝒞ℬ(X).


## 3. Main Result

Now we prove a related fixed point theorem for set-valued mappings in three complete Menger spaces.Theorem 8.
Let(X,ℱ,Δ), (Y,𝒢,Δ), and (Z,H,Δ) be three complete Menger spaces, where Δ is a continuous t-norm. If P is a continuous mapping of X into 𝒞ℬ(Y), Q is a continuous mapping of Y into 𝒞ℬ(Z), and R is a mapping of Z into 𝒞ℬ(X) satisfying the inequalities
(5)FδRQPx,RQPx′(kt)≥min{Fx,x′(t),Fδx,RQPx(t),Fδx′,RQPx′(t),GδPx,Px′(t),HδQPx,QPx′(t)},(6)GδPRQy,PRQy′(kt)≥min{Gy,y′(t),Gδy,PRQy(t),Gδy′,PRQy′(t),HδQy,Qy′(t),FδRQy,RQy′(t)},(7)HδQPRz,QPRz′(kt)≥min{Hz,z′(t),Hδz,QPRz(t),Hδz′,QPRz′(t),FδRz,Rz′(t),GδPRz,PRz′(t)}
for all x,x′∈X, y,y′∈Y, z,z′∈Z, k∈(0,1) and t>0, Then RQP has a unique fixed point u in X, PRQ has a unique fixed point v in Y, and QPR has a unique fixed point w in Z. Further Pu={v}, Qv={w}, and Rw={u}.Proof.
Letx=x1 be an arbitrary point in X. We define the sequences {xn} in X, {yn} in Y, and {zn} in Z inductively as follows. We choose a point y1 in Px1, then a point z1 in Qy1, and then a point x2 in Rz1. In general, having chosen xn in X, yn in Y, and zn in Z, choose a point xn+1 in Rzn, then a point yn+1 in Pxn+1, and then a point zn+1 in Qyn+1 for n=1,2,…. Applying inequality (5), we get
(8)Fxn+1,xn+2(kt)≥FδRQPxn,RQPxn+1(kt)≥min{Fxn,xn+1(t),Fδxn,RQPxn(t),Fδxn+1,RQPxn+1(t),GδPxn,Pxn+1(t),HδQPxn,QPxn+1(t)},≥min{FδRQPxn-1,RQPxn(t),FδRQPxn,RQPxn+1(t),GδPRQyn-1,PRQyn(t),HδQPRzn-1,QPRzn(t)},≥min{FδRQPxn-1,RQPxn(t),GδPRQyn-1,PRQyn(t),HδQPRzn-1,QPRzn(t)}.
Using inequality (6), we get
(9)Gyn+1,yn+2(kt)≥GδPRQyn,PRQyn+1(kt)≥min{Gyn,yn+1(t),Gδyn,PRQyn(t),Gδyn+1,PRQyn+1(t),HδQyn,Qyn+1(t),FδRQyn,RQyn+1(t)},≥min{GδPRQyn-1,PRQyn(t),GδPRQyn,PRQyn+1(t),HδQPRzn-1,QPRzn(t),FδRQPxn,RQPxn+1(t)},≥min{GδPRQyn-1,PRQyn(t),HδQPRzn-1,QPRzn(t),FδRQPxn,RQPxn+1(t)},≥min{GδPRQyn-1,PRQyn(t),HδQPRzn-1,QPRzn(t),FδRQPxn-1,RQPxn(t)},
on using inequality (8). Further, on using inequality (7), we have
(10)Hzn+1,zn+2(kt)≥HδQPRzn,QPRzn+1(kt)≥min{Hzn,zn+1(t),Hδzn,QPRzn(t),Hδzn+1,QPRzn+1(t),FδRzn,Rzn+1(t),GδPRzn,PRzn+1(t)},≥min{HδQPRzn-1,QPRzn(t),HδQPRzn,QPRzn+1(t),FδRQPxn,RQPxn+1(t),GδPRQyn,PRQyn+1(t)},≥min{HδQPRzn-1,QPRzn(t),FδRQPxn-1,RQPxn(t),GδPRQyn-1,PRQyn(t)},
on using inequalities (8) and (9). It follows easily by induction on using inequalities (8), (9), and (10) that
(11)Fxn+1,xn+2(t)≥min{FδRQPx1,RQPx2(tkn-1),GδPRQy1,PRQy2(tkn-1),HδQPRz1,QPRz2(tkn-1)},Gyn+1,yn+2(t)≥min{FδRQPx1,RQPx2(tkn-1),GδPRQy1,PRQy2(tkn-1),HδQPRz1,QPRz2(tkn-1)},Hzn+1,zn+2(t)≥min{FδRQPx1,RQPx2(tkn-1),GδPRQy1,PRQy2(tkn-1),HδQPRz1,QPRz2(tkn-1)}
for all n=1,2,…. Since FδRQPx1,RQPx2(t/kn-1)→1, GδPRQy1,PRQy2(t/kn-1)→1, and HδQPRz1,QPRz2(t/kn-1)→1 as n→∞, it follows that {xn}, {yn}, and {zn} are Cauchy sequences in complete Menger space X and so one has a limit u in X. Similarly, the sequences {yn} and {zn} are also Cauchy sequences with limits v and w in the complete Menger spaces Y and Z, respectively. Then we have
(12)limn→∞RQPxn={u}=limn→∞RQyn,(13)limn→∞PRQyn={v}=limn→∞PRzn,limn→∞QPRzn={w}=limn→∞QPxn.
Using the continuity of P and Q, we have
(14)limn→∞yn=limn→∞Pxn=Pu={v},limn→∞zn=limn→∞Qyn=Qv={w},
and then we see that
(15)QPu={w}.
Now we show that u is a fixed point of RQP. Applying inequality (5), we have
(16)FδRQPu,RQPxn(kt)≥FδRQPu,RQPxn-1(kt)≥min{Fu,xn-1(t),Fδu,RQPu(t),Fδxn-1,RQPxn-1(t),GδPu,Pxn-1(t),HδQPu,QPxn-1(t)}.
Since P and Q are continuous, it follows on letting n→∞ and using (12) that
(17)FδRQPu,u(kt)≥Fδu,RQPu(t).
Owing to Lemma 6, RQP={u} and so u is a fixed point of RQP. We now have
(18)RQPu=Rw={u},
on using (15) and then
(19)PRQv=PRQPu={v},QPRw=QPRQv=Qv={w},
on using (14). Hence v and w are fixed points of PRQ and QPR, respectively. Further, we see that
(20)RQv={u},PRw={v}.
To prove the uniqueness of u, suppose that RQP has another fixed point u′. Then, using inequality (5), we have
(21)Fδu′,RQPu′(kt)≥FδRQPu′,RQPu′(kt)≥min{Fu′,u′(t),Fδu′,RQPu′(t),GδPu′,Pu′(t),HδQPu′,QPu′(t)}≥min{GδPu′,Pu′(t),HδQPu′,QPu′(t)}.
Next, using inequality (6), we have
(22)GδPu′,Pu′(kt)≥GδPRQPu′,PRQPu′(kt)≥min{GPu′,Pu′(t),GδPu′,PRQPu′(t),HδQPu′,QPu′(t),FδRQPu′,RQPu′(t)}≥min{HδQPu′,QPu′(t),FδRQPu′,RQPu′(t)},
and using inequality (7), we obtain
(23)HδQPu′,QPu′(kt)≥HδQPRQPu′,QPRQPu′(kt)≥min{HQPu′,QPu′(t),HδQPu′,QPRQPu′(t),FδRQPu′,RQPu′(t),GδPRQPu′,PRQPu′(t)}≥min{FδRQPu′,RQPu′(t),GδPRQPu′,PRQPu′(t)}.
It now follows easily from inequalities (22) and (23) that
(24)GδPRQPu′,PRQPu′(t)≥FδRQPu′,RQPu′(tk),
and then
(25)HδQPu′,QPu′(t)≥FδRQPu′,RQPu′(tk).
Using inequalities (21), (24), and (25), we obtain
(26)Fδu′,RQPu′(t)≥FδRQPu′,RQPu′(t)≥FδRQPu′,RQPu′(tk2)≥⋯≥FδRQPu′,RQPu′(tk2n),
for all n=1,2,…. Since FδRQPu′,RQPu′(t/k2n)→1 as n→∞, it follows that RQPu′=u′ and so RQPu′ is a singleton. It then follows from inequality (25) that QPu′ is a singleton and from inequality (24) that Pu′ is a singleton. Using inequality (5) again, we have
(27)Fu,u′(kt)=FδRQPu,RQPu′(kt)≥min{Fu,u′(t),Fu,u(t),Fu′,RQPu′(t),GPu,Pu′(t),HQPu,QPu′(t)}≥min{GPu,Pu′(t),HQPu,QPu′(t)}.
Next using inequality (6), we have
(28)GPu,Pu′(kt)=GδPRQPu,PRQPu′(kt)≥min{GPu,Pu′(t),GPu,PRQPu(t),GPu′,PRQPu′(t),HQPu,QPu′(t),FRQPu,RQPu′(t)}=min{HQPu,QPu′(t),Fu,u′(t)},
and using inequality (7), we have
(29)HQPu,QPu′(kt)≥HδQPRQPu,QPRQPu′(kt)≥min{HQPu,QPu′(t),HQPu,QPRQPu(t),HQPu′,QPRQPu′(t),FRQPu,RQPu′(t),GPRQPu,PRQPu′(t)}=min{Fu,u′(t),GPu,Pu′(t)}.
It follows from inequalities (27), (28), and (29) that Fu,u′(t)=1, which shows the uniqueness of u. The uniqueness of v and w can be verified similarly.Example 9.
ConsiderX=[0,2], Y=[1,5], and Z=[0,10] and define
(30)Fx,y(t)=Gx,y(t)=Hx,y(t)={tt+|x-y|,ift>0;0,ift=0,
for all x,y∈X. Then (X,ℱ,Δ), (Y,𝒢,Δ), and (Z,H,Δ) are three complete Menger spaces with continuous t-norm Δ(a,b)=min{a,b} for all a,b∈[0,1]. Define the set-valued mappings P:X→𝒞ℬ(Y), Q:Y→𝒞ℬ(Z) and R:Z→𝒞ℬ(X) by Px=[1+x,2] for all x∈X, Qy=[2y+1,5] for all y∈Y and
(31)Rz={[z5,1],ifz∈[0,5];(1,z5],ifz∈(5,10].
We can easily verify the inequalities with k=1/2∈(0,1). Thus all the conditions of Theorem 8 are satisfied and RQP has a unique fixed point {1} in 𝒞ℬ(X), PRQ has a unique fixed point {2} in 𝒞ℬ(Y), and QPR has a unique fixed point {5} in 𝒞ℬ(Z). Hence, P(1)={2}, Q(2)={5}, and R(5)={1}, and so RQP(1)=RQ(2)=R(5)={1}, PRQ(2)=PR(5)=P(1)={2}, and QPR(5)=QP(1)=Q(2)={5}.Corollary 10.
Let(X,ℱ,Δ), (Y,𝒢,Δ) and (Z,H,Δ) be three complete Menger spaces, where Δ is a continuous t-norm. If P is a continuous mapping of X into Y, Q is a continuous mapping of Y into Z and R is a mapping of Z into X satisfying the inequalities
(32)FRQPx,RQPx′(kt)≥min{Fx,x′(t),Fx,RQPx(t),Fx′,RQPx′(t),GPx,Px′(t),HQPx,QPx′(t)}GPRQy,PRQy′(kt)≥min{Gy,y′(t),Gy,PRQy(t),Gy′,PRQy′(t),HQy,Qy′(t),FRQy,RQy′(t)}HQPRz,QPRz′(kt)≥min{Hz,z′(t),Hz,QPRz(t),Hz′,QPRz′(t),FRz,Rz′(t),GPRz,PRz′(t)}
for all x,x′∈X, y,y′∈Y, z,z′∈Z, k∈(0,1), and t>0. Then RQP has a unique fixed point u in X, PRQ has a unique fixed point v in Y and QPR has a unique fixed point w in Z. Further, Pu=v, Qv=w and Rw=u.Example 11.
LetX=[0,2], Y=[1,3], Z=[3,5] and Fx,y(t), Gx,y(t) and Hx,y(t) be defined as Example 9. Then (X,ℱ,Δ), (Y,𝒢,Δ), and (Z,H,Δ) be three complete Menger spaces with continuous t-norm Δ(a,b)=min{a,b} for all a,b∈[0,1]. Define the mappings P:X→Y, Q:Y→Z and R:Z→X by Px=x+1 for all x∈X, Qy=y+2 for all y∈Y and
(33)Rz={z4,ifz∈[3,4];z+13,ifz∈(4,5].
Also the inequalities (32) can be verified with k=1/2∈(0,1). Thus all the conditions of Corollary 10 are satisfied and RQP has a unique fixed point 1 in X, PRQ has a unique fixed point 2 in Y, and QPR has a unique fixed point 4 in Z. Further, P(1)=2, Q(2)=4, and R(4)=1; that is, RQP(1)=RQ(2)=R(4)=1, PRQ(2)=PR(4)=P(1)=2, and QPR(4)=QP(1)=Q(2)=4.The following corollaries are the natural results due to Theorem8 and Corollary 10.Corollary 12.
Let(X,ℱ,Δ) and (Y,𝒢,Δ) be two complete Menger spaces, where Δ is a continuous t-norm. If P is a continuous mapping of X into 𝒞ℬ(Y), Q is a mapping of Y into 𝒞ℬ(X) satisfying the inequalities
(34)FδQPx,QPx′(kt)≥min{Fx,x′(t),Fδx,QPx(t),Fδx′,QPx′(t),GδPx,Px′(t)}GδPQy,PQy′(kt)≥min{Gy,y′(t),Gδy,PQy(t),Gδy′,PQy′(t),FδQy,Qy′(t)}
for allx,x′∈X, y,y′∈Y, k∈(0,1), and t>0. Then QP has a unique fixed point u in X and PQ has a unique fixed point v in Y.Corollary 13.
Let(X,ℱ,Δ) and (Y,𝒢,Δ) be two complete Menger spaces, where Δ is a continuous t-norm. If P is a continuous mapping of X into Y, Q is a mapping of Y into X satisfying the inequalities
(35)FQPx,QPx′(kt)≥min{Fx,x′(t),Fx,QPx(t),Fx′,QPx′(t),GPx,Px′(t)},GPQy,PQy′(kt)≥min{Gy,y′(t),Gy,PQy(t),Gy′,PQy′(t),FQy,Qy′(t)}
for all x,x′∈X, y,y′∈Y, k∈(0,1), and t>0. Then QP has a unique fixed point u in X and PQ has a unique fixed point v in Y.Remark 14.
Theorem8 generalizes the result of Jain et al. [6] and Fisher [2]. Corollaries 10–13 extend the results of Fisher and Tűrkoğlu [3, Theorem 1, Corollary 1] to PM-spaces.



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