# n-Tupled Fixed Points Theorem in Fuzzy Metric Spaces with Application

Authors: P. P. Murthy; Rashmi Kenvat
Journal: Advances in Fuzzy Systems
 (2015)

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## Abstract

We will introduce the concept ofn-tupled fixed points (for positive integer n) in fuzzy metric space by mild modification of the concept of n-tupled fixed points (for even positive interger n) introduced by Imdad et al. (2013) in metric spaces. As application of the above-mentioned concept, we will establish some n-tupled fixed point theorems for contractive type mappings in fuzzy metric space which extends the result of Roldán et al. (2013). Also we have given an application to solve a kind of Lipschitzian systems for n variables and an integral system.


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## Body

## 1. Introduction

The concept of coupled fixed point was introduced by Bhaskar and Lakshmikantham [1] and it motivated the fixed point theorists to work in the area of multidimensional fixed points; for example, see [2–9].In recent years some of the fixed point theorists tried to establish the existence ofn-tupled fixed points and common n-tupled fixed points for some contractions in metric spaces, partially ordered metric spaces and asymptotically regular metric spaces. In particular, Imdad et al. [9] introduced the concept of n-tupled coincidence points as well as n-tupled fixed point (for even positive integer) and utilize these two definitions to obtain n-tupled coincidence as well as n-tupled common fixed point theorems. Very recently, Soliman et al. [10] proved some n-tupled coincident point theorems in partially ordered complete asymptotically regular metric spaces.The purpose of our results is to introducen-tupled fixed points (for all positive integers) and to prove n-tupled fixed points theorems for contractive type mappings in fuzzy metric spaces. Results obtained by us in Section 2 extents the work of Roldán et al. [11]. At the end we have given an application to solve a kind of Lipschitzian systems for n variables and an integral system. Results proved in this paper follow the lines of the proof of Roldán et al. [11] for obtaining coincidence and fixed points.A triangular norm (also called at-norm) is a mapping *:[0,1]×[0,1]→[0,1] is associative, commutative, and nondecreasing in both arguments and has 1 as an identity element.Definition 1 (see [11, 12]).
For anya∈[0,1], let the sequence {*na}n=1∞ be defined by *1a=a and *na=(*n-1a)*a. Then a t-norm * is said to be of H-type if the sequence {*na}n=1∞ is equicontinuous at a=1, that is, for all ϵ∈(0,1), there exists η∈(0,1) such that if a∈(1-η,1], then *ma>1-ϵ for all m∈N.The most important and well-known continuoust-norm of H-type is *=min⁡, which verifies min⁡(a,b)≥ab for all a,b∈[0,1]. The following result presents a wide range of t-norms of H-type.Lemma 2.
Letδ∈(0,1] be a real number and let * be a t-norm. Define *δ as x*δy=x*y, if max⁡(x,y)≤1-δ and x*δy=min⁡(x,y), if max⁡(x,y)>1-δ. Then *δ is a t-norm of H-type.Definition 3 (see [13]).
A triplet(X,M,*) is said to be a fuzzy metric space (in the sense of Kramosil and Michalek; briefly, a FMS), if X is a nonempty set, * is a continuous t-norm, and M:X×X×[0,∞)→[0,1] is a fuzzy set satisfying the following conditions for each x,y,z∈X and t,s>0: (FM1)
M(x,y,t)>0,(FM2)
M(x,y,t)=1 if and only if x=y,(FM3)
M(x,y,t)=M(y,x,t),FM4
M(x,y,·):[0,∞)→[0,1] is left continuous,(FM5)
M(x,y,t)*M(y,z,s)≤M(x,z,t+s),in this case, we also say that(X,M) is a FMS under *; in the sequel, we will only consider FMS verifying(FM6)
limt→∞M(x,y,t)=1 for all x,y∈X.Lemma 4.
M
(
x
,
y
,
·
) is a nondecreasing function on [0,∞).Definition 5 (see [14]).
Let(X,M,*) be a fuzzy metric space; then (1)
a sequence{xn} in X is said to be convergent to x if (1)limn→∞⁡M(xn,x,t)=1,for all t>0;(2)
a sequence{xn} in X is said to be a Cauchy sequence if, for any ϵ>0, there exists n0∈N, such that (2)M(xn,xm,t)>1-ϵ,for all t>0 and n,m≥n0;(3)
a fuzzy metric space(X,M,*) is said to be complete if and only if every Cauchy sequence in X is convergent.Given anyt-norm *, it is easy to prove that *≤min⁡. Therefore, if (X,M) is a FMS under min⁡, then (X,M) is a FMS under any (continuous or not) t-norm. This is the case in the following examples (in which, obviously, we only define M(x,y,t) for t>0 and x≠y).Example 6.
From a metric space(X,d), we can consider a FMS in different ways. For t>0 and x≠y, define (i)
Md(x,y,t)=t/t+d(x,y);(ii)
Me(x,y,t)=e-d(x,y)/t;(iii)
Mcx,y,t=0,ift≤d(x,y)1ift>dx,y.
It is well known that(X,Md) is a FMS under the product *=·, called the standard FMS on (X,d), since it is the standard way of viewing the metric space (X,d) as a FMS. However, it is also true (though lesser known) that (X,Md), (X,Me), and (X,Mc) are FMS under min⁡.
Furthermore,(X,d) is a complete metric space if and only if (X,Md) (or (X,Mc) or (X,Me)) is a complete FMS. For instance, this is the case of any nonempty and closed subset (or subinterval) of R provided with its Euclidean metric.Definition 7.
A functiong:X→X on a fuzzy metric space is said to be continuous at a point x0∈X if, for any sequence {xn} in X converging to x0, the sequence {gxn} converges to gx0. If g is continuous at each x∈X, then g is said to be continuous on X. As usual, if x0∈X, we will denote g-1(x0)={x∈X:gx=x0}.Remark 8.
Ifx∈[0,1] and a,b∈(0,∞), then a≤b implies that xa≥xb. We will use this fact in the following way: 0<a≤b≤1 implies that [M(x,y,t)]a≥[M(x,y,t)]b≥M(x,y,t).


## 2. Main Results

Henceforth,X will denote a nonempty set and Xn=X×X×⋯×X︸n-times.Here, we introducen-tupled fixed points (for positive integer n) in fuzzy metric space by slightly modified concept of n-tupled fixed points (for natural number n) introduced by Imdad et al. [9] in metric space.Definition 9.
LetF:Xn→X and g:X→X be two mappings. (a)
One says thatF and g are commuting if (3)gF(x(1),x(2),…,x(n))=F(gx(1),gx(2),…,gx(n))for all x(1),x(2),…,x(n)∈X.(b)
An element(x(1),x(2),x(3),…,x(n))∈Xn is called n-tupled coincidence points of the mappings F and g if (4)F(x(1),x(2),x(3),…,x(n))=gx(1)F(x(2),x(3),…,x(n),x(1))=gx(2)F(x(3),…,x(n),x(1),x(2))=gx(3)⋮Fxn,x1,x2,…,xn-1=gxn.Definition 10.
An element(x(1),x(2),…,x(n))∈Xn is called an n-tupled fixed point of map F:Xn→X and g:X→X if (5)F(x(1),x(2),x(3),…,x(n))=gx(1)=x(1)F(x(2),x(3),…,x(n),x(1))=gx(2)=x(2)F(x(3),…,x(n),x(1),x(2))=gx(3)=x(3)⋮Fxn,x1,x2,…,xn-1=gxn=xn.Now, we present our main theorem.Theorem 11.
Let* be a t-norm of H-type such that s*t≥st for all s,t∈[0,1]. Let k∈(0,1) and a1,a2,…,an∈[0,1] be real numbers such that a1+a2+⋯+an≤1, let (X,M,*) be a complete FMS, and let F:Xn→X and g:X→X be two mappings such that F(Xn)⊆g(X) and g is continuous and commuting with F. Suppose that, for all x(1),x(2),…,x(n),y(1),y(2),…,y(n)∈X and all t>0,(6)MF(x(1),x(2),…,x(n)),F(y1,y2,…,yn),kt≥Mgx1,gy1,ta1*Mgx2,gy2,ta2*⋯*Mgxn,gyn,tan.Then there exists a unique x(1)∈X such that x(1)=gx(1)=F(x(1),x(1),…,x(1)). In particular, F and g have, at least, one n-tupled coincidence point. Furthermore, (x(1),x(1),…,x(1)) is the unique n-tupled coincidence point of F and g if we assume that g-1(x0(1))={x0(1)} only in the case that F≡x0(1) is constant on Xn.
In this result, in order to avoid the indetermination00, we assume that [M(gx(1),gx(2),t)]0=1 for all t>0 and all x(1),x(2)∈X.Proof.
Suppose thatF is constant in Xn; that is, there exists x0(1)∈X such that (7)F(x(1),x(2),…,x(n))=x0(1)∀x(1),x(2),…,x(n)∈X.As F and g are commuting, we deduce that (8)gx01=gFx1,x2,…,xn=F(gx(1)gx(2)⋯gx(n))=x0(1).Therefore, x0(1)=gx0(1)=Fx01,x01,…,x01 and (x0(1),x0(1),…,x0(1)) is n-tupled coincidence point of F and g.
Now, suppose thatg-1(x0(1))={x0(1)} and x1,x2,…,xn∈Xn is another n-tupled coincidence point of F and g. Then, (9)gx1=Fx1,x2,…,xn=x01,sox(1)∈g-1x01=x01.Similarly, x(1)=x(2)=⋯=x(n)=x0(1) and (x0(1),x0(1),…,x0(1)) is the unique n-tupled coincidence points of F and g.
Next, suppose thatF is not constant in Xn.
In this case,(x(1),x(2),…,x(n))≠(0,0,0) and the proof is divided into five steps. In the entire proof, n and p stand to be nonnegative integers and t∈[0,∞).
Step  1. First we define of the sequences {xn(1)},{xn(2)},…,{xn(n)}.
Letx0(1),x0(2),…,x0(n)∈X be arbitrary points in X.
SinceF(Xn)⊆g(X). we can choose x1(1),x1(2),…,x1(n)∈X such that (10)g(x1(1))=F(x0(1),x0(2),…,x0(n)),g(x12)=F(x02,x03,…,x0(n),x0(1)),⋮gx1n=Fx0n,x01,…,x0n-1,Again from F(Xn)⊆g(X). we can choose x2(1),x2(2),…,x2(n)∈X such that (11)g(x2(1))=F(x1(1),x1(2),…,x1(n)),g(x22)=F(x12,x13,…,x1(n),x1(1)),⋮gx2n=Fx1n,x11,…,x1n-1,Continuing in this way, we can construct sequences {xn(1)},{xn(2)},…,{xn(n)} in X such that (12)g(xn(1))=F(xn-1(1),xn-1(2),…,xn-1(n)),g(xn2)=F(xn-12,xn-13,…,xn-1(n),xn-1(1)),⋮gxnn=Fxn-1n,xn-11,…,xn-1n-1,for n≥1.
Step  2. Now, we will show that the sequences {gxn(1)},{gxn(2)},…,{gxn(n)} are Cauchy sequences. Define, for n≥1 and all t≥0,(13)δnt=Mgxn1,gxn+11,t*Mgxn2,gxn+12,t*⋯*Mgxnn,gxn+1n,t.Since δn is a nondecreasing function and t-kt≤t≤t/k, we have that(14)δn(t-kt)≤δn(t)≤δntk,∀t>0,n≥1.From inequality (6) we deduce, for all n∈N and all t≥0,(15)Mgxn1,gxn+11,t=MFxn-11,xn-12,…,xn-1n,Fxn1,xn2,…,xnn,t≥Mgxn-11,gxn1,tka1*Mgxn-12gxn2,tka2*⋯*Mgxn-1n,gxnn,tkanMgxn2,gxn+12,t=MF(xn-12,xn-13,…,xn-11),Fxn2,xn3,…,xn1,t≥Mgxn-12,gxn2,tka1*Mgxn-13gxn3,tka2*⋯*Mgxn-11,gxn1,tkan⋮Mgxnn,gxn+1n,t=MF(xn-1n,xn-11,…,xn-1n-1),Fxnn,xn1,…,xnn-1,t≥Mgxn-1n,gxnn,tka1*Mgxn-11gxn1,tka2*⋯*Mgxnn-1,gxn1,tkan.Using (15) and Remark 8, we have (16)Mgxn1,gxn+11,t≥Mgxn-11,gxn1,tka1*Mgxn-12gxn2,tka2*⋯*Mgxn-1n,gxnn,tkan≥Mgxn-11,gxn1,tk*Mgxn-12,gxn2,tk*⋯*Mgxn-1n,gxnn,tk=δn-1tkMgxn2,gxn+12,t≥Mgxn-12,gxn2,tka1*Mgxn-13,gxn3,tka2*⋯*Mgxn-11,gxn1,tkan≥Mgxn-12,gxn2,tk*Mgxn-13,gxn3,tk*⋯*Mgxn-11,gxn1,tk=δn-1tkMgxnn,gxn+1n,t≥Mgxn-1n,gxnn,tka1*Mgxn-11,gxn1,tka2*⋯*Mgxn-1n-1,gxnn-1,tkan≥Mgxn-1n,gxnn,tk*Mgxn-11,gxn1,tk*⋯*Mgxn-1n-1,gxnn-1,tk=δn-1tk.This proves that, for all t>0 and all n≥0,(17)Mgxn1,gxn+11,t*Mgxn2,gxn+12,t*⋯*Mgxnn,gxn+1n,t≥δn-1tk≥δn-1t.Swapping t by t-kt, we deduce, for all t>0 and n≥0, that (18)Mgxn1,gxn+11,t-kt*Mgxn2,gxn+12,t-kt*⋯*Mgxnn,gxn+1n,t-kt≥δn-1t-kt.Taking into account that * is commutative and *≥· and (15), we observe that (19)δnt=Mgxn1,gxn+11,t*Mgxn2,gxn+12,t*⋯*Mgxnn,gxn+1n,t≥Mgxn-11,gxn1,tka1*Mgxn-12,gxn2,tka2*⋯*Mgxn-1n,gxnn,tkan*Mgxn-12,gxn2,tka1*Mgxn-13,gxn3,tka2*⋯*Mgxn-11,gxn1,tkan*⋯*Mgxn-1n,gxnn,tka1*Mgxn-11,gxn1,tka2*⋯*Mgxn-1n-1,gxnn-1,tkan=Mgxn-11,gxn1,tka1*Mgxn-11,gxn1,tka2*⋯*Mgxn-11,gxn1,tkan*Mgxn-12,gxn2,tka1*Mgxn-12,gxn2,tka2*⋯*Mgxn-12,gxn2,tkan*⋯*Mgxn-1n,gxnn,tka1*Mgxn-1n,gxnn,tka2*⋯*Mgxn-1n,gxnn,tkan≥Mgxn-11,gxn1,tka1·Mgxn-11,gxn1,tka2·…·Mgxn-11,gxn1,tkan*Mgxn-12,gxn2,tka1·Mgxn-12,gxn2,tka2·…·Mgxn-12,gxn2,tkan*⋯*Mgxn-1n,gxnn,tka1·Mgxn-1n,gxnn,tka2·…·Mgxn-1n,gxnn,tkan=Mgxn-11,gxn1,tka1+a2+⋯+an*Mgxn-12,gxn2,tka1+a2+⋯+an*⋯*Mgxn-1n,gxnn,tka1+a2+⋯+an≥Mgxn-11,gxn1,tk*Mgxn-12,gxn2,tk*⋯*Mgxn-1n,gxnn,tk=δn-1tk.If we join this property to (14), (20)δnt≥δn-1tk≥δn-1t≥δn-1t-kt,∀t>0,n≥1.Repeatedly applying the first inequality, we deduce that (21)δn(t)≥δn-1(tk)≥δn-2(tk2)≥⋯≥δ0(tkn)for all t>0 and n≥1. This means that, for all t>0,(22)limn→∞⁡δnt≥limn→∞⁡δ0tkn=1⟹limn→∞⁡δnt=1.Properties (17) and (20) imply that (23)Mgxn1,gxn+11,t*Mgxn2,gxn+12,t*⋯*Mgxnn,gxn+1n,t≥δnt≥δn-1t-kt.Next, we claim that (24)Mgxn1,gxn+p1,t*Mgxn2,gxn+p2,t*⋯*Mgxnn,gxn+pn,t≥*pδn-1(t-kt)for all t>0 and n,p≥1. We prove it by using mathematical induction on p≥1. If p=1, (24) is true for all n≥1 and all t>0 by (23). Suppose that (24) is true for all n≥1 and all t>0 for some p, and we are going to prove it is also true for p+1. Applying (6), the induction hypothesis, and that *≥·,(25)Mgxn+11,gxn+p+11,kt=MFxn1,xn2,…,xnn,Fxn+p1,xn+p2,…,xn+pn,kt≥Mgxn1,gxn+p1,ta1*Mgxn2,gxn+p2,ta2*⋯*Mgxnn,gxn+pn,tan≥*pδn-1t-kta1**pδn-1t-kta2*⋯**pδn-1t-ktan≥*pδn-1t-kta1·*pδn-1t-kta2·…·*pδn-1t-ktan≥*pδn-1t-kta1+a2+⋯+an≥*pδn-1(t-kt).Arguing in the same way, we arrive at the following: (26)Mgxn+12,gxn+p+12,kt,Mgxn+13,gxn+p+13,kt,…,Mgxn+1n,gxn+p+1n,kt≥*pδn-1(t-kt).Applying the axiom (FM5) of a FMS, (18), and the induction hypothesis, (27)Mgxn1,gxn+p+11,t=Mgxn1,gxn+p+11,t+kt-kt≥Mgxn1,gxn+11,t-kt*Mgxn+11,gxn+p+11,kt≥δn-1t-kt**pδn-1t-kt=*p+1δn-1t-kt.Arguing in the same manner, we have(28)Mgxn2,gxn+p+12,t,Mgxn3,gxn+p+13,t,…,Mgxnn,gxn+p+1n,t.Therefore, (24) is true.
This permits us to show that{gxn(1)} is Cauchy. Suppose that t>0 for a given ϵ∈(0,1).
By the hypothesis, as* is a t-norm of H-type, there exists 0<η<1 such that *pa>1-ϵ for all a∈(1-η,1] and for all p≥1.
By (22), limn→∞δn(t)=1, so there exists n0∈N such that δn(t-kt)>1-η for all n≥n0.
Hence (24), we get(29)Mgxn1,gxn+p+11,t,Mgxn2,gxn+p+12,t,…,M(g(xn(n)),g(xn+p+1(n)),t)>1-ϵfor all n≥n0 and p≥1.
Therefore,{gxn(1)} is a Cauchy sequence. Similarly, we can show that {gxn(2)},{gxn(3)},…,{gxn(n)} are also Cauchy sequences.
Step  3. Now, we claim that g and F have a tripled coincidence point. Since X is complete, there exists x(1),x(2),…,x(n)∈X such that (30)limn→∞⁡gxn1=x1,limn→∞⁡gxn2=x(2),…,limn→∞⁡gxnn=x(n).As g is continuous, we have that (31)limn→∞⁡ggxn1=gx1,limn→∞⁡ggxn2=gx2,…,limn→∞⁡ggxnn=gx(n).The commutativity of F with g implies that (32)ggxn+11=gFxn1,xn2,…,xnn=F(gxn(1),gxn(2),…,gxn(n)).By (6) and Remark 8, we get (33)Mggxn+11,Fx1,x2,…,xn,kt=MF(gxn1,gxn2,…,gxnn),F(x1,x2,…,xn),kt≥Mggxn1,gx1,ta1*Mggxn2,gx2,ta2*⋯*Mggxnn,gxn,tan≥Mggxn1,gx1,t*Mggxn2,gx2,t*⋯*Mggxnn,gxn,t.Letting n→∞, we deduce that (34)limn→∞⁡ggxn1=F(x(1),x(2),…,x(n)).Hence, F(x(1),x(2),…,x(n))=g(x(1)).
Similarly, we can show that(35)F(x(2),x(3),…,x(1))=g(x(2)),F(x(3),x(4),…x(2))=g(x(3)),⋮F(x(n),x(1),…,x(n-1))=g(x(n)).So (x(1),x(2),…,x(n)) is n-tupled coincidence point of the mappings F and g:(36)F(x(1),x(2),…,x(n))=g(x(1))F(x(2),x(3),…,x(1))=g(x(2))⋮F(x(n),x(1),…,x(n-1))=g(x(n)).
Step  4. Now, we claim that(37)F(x(1),x(2),…,x(n))=x(2)F(x(2),x(3),…,x(1))=x(3)⋮F(x(n),x(1),…,x(n-1))=x(1).We note that, by condition (6), (38)Mgx1,gxn2,kt=MFx1,x2,…,xn,Fxn-12,xn-13,…,xn-11,kt≥Mgx1,gxn-12,ta1*Mgx2,gxn-13,ta2*⋯*Mgxn,gxn-11,tanMgx2,gxn3,kt=MFx2,x3,…,x1,F(xn-13,xn-14,…,xn-12),kt≥Mgx2,gxn-13,ta1*Mgx3,gxn-14,ta2*⋯*Mgx1,gxn-12,tan⋮Mgxn,gxn1,kt=MFxn,x1,…,xn-1,F(xn-11,xn-12,…,xn-1n),kt≥Mgxn,gxn-11,ta1*Mgx1,gxn-12,ta2*⋯*Mgxn-1,gxn-1n,tan.Let βn(t)=M(gx(1),gxn(2),t)*M(gx(2),gxn(3),t)*⋯*M(gx(n),gxn(1),t) for all t>0 and n≥0.
By (38),(39)βn+1kt=Mgx1,gxn+12,kt*Mgx2,gxn+13,kt*⋯*M(gx(n),gxn+1(1),kt)≥Mgx1,gxn2,ta1*Mgx2,gxn3,ta2*⋯*Mgxn,gxn1,tan*Mgx2,gxn3,ta1*Mgx3,gxn4,ta2*⋯*Mgx1,gxn2,tan*Mgxn,gxn1,ta1*Mgx1,gxn2,ta2*⋯*Mgxn-1,gxnn,tan*⋯*=Mgx1,gxn2,ta1*Mgx1,gxn2,ta2*⋯Mgx1,gxn2,tan*Mgx2,gxn3,ta1*Mgx2,gxn3,ta2*⋯Mgx2,gxn3,tan*⋯*Mgxn,gxn1,ta1*Mgxn,gxn1,ta2*⋯Mgxn,gxn1,tan≥Mgx1,gxn2,ta1·Mgx1,gxn2,ta2·…Mgx1,gxn2,tan*Mgx2,gxn3,ta1·Mgx2,gxn3,ta2·…Mgx2,gxn3,tan*⋯*Mgxn,gxn1,ta1·Mgxn,gxn1,ta2·…Mgxn,gxn1,tan=Mgx1,gxn2,ta1+a2+⋯+an*Mgx1,gxn2,ta1+a2+⋯+an*⋯*Mgx1,gxn2,ta1+a2+⋯+an≥Mgx1,gxn2,t*Mgx2,gxn3,t*M(gx(n),gxn(1),t)=βn(t).This proves that βn+1(kt)≥βn(t) for all t>0 and n≥0. Repeating this process,(40)βn(t)≥βn-1tk≥βn-2tk2≥⋯≥β0tknfor all t>0 and n≥0.
Now, by (40) and (38), (41)Mgx1,gxn+12,kt≥Mgx1,gxn2,ta1*Mgx2,gxn3,ta2*⋯*Mgxn,gxn1,tan≥βn(t)≥β0(tkn)Mgx2,gxn+13,kt≥Mgx2,gxn3,ta1*Mgx3,gxn4,ta2*⋯*Mgx1,gxn2,tan≥βn(t)≥β0(tkn)⋮Mgxn,gxn+11,kt≥Mgxn,gxn1,ta1*Mgx1,gxn2,ta2*⋯*Mgxn-1,gxnn,tan≥βnt≥β0tkn.Therefore, (42)Mgx1,gxn+12,kt,Mgx2,gxn+13,kt,…,Mgxn,gxn+11,kt≥β0(tkn)for all t>0 and n≥1.
Sincelimn→∞β0(t/kn)=1 for all t>0, we have, taking limit in (41), that(43)limn→∞⁡gxn+12=gx1,limn→∞⁡gxn+13=gx(2),…,limn→∞⁡gxn+11=gx(n).This shows, using (36), that (44)F(x(1),x(2),…,x(n))=g(x(1))=limn→∞⁡gxn2=x(2),F(x(2),x(3),…,x(1))=g(x(2))=limn→∞⁡gxn3=x(3),⋮Fxn-1,xn,…,xn-2=gxn-1=limn→∞⁡gxnn=x(n).
Step  5. Now, we will prove that x(1)=x(2)=⋯=x(n).
Let(45)θt=Mx1,x2,t*Mx2,x3,t*⋯*M(x(n),x(1),t)for all t>0.
Using condition (6), we have (46)Mx1,x2,kt=MF(xn,x1,…,xn-1),F(x(1),x(2),…,x(n)),kt≥Mgxn,gx1,ta1*Mgx1,gx2,ta2*⋯*Mgxn-1,gxn,tan=Mx1,x2,ta1*Mx2,x3,ta2*⋯*Mxn,x1,tanMx2,x3,kt=MFx1,x2,…,xn,F(x2,x3,…,x1),kt≥Mgx1,gx2,ta1*Mgx2,gx3,ta2*⋯*Mgxn,gx1,tan=Mx2,x3,ta1*Mx3,x4,ta2*⋯*Mx1,x2,tan⋮M(xn,x1,kt)=MFxn-1,xn,…,xn-2,F(xn,x1,…,xn-1),kt≥Mgxn-1,gxn,ta1*Mgxn,gx1,ta2*⋯*Mgxn-2,gxn-1,tan=Mxn,x1,ta1*Mx1,x2,ta2*⋯*Mxn-1,xn,tan.Combining all the inequalities, we will have the following:(47)θkt=Mx1,x2,kt*Mx2,x3,kt*⋯*Mxn,x1,kt≥Mx1,x2,ta1*Mx2,x3,ta2*⋯*Mxn,x1,tan*Mx2,x3,ta1*Mx3,x4,ta2*⋯*Mx1,x2,tan*⋯*Mxn,x1,ta1*Mx1,x2,ta2*⋯*Mxn-1,xn,tan=Mx1,x2,ta1*Mx1,x2,ta2*⋯*Mx1,x2,tan*Mx2,x3,ta1*Mx2,x3,ta2*⋯*Mx2,x3,tan*⋯*Mxn,x1,ta1*Mxn,x1,ta2*⋯*Mxn,x1,tan≥Mx1,x2,ta1·Mx1,x2,ta2·…·Mx1,x2,tan*Mx2,x3,ta1·Mx2,x3,ta2·…·Mx2,x3,tan*⋯*Mxn,x1,ta1·Mxn,x1,ta2·…·Mxn,x1,tan=Mx1,x2,ta1+a2+⋯+an*Mx2,x3,ta1+a2+⋯+an*⋯*Mxn,x1,ta1+a2+⋯+an≥Mx1,x2,t*Mx2,x3,t*⋯*Mxn,x1,t=θt.We conclude that θ(kt)≥θ(t) implies that θ(t)≥θ(t/k)≥θ(t/k2)≥⋯≥θ(t/kn) for all t>0 and n≥1. From (46),(48)Mx1,x2,kt≥Mx1,x2,ta1*Mx2,x3,ta2*⋯*Mxn,x1,tan≥Mx1,x2,t*Mx2,x3,t*⋯*M(x(n),x(1),t)=θ(t)≥θ(tkn)Mx2,x3,kt≥Mx2,x3,ta1*Mx3,x4,ta2*⋯*Mx1,x2,tan≥Mx2,x3,t*Mx3,x4,t*⋯*Mx1,x2,t=θ(t)≥θ(tkn)⋮Mxn,x1,kt≥Mxn,x1,ta1*Mx1,x2,ta2*⋯*Mxn-1,xn,tan≥Mxn,x1,t*Mx1,x2,t*⋯*Mxn-1,xn,t=θt≥θtkn.Letting n→∞, we have limn→∞θ(t/kn)=1 for all t>0, and it means that M(x(1),x(2),kt)=M(x(2),x(3),kt)=⋯=M(x(n),x(1),kt)=1 for all t>0; that is, (49)x(1)=x(2)=⋯=x(n).The uniqueness of x(1) follows directly from (6).Remark 12.
The uniqueness of the coincidence point ofF and g is not always true. For instance, if F≡x0 is constant and g≡x0 is also constant, then every (x(1),x(2),…,x(n))∈Xn is a coincidence point of F and g.Consequences. In the proof of the next result, the view of (X,d) as the crisp FMS (X,Mc,min⁡) is used (see Example 6). Here we deduce results for n-tupled coincidence point in metric spaces which are generalization of the results of Roldán et al. [11] which are not in a partially ordered set.Theorem 13.
Let(X,d) be a complete metric space and let F:Xn→X and g:X→X be two mappings such that F(Xn)⊆g(X) and g is continuous and commuting with F. Suppose that F and g satisfy some of the following conditions for all x(1),x(2),…,xn,y1,y2,…,yn∈X;(1)
(50)dFx1,x2,…,xn,Fy1,y2,…,yn≤kmax⁡d(gx1,gy1),dgx2,y2,…,d(gx(n),gy(n))for somek∈(0,1),(2)
(51)dFx1,x2,…,xn,Fy1,y2,…,yn≤kmax⁡a1d(gx1,gy1)+a2dgx2,y2+⋯+and(gx(n),gy(n))for somek∈(0,1) and some a1,a2,…,an∈0,1/n,(3)
(52)dFx1,x2,…,xn,Fy1,y2,…,yn≤a1dgx1,gy1+a2dgx2,y2+⋯+and(gx(n),gy(n))for somea1,a2,…,an∈[0,1], such that ∑i=1nai<1.
Then there exists a uniquex∈X such that x=gx=F(x,x,…,x).Proof.
(
1
) Consider Mc defined as in Example 6. As (X,d) is complete, then (X,Mc,min⁡) is a complete FMS. Fix x(1),x(2),…,xn,y1,y2,…,y(n)∈X and t>0.
Now we are ready to prove (6).
Puta(1)=a(2)=⋯=a(n)=1/n and *=min⁡. If Mc(g(x(1)),gy(1),t)=0 or Mc(g(x(2)),gy(2),t)=0 or ⋯ or Mc(g(x(n)),gy(n),t)=0, then (6) is obvious.
Suppose that(53)Mcgx1,gy1,t=1,Mcgx2,gy2,t=1,…,Mcgxn,gyn,t=1.This means that(54)dgx1,gy1<t,dgx2,gy2<t,…,dgxn,gyn<t.Therefore,(55)t>max⁡dgx1,gy1,dgx2,gy2,…,0000000dgxn,gyn,kt>kmax⁡dgx1,gy1,dgx2,gy2,…,000000000d(gx(n),gy(n))≥dFx1,x2,…,xn,Fy1,y2,…,yn.Hence, Mc(F(x1,x2,…,xn),F(y1,y2,…,y(n)),kt)=1 and (6) is also true.
(
2
) In this case,(56)d(F(x1,x2,…,xn),F(y1,y2,…,y(n)))≤ka1dgx1,gy1+a2dgx2,gy2+⋯+andgxn,gyn≤k1ndgx1,gy1+1ndgx2,gy2+⋯+1ndgxn,gyn=kndgx1,gy1+dgx2,gy2+⋯+dgxn,gyn≤knnmax⁡dgx1,gy1,dgx2,gy2,…,dgxn,gyn=kmax⁡dgx1,gy1,dgx2,gy2,…,dgxn,gyn.(3) If k=∑i=1n‍ai<1.(57)dFx1,x2,…,xn,Fy1,y2,…,yn≤a1dgx1,gy1+a2dgx2,gy2+⋯+andgxn,gyn≤a1max⁡⁡dgx1,gy1,dgx2,gy2,…,d(gxn,gyn)+a2max⁡dgx1,gy1,dgx2,gy2,…,⁡d(gxn,gyn)+⋯+anmax⁡⁡dgx1,gy1,dgx2,gy2,…,d(gxn,gyn)=a1+a2+⋯+an·max⁡⁡dgx1,gy1,dgx2,gy2,…,dgxn,gyn=kmax⁡dgx1,gy1,dgx2,gy2,…,d(gxn,gyn).Remark 14.
We can obtain Theorems 11 and 15 due to Roldán et al. [11] as corollaries, if we establish our main result for the mappings F:X×X×X→X and g:X→X.Applications. Now we are ready to establish application.Lipschitzian Systems. Let f1,f2,…,fn:R→R be Lipschitzian mappings and let β1,β2,…,βn∈R be real numbers. Define h:R→R as h(x)=β1f1(x)+β2f2(x)+⋯+βnfn(x) for all x∈R. Then h is another Lipschitzian mapping and kh≤|β1|kf1+|β2|kf2+⋯+|βn|kfn. Obviously, if K=|β1|kf1+|β2|kf2+⋯+|βn|kfn<1, then h is a contraction, so there exists a unique x(1)∈R such that hx(1)=x(1).Next, defineF:Rn→R as(58)Fx1,x2,…,xn=β1f1(x1)+β2f2(x2)+⋯+βnfnxnfor all x(1),x(2),…,x(n)∈R. It is clear that F(x1,x1,…,x(1))=hx(1) for all x(1)∈R. Furthermore,(59)dFx1,x2,…,xn,Fy1,y2,…,yn≤∑i=1n‍βifixi-fiyi≤∑i=1n‍βikfixi-yi≤Kmax⁡1≤j≤n⁡dxj,yj.If K<1, then F verifies (6) with g(x(1))=x(1) for all x(1)∈R.Corollary 15.
Letf1,f2,…,fn:R→R be Lipschitzian mappings on R (provided with the Euclidean metric) and let β1,β2,…,βn∈R such that |β1|kf1+|β2|kf2+⋯+|βn|kfn<1. Then the system (60)β1f1(x(1))+β2f2(x(2))+⋯+βnfn(x(n))=(x(1)),β1f1(x(2))+β2f2(x(3))+⋯+βnfn(x(1))=(x(2))⋮β1f1(x(n))+β2f2(x(1))+⋯+βnfn(x(n-1))=(x(n))has a unique solution, which is (x(1),x(1),…,x(1)), where x(1) is the only real solution of β1f1(x(1))+β2f2(x(2))+⋯+βnfn(x(n))=(x(1)).An Integral System. Let a,b∈R with a<b and let I=[a,b]. Consider X=Ł1(I) with the distance d1·(f,g)=∫I‍|f(t)-g(t)|dt, where ∫‍ represents the Lebesgue integral. It is well known that (Ł(I),d1) is a complete metric space. Let k,β1,β2,…,βn∈R, be real numbers and let G:Rn→R be a mapping verifying G(0,0,…,0)=0, and(61)Gx1,x2,…,xn-Gy1,y2,…,yn≤k∑i=1n‍βixi-yifor all (x(1),x(2),…,x(n)),(y(1),y(2),…,y(n))∈Rn.IfA∈R, we want to find functions f1,f2,…,fn∈L1(I) such that(62)fi(x)=A+∫a,x‍Gfit,fi+1t,…,fi+n-ntdtholds for all x∈I,i=1,2,…,n.For allf1,f2,…,fn∈Ł1(I) and all x∈I, define(63)Ff1f2⋯fnx=A+∫a,x‍Gf1t,f2t⋯fntdt.On the one hand, it is not difficult to prove that Ff1f2⋯fn∈Ł1(I); hence F:L1(I)n→L1(I) is well defined. On the other hand,(64)d1Ff1f2⋯fn,Fg1g2⋯gn=∫I‍Ff1f2⋯fnx-Fg1g2⋯gnxdx≤∫I‍∫a,x‍Gf1t,f2t⋯fnt-Gg1t,g2t⋯gntdt∫a,x‍dx≤∫I‍∫a,x‍k∑i=1n‍βifit-gitdtdx≤k∑i=1n‍βi∫I‍∫I‍fit-gitdtdx=k∑i=1n‍βi∫I‍d1fi,gidx=kb-a∑i=1n‍βid1fi,gi.If we suppose that K=k(b-a)(β1,β2,…,βn)<1, then F verifies (6) with g(f)=f for all f∈Ł1(I). Then system (62) has a unique solution, which is of the form (f0,f0,…,f0), where f0∈Ł1(I) is the only solution of the equation(65)f0(x)=A+∫a,x‍Gf0t,f0t,…,f0tdtfor all x∈I.



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*Source: 285149-2015-03-09.xml*