# Some Results on N-Tupled Coincidence and Fixed Points of Graphs on Metric Spaces and an Application to Integral Equations

Authors: Ahmed H. Soliman; Tamer Nabil
Journal: Journal of Function Spaces
 (2018)

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## Abstract

In this work, we establish some N-tupled common coincidence and N-tupled common fixed points for the mappings satisfying a(φ-ψ)-type contractive condition in a complete metric space endowed with a directed graph (for short digraph). Also, we apply our theoretical results to study the existence and uniqueness of solutions for systems of integral equations.


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## Body

## 1. Introduction

The theory of fixed point has become very important and useful to solve many mathematical problems in different subjects of sciences such as dynamical programming [1], optimization theory [2], signal processing [3], and iterative process [4], and other times it is used to prove the existence of solution of differential and integral equations [5, 6].Coupled fixed point was started by Guo and Lakshmikantham [7] in 1987 which is additionally trailed by Bhasker and Lakshmikantham [8] wherein authors gave the notion of mixed monotone property for a generalized contraction mapping F:X2→X (where X is an ordered metric space) and utilized the same to introduce some results on existence of coupled fixed points which are seen as coupled definition of specific results of Nieto and Lopez [9]. In 2009, Lakshmikantham and Ciric [10] extended these results for nonlinear contraction mappings by giving the notions of mixed g-monotone property and coupled coincidence point. Recently, the idea of coupled fixed point is reached out to higher dimensions by numerous authors. In fact, in 2010, Samet and Vetro [11] defined the notion of fixed point of n-tupled fixed point, n∈N and n≥2, as natural extension of coupled fixed point and proved some theorems for n-tupled fixed point in complete metric spaces. In 2013, Imdad et al. [12] extended the notion of n-tupled fixed point for a pair of mappings F and g by defining n-tupled coincidence point and also introduced the general concept of mixed g-monotone property on Xn for merely even n. Also, they employed the same to prove an even-tupled coincidence theorem for nonlinear contractions satisfying mixed g-monotone property. In fact, this result is true for only even n but not for odd ones. Recently, Kir et al. [13] considered the existence of a coupled fixed point of (φ,ψ)-contractive mappings endowed with a digraph and applied the obtained results to the system of two integral equations. There already exists an extensive literature on this topic. Keeping in view the relevance of this paper, we refer to [14–21].In this paper, we will obtain more general results of Kir et al. [13], which are common N-tupled coincidence points and common N-tupled fixed points of operators satisfying (φ,ψ)-contractive condition in metric spaces endowed with a digraph.Our paper is organized as follows. In Section2, some basic definitions are introduced. Section 3 contains some results on N-tupled coincidence points. In Section 4, the existence of a common N-tupled fixed point of operators satisfying (φ,ψ)-contractive conditions in metric spaces endowed with a digraph is considered. Section 5 gives an application to integral equations to support availability of the obtained results.


## 2. Preliminaries

Through all of this paper let(X,d) be a complete metric space. We recall the following definition which is given in [22].Definition 1.
An element(x1,x2,…,xN)∈XN is said to be an N-tupled fixed point of the function F:XN→X if (1)Fx1,x2,…,xN=x1,Fx2,x3,…,x1=x2,⋮FxN,x2,…,xN-1=xN.
Therefore, an element(x1,x2,…,xN)∈XN is said to be a common N-tupled coincidence point of the functions F:XN→X and g:X→X if (2)Fx1,x2,…,xN=gx1,Fx2,x3,…,x1=gx2,⋮FxN,x2,…,xN-1=gxN.Also, an element (x1,x2,…,xN)∈XN is said to be a common N-tupled fixed point of the functions F:XN→X and g:X→X if (x1,x2,…,xN) is N-tupled fixed point for both F and g.The following definition of commutative mappings inXN is the generalization of the definition of Lakshmikntham and Ciric [10].Definition 2.
The two mappingsF:XN→X and g:X→X are said to be commutative if (3)gFx1,x2,…,xN=Fgx1,gx2,…,gxNfor all (x1,x2,…,xN)∈XN.Inspired by the definition of compatible mappings inX2 of Choudhury and Kundu [23], we introduce the following definition.Definition 3.
The mappingsF:XN→X and g:X→X are called compatible if (4)limn→∞dgFxn1,xn2,…,xnN,Fgxn1,gxn2,…,gxnN=0,limn→∞dgFxn2,xn3,…,xnN,xn1,Fgxn2,gxn3,…,gxnN,gxn1=0,⋮limn→∞dgFxnN,xn1,…,xnN-1,FgxnN,gxn1,…,gxnN-1=0,where xn1,xn2,…,xnN are sequences in X such that (5)limn→∞Fxn1,xn2,…,xnN=limn→∞gxn1=x1,⋮limn→∞FxnN,xn1,…,xnN-1=limn→∞gxnN=xN,where (x1,x2,…,xN)∈XN.Definition 4 (see [24]).
LetΦ be the set of all functions φ:[0,∞)→[0,∞) such that, for any φ∈Φ, all following conditions are satisfied:
( 1 )   φ is lower semi-continuous and (strictly) increasing;
( 2 )   φ ( t ) < t for all t>0;
( 3 )   φ ( t + s ) ≤ φ ( t ) + φ ( s ) for all t,s∈[0,∞);
one can note thatlimn→∞φ(tn)=0 if and only if limn→∞tn=0 for tn∈[0,∞);
( 4 ) for φ∈Φ,Ψφ denote all functions ψ:[0,∞)→[0,∞) which satisfy the following:(4A)
limsupn→∞ψ(tn)<φ(r) if limn→∞tn=r>0;(4B)
limn→∞ψ(tn)=0 if limn→∞tn=0 for a sequence tn∈[0,∞).Now, let(X,d) be a metric space. Define △=x,x:x∈X to be a diagonal of X2 and G=(V(G),E(G)) is digraph which does not have parallel edges, where V(G) is the set of vertices of G and E(G) is the set of edges of G.Through all of this paper, letV(G) coincide with X and △∈E(G). Chifu and Petrusel [25] considered the concept of G-continuous mappings which is extended in XN by the following definition.Definition 5.
LetF:XN→X and g:X→X be two mappings. Then (I) F is said to be G-N-continuous if for all (x1∗,x2∗,…,xN∗)∈XN and any positive sequence nii∈N(6)Fxni1,xni2,…,xniN→x1∗,Fxni2,xni3,…,xni1→x2∗,⋮FxniN,xni1,…,xniN-1→xN∗,as i→∞ and (7)Fxni1,xni2,…,xniN,Fxni+11,xni+12,…,xni+1N∈EG;Fxni2,xni3,…,xni1,Fxni+12,xni+13,…,xni+11∈EG,⋮FxniN,xni1,…,xniN-1,Fxni+1N,xni+!1,…,xni+1N-1∈EG.We have (8)FFxni1,xni2,…,xniN,Fxni2,xni3,…,xni1,…,FxniN,xni1,…,xniN-1=Fx1∗,x2∗,…,xN∗,FFxni2,xni3,…,xni1,Fxni3,xni4,…,xni2,…,Fxni1,xni2,…,xniN=Fx2∗,x3∗,…,x1∗,⋮FFxniN,xni1,…,xniN-1,Fxni1,xni2,…,xniN,…,FxniN-1,xniN,…,xni1=FxN∗,x1∗,…,xN-1∗.(II) (X,d, G) has property A if any sequence xn⊆X with xn→x as n→∞ and (xn,xn+1)∈E(G), then (xn,x)∈E(G).Consider the setNTCP(F,g) of all N-tupled coincidence points of the mappings F:XN→X and g:X→X, i.e.,(9)NTCPF,g=x1,x2,…,xN∈XN:Fx1,x2,…,xN=gx1,Fx2,x3,…,x1=gx2,…,FxN,x1,…,xN-1=gxN.Therefore, (XN)gF is defined as(10)XNgF=x1,x2,…,xN∈XN:gx1,Fx1,x2,…,xN∈EG,…,gxN,FxN,x1,…,xN-1∈EG.In 2015, Suantai et al. [26] considered the definition of G- edge preserving which is extended by the following definition.Definition 6.
The mappingsF:XN→X and g:X→X are said to be N-G-edge preserving if (11)gx1,gu1,…,gxN,guN∈EGimplies that (12)Fx1,…,xN,Fu1,…,uN,…,FxN,…,XN-1,FuN,…,uN-1∈EG.Definition 7 (see [26]).
E ( G ) satisfies the transitivity property in X if and only if ∀x,y,t∈X, where (x,y),(y,t)∈E(G)→(x,t)∈E(G).


## 3. An N-Tupled Coincidence Point Theorem

Firstly, we give the following definition which will be used in our main results.Definition 8.
Let(X,d) be a complete metric space endowed with digraph G. The mappings F:XN→X and g:X→X are called (ϕ,ψ)-contractive if
( i )   F and f are N-G- edge preserving,
( i i ) there exist φ∈Φ and ψ∈Ψφ such that, for all (x1,…,xN) and (u1,…,uN)∈XN, (13)gx1,gu1,…,gxN,guN∈EG,then(14)φ2-1dFx1,…,xN,Fu1,…,uN+dFx2,…,x1,Fu2,…,u1+⋯+dFxN,…,xN-1,FuN,…,uN-1≤ψdgx1,gu1+dgx2,gu2+⋯+dgxN,guN.We study the existence of N-tupled coincidence point of the mappingsF:XN→X and g:X→X under the following hypotheses:( H 1 )   g is continuous and g(X) is closed;( H 2 )   F ( X N ) ⊆ g ( X ), and (F,g) is compatible;( H 3 )   F is N-G-continuous;( H 4 ) the tripled (X,d,G) has a property A.Now, we are ready to introduce ourN-tupled coincidence point theorem.Theorem 9.
LetF:XN→X and g:X→X be (φ,ψ)-contractive mappings, where (X,d) is a complete metric space and G is the digraph endowed by X. If the hypotheses H1,H2, and (H3) hold, then NTCP(F,g)≠∅ if and only if (XN)gF≠∅.Proof.
Consider(x01,x02,…,x0N),(y01,y02,…,y0N)∈XN. Since F(XN)⊆g(X), then there exist
x 1 1 , x 1 2 , … , x 1 N , y 1 1 , y 1 2 , … , y 1 N ∈ X such that (15)Fx01,x02,…,x0N=gx11Fx02,x03,…,x01=gx12⋮Fx0N,x01,…,x0N-1=gx1NAlso, we have that (16)Fy01,y02,…,y0N=gy11Fy02,y03,…,y01=gy12⋮Fy0N,y01,…,y0N-1=gy1N.By doing this process, then we get the sequences xn1,xn2,…,xnN, yn1,yn2,…,ynN such that(17)Fxn1,xn2,…,xnN=gxn+11;Fxn2,xn3,…,xn1=gxn+12;⋮FxnN,xn1,…,xnN-1=gxn+1N.Also, we have that(18)Fyn1,yn2,…,ynN=gyn+11;Fyn2,yn3,…,yn1=gyn+12;⋮FynN,yn1,…,ynN-1=gyn+1N.
Now, suppose thatNTCP(F,g)≠∅. Let (u1,u2,…,uN)∈NTCP(F,g). Thus, we have that (19)Fu1,u2,…,uN=gu1Fu2,u3,…,u1=gu2⋮FuN,u1,…,uN-1=guN.Hence, we get that (20)gu1,Fu1,u2,…,uN=gu1,gu1∈EG;gu2,Fu2,u3,…,u1=gu2,gu2∈EG;⋮guN,FuN,u1,…,uN-1=guN,guN∈EG.Hence, (u1,u2,…,uN)∈(XN)gF. Then, we obtain (XN)gF≠∅.
Let(x01,…,x0N)∈(XN)gF, then we have that (21)gx01,Fx01,x02,…,x0N∈EG;gx02,Fx02,x03,…,x01∈EG;⋮gx0N,Fx0N,x01,…,x0N-1∈EG.By using (15), we get that (22)gx01,Fx01,x02,…,x0N=gx01,gx11∈EG;gx02,Fx02,x03,…,x01=gx02,gx12∈EG;⋮gx0N,Fx0N,x01,…,x0N-1=gx0N,gx1N∈EG.Since F and g are N-G-edge preserving, we have that (23)Fx01,x02,…,x0N,Fx11,x12,…,x1N=gx11,gx21∈EG;Fx02,x03,…,x01,Fx12,x13,…,x11=gx12,gx22∈EG;⋮Fx0N,x01,…,x0N-1,Fx1N,x11,…,x1N-1=gx1N,gx2N∈EG.By doing this process, we have (24)gxn1,gxn+11,gxn2,gxn+12,…,gxnN,gxn+1N∈EG. Let rn=((dgxn+11,gxn+21+dgxn+12,gxn+22+⋯+dgxn+1N,gxn+2N)/2). We get that(25)φdgxn+11,gxn+21+dgxn+12,gxn+22+⋯+dgxn+1N,gxn+2N2=φdFxn1,…,xnN,Fxn+11,…,xn+1N+dFxn2,…,xn1,Fxn+12,…,xn+11+⋯+dFxnN,…,xnN-1,Fxn+1N,…,xn+1N-12≤ψdgxn1,gxn+11+dgxn2,gxn+12+⋯+dgxnN,gxn+1N2, for all n∈N, then we have that(26)φrn+1≤ψrn<φrn By using (26) and because φ is monotonic function, we obtain rn is a positive decreasing sequence. Then, rn→r as n→∞ for some r∈[0,∞). Consider r>0. In (26), taking the limit as n→∞ and from the properties of φ,ψ, we have that(27)φr≤limsupn→∞φrn+1≤limsupn→∞ψrn<φr We obtain contradiction. Thus r=0 and then we get (28)limn→∞rn=limn→∞dgxn1,gxn+11+dgxn2,gxn+12+⋯+dgxnN,gxn+1N2=0 Now, we will show that gxn1,gxn2,…,gxnN are Cauchy sequences. Let gxn1,gxn2,…,gxnN be not Cauchy sequences. Then there exist ϵ>0, positive integers n(k)>m(k)≥k and we can find gxnk1,gxmk1,gxnk2,gxmk2,…,gxnkN,gxmkN such that(29)ηk=dgxnk1,gxmk1+dgxnk2,gxmk2+⋯+dgxnkN,gxmkN2≥ϵ. Therefore, we can find n(k) is the smallest integer for which (29) holds, and we have that(30)dgxnk-11,gxmk1+dgxnk-12,gxmk2+⋯+dgxnk-1N,gxmkN2≥ϵ. Using (29) and (30), we get(31)ϵ≤ηk<ϵ+dgxnk1,gxnk-11+dgxnk2,gxnk-12+⋯+dgxnkN,gxnk-1N2. From (27) and (31), we get that(32)ηk=dgxnk1,gxmk1+dgxnk2,gxmk2+⋯+dgxnkN,gxmkN2→0,ask→∞. Therefore, we obtain that(33)ηk=dgxnk1,gxmk1+dgxnk2,gxmk2+⋯+dgxnkN,gxmkN2≤rnk+rmk+dgxnk+11,gxmk+11+dgxnk+12,gxmk+12+⋯+dgxnk+1N,gxmk+1N2. Using the properties of φ, we get that(34)φηk≤φrnk+φrmk+φdgxnk+11,gxmk+11+dgxnk+12,gxmk+12+⋯+dgxnk+1N,gxmk+1N2≤φrnk+φrmk+φdFxnk1,…,xnkN,Fxmk1,…,xmkN+⋯+dFxnkN,…,xnkN-1,FxmkN,…,xmkN-12≤φrnk+φrmk+ψdgxnk1,gxmk1+⋯+dgxnkN,gxmkN)2≤φrnk+φrmk+φηk.Again from the properties of φ which is lower semi-continuous, if we take the limit as k→∞, we have that(35)φϵ≤limsupk→∞φηnk≤limk→∞φrnk+limk→∞φrmk+limsupk→∞⁡ψηnk<φϵ.That is a contradiction. Hence, gxn1,gxn2,…,gxnN are Cauchy sequences. Then, there exist u1,u2,…,uN∈g(X) such that (36)limn→∞Fxn1,xn2,…,xnN=limn→∞gxn1=u1;limn→∞Fxn2,xn3,…,xn1=limn→∞gxn2=u2;⋮limn→∞FxnN,xn1,…,xnN-1=limn→∞gxnN=uN.Since (F,g) is compatible, we get (37)limn→∞dgFxn1,xn2,…,xnN,Fgxn1,xn2,…,gxnN=0;limn→∞dgFxn2,xn3,…,xn1,Fgxn2,xn3,…,gxn1=0;⋮limn→∞dgFxnN,xn1,…,xnN-1,FgxnN,xn1,…,gxnN-1=0. Since F is N-G-continuous, we get, for all n≥0, (38)dFgxn1,xn2,…,gxnN,gu1)≤dFgxn1,xn2,…,gxnN,gFxn1,xn2,…,xnN)+d(gFxn1,xn2,…,xnN,gu1.dFgxn2,xn3,…,gxn1,gu2)≤dFgxn2,xn3,…,gxn1,gFxn2,xn3,…,xn1)+dgFxn2,xn3,…,xn1,gu2.⋮dFgxnN,xn1,…,gxnN-1,guN)≤dFgxnN,xn1,…,gxnN-1,gFxnN,xn1,…,xnN-1)+dgFxnN,xn1,…,xnN-1,guN.Taking the limit as n→∞ and using (H1) and (H2), we obtain that (39)dFu1,u2,…,uN,gu1=0,dFu2,u3,…,u1,gu2=0,…,dFuN,u1,…,uN-1,guN=0.Thus, we have that (40)Fu1,u2,…,uN=gu1,Fu2,u3,…,u1=gu2,…,FuN,u1,…,uN-1=guN.Hence, NTCP(F,g)≠∅.Theorem 10.
LetF:XN→X and g:X→X be (φ,ψ)-contractive mappings, where (X,d) is a complete metric space and G is the digraph endowed by X. If the hypotheses (H1),(H2), and (H4) hold, then NTCP(F,g)≠∅ if and only if (XN)gF≠∅.Proof.
Let(u1,…,uN),(x1,…,xN)∈XN such that u1=gx1,…,uN=gxN. Following the same argument as in the proof of Theorem 9, we get sequences xn1,…,xnN such that (41)gxn1,x1,…,gxnN,xN∈EG.Thus, we have that (42)dgx1,Fx1,…,xN≤dgx1,gxn+11+dgxn+11,Fx1,…,xN=dgx1,gxn+11+dFxn1,…,xnN,Fx1,…,xN;dgx2,Fx2,…,x1≤dgx2,gxn+12+dgxn+12,Fx2,…,x1=dgx2,gxn+12+dFxn2,…,xn1,Fx2,…,x1;⋮dgxN,FxN,…,xN-1≤dgxN,gxn+1N+dgxn+1N,FxN,…,xN-1=dgxN,gxn+1N+dFxnN,…,xnN-1,FxN,…,xN-1.Thus, we obtain that (43)dgx1,Fx1,…,xN-dgx1,gxn+11≤dFxn1,…,xnN,Fx1,…,xN;dgx2,Fx2,…,x1-dgx2,gxn+12≤dFxn2,…,xn1,Fx2,…,x1;⋮dgxN,FxN,…,xN-1-dgxN,gxn+1N≤dFxnN,…,xnN-1,FxN,…,xN-1,which implies that(44)dgx1,Fx1,…,xN-dgx1,,gxn+11+⋯+dgxN,FxN,…,xN-1-dgxN,gxn+1N2≤dFxn1,…,xnN,Fx1,…,xN+⋯+dFxnN,…,xnN-1,FxN,…,xN-12. Again using the monotonicity property of φ, we get(45)φdgx1,Fx1,…,xN-dgx1,,gxn+11+⋯+dgxN,FxN,…,xN-1-dgxN,gxn+1N2≤φdFxn1,…,xnN,Fx1,…,xN+⋯+dFxnN,…,xnN-1,FxN,…,xN-12≤ψdgxn1,gx1+⋯+dgxnN,gxN2. By taking the limit as n→∞, we obtain that (46)gx1=Fx1,…,xN,…,gxN=FxN,…,x1. Hence, NTCP(F,g)≠∅.


## 4. The Existence and Uniqueness of N-Tupled Fixed Points

Let the set of all common N-tupled fixed points of the mappingsF:XN→X and g:X→X denoted by NTF(F,g) and defined as (47)NTFF,g=x1,…,xN∈XN:Fx1,…,XN=gx1=x1,…,FxN,…,xN-1=gxN=xN.Now, we consider the following fixed point property:( F P ) Whenever F:XN→X and g:X→X are two mappings, then F,g have a unique common N-tupled fixed point.Theorem 11.
LetF:XN→X and g:X→X be commutative (φ,ψ)-contractive mappings, where (X,d) is a complete metric space and G is the digraph endowed by X. If the hypotheses (H1),(H2), and (H3) hold and for any two elements (x1,…,xN),(y1,…,yN)∈XN there exists (z1,…,zN)∈XN such that [(gx1,gz1),(gy1,gz1),…,(gxN,gzN),(gyN,gzN)]∈E(G). Then (XN)gF≠∅ if and only if F,g have the fixed point property (FP).Proof.
By doing the same argument as in the proof of Theorem9, there exists (x1,x2,…,xN)∈XN such that (48)Fx1,x2,…,xN=gx1;Fx2,x3,…,x1=gx2;⋮FxN,x1,…,xN-1=gxN.Assume there exists another element (y1,y2,…,yN)∈XN such that (49)Fy1,y2,…,yN=gy1;Fy2,y3,…,y1=gy2;⋮FyN,y1,…,yN-1=gyN.We will show that gx1=gy1,gx2=gy2,…,gxN=gyN. We can find (z1,z2,…,zN)∈XN such that (50)gx1,gz1,gy1,gz1,…,gxN,gzN,gyN,gzN∈EG. By doing the same process as in (17) and (18), we get the sequences xn1,…,xnN,yn1,…,ynN, zn1,…,znN such that (51)x1=x01,x2=x02,…,xN=x0N;y1=y01,y2=y02,…,yN=y0N;and (52)z1=z01,z2=z02,…,zN=z0N. Therefore, we obtain that (53)Fxn1,xn2,…,xnN=gxn+11;Fxn2,xn3,…,xn1=gxn+12;⋮FxnN,xn1,…,xnN-1=gxn+1N;Fyn1,yn2,…,ynN=gyn+11;Fyn2,yn3,…,yn1=gyn+12;⋮FynN,yn1,…,ynN-1=gyn+1N;Fzn1,zn2,…,znN=gzn+11;Fzn2,zn3,…,zn1=gzn+12;⋮FznN,zn1,…,znN-1=gzn+1N;and we obtain that (54)Fx1,x2,…,xN=gxn1;Fx2,x3,…,x1=gxn2;⋮FxN,x1,…,xN-1=gxnN;Fy1,y2,…,yN=gyn1;Fy2,y3,…,y1=gyn2;⋮FyN,y1,…,yN-1=gynNfor all n∈N. Then we have that (55)gx1,gz1,gy1,gz1∈EG⇒gx1,gz01∈EG;gx2,gz2,gy2,gz2∈EG⇒gx2,gz02∈EG;⋮gxN,gzN,gyN,gzN∈EG⇒(56)gxN,gz0N∈EG;therefore, we get that (57)Fx1,x2,…,xN,Fz01,z02,…,z0N=gx1,gz11∈EG;Fx2,x3,…,x1,Fz03,z02,…,z01=gx2,gz12∈EG;⋮FxN,x1,…,xN-1,Fz0N,z01,…,z0N-1=gxN,gz1N∈EG.Similarly, one can get that [(gx1,gzn1),…,(gxN,gznN)]∈E(G). Using the properties of φ,ψ, we get that(58)φdgzn+11,gx1+⋯+dgzn+1N,gxN2≤φdFzn1,…,znN,Fx1,…,xN+⋯+dFznN,…,znN-1,FxN,…,xN-12≤ψdgzn1,gx1+⋯+dgznN,gxN2. We obtain that(59)φξn+1≤ψξn, where (60)ξn=dgzn1,gx1+⋯+dgznN,gxN2. The sequence ξn is monotonic decreasing sequence of positive real numbers; then ∃ξ≥0 such that (61)ξn→∞asn→∞. Let ξ>0. Take the limit as n→∞ for (59), (62)φξ≤limsupn→∞φξn+1<φξ. This is a contradiction; hence ξ=0. Therefore, (63)limn→∞dgzn1,gx1+⋯dgznN,gxN2=0, which gives that (64)limn→∞dgzn1,gx1=limn→∞dgzn2,gx2=⋯=limn→∞dgznN,gxN=0; and according to [(gz1,gy1),…,(gzN,gyN)]∈E(G), then we get that (65)limn→∞dgzn1,gy1=limn→∞dgzn2,gy2=⋯=limn→∞dgznN,gyN=0.
Since(66)dgy1,gx1≤dgy1,gzn1+dgzn1,gx1 for all n∈N, then by taking the limit as n→∞, we have that (67)dgy1,gx1=dgy2,x2=⋯=dgyN,xN=0. Thus, gx1=gy1, gx1=gy2,…,gxN=gyN. Hence the uniqueness of N-tupled common coincidence point for F and g is obtained. Let (w1,w2,…,wN)∈XN be the unique N-tupled common coincidence point for F and g, then it is clear that (68)w1=Fw1,w2,…,wN,w2=Fw2,w3,…,w1,…,wN=FwN,w1,…,wN-1. Since (F,g) are commutative, we have (69)gw1=gFw1,w2,…,wN=Fgw1,gw2,…,gwN,gw2=gFw2,w3,…,w1=Fgw2,gw3,…,g,w1,…,gwN=gFwN,w1,…,wN-1=FgwN,gw1,…,gwN-1.
Hence,(gw1,gw2,…,gwN) is another N-tupled common coincidence point for F and g. Then, we obtain that (70)w1=gw1=Fw1,w2,…,wN,w2=gw2=Fw2,w3,…,w1,…,wN=gwN=FwN,w1,…,wN-1. To prove the uniqueness of N-tupled fixed point, let (w1^,w2^,…,wN^)∈XN be another fixed point. Then there exists (u1,u2,…,uN)∈XN such that (71)u01=u1,u02=u2,…,u0N=uN, and we obtain that (72)Fun1,un2,…,unN=gun+11;Fun2,un3,…,un1=gun+12;⋮FunN,un1,…,unN-1=gun+1Nfor all n∈N. Then we have that (73)gw1,gun1,gw2,gun2,…,gwN,unN∈EG. Then we have that(74)φdw1,gun+11+⋯+dw2,gun+1N2=φdFw1,…,wN,Fun+11,…,un+1N+⋯+dFwN,…,wN-1,Fun+1N,…,un+1N-12≤ψdwn1,gun1+⋯+dwnN,gunN2. Therefore, by putting (75)ρn=dwn1,gun1+⋯+dwnN,gunN2,we obtain (76)φρn+1≤ψρn By doing the same steps above, we can show that there exist ρ≥0 such that ρn→ρ as n→∞. If ρ>0, (77)φρ≤limsupn→∞φρn+1≤limn→∞ψρn<φρ, which is a contraction. Hence it must be ρ=0. Then we have that (78)limn→∞dw1,un1=limn→∞dw2,un2=⋯=limn→∞dwN,unN=0. Similarly, we can prove (79)limn→∞dw1^,un1=limn→∞dw2^,un2=⋯=limn→∞dwN^,unN=0. Hence, we get w1^=w1,w2^=w2,…,wN^=wN.Theorem 12.
LetF:XN→X and g:X→X be commutative (φ,ψ)-contractive mappings, where (X,d) is a complete metric space and G is the digraph endowed by X. If the hypotheses (H1),(H2), and (H4) hold and for any two elements (x1,…,xN),(y1,…,yN)∈XN there exist (z1,…,zN)∈XN such that [(gx1,gz1),(gy1,gz1),…,(gxN,gzN),(gyN,gzN)]∈E(G), then (XN)gF≠∅ if and only if F,g have the fixed point property (FP).


## 5. An application

Now, we apply the N-tupled fixed point theorem to study the existence and uniqueness of the solution of the system of nonlinear integral equations. This system can be written as(80)x1t=At+∫0aBt,sKs,x1s,x2s,…,xNsds;x2t=At+∫0aBt,sKs,x2s,x3s,…,x1sds;⋮xNt=At+∫0aBt,sKs,xNs,x1s,…,xN-1sds,where a>0,t∈[0,a]. Define the space X=C([0,a],Rn)=x:[0,a]→Rn:x is continuous , where, for all x∈X, x=maxt∈[0,a]xt. Therefore, d(x,y)=x-y and (X,d) is a complete metric space. Let G be a digraph which V(G) coincides with X and E(G) can be obtained by the the partial order relation (81)x,y∈EG⇔x≤y⇔xt≤ytforallt∈0,a. Thus (X,d) is endowed with a digraph G. It is clear that the transitivity property is satisfied on E(G) and (x,d,G) has property A. To prove the existence of unique solution of (80), we consider( C 1 )   K : [ 0 , a ] × R N × R N ⋯ × R N ︸ N - t i m e s → R N and A:[0,a]→RN are continuous,( C 2 ) there exists a 0≤θ≤1 such that(82)Ks,x1,x2,…,xN-Ks,y1,y2,…,yN≤θx1-y1+x2-y2+⋯+xN-yN,( C 3 )   ∀ s , t ∈ [ 0 , a ], there exists a continuous mapping B:[0,a]×R→R such that(83)supt∈0,a∫0aBt,sds<1,( C 4 ) there exists (x10,x20,…,xN0)∈XN such that(84)x10t≤At+∫0aBt,sKs,x10s,x20s,…,xN0sds;x20t≤At+∫0aBt,sKs,x20s,x30s,…,x10sds;⋮xN0t≤At+∫0aBt,sKs,xN0s,x10s,…,xN-10sds,for all t∈[0,a].Theorem 13.
If conditions (C1), (C2), (C3), and (C4) are satisfied, then system (80) has a unique solution in C([0,a],Rn).Proof.
DefineF:XN→X, where(85)Fx1,x2,…,xNt=At+∫0aBt,sKs,x1s,x2s,…,xNsds;t∈0,a. Define g:X→X as gx(t)=2x(t). System (80) can be written as (86)x1=Fx1,x2,…,xN,x2=Fx2,x3,…,x1,…,xN=FxN,x1,…,xN-1.Let x1,x2,…,xN,u1,u2,…,uN∈X such that (87)gx1≤gu1,gx2≤gu2,…,gxN≤guN. Then we get that x1≤u1,x2≤u2,…,xN≤uN, and(88)Fx1,x2,…,xNt=At+∫0aBt,sKs,x1s,x2s,…,xNsds≤At+∫0aBt,sKs,u1s,u2s,…,uNsds=Fu1,u2,…,uNt;Fx2,x3,…,x1t=At+∫0aBt,sKs,x2s,x3s,…,x1sds≤At+∫0aBt,sKs,u2s,u3s,…,u1sds=Fu2,u3,…,u1t;⋮FxN,x1,…,xN-1t=At+∫0aBt,sKs,xNs,x1s,…,xN-1sds≤At+∫0aBt,sKs,uNs,u1s,…,uN-1sds=FuN,u1,…,uN-1t;forallt∈0,aHence, (F,g) are N-G-edge preserving. By using conditions (C2) and (C3), and for all t∈[0,a], we get that(89)Fx1,…,xNt-Fu1,…,uNt+⋯+FxN,…,xN-1t-FuN,…,uN-1t2=∫0aBt,sKs,x1s,x2s,…,xNs-Ks,u1s,u2s,…,uNsds2+⋯+∫0aBt,sKs,xNs,x1s,…,xN-1s-Ks,uNs,u1s,…,uN-1sds2≤Fx1,…,xNt-Fu1,…,uNt+⋯+FxN,…,xN-1t-FuN,…,uN-1t2=∫0aBt,sKs,x1s,x2s,…,xNs-Ks,u1s,u2s,…,uNsds2+⋯+∫0aBt,sKs,xNs,x1s,…,xN-1s-Ks,uNs,u1s,…,uN-1sds2≤θ2x1s-u1s+⋯+2xNs-uNs2≤θgx1s-gu1s+⋯+gxNs-guNs2=θdgx1,gu1+⋯+dgxN,guN2 Take φ(t)=t and ψ(t)=θt. We get(90)φdFx1,…,xN,Fu1,…,uN+⋯+dFxN,…,xN-1,FuN,…,uN-12≤ψdgx1,gu1+⋯+dgxN,guN2 Thus F and g are (φ-ψ)-contractive. Therefore by using (C4) there exists (x10,x20,…,xN0)∈XN such that (91)gx10≤Fx10,x20,…,xN0,gx20≤Fx20,x30,…,x10,…,gxN0≤FxN0,x10,…,xN-10. Hence (XN)gF≠∅. Hence there exists a unique N-tupled common fixed point (x1∗,x2∗,…,xN∗)∈XN of F and g is the unique solution of system (80).



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*Source: 9839627-2018-11-05.xml*