IJMMS International Journal of Mathematics and Mathematical Sciences 1687-0425 0161-1712 Hindawi Publishing Corporation 329298 10.1155/2012/329298 329298 Research Article Tripled Coincidence Point Theorems for Nonlinear Contractions in Partially Ordered Metric Spaces Choudhury Binayak S. 1 Karapınar Erdal 2 Kundu Amaresh 3 Ricci Paolo 1 Department of Mathematics Bengal Engineering and Science University Shibpur Howrah 711103 India becs.ac.in 2 Department of Mathematics Atilim University İncek 06836 Ankara Turkey atilim.edu.tr 3 Department of Mathematics Siliguri Institute of Technology Darjeeling 734009 India sittechno.org 2012 19 07 2012 2012 12 04 2012 19 05 2012 2012 Copyright © 2012 Binayak S. Choudhury et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Tripled fixed points are extensions of the idea of coupled fixed points introduced in a recent paper by Berinde and Borcut, 2011. Here using a separate methodology we extend this result to a triple coincidence point theorem in partially ordered metric spaces. We have defined several concepts pertaining to our results. The main results have several corollaries and an illustrative example. The example shows that the extension proved here is actual and also the main theorem properly contains all its corollaries.

1. Introduction and Preliminaries

In recent times coupled fixed point theory has experienced a rapid growth in partially ordered metric spaces. The speciality of this line of research is that the problems herein utilize both order theoretic and analytic methods. References  are some instances of these works.

Definition 1.1 (see [<xref ref-type="bibr" rid="B15">14</xref>]).

A function g:RR is said to be monotone nondecreasing (or increasing) if xy implies g(x)g(y).

Definition 1.2 (see [<xref ref-type="bibr" rid="B15">14</xref>]).

Let X be a nonempty set. Let F:X×XX be a mapping. An element (x,y) is called a coupled fixed point of F if (1.1)F(x,y)=x,F(y,x)=y.

Recently, Berinde and Borcut  extended the idea of coupled fixed points to tripled fixed points. The definition is as follows.

Definition 1.3 (see [<xref ref-type="bibr" rid="B5">20</xref>]).

Let X be a nonempty set. Let F:X×X×XX be a mapping. An element (x,y,z) is called a tripled fixed point of F if (1.2)F(x,y,z)=x,F(y,x,y)=y,F(z,y,x)=z.

They also extended the mixed monotone property to functions with three arguments.

Definition 1.4 (see [<xref ref-type="bibr" rid="B5">20</xref>]).

Let (X,) be a partially ordered set and F:X×X×XX. The mapping F is said to have the mixed monotone property if for any x,y,zX(1.3)x1,x2X,x1x2F(x1,y,z)F(x2,y,z),y1,y2X,y1y2F(x,y1,z)F(x,y2,z),z1,z2X,z1z2F(x,y,z1)F(x,y,z2).

Our purpose here is to establish tripled coincidence point results in metric spaces with partial ordering. For that purpose we define mixed g-monotone property in the following. Mixed g-monotone property was already defined in the context of coupled fixed points . Here in the spirit of Definition 1.4 we have made an extension of that.

Definition 1.5.

Let (X,) be a partially ordered set. Let g:XX and F:X×X×XX. The mapping F is said to have the mixed g-monotone property if for any x,y,zX. (1.4)x1,x2X,gx1gx2F(x1,y,z)F(x2,y,z),y1,y2X,gy1gy2F(x,y1,z)F(x,y2,z),z1,z2X,gz1gz2F(x,y,z1)F(x,y,z2).

Coupled coincidence point was defined by Lakshmikantham and Ćirić . We also extend the concept of coupled coincidence point to tripled coincidence point in the following.

Definition 1.6.

Let X be any nonempty set. Let g:XX and F:X×X×XX. An element (x,y,z) is called a tripled coincidence point of g and F if (1.5)F(x,y,z)=gx,F(y,x,y)=gy,F(z,y,x)=gz.

We extend the concept of commuting mappings given by Lakshmikantham and Ćirić , in the following definition.

Definition 1.7.

Let X be a nonempty set. Then one says that the mappings g:XX and F:X×X×XX are commuting if for all x,y,zX(1.6)g(F(x,y,z))=F(gx,gy,gz).

The following is the definition of compatible mappings which is an extension of the compatibility defined by Choudhury and Kundu in .

Definition 1.8 (see [<xref ref-type="bibr" rid="B9">8</xref>]).

Let (X,d) be a metric space. The mappings g and F, where g:XX and F:X×X×XX are said to be compatible if (1.7)limnd(gF(xn,yn,zn),F(gxn,gyn,gzn))=0,limnd(gF(yn,xn,yn),F(gyn,gxn,gyn))=0,limnd(gF(zn,yn,xn),F(gzn,gyn,gxn))=0, whenever {xn},{yn},{zn} are sequences in X such that (1.8)limnF(xn,yn,zn)=gxn=x,limnF(yn,xn,yn)=gyn=y,limnF(zn,yn,xn)=gzn=z.

2. Main Results Theorem 2.1.

Let (X,) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Suppose F:X×X×XX and g:XX are such that, g is monotone increasing, F has the mixed g-monotone property and (2.1)d(F(x,y,z),F(u,v,w))ψ(max{d(gx,gu),d(gy,gv),d(gz,gw)}) for all x,y,zX for which gxgu, gygv and gzgw, where ψ:[0,+)[0,+) is such that ψ(t) is monotone, ψ(t)<t and limrt+ψ(r)<t for all t>0. Suppose F(X×X×X)g(X), g is continuous, and {g,F} is a compatible pair. Suppose either

F  is continuous or

X has the following properties:

if a nondecreasing sequence {αn}α, then αnα for all n,

if a nonincreasing sequence {βn}β, then βnβ for all n.

If there exist x0,y0,z0X such that gx0F(x0,y0,z0), gy0F(y0,x0,y0), and gz0F(z0,y0,x0), then there exist x,y,zX such that (2.2)F(x,y,z)=gx,F(y,x,y)=gy,F(z,y,x)=gz, that is, g and F have a tripled coincidence point.

Proof.

By a condition of the theorem, there exist x0,y0,z0X such that gx0F(x0,y0,z0), gy0F(y0,x0,y0), and gz0F(z0,y0,x0). Since F(X×X×X)g(X), we can choose x1,y1,z1X such that (2.3)gx1=F(x0,y0,z0),gy1=F(y0,x0,y0),gz1=F(z0,y0,x0). Continuing this process, we can construct sequences {xn},{yn}, and {zn} in X such that (2.4)gxn+1=F(xn,yn,zn),gyn+1=F(yn,xn,yn),gzn+1=F(zn,yn,xn). Next we will show that, for n0, (2.5)gxngxn+1,gyngyn+1,gzngzn+1. Since, gx0F(x0,y0,z0),gy0F(y0,x0,y0), and gz0F(z0,y0,x0), by (2.3), we get (2.6)gx0gx1,gy0gy1,gz0gz1, that is, (2.5) holds for n=0.

We presume that (2.5) holds for some n=m>0. As F has the mixed g-monotone property and gxmgxm+1,gymgym+1 and gzmgzm+1, we obtain (2.7)gxm+1=F(xm,ym,zm)F(xm+1,ym,zm)F(xm+1,ym,zm+1)F(xm+1,ym+1,zm+1)=gxm+2,(2.8)gym+1=F(ym,xm,ym)F(ym,xm,ym+1)F(ym+1,xm,ym+1)F(ym+1,xm+1,ym+1)=gym+2,(2.9)gzm+1=F(zm,ym,xm)F(zm+1,ym,xm)F(zm+1,ym+1,xm)F(zm+1,ym+1,xm+1)=gzm+2. Thus, (2.5) holds for n=m+1. Then, by induction, we conclude that (2.5) holds for n1.

If for some n, (2.10)gxn=gxn+1,gyn=gyn+1,gzn=gzn+1, then, by (2.4), (xn,yn,zn) is a tripled coincidence point of g and F. Therefore we assume, for any n, (2.11)(gxn,gyn,gzn)(gxn+1,gyn+1,gzn+1). Set δn=max{d(gxn,gxn+1),d(gyn,gyn+1),d(gzn,gzn+1)}.

Then (2.12)δn>0n0. Then, by (2.1), (2.4) and (2.5), we have (2.13)d(gxn,gxn+1)=d(F(xn-1,yn-1,zn-1),F(xn,yn,zn))ψ(max{d(gxn-1,gxn),d(gyn-1,gyn),d(gzn-1,gzn)}),d(gyn,gyn+1)=d(F(yn-1,xn-1,yn-1),F(yn,xn,yn))ψ(max{d(gyn-1,gyn),d(gxn-1,gxn),d(gyn-1,gyn)}),d(gzn,gzn+1)=d(F(zn-1,yn-1,xn-1),F(zn,yn,xn))ψ(max{d(gzn-1,gzn),d(gyn-1,gyn),d(gxn-1,gxn)}). Thus, from (2.13) we obtain that (2.14)δn=max{d(gxn,gxn+1),d(gyn,gyn+1),d(gzn,gzn+1)}ψ(max{d(gxn-1,gxn),d(gyn-1,gyn),d(gzn-1,gzn)}). It then follows from (2.12) and a property ψ, that for all n1, (2.15)δnψ(δn-1)<δn-1. Thus, {δn} is a monotone decreasing sequence of nonnegative real numbers. So, there exist a δ0 such that (2.16)limnδn=δ. Suppose δ>0. Letting n in (2.14), using (2.15), (2.16), and a property of ψ, we get (2.17)δψ(δ)<δ, which is a contradiction. Thus δ=0, or (2.18)limnδn=0, or (2.19)limnd(gxn+1,gxn)=0,limnd(gyn+1,gyn)=0,limnd(gzn+1,gzn)=0. Now, we will prove that {gxn}, {gyn}, and {gzn} are Cauchy sequences. Suppose, to the contrary, that at least one of {gxn}, {gyn}, and {gzn} is not a Cauchy sequence. So, there exists an ɛ>0 for which we can find subsequences {gxn(k)} of {gxn}, {gyn(k)} of {gyn}, and {gzn(k)} of {gzn} with n(k)>m(k)k such that (2.20)αk=max{d(gxn(k),gxm(k)),d(gyn(k),gym(k)),d(gzn(k),gzm(k))}ɛ. Additionally, corresponding to m(k), we may choose n(k) such that it is the smallest integer satisfying (2.20). Then, for all k0, (2.21)max{d(gxn(k)-1,gxm(k)),d(gyn(k)-1,gym(k)),d(gzn(k)-1,gzm(k))}<ɛ. By using (2.20) and (2.21) we have for k0, (2.22)ɛαk=max{d(gxn(k),gxm(k)),d(gyn(k),gym(k)),d(gzn(k),gzm(k))}max{d(gxn(k),gxn(k)-1)+d(gxn(k)-1,gxm(k)),d(gyn(k),gyn(k)-1)+d(gyn(k)-1,gym(k)),d(gzn(k),gzn(k)-1)+d(gzn(k)-1,gzm(k))}max{d(gxn(k),gxn(k)-1),d(gyn(k),gyn(k)-1),d(gzn(k),gzn(k)-1)}+ɛδn(k)-1+ɛ. Letting k in (2.22), and using (2.19), we get (2.23)limkαk=limkmax{d(gxn(k),gxm(k)),d(gyn(k),gym(k)),d(gzn(k),gzm(k))}=ɛ. Let, for k0, (2.24)βk=max{d(gxn(k)+1,gxm(k)+1),d(gyn(k)+1,gym(k)+1),d(gzn(k)+1,gzm(k)+1)}.

Again, for all k0, (2.25)αk=max{d(gxn(k),gxm(k)),d(gyn(k),gym(k)),d(gzn(k),gzm(k))}max{d(gxn(k),gxn(k)+1)+d(gxn(k)+1,gxm(k)+1)+d(gxm(k)+1,gxm(k)),d(gyn(k),gyn(k)+1)+d(gyn(k)+1,gym(k)+1)+d(gym(k)+1,gym(k)),d(gzn(k),gzn(k)+1)+d(gzn(k)+1,gzm(k)+1)+d(gzm(k)+1,gzm(k))}max{d(gxn(k),gxn(k)+1),d(gyn(k),gyn(k)+1),d(gzn(k),gzn(k)+1)}+max{d(gxn(k)+1,gxm(k)+1),d(gyn(k)+1,gym(k)+1),d(gzn(k)+1,gzm(k)+1)}+max{d(gxm(k),gxm(k)+1),d(gym(k),gym(k)+1),d(gzm(k),gzm(k)+1)}δn(k)+1+βk+δm(k)+1. Analogously we have for k0,(2.26)βk=max{d(gxn(k)+1,gxm(k)+1),d(gyn(k)+,gym(k)+1),d(gzn(k)+1,gzm(k)+1)}max{d(gxn(k)+1,gxn(k))+d(gxn(k),gxm(k))+d(gxm(k),gxm(k)+1),d(gyn(k)+1,gyn(k))+d(gyn(k),gym(k))+d(gym(k),gym(k)+1),d(gzn(k)+1,gzn(k))+d(gzn(k),gzm(k))+d(gzm(k),gzm(k)+1)}max{d(gxn(k),gxn(k)+1),d(gyn(k),gyn(k)+1),d(gzn(k),gzn(k)+1)}+max{d(gxn(k),gxm(k)),d(gyn(k),gym(k)),d(gzn(k),gzm(k))}+max{d(gxm(k),gxm(k)+1),d(gym(k),gym(k)+1),d(gzm(k),gzm(k)+1)}δn(k)+1+αk+δm(k)+1. Letting k in (2.25) and (2.26), we get that (2.27)limkmax{d(gxn(k)+1,gxm(k)+1),d(gyn(k)+1,gym(k)+1),d(gzn(k)+1,gzm(k)+1)}=limkβk=ɛ=limkαk. Since n(k)>m(k), for k0, we have (2.28)gxn(k)gxm(k),gyn(k)gym(k),gzn(k)gzm(k). Then from (2.1), (2.4), and (2.28), we have for k0, (2.29)d(gxn(k)+1,gxm(k)+1)=d(F(xn(k),yn(k),zn(k)),F(xm(k),ym(k),zm(k)))ψ(max{d(gxn(k),gxm(k)),d(gyn(k),gym(k)),d(gzn(k),gzm(k))}),d(gyn(k)+1,gym(k)+1)=d(F(yn(k),xn(k),yn(k)),F(ym(k),xm(k),ym(k)))ψ(max{d(gyn(k),gym(k)),d(gxn(k),gxm(k)),d(gyn(k),gym(k))}),d(gzn(k)+1,gzm(k)+1)=d(F(zn(k),yn(k),xn(k)),F(zm(k),ym(k),xm(k)))ψ(max{d(gzn(k),gzm(k)),d(gyn(k),gym(k)),d(gxn(k),gxm(k))}). From (2.29) for k0, we get (2.30)βkψ(max{d(gxn(k),gxm(k)),d(gyn(k),gym(k)),d(gzn(k),gzm(k))})=ψ(αk). Letting k in (2.30), using (2.20), (2.27), and a property of ψ, we get (2.31)ɛψ(ɛ)<ɛ, which is a contradiction. This shows that {gxn}, {gyn}, and {gzn} are Cauchy sequences.

Since X is complete, there exist x,y,zX such that (2.32)limngxn=x,limngyn=y,limngzn=z. From (2.4) and (2.32), using the continuity of g, we have (2.33)gx=limng(gxn+1)=limng(F(xn,yn,zn)),(2.34)gy=limng(gyn+1)=limng(F(yn,xn,yn)),(2.35)gz=limng(gzn+1)=limng(F(zn,yn,xn)). Now we will show that gx=F(x,y,z),gy=F(y,x,y), and gz=F(z,y,x).

Since g and F are compatible, in addition with (2.33), (2.34), and (2.35), respectively imply (2.36)limnd(g(F(xn,yn,zn)),F(g(xn),g(yn),g(zn)))=0,(2.37)limnd(g(F(yn,xn,yn)),F(g(yn),g(xn),g(yn)))=0,(2.38)limnd(g(F(zn,yn,xn)),F(g(zn),g(yn),g(xn)))=0. Suppose now the assumption (a) holds, that is, F is continuous.

For all n0, we have (2.39)d(gx,F(gxn,gyn,gzn))d(gx,g(F(xn,yn,zn)))+d(g(F(xn,yn,zn)),F(gxn,gyn,gzn)). Taking the limit as n, using (2.32), (2.33), (2.36), and the facts that g and F are continuous, we have d(gx,F(x,y,z))=0.

Similarly, by using (2.32), (2.34), and (2.37) and (2.32), (2.35), and (2.38), respectively, and also the facts that g and F are continuous, we have d(gy,F(y,x,y))=0 and d(gz,F(z,y,x))=0.

Thus we have proved that g and F have a tripled coincidence point.

Suppose that the assumption (b) holds. Since {gxn},  {gzn} are nondecreasing and gxnx with gznz and also {gyn} is nonincreasing with gyny, by assumption (b) we have for all n(2.40)gxnx,gyny,gznz. By virtue of monotone increasing property of g we have (2.41)ggxngx,ggyngy,ggzngz. Now using (2.4) we have (2.42)d(gx,F(x,y,z))d(gx,g(gxn+1))+d(g(g(xn+1)),F(x,y,z))d(gx,g(gxn+1))+d(g(F(xn,yn,zn)),F(gxn,gyn,gzn))+d(F(gxn,gyn,gzn),F(x,y,z))d(gx,g(gxn+1))+d(g(F(xn,yn,zn)),F(gxn,gyn,gzn))+ψ(max{d(ggxn,gx),d(ggyn,gy),d(ggzn,gz)}),(by(2.1),(2.41)). Taking the limit as n in the above inequality, using (2.33), (2.36), and (2.41) we have (2.43)d(gx,F(x,y,z))limnψ(max{d(ggxn,gx),d(ggyn,gy),d(ggzn,gz)}). By (2.33), (2.34), (2.35), and the property of ψ, we have (2.44)d(gx,F(x,y,z))ψ(0)=0, that is (2.45)gx=F(x,y,z). In a similar manner using (2.33), (2.34), (2.35), and (2.36), (2.37), (2.38), respectively, we obtain (2.46)gy=F(y,x,y),gz=F(z,y,x). Thus, we proved that g and F have a tripled coincidence point.

This completes the proof of the theorem.

Corollary 2.2.

Let (X,) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Suppose F:X×X×XX and g:XX are such that F has the mixed g-monotone property and (2.47)d(F(x,y,z),F(u,v,w))ψ(max{d(gx,gu),d(gy,gv),d(gz,gw)}) for any x,y,zX for which gxgu, gygv and gzgw, where ψ:[0,+)[0,+) be such that ψ(t) is monotone, ψ(t)<t and limrt+ψ(r)<t for all t>0. Suppose F(X×X×X)g(X), g is continuous, and F and g are commuting. Suppose either

F is continuous, or

X has the  following property:

if a nondecreasing sequence {xn}x,  then xnx for all n,

if a nonincreasing sequence  {yn}y, then yny for all n.

If there exist x0,y0,z0X such that gx0F(x0,y0,z0), gy0F(y0,x0,y0), and gz0F(z0,y0,x0), then there exist x,y,zX such that (2.48)F(x,y,z)=gx,F(y,x,y)=gy,F(z,y,x)=gz, that is, F and g have a tripled coincidence point.

Proof.

Since a commuting pair is also a compatible pair, the result of the Corollary 2.2 follows from Theorem 2.1.

Later, by an example, we will show that the Corollary 2.2 is properly contained in Theorem 2.1.

Corollary 2.3.

Let (X,) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Suppose F:X×X×XX be such that F has the mixed monotone property and (2.49)d(F(x,y,z),F(u,v,w))ψ(max{d(x,u),d(y,v),d(z,w)}) for any x,y,zX for which xu,yv and zw, where ψ:[0,+)[0,+) be such that ψ(t) is monotone, ψ(t)<t and limrt+ψ(r)<t for all t>0. Suppose

F is continuous, or

X has the following property:

if a nondecreasing sequence {xn}x, then xnx for all n,

if a nonincreasing sequence {yn}y, then yny for all n.

If there exist x0,y0,z0X such that x0F(x0,y0,z0), y0F(y0,x0,y0), and z0F(z0,y0,x0), then there exist x,y,zX such that (2.50)F(x,y,z)=x,F(y,x,y)=y,F(z,y,x)=z, that is, F has a tripled fixed point.

Proof.

Taking g(x)=x in Theorem 2.1 we obtain Corollary 2.3.

Corollary 2.4.

Let (X,) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Suppose F:X×X×XX and g:XX are such that F has the mixed monotone property and (2.51)d(F(x,y,z),F(u,v,w))kmax{d(x,u),d(y,v),d(z,w)} for any x,y,zX for which xu,yv and zw, where 0<k<1. Suppose either

F is continuous,  or

X has the following  property:

if a nondecreasing sequence {xn}x, then  xnx for all n,

if a nonincreasing sequence {yn}y, then  yny for all n.

If there exist x0,y0,z0X such that x0F(x0,y0,z0), y0F(y0,x0,y0), and z0F(z0,y0,x0), then there exist x,y,zX such that (2.52)F(x,y,z)=x,F(y,x,y)=y,F(z,y,x)=z, that is, F has a tripled coincidence point.

Proof.

Taking ψ(t)=kt, t>0 where 0<k<1, in Corollary 2.3 we obtain Corollary 2.4.

The following corollary is the result of Berinde and Borcut in .

Corollary 2.5.

Let (X,) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Suppose F:X×X×XX be such that F has the mixed monotone property and (2.53)d(F(x,y,z),F(u,v,w))a1d(x,u)+a2d(y,v)+a3d(z,w) for any x,y,zX for which xu,yv and zw, where a1+a2+a3<1. Suppose either

F is continuous, or

X has the following property:

if a nondecreasing sequence {xn}x, then xnx for all n

if a nonincreasing sequence {yn}y, then yny for all n.

If there exist x0,y0,z0X such that x0F(x0,y0,z0), y0F(y0,x0,y0), and z0F(z0,y0,x0), then there exist x,y,zX such that (2.54)F(x,y,z)=x,F(y,x,y)=y,F(z,y,x)=z, that is, F has a tripled fixed point.

Proof.

The proof follows from Corollary 2.4, since the inequality in Corollary 2.5 implies that Corollary 2.4.

Remark 2.6.

The method used in the proof of Corollary 2.5 is different from that used by Berinde and Borcut .

Next we discuss an example.

Example 2.7.

Let X=. Then (X,) is a partially ordered set with the partial ordering defined by xy if and only if |x||y| and x·y0.

Let d(x,y)=|x-y| for x,y. Then (X,d) is a complete metric space.

Let g:XX be defined as g(x)=x2/10,    for all xX.

Let F:X×X×XX be defined as (2.55)F(x,y,z)=x2-y2+z29,x,y,zX. Then F obeys the mixed g-monotone property.

Let ψ:[0,)[0,) be defined as ψ(t)=(1/3)t for all t[0,).

Let, {xn},{yn}, and {zn} be three sequences in X such that (2.56)limnF(xn,yn,zn)=limng(xn)=a,limnF(yn,xn,yn)=limng(yn)=b,limnF(zn,yn,xn)=limng(zn)=c. Then explicitly, (2.57)limnxn2-yn2+zn29=limnxn210,x,y,zX,or,10a-10b+10c9=aimplya-10b+10c=0. Again, (2.58)limnyn2-xn2+yn29=limnyn210,x,y,zX,or,10b-10a+10b9=bimply11b-10a=0. And (2.59)limnzn2-yn2+xn29=limnzn210,x,y,zX,or,10c-10b+10a9=cimplyc-10b+10a=0. Then from the above relations we have, a=0,b=0, and c=0.

Therefore, (2.60)d(g(F(xn,yn,zn)),F(gxn,gyn,gzn))0as  n,d(g(F(yn,xn,yn)),F(gyn,gxn,gyn))0as  n,d(g(F(zn,yn,xn)),F(gzn,gyn,gxn))0as  n. Hence, the pair (g,F) is compatible in X.

Also, x0=0,z0=c(>0), and y0=0 are three points in X such that g(x0)=g(0)=0<c2/9=F(0,0,c)=F(x0,y0,z0),g(y0)=g(0)=0=F(0,0,0)=F(y0,x0,y0), and g(z0)=g(c)=c2/10<c2/9=F(c,0,0)=F(z0,y0,x0).

We next verify inequality (2.1) of Theorem 2.1. We take x,y,z,u,v,wX, such that gxgu,gzgw and gygv, that is, x2u2,z2w2, and y2v2.

Let A=max{d(gx,gu),d(gy,gv),d(gz,gw)}=max{|(x2-u2)|,|(y2-v2)|,|(z2-w2)|}.

Then d(F(x,y,z), F(u,v,w))=d((x2-y2+z2)/9,(u2-v2+w2)/9)=(|(x2-u2)-(y2-v2)+(z2-w2))/3|(|(x2-u2)|+|(y2-v2)|+|(z2-w2)|)/93A/9=A/3=ψ(A)=ψ(max{d(gx,gu),d(gy,gv),d(gz,gw)}).

Thus it is verified that the functions g,F, and ψ satisfy all the conditions of Theorem 2.1. Here (0,0,0) is the tripled coincidence point of g and F in X.

Remark 2.8.

It is observed that in Example 2.7 the function F and g do not commute, but they are compatible. Hence Corollary 2.2 cannot be applied to this example. This shows that Theorem 2.1 properly contains Corollary 2.2. Also gI, so the results of Berinde and Borcut  cannot be applied to this example. This shows that result in  is effectively generalised.

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