# Remarks on Some Recent Coupled Coincidence Point Results in SymmetricG-Metric Spaces

Authors: Stojan Radenović
Journal: Journal of Operators
 (2013)

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## Abstract

We use a method of reducing coupled coincidence point results in (ordered) symmetricG-metric spaces to the respective results for mappings with one variable, even obtaining (in some cases) more general theorems. Our results generalize, extend, unify, and complement recent coupled coincidence point theorems in this frame, established by Cho et al. (2012), Aydi et al. (2011), and Choudhury and Maity (2011). Also, by using our method several recent tripled coincidence point results in ordered symmetric G-metric spaces can be reduced to the coincidence point results with one variable.


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## Body

## 1. Introduction and Preliminaries

In 2004, Mustafa and Sims introduced a new notion of generalized metric space called G-metric space, where to every triplet of elements a nonnegative real number is assigned [1]. Fixed point theory, as well as coupled and tripled cases, in such spaces were studied in [2–6]. In particular, Banach contraction mapping principle was established in these works.Fixed point theory has also developed rapidly in metric and cone metric spaces endowed with a partial ordering (see [7, 8] and references therein). Fixed point problems have also been considered in partially ordered G-metric spaces [9–11].For more details on the following definitions and results concerning G-metric spaces, we refer the reader to [1, 9, 12–20].Definition 1.
LetX be a nonempty set, and let G:X3→ℝ+ be a function satisfying the following properties:(a)
G(x,y,z)=0 if x=y=z;(b)
0<G(x,y,z) for all x, y, and z∈X with x≠y;(c)
G(x,x,y)≤G(x,y,z) for all x, y, and z∈X, with y≠z;(d)
G(x,y,z)=G(x,z,y)=G(y,z,x)=⋯ (symmetry in all three variables); and(e)
G(x,y,z)≤G(x,a,a)+G(a,y,z) for all x, y, z, and a∈X.
Then the functionG is called a G-metric on X and the pair (X,G) is called a G-metric space.Definition 2.
Let(X,G) be a G-metric space and let {xn} be a sequence of points in X.(i)
A pointx∈X is said to be the limit of a sequence {xn} if limn,m→∞G(x,xn,xm)=0, and one says that the sequence {xn} is G-convergent to x.(ii)
The sequence{xn} is said to be a G-Cauchy sequence if, for every ε>0, there is a positive integer N such that G(xn,xm,xl)<ε, for all n,m,l≥N; that is, G(xn,xm,xl)→0, as n,m,l→∞.(iii)
(X,G) is said to be G-complete (or a complete G-metric space) if every G-Cauchy sequence in (X,G) is G-convergent in X.Proposition 3 (see [1]).
Let(X,G) be a G-metric space, and let {xn} be a sequence of points in X. Then the following are equivalent.(1)
The sequence{xn}  is G-convergent tox.(2)
G(xn,xn,x)→0  as  n→∞.(3)
G(xn,x,x)→0  as  n→∞.(4)
G(xn,xm,x)→0  as  n,m→∞.Definition 4 (see [1, 10]).
A G-metric onX is said to be symmetric if G(x,y,y)=G(y,x,x) for all x,y∈X. Every G-metric on X defines a metric dG on X by
(1)dG(x,y)=G(x,y,y)+G(y,x,x),∀x,y∈X.
For a symmetric G-metric space, one obtains
(2)dG(x,y)=2G(x,y,y),∀x,y∈X.
However, for an arbitrary G-metric G on X, just the following inequality holds: (3/2)G(x,y,y)≤dG(x,y)≤3G(x,y,y), for all x,y∈X.Definition 5.
In this work, one will consider the following three classes of mappings [3, 21]:Ψ={ψ∣ψ:[0,∞)→[0,∞) is continuous and nondecreasing and ψ-1({0})={0}},Φ={ϕ∣ϕ:[0,∞)→[0,∞) is lower semicontinuous and ϕ-1({0})={0}}, andΘ={φ∣φ:[0,∞)→[0,∞),φ(t)<tfort>0andlimr→t+φ(r)<t,foreacht>0}.
For weakϕ-contractions in the frame of metric spaces see [19, 21].
At first we need the following well-known definitions and results (see, e.g., [9, 15, 22]).Definition 6.
Let(X,⪯) be a partially ordered nonempty set, and let F:X2→X,g:X→X be two mappings. The mapping F has the mixed g-monotone property if for any x,y∈X the following hold:
(3)gx1⪯gx2⟹F(x1,y)⪯F(x2,y),forx1,x2∈X,gy1⪯gy2⟹F(x,y1)⪰F(x,y2),fory1,y2∈X.
Note that, if g=iX, the identity mapping, then F is said to have the mixed monotone property.Remark 7.
If(X,⪯) is a partially ordered set, then (X2,⊑) is also partially ordered with
(4)(x,y)⊑(a,b)⟺x⪯a∧y⪰b.Definition 8.
LetF:X2→X and g:X→X. An element (x,y)∈X2 is called a coupled coincidence point of F and g if
(5)F(x,y)=gx,F(y,x)=gy,
while (gx,gy)∈X2 is called a coupled point of coincidence of mappings F and g. Moreover, (x,y) is called a coupled common fixed point of F and g if
(6)F(x,y)=gx=x,F(y,x)=gy=y.Remark 9.
Otherwise,(x,y) is a coupled coincidence point of F and g if and only if (x,y) is a coincidence point of the mappings TF:X2→X2 and Tg:X2→X2 which are defined by
(7)TF(x,y)=(F(x,y),F(y,x)),Tg(x,y)=(gx,gy).Definition 10.
Mappingsf,g:X→X are said to be compatible in a G-metric space (X,G) if
(8)G(fgxn,gfxn,gfxn)⟶0asn⟶∞,
whenever {xn} is a sequence in X such that limn→∞fxn=limn→∞gxn in (X,G).
It is easy to prove thatf and g are compatible in (X,G) if and only if they are compatible in the associated metric space (X,dG).Definition 11.
Let(X,G) be a G-metric space, and let F:X2→X and g:X→X be two mappings. One says that F and g are compatible if
(9)G(gF(xn,yn),F(gxn,gyn),F(gxn,gyn))⟶0,G(gF(yn,xn),F(gyn,gxn),F(gyn,gxn))⟶0,asn⟶∞,
whenever xn and yn are such that
(10)limn→∞F(xn,yn)=limn→∞gxn,limn→∞F(yn,xn)=limn→∞gyn.
The proof of the following lemma is immediate (for (2) see [18]).Lemma 12.
(
1
) Let (X,G) be a symmetric G-metric space. Define G1:X2×X2×X2→ℝ+ by
(11)G1((x,y),(u,v),(a,b))=max{G(x,u,a),G(y,v,b)}.
Then (X2,G1) is a symmetric G-metric space.
(
2
) Let X={a,b}. Define
(12)G(a,a,a)=G(b,b,b)=0,G(a,a,b)=1,G(a,b,b)=2,
and extend G to X3 by using the symmetry in the variables. Then it is clear that (X,G) is an asymmetric G-metric space. It is not hard to see that (X3,G1) is not a G-metric space.Remark 13.
One can prove that mappingsF and g are compatible in (X,G) if and only if TF and Tg are compatible in (X2,G1).Definition 14.
(1) LetX be a nonempty set. Then (X,G,⪯) is called an ordered G-metric space if(i)
(X,G) is a G-metric space and(ii)
(X,⪯) is a partially ordered set.
(2) Let(X,G,⪯) be a partially ordered G-metric space. One says that (X,G,⪯) is regular if the following hypotheses hold:(i)
if a nondecreasing sequence{xn} is such that xn→x as n→∞, then xn⪯x⪯x for all n∈ℕ,(ii)
if a nonincreasing sequence{yn} is such that yn→y as n→∞, then yn⪰y⪰y for all n∈ℕ.


## 2. Main Results

Now, we are ready to state and prove our first result.Theorem 15.
Let(X,G,⪯) be a partially ordered symmetric G-metric space, F:X2→X, and g:X→X. Assume that there exist ψ∈Ψ and ϕ∈Φ such that
(13)ψ(max{G(F(x,y),F(u,v),F(a,b)),G(F(y,x),F(v,u),F(b,a))})≤ψ(max{G(gx,gu,ga),G(gy,gv,gb)})-ϕ(max{G(gx,gu,ga),G(gy,gv,gb)})
for all x, y, u, v, a, and b∈X for which gx⪯gu⪯ga∧gy⪰gv⪰gb or gx⪰gu⪰ga∧gy⪯gv⪯gb. Assume that F and g satisfy the following conditions:(1)
F(X2)⊂g(X);(2)
Fhas the mixedg-monotone property;(3)
Fandgare continuous and compatible and(X,G)is G-complete, or(3′)
(X,G,⪯)is regular and one ofF(X2)org(X)is G-complete;(4)
there existx0,y0∈Xsuch that(14)gx0⪯F(x0,y0)∧gy0⪰F(y0,x0)orgx0⪰F(x0,y0)∧gy0⪯F(y0,x0).Then F and g have a coupled coincidence point.Remark 16.
(a) Obviously, the condition (1) from [23] is equivalent to the condition (13). Hence, by using a new symmetric G-metric space (X2,G1) we have obtained a method of reducing coupled coincidence and coupled fixed point results in (ordered) symmetric G-metric spaces to the respective results for mappings with one variable, even obtaining (in some cases) more general theorems. We note that this method cannot be applied in the case of asymmetric G-metric spaces (see (2) of Lemma 12). For other details of coupled case in ordered metric spaces see also [22].
(b) Also, we note that Theorem 3.1. from [23] holds if F and g are compatible instead of commuting (see Step 3 in [23]). Indeed, since
(15)limn→∞gxn=limn→∞F(xn,yn)=x,limn→∞gyn=limn→∞F(yn,xn)=y,
then
(16)limn→∞G(gF(xn,yn),F(gxn,gyn),F(gxn,gyn))=0,limn→∞G(gF(yn,xn),F(gyn,gxn),F(gyn,gxn))=0,
because F and g are compatible.
Therefore, now we have(17)G(gx,F(x,y))≤G(gx,ggxn+1,ggxn+1)+G(ggxn+1,F(x,y),F(x,y))≤G(gx,ggxn+1,ggxn+1)+G(ggxn+1,F(gxn,gyn),F(gxn,gyn))+G(F(gxn,gyn),F(x,y),F(x,y))⟶G(gx,gx,gx)+0+G(F(x,y),F(x,y),F(x,y))=0,asn⟶∞,
that is, F(x,y)=gx. Similarly, we obtain that F(y,x)=gy.Assertions similar to the following lemma were used in the frame of metric spaces in the course of proofs of several fixed point results in various papers (see, e.g., [9, 21]). This lemma holds in every G-metric space.Lemma 17.
Let(X,G) be a G-metric space, and let {xn} be a sequence in X such that limn→∞G(xn,xn+1,xn+1)=0. If {xn} is not a G-Cauchy sequence in (X,G), then there exist ε>0 and two sequences {mk} and {nk} of positive integers such that the following four sequences tend to ε+ when k→∞:
(18)G(xmk,xnk,xnk),G(xmk,xnk-1,xnk-1),G(xmk+1,xnk,xnk),G(xnk-1,xmk+1,xmk+1).The following lemma is crucial for the proof of Theorem15, and it holds in every G-metric space.Lemma 18.
Let(X,G,⪯) be a partially ordered G-metric space, and let f and g be two self-mappings on X. Assume that there exist ψ∈Ψ and ϕ∈Φ such that
(19)ψ(G(fx,fy,fz))≤ψ(max{G(gx,gy,gz),G(gu,gv,gw),G(ga,gb,gc)})-ϕ(max{G(gx,gy,gz),G(gu,gv,gw),G(ga,gb,gc)})
for all x, y, z, u, v, w, a, b, and c∈X for which gx⪯gu⪯ga∧gy⪰gv≥gb∧gz⪯gw⪯gc or gx⪰gu⪰ga∧gy⪯gv⪯gb∧gz⪰gw⪰gc. If the following conditions hold:(i)
f is a g-nondecreasing with respect to ⪯ and fX⊂gX;(ii)
there existsx0∈X such that gx0⪯fx0;(iii)
f and g are continuous and compatible, and (X,G) is G-complete or(iii′)
(X,G,⪯) is regular, and one of fX or gX is G-complete.
Thenf and g have a coincidence point in X.Proof.
Ifgx0=fx0, then x0 is a coincidence point of f and g. Therefore, let gx0≺fx0. Since fX⊂gX, we obtain a Jungck sequence yn=fxn=gxn+1 for all n=0,1,2,…, where xn∈X, and by induction we get that yn⪯yn+1. If yn=yn+1 for some n∈ℕ, then xn+1 is a coincidence point of f and g. Therefore, suppose that yn≠yn+1 for each n. Now, we will prove the following:(1)
G(yn,yn+1,yn+1)→0 as n→∞;(2)
{yn} is a G-Cauchy sequence.
Indeed, by puttingx=u=a=xn, y=v=b=xn+1, and z=u=c=xn+1 in (19) we get
(20)ψ(G(yn,yn+1,yn+1))=ψ(G(fxn,fxn+1,fxn+1))≤ψ(G(gxn,gxn+1,gxn+1))-ϕ(G(gxn,gxn+1,gxn+1))=ψ(G(yn-1,yn,yn))-ϕ(G(yn-1,yn,yn))<ψ(G(yn-1,yn,yn)),
and since the function ψ is nondecreasing, it follows that G(yn,yn+1,yn+1)≤G(yn-1,yn,yn); that is, there exists limn→∞G(yn,yn+1,yn+1)=G*≥0. If G*>0, we get from the previous relation ψ(G*)≤ψ(G*)-ϕ(G*); that is, G*=0 which is a contradiction. Hence, we obtain that limn→∞G(yn,yn+1,yn+1)=0.
Further, using Lemma17 we shall prove that {yn} is a G-Cauchy sequence. Suppose this is not the case. Then, by Lemma 17 there exist ε>0 and two sequences {mk} and {nk} of positive integers such that the following sequences tend to ε+ when k→∞:
(21)G(xmk,xnk,xnk),G(xmk,xnk-1,xnk-1),G(xmk+1,xnk,xnk),G(xnk-1,xmk+1,xmk+1).
Putting x=xmk+1, y=xnk, andz=xnk in (19) we have
(22)ψ(G(fxmk+1,fxnk,fxnk))≤ψ(G(gxmk+1,gxnk,gxnk))-ϕ(G(gxmk+1,gxnk,gxnk)),
that is,
(23)ψ(G(ymk+1,ynk,ynk))≤ψ(G(ymk,ynk-1,ynk-1))-ϕ(G(ymk,ynk-1,ynk-1)).
Letting k→∞, we get ψ(ε)≤ψ(ε)-ϕ(ε); that is, ϕ(ε)=0. Since ϕ∈Φ, we get ε=0, which is a contradiction. We have proved that {yn} is a G-Cauchy sequence in (X,G).
In case (iii), since(X,G) is G-complete, there exists z∈X such that yn→z. Then we have
(24)G(fxn,z,z)⟶0,G(gxn,z,z)⟶0asn⟶∞.
Further, according to Definition 1 (e) and since f and g are continuous and compatible, we get
(25)G(fz,gz,gz)≤G(fz,fgxn,fgxn)+G(fgxn,gz,gz)≤G(fz,fgxn,fgxn)+G(fgxn,gfxn,gfxn)+G(gfxn,gz,gz)⟶0,asn⟶∞.
It follows that z is a coincidence point for f and g.
In case(iii′), it follows that yn=fxn=gxn+1→gz, z∈X (in both cases when fX or gX is G-complete), and then gxn⪯gz⪯gz, and by the contractive condition (19) we have
(26)ψ(G(fxn,fz,fz))≤ψ(G(gxn,gz,gz))-ϕ(G(gxn,gz,gz)).
By taking limit as n→∞ in the above inequality, we obtain
(27)ψ(G(gz,fz,fz))≤ψ(G(gz,gz,gz))-ϕ(G(gz,gz,gz))=0-0=0,
and hence fz=gz.Proof of Theorem15.
Firstly, (13) implies
(28)ψ(G1(TF(Y),TF(V),TF(A)))≤ψ(G1(Tg(Y),Tg(V),Tg(A)))-ϕ(G1(Tg(Y),Tg(V),Tg(A))),
for all Y=(x,y),V=(u,v), and A=(a,b) from X2 for which Tg(Y)⊑Tg(V)⊑Tg(A) or Tg(A)⊑Tg(V)⊑Tg(Y).
Further,(1)
implies thatTF(X2)⊂Tg(X2);(2)
implies thatTF is Tg-nondecreasing with respect to ⊑ and TF(X2)⊂Tg(X2);(3)
implies thatTF and Tg are continuous and compatible and (X2,G1) is G-complete,
or(3′)
implies that(X2,G1,⊑) is regular and one of TF(X2) or Tg(X2) is G-complete;(4)
implies that there existsY0=(x0,y0)∈X2 such that Tg(Y0)⊑TF(Y0) or TF(Y0)⊑Tg(Y0).
All conditions of Lemma18 for the ordered G-metric space (X2,G1) are satisfied. Therefore, the mappings TF and Tg have a coincidence point in X2. According to Remark 9 the mappings F and g have a coupled coincidence point. The proof of Theorem 15 is complete.Our second main result is the following.Theorem 19.
Let(X,G,⪯) be a partially ordered symmetric G-metric space, F:X2→X, and g:X→X. Assume that there exists φ∈Θ such that
(29)max{G(F(x,y),F(u,v),F(a,b)),G(F(y,x),F(v,u),F(b,a))}≤φ(max{G(gx,gu,ga),G(gy,gv,gb)}),
for all x, y, u, v, a, and b∈X for which gx⪯gu⪯ga∧gy⪰gv≥gb or gx⪰gu⪰ga∧gy⪯gv⪯gb. Assume that F and g satisfy the following conditions:(1)
F(X2)⊂g(X);(2)
Fhas the mixedg-monotone property;(3)
Fandgare continuous and compatible and(X,G)is G-complete, or(3′)
(X,G,⪯)is regular and one ofF(X2)org(X)is G-complete;(4)
there existx0,y0∈Xsuch that(30)gx0⪯F(x0,y0)∧gy0⪰F(y0,x0)orgx0⪰F(x0,y0)∧gy0⪯F(y0,x0).Then F and g have a coupled coincidence point.For the proof of Theorem19 we use the following.Lemma 20.
Let(X,G,⪯) be a partially ordered G-metric space, and let f and g be two self-mappings on X. Assume that there exists φ∈Θ such that
(31)G(fx,fy,fz)≤φ(max{G(gx,gy,gz),G(gu,gv,gw),G(ga,gb,gc)})
for all x, y, z, u, v, w, a, b, and c∈X for which gx⪯gu⪯ga∧gy⪰gv≥gb∧gz⪯gw⪯gc or gx⪰gu⪰ga∧gy⪯gv⪯gb∧gz⪰gw⪰gc. If the following conditions hold:(i)
f is g-nondecreasing with respect to ⪯ and fX⊂gX;(ii)
there existsx0∈X such that gx0⪯fx0;(iii)
f and g are continuous and compatible, and (X,G) is G-complete or(iii′)
(X,G,⪯) is regular and one of fX or gX is G-complete,
Thenf and g have a coincidence point in X.Proof.
Ifgx0=fx0, then x0 is a coincidence point of f and g. Therefore, let gx0≺fx0. Since fX⊂gX, we obtain a Jungck sequence yn=fxn=gxn+1 for all n=0,1,2,…, where xn∈X, and by induction we get that yn⪯yn+1. If yn=yn+1 for some n∈ℕ, then xn+1 is a coincidence point of f and g. Therefore, suppose that yn≠yn+1 for each n. Now, we will prove that G(yn,yn+1,yn+1)→0 as n→∞.
Indeed, by puttingx=u=a=xn, y=v=b=xn+1, and z=u=c=xn+1 in (31) we get
(32)G(yn,yn+1,yn+1)=G(fxn,fxn+1,fxn+1)≤φ(G(gxn,gxn+1,gxn+1))=φ(G(yn-1,yn,yn))<G(yn-1,yn,yn).
That is, there exists limn→∞G(yn,yn+1,yn+1)=G*≥0. If G*>0, we get
(33)G*=limn→∞G(yn,yn+1,yn+1)≤limn→∞φ(G(yn-1,yn,yn))=limG(yn-1,yn,yn)→(G*)+φ(G(yn-1,yn,yn))<G*,
which is a contradiction. Hence, we obtain that limn→∞G(yn,yn+1,yn+1)=0.
Further, by using Lemma17, we shall prove that {yn} is a G-Cauchy sequence. Suppose this is not the case. Then, by Lemma 17 there exist ε>0 and two sequences {mk} and {nk} of positive integers such that the following sequences tend to ε+ when k→∞:
(34)G(ymk,ynk,ynk),G(ymk,ynk-1,ynk-1),G(ymk+1,ynk,ynk),G(ynk-1,ymk+1,ymk+1).
Putting x=xmk+1, y=xnk, and z=xnk in (31) we have
(35)G(fxmk+1,fxnk,fxnk)≤φ(G(gxmk+1,gxnk,gxnk)),
that is,
(36)G(ymk+1,ynk,ynk)≤φ(G(ymk,ynk-1,ynk-1)).
Letting k→∞, we obtain
(37)ε≤limk→∞φ(G(ymk,ynk-1,ynk-1))=limG(ymk,ynk-1,ynk-1)→ε+φ(G(ymk,ynk-1,ynk-1))<ε.
Hence, we get ε<ε, which is a contradiction. We have proved that {yn} is a G-Cauchy sequence in (X,G).
Now, in case(iii′), since (X,G) is G-complete, there exists z∈X such that yn→z. Then we have
(38)G(fxn,z,z)⟶0,G(gxn,z,z)⟶0asn⟶∞,
and because
(39)G(fz,gz,gz)≤G(fz,fgxn,fgxn)+G(fgxn,gz,gz)≤G(fz,fgxn,fgxn)+G(fgxn,gfxn,gfxn)+G(gfxn,gz,gz)⟶G(fz,fz,fz)+0+G(gz,gz,gz)=0,asn⟶∞,
it follows that z is a coincidence point for f and g.
In case(iii′), it follows that yn=fxn=gxn+1→gz, z∈X (in both cases when fX or gX is G-complete), and then gxn⪯gz⪯gz, and by the contractive condition (31) we have
(40)G(fxn,fz,fz)≤φ(G(gxn,gz,gz)).
By taking the limit as n→∞ in the above inequality we obtain
(41)G(gz,fz,fz)≤limn→∞φ(G(gxn,gz,gz))≤limn→∞G(gxn,gz,gz)=0,
and hence fz=gz.Proof of Theorem19.
The proof is very similar to the proof of Theorem15. Namely, the contractive condition (29) for the mappings F and g is equivalent to the following condition:
(42)G1(TF(Y),TF(V),TF(A))≤φ(G1(Tg(Y),Tg(V),Tg(A))),
for the mappings TF and Tg. The proof is further an immediate consequence of Lemma 20.Remark 21.
We have obtained that, in case of symmetric G-metric spaces, Theorem19 generalizes both results (Theorems 3.1 and 3.2) from [3]. Also, Example 22 shows that this generalization is proper. It is clear that our new method with ordered symmetric G-metric spaces (X2,G1) and with mappings TF and Tg implies that all results from [3] can be reduced to known results with one variable.
The following example supports both of our theorems, the first withψ(t)=t and ϕ(t)=(1/2)t and the second with φ(t)=(1/2)t.Example 22.
LetX=ℝ be endowed with the complete G-metric
(43)G(x,y,z)=|x-y|+|y-z|+|z-x|,
for all x, y, and z∈X and with the usual order. Consider the mappings F(x,y)=(x3-2y3)/8 and g(x)=x3. All the conditions of Theorems 15 and 19 are satisfied. In particular, the mapping F has the mixed g-monotone property, and we will check that F and g are compatible.
Let{xn} and {yn} be two sequences in X such that
(44)limn→∞F(xn,yn)=limgxn=A,limn→∞F(yn,xn)=limgyn=B.
Then (A-2B)/8=A and (B-2A)/8=B, wherefrom it follows that A=B=0. Then
(45)G(gF(xn,yn),F(gxn,gyn),F(gxn,gyn))=|(xn3-2yn38)3-xn9-2yn98|+0+|xn9-2yn98-(xn3-2yn38)3|=2|(xn3-2yn38)3-xn9-2yn98|⟶2·|0-0|=0,asn⟶∞,
and similarly
(46)G(gF(yn,xn),F(gyn,gxn),F(gyn,gxn))⟶0,asn⟶∞.
Also, F and g do not commute, and therefore a coupled coincidence point of F and g cannot be obtained by Theorem 3.1 from [3].
The contractive condition (13) is satisfied with ψ(t)=t and ϕ(t)=(1/2)t which follows from
(47)G(F(x,y),F(u,v),F(a,b))=|x3-2y38-u3-2v38|+|u3-2v38-a3-2b38|+|a3-2b38-x3-2y38|=18(|(x3-u3)-2(y3-v3)|+|(u3-a3)-2(v3-b3)|+|(a3-x3)-2(b3-y3)|)≤18(G(gx,gu,ga)+2G(gy,gv,gb))≤48·G(gx,gu,ga)+G(gy,gv,gb)2≤12max{G(gx,gu,ga),G(gy,gv,gb)}=maxG(gx,gu,ga),G(gy,gv,gb)-12max{G(gx,gu,ga),G(gy,gv,gb)},G(F(y,x),F(v,u),F(b,a))=|y3-2x38-v3-2u28|+|v3-2u38-b3-2a38|+|b3-2a38-y3-2x38|=18(|(y3-v3)-2(x3-u3)|+|(v3-b3)-2(u3-a3)|+|(b3-y3)-2(a3-x3)|)≤18(2G(gx,gu,ga)+G(gy,gv,gb))≤48·G(gx,gu,ga)+G(gy,gv,gb)2≤12max{G(gx,gu,ga),G(gy,gv,gb)}=max{G(gx,gu,ga),G(gy,gv,gb)}-12max{G(gx,gu,ga),G(gy,gv,gb)},
for all x, y, u, v, a, and b∈X for which gx≤gu≤ga∧gy≥gv≥gb or gx≥gu≥ga∧gy≤gv≤gb.
Hence,(48)ψ(max{G(F(x,y),F(u,v),F(a,b)),G(F(y,x),F(v,u),F(b,a))})≤ψ(max{G(gx,gu,ga),G(gy,gv,gb)})-ϕ(max{G(gx,gu,ga),G(gy,gv,gb)}).
Similarly, the condition (29) is satisfied with φ(t)=(1/2)t; that is,
(49)max{G(F(x,y),F(u,v),F(a,b)),G(F(y,x),F(v,u),F(b,a))}≤φ(max{G(gx,gu,ga),G(gy,gv,gb)}),
for all x, y, u, v, a, and b∈X for which gx≤gu≤ga∧gy≥gv≥gborgx≥gu≥ga∧gy≤gv≤gb. There exists a coupled coincidence point (0,0) of the mappings F and g.



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*Source: 290525-2013-05-16.xml*