# A Coincidence Best Proximity Point Problem inG-Metric Spaces

Authors: M. Abbas; A. Hussain; P. Kumam
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

The aim of this paper is to initiate the study of coincidence best proximity point problem in the setup of generalized metric spaces. Some results dealing with existence and uniqueness of a coincidence best proximity point of mappings satisfying certain contractive conditions in such spaces are obtained. An example is provided to support the result proved herein. Our results generalize, extend, and unify various results in the existing literature.


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## Body

## 1. Introduction and Preliminaries

LetY be any nonempty subset of a metric space X and T:Y→X. A fixed point problem Fix⁡(X,Y,T) defined by X,Y and T is to find a point x* in Y such that d(x*,Tx*)=0. A point x* in Y, where inf⁡{d(y,Tx*):y∈Y}, is attained; that is, d(x*,Tx*)=inf⁡{d(y,Tx*):y∈Y} holds and is called an approximate fixed point of T. In case it is not possible to solve Fix⁡(X,Y,T), it could be interesting to study the conditions that assure existence and uniqueness of approximate fixed point of a mapping T.LetA and B be two nonempty subsets of X and T:A→B. Suppose that ΔAB=d(A,B)=inf⁡({d(a,b):a∈A,b∈B}) is the measure of a distance between two sets A and B. A point x* is called the best proximity point of T if d(x*,Tx*)=ΔAB. Thus the best proximity point problem defined by a mapping T and a pair of sets (A,B) is to find a point x* in A such that d(x*,Tx*)=ΔAB. If A∩B=ϕ, the fixed point problem defined by A,B and T has no solution. If A=B, the best proximity point problem reduces to a fixed point problem. In this way, the best proximity point problem can be viewed as a natural generalization of a fixed point problem. Furthermore, results dealing with existence and uniqueness of the best proximity point of certain mappings are more general than the ones dealing with fixed point problem of those mappings. A coincidence best proximity point problem is defined as follows: find a point x* in A such that d(gx*,Tx*)=ΔAB, where g is a self-mapping on A. This is an extension of the best proximity point problem. There are several results dealing with proximity point problem in the setup of metric spaces (see, e.g., [1–11] and references mentioned therein).Mustafa and Sims [12] introduced the concept of a G-metric space as a substantial generalization of metric space. They [13] obtained some fixed point theorems for mappings satisfying different contractive conditions in such spaces. Based on the notion of generalized metric spaces, Mustafa et al. [14–16] obtained several fixed point theorems for mappings satisfying different contractive conditions. Mustafa et al. [17–19] obtained some fixed point theorems for mappings satisfying different contractive conditions. Chugh et al. [20] obtained some fixed point results for maps satisfying property P in G-metric spaces. Saadati et al. [21] studied fixed point of contractive mappings in partially ordered G-metric spaces. Shatanawi [22] obtained fixed points of Φ-maps in G-metric spaces. For more details, we refer to, for example, [22–39] and references therein.A study of the best proximity point problem in the setup ofG-metric space is a recent development by Hussain et al. [40]. This motivates us to extend the scope of this investigation and extend this study to coincidence proximity point problem of certain mappings in the framework of generalized metric spaces.Consistent with Mustafa and Sims [12], the following definitions and results will be needed in the sequel.Definition 1.
LetX be a nonempty set. Suppose that a mapping G:X×X×X→R+ satisfies(G1)
0≤G(x,y,z) for all x,y,z∈X and G(x,y,z)=0 if and only if x=y=z,(G2)
0<G(x,x,y) for all x,y∈X, with x≠y,(G3)
G(x,x,y)≤G(x,y,z) for all x,y,z∈X, with y≠z,(G4)
Gx,y,z=Gx,z,y=Gy,z,x=⋯ (symmetric in all three variables),(G5)
G(x,y,z)≤G(x,a,a)+G(a,y,z) for all x,y,z,a∈X (rectangle inequality).Then G is called a generalized metric on X or G-metric on X and (X,G) is called a G-metric space.Definition 2.
Let(X,G) be a G-metric space, {xn} a sequence in X, and x∈X. One says that {xn} is(i)
aG-Cauchy sequence if, for any ɛ>0, there exists a natural number N such that, for all n,m,l≥N,Gxn,xm,xl<ɛ;(ii)
aG-convergent sequence if, for any ɛ>0, there exists a natural number N such that, for all n,m≥N,G(xn,xm,x)<ɛ for some x in X.A G-metric space X is said to be complete if every G-Cauchy sequence in X is convergent in X. It is known that {xn} converges to x∈(X,G) if and only if G(xm,xn,x)→0 as n,m→∞.Proposition 3.
Let(X,G) be a G-metric space; then the following are equivalent.(1)
{xn} converges to x∈X.(2)
G(xn,xm,x)→0, as m,n→∞.(3)
G(xn,xn,x)→0, as n→∞.(4)
G(xn,x,x)→0, as n→∞.Definition 4.
AG-metric on X is said to be symmetric if G(x,y,y)=G(y,x,x) for all x,y∈X.Proposition 5.
EveryG-metric on X will define a metric dG on X by
(1)dG(x,y)=G(x,y,y)+G(y,x,x),∀x,y∈X.Remark 6.
Let{xn} be a sequence in G-metric space X. If {G(xn,xn+1,xn+1)}→0 and {xn} is not a Cauchy sequence, then there exist ϵ0>0 and two subsequences {xm(k)} and {xn(k)} such that, for all k∈N, k≤m(k)<n(k), G(xm(k),xn(k),xn(k))≥ϵ0, and G(xm(k),xl,xl)<ϵ0 for all l∈{m(k)+1,m(k)+2,…,n(k)-2,n(k)-1}. If lim⁡k→∞G(xm(k),xn(k),xn(k))=ϵ0, then
(2)lim⁡k→∞Gxmk-1,xnk-1,xnk-1=ϵ0,lim⁡k→∞Gxmk,xnk+l,xnk+l=ϵ0,
for all l≥0. Indeed, if {G(xn,xn+1,xn+1)}→0, then, for all k∈N, we have
(3)G(xm(k)-1,xn(k)-1,xn(k)-1)≤G(xmk-1,xmk,xmk)+G(xmk,xnk-1,xnk-1)≤G(xm(k)-1,xm(k),xm(k))+G(xm(k),xn(k),xn(k))+Gxnk,xnk-1,xnk-1,Gxmk,xnk,xnk≤G(xm(k)-1,xm(k),xm(k))+G(xm(k)-1,xn(k)-1,xn(k)-1)+G(xn(k),xn(k)-1,xn(k)-1).
From (3) we have
(4)G(xm(k),xn(k),xn(k))-G(xm(k)-1,xm(k),xm(k))-G(xn(k),xn(k)-1,xn(k)-1)≤G(xm(k)-1,xn(k)-1,xn(k)-1)≤G(xm(k)-1,xm(k),xm(k))+G(xm(k),xn(k),xn(k))+G(xn(k),xn(k)-1,xn(k)-1).
Taking limit as k→∞, we obtain that lim⁡k→∞G(xm(k)-1,xn(k)-1,xn(k)-1)=ϵ0. To prove
(5)lim⁡k→∞G(xm(k),xn(k)+l+1,xn(k)+l+1)=ϵ0
for all l≥0, we use induction on l. Equation (5) for l=0 holds obviously. Suppose that (5) holds for some l>0. Consider
(6)Gxmk,xnk+l+1,xnk+l+1≤G(xm(k),xn(k)+l,xn(k)+l)+G(xn(k)+l,xn(k)+l+1,xn(k)+l+1).
Also,
(7)Gxmk,xnk+l,xnk+l≤G(xm(k),xn(k)+l+1,xn(k)+l+1)+Gxnk+l+1,xnk+1,xnk+1.
From (6) and (7), we obtain that
(8)G(xm(k),xn(k)+l,xn(k)+l)-G(xn(k)+l+1,xn(k)+1,xn(k)+1)≤G(xm(k),xn(k)+l+1,xn(k)+l+1)≤Gxmk,xnk+l,xnk+l+Gxnk+l+1,xnk+1,xnk+1.
Taking limit as k→∞, we have lim⁡k→∞G(xm(k),xn(k)+l+1,xn(k)+l+1)=ϵ0.Definition 7.
LetX be a G-metric space and A and B two nonempty subsets of X. Define
(9)ΔABG=G(A,B,B)=inf⁡{G(a,b,b):a∈A,b∈B},A0=ΔABGa∈A:thereexistssomeb∈BsuchthatG(a,b,b),vvvvvvvv.vvv=ΔABG,B0=ΔABGb∈B:thereexistssomea∈AsuchthatG(a,b,b)vvvvvvvvvv=ΔABG.
Now we define the concept ofg-best proximity point of a mapping in the setup of G-metric spaces.Definition 8.
LetX be a G-metric space and A and B two nonempty subsets of X. Suppose that T:A→B, and g:A→A. A point x∈A is called g-best proximity point of T if G(gx,Tx,Tx)=ΔABG.
Note that ifg is an identity mapping on A, then x in above definition becomes the best proximity point of T.
Consistent with [41], we consider the following classes of mappings.
Ψ
=
{
φ
:
[
0
,
∞
)
→
[
0
,
∞
) such that, for all t>0, the series ∑n≥1φn(t) converges}. Elements in Ψ are called (c)-comparison functions.
Φ
=
{
ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) such that ϕ(t)<t and lim⁡r→t+ϕ(r)<t for all t>0}.
Θ
=
{
θ
:
[
0
,
∞
)
4
→
[
0
,
∞
) such that θ(a,b,c,d)=0 if one or more arguments take the value zero and θ is continuous}.
Ω
1
=
{
θ
:
[
0
,
∞
)
4
→
[
0
,
∞
) such that θ(a,b,c,d)=0 if one or more arguments take the value zero}.
Ω
1
=
{
θ
:
[
0
,
∞
)
4
→
[
0
,
∞
) such that θ(0,b,c,d)=0}.
Ω
2
=
{
θ
:
[
0
,
∞
)
4
→
[
0
,
∞
) such that lim⁡n→∞θ(tn1,tn2,tn3,tn4)=0, whenever the sequences {tn1}, {tn1}, {tn1}, {tn1}⊂[0,∞) are such that at least one of them is convergent to zero}.Definition 9.
LetX be a G-metric space and A and B two nonempty subsets of X, g:A→A, and α:X×X→[0,∞). A mapping T:A→B is said to be (φ,θ,α,g)-contraction if, for all x,y∈A0 with G(gy,Tx,Tx)=ΔABG and α(gx,gy)≥1, one has
(10)α(gx,gy)G(Tx,Ty,Ty)≤φ(Mg(x,y,y))+θG(gy,Tx,Tx)-ΔABG,G(gx,Ty,Ty)-ΔABG,vvvvvvvG(gx,Tx,Tx)-ΔABG,G(gy,Ty,Ty)-ΔABG,
where
(11)Mgx,y,y=max⁡G(gx,Ty,Ty)+G(gy,Ty,Ty)2G(gx,gy,gy),G(gx,Tx,Tx)-ΔAB,vvvvvvvGgy,Ty,Ty-ΔABG,vvvvvvvGgx,Ty,Ty+Ggy,Ty,Ty2-ΔABG,ψ∈Ψ and θ∈Θ.Definition 10.
LetX be a G-metric space and A and B two nonempty subsets of X, g:A→A, and α:X×X→[0,∞). A mapping T:A→B is said to be (α,g)-proximinal and admissible if a1,a2,b1,b2∈A0, α(gb1,gb2)≥1, G(ga1,Tb1,Tb1)=ΔABG, G(ga2,Tb2,Tb2)=ΔABG, and
(12)⟹α(ga1,ga2)≥1.Definition 11.
LetX be a G-metric space and A and B two subsets of X such that A0 is nonempty, T:A→B, and g:A→A. For a1,a2,a3,a4∈A0, the quadruple (A,B,T,g) has(1)
weakP-property of the first kind if
(13)Gga1,Ta3,Ta3=ΔABG,Gga2,Ta4,Ta4=ΔABGimpliesthatGga1,ga2,ga2≤GTa3,Ta4,Ta4;(2)
weakP-property of the second kind if
(14)G(ga1,Ta3,Ta3)=ΔABG,G(ga2,Ta4,Ta4)=ΔABGimpliesthatGga1,ga2,ga2=GTa3,Ta4,Ta4;(3)
weakP-property of the third kind if
(15)G(ga1,Ta3,Ta3)=ΔABG,G(ga2,Ta4,Ta4)=ΔABGimpliesthatG(ga1,ga2,ga2)≤G(Tb1,Tb2,Tb2).Definition 12 (see [41]).
Letg:A→A and α:X×X→[0,1) be two mappings and let N∈N, N≥2. One will say that α is (N,g)-transitive on A0 if x1,x2,…,xN+1∈A0, α(gxi,gxi+1)≥1∀i∈{1,2,…,N}⇒α(gx1,gxN+1)≥1.
Indeed, we will only use the notion of(2,g)-transitive mapping on A0; that is, x1,x2,x3∈A0, α(gx1,gx2)≥1, α(gx2,gx3)≥1, and
(16)⟹α(gx1,gx3)≥1.


## 2. Coincidence Best Proximity Point Results

In this section, we obtain several coincidence best proximity results in the setup of generalized metric spaces.Theorem 13.
LetX be a complete G-metric space, A and B two closed subsets of X, and g a continuous self-mapping on A such that ϕ≠A0⊆gA0. Suppose that T:A→B is continuous (α,g)-proximal and admissible and (φ,θ,α,g)-contraction, where φ∈Ψ, θ∈Ω1, and T(A0)⊆B0. If the following conditions hold:(a)
quadruple(A,B,T,g) satisfies weak P-property of the first kind;(b)
if a sequence{zn} in A0 such that {gzn}⊆A0 is Cauchy, then {zn} is also a Cauchy;(c)
there exists(x0,x1)∈A0×A0 such that G(gx1,Tx0,Tx0)=ΔABG and α(gx0,gx1)≥1.
Then there exists a convergent sequence{xn}⊆A0 which satisfies
(17)Ggxn+1,Txn,Txn=ΔABG∀n≥0,
and the limit of {xn} is a g-best proximity point of T.Proof.
Letx1∈A0. Then Tx1∈T(A0)⊆B0. Hence there is z2∈A such that G(z2,Tx1,Tx1)=ΔABG which implies that z2∈A0. As A0⊆gA0, there is x2∈A0 such that g(x2)=z2, so G(gx2,Tx1,Tx1)=G(z2,Tx1,Tx1)=ΔABG. In a similar way, there is x3∈A0 such that G(gx3,Tx2,Tx2)=ΔABG. Inductively we construct a sequence {xn}⊆A0 such that
(18)G(gxn+1,Txn,Txn)=ΔABG∀n≥0.
If there exists some n0∈N, such that gxn0=gxn0+1, then G(gxn0,Txn0,Txn0)=G(gxn0+1,Txn0,Txn0)=ΔABG implies that xn0 is a g-best proximity point of T. If we define xm=xn0 for all m≥n0, then {xn} converges to a g-best proximity point of T. The proof is complete. Assume that
(19)G(gxn,gxn+1,gxn+1)>0∀n≥0.
Note that xn,gxn+1∈A0 and Txn∈B0 for all n≥0. We claim that
(20)α(gxn,gxn+1)≥1∀n≥0.
If n=0, then α(gxn,gxn+1)≥1 holds by given hypothesis. Suppose that α(gxn,gxn+1)≥1 for some n>0. As T is (α,g)-proximal and admissible, for xn,xn+1,xn+2∈A0, α(gxn,gxn+1)≥1, G(gxn+1,Txn,Txn)=ΔABG, and G(gxn+2,Txn+1,Txn+1)=ΔABG, we have α(gxn+1,gxn+2)≥1. Thus (20) holds.
Use weakP-property of the first kind, for all n>0, xn,xn+1,xn+2∈A0, G(gxn+1,Txn,Txn)=ΔABG, G(gxn+2,Txn+1,Txn+1)=ΔABG imply the following inequality:
(21)⟹G(gxn+1,gxn+2,gxn+2)≤G(Txn,Txn+1,Txn+1).
Now by (20), (21), and (φ,θ,α,g)-contractive property of T, we have
(22)Ggxn+1,gxn+2,gxn+2≤G(Txn,Txn+1,Txn+1)≤α(gxn,gxn+1)G(Txn,Txn+1,Txn+1)≤φ(Mg(xn,xn+1,xn+1))+θΔABGΔABGG(gxn+1,Txn,Txn)vvvvvv.-ΔABG,G(gxn,Txn+1,Txn+1)vvvvvv.-ΔABG,G(gxn,Txn,Txn)-ΔABG,vvvvvv.G(gxn+1,Txn+1,Txn+1)-ΔABG=φ(Mg(xn,xn+1,xn+1))
for all n>0, where
(23)φMgxn,xn+1,xn+1=max⁡Ggxn,Txn+1,Txn+1+Gxn+1,Txn,Txn2Ggxn,gxn+1,gxn+1,Ggxn,Txn,Txn-ΔABG,vvvvvvvGgxn+1,Txn+1,Txn+1-ΔABG,vvvvvvvGgxn,Txn+1,Txn+1+Gxn+1,Txn,Txn2vvvvvvv-ΔABGGgxn,Txn+1,Txn+1+Gxn+1,Txn,Txn2≤max⁡12Ggxn,gxn+1,gxn+1,Gxn,xn+1,xn+1vvvv+Ggxn+1,Txn,Txn-ΔABG,vvvvGgxn+1,gxn+2,gxn+2vvvv+Ggxn+2,Txn+1,Txn+1-ΔABG,vvvv12Gxn,gxn+1,gxn+1vvvvvvvvvv.+Ggxn+1,Txn+1,Txn+1vvvvvvvvvv.+G(xn+1,gxn+2,gxn+2)vvvvvvvvvv.v+G(gxn+2,Txn,Txn)-ΔABG12=max⁡-ΔABGG(gxn,gxn+1,gxn+1),G(xn,xn+1,xn+1)vvvvvv+ΔABG-ΔABG,vvvvvvGgxn+1,gxn+2,gxn+2+ΔABG-ΔABG,vvvvvv2ΔABGG(xn,gxn+1,gxn+1)+G(gxn+1,gxn+2,gxn+2)vvvvvvv+2ΔABG×2-1-ΔABG=max⁡Gxn,gxn+1,gxn+1+Ggxn+1,gxn+2,gxn+22G(gxn,gxn+1,gxn+1),G(xn,xn+1,xn+1),vvvvvvvvGgxn+1,gxn+2,gxn+2,vvvvvvvvGxn,gxn+1,gxn+1+Ggxn+1,gxn+2,gxn+22=max⁡Ggxn,gxn+1,gxn+1,Ggxn+1,gxn+2,gxn+2.
That is,
(24)φMgxn,xn+1,xn+1≤max⁡Ggxn,gxn+1,gxn+1,Ggxn+1,gxn+2,gxn+2.
From (22) and (24), we have
(25)Ggxn+1,gxn+2,gxn+2≤φmax⁡Ggxn,gxn+1,gxn+1,Ggxn+1,gxn+2,gxn+2
for all n>0.
If there exists somen0∈N such that
(26)max⁡(G(gxn0,gxn0+1,gxn0+1),G(gxn0+1,gxn0+2,gxn0+2))=G(gxn0+1,gxn0+2,gxn0+2),
then, using (19) and the fact that φ(t)<t for all t>0, we have
(27)Ggxn0+1,gxn0+2,gxn0+2≤φ(G(gxn0+1,gxn0+2,gxn0+2))<G(gxn0+1,gxn0+2,gxn0+2),
which is a contradiction. Hence
(28)max⁡(G(gxn,gxn+1,gxn+1),G(gxn+1,gxn+2,gxn+2))=G(gxn,gxn+1,gxn+1)
for all n>0. Now (25) implies that
(29)G(gxn+1,gxn+2,gxn+2)≤φ(G(gxn,gxn+1,gxn+1))
for all n>0.
In particular, for alln≥1, we have
(30)G(gxn,gxn+1,gxn+1)≤φ(Ggxn-1,gxn,gxn)≤φ2Ggxn-2,gxn-1,gxn-1≤⋯≤φn(Ggx0,gx1,gx1).
Fix ϵ>0 and t0=G(gx0,gx1,gx1)>0. Since φ∈Ψ, ∑n≥1φn(t0) converges. In particular, there exists some m0∈N such that ∑k=m0∞φn(t0)<ϵ. Hence, for m>n≥m0, we have
(31)Ggxn,gxm,gxm≤∑k=nm-1‍Ggxk,gxk+1,gxk+1≤∑k=nm-1‍φkG(gx0,gx1,gx1)≤∑k=m0∞‍φn(t0)<ϵ.
This implies that {gxn} is a Cauchy sequence. By given hypothesis, {xn} is a Cauchy sequence. By completeness of X, there exists z∈X such that {xn}→z. As xn∈A0⊆A for all n, so z∈A. Since T and g are continuous mappings, {Txn}→Tz and {gxn}→gz. Taking limit in (18) as n→∞, we conclude that z is a g-best proximity point of T.Remark 14.
Ifg is an identity map in Theorem 13, then we obtain the best proximity point of mapping T.Corollary 15.
LetX be a complete G-metric space, A and B two closed subsets of X, and g a continuous self-mapping on A such that ϕ≠A0⊆gA0. Suppose that T:A→B is continuous (α,g)-proximal and admissible and (φ,θ,α,g)-contraction, where φ∈Ψ, θ∈Ω1, and T(A0)⊆B0. If following conditions hold:(a)
quadruple(A,B,T,g) satisfies weak P-property of the first kind,(b)
forx,y,z∈A0 with G(gy,Tx,Tx)=ΔABG and α(gx,gy)≥1, the following holds:
(32)αgx,gyGTx,Ty,Tz≤kMg(x,y,z)+θGgy,Tx,Tx-ΔABG,vvvvvvvvGgx,Ty,Ty-ΔABG,Ggx,Tx,Tx-ΔABG,vvvvvvvvG(gy,Ty,Ty)-ΔABG,(c)
if a sequence{zn} in A0 with {gzn}⊆A0 is Cauchy, then {zn} is Cauchy,(d)
there is(x0,x1)∈A0×A0 such that G(gx1,Tx0,Tx0)=ΔABG and α(gx0,gx1)≥1.
Then there exists a convergent sequence{xn}⊆A0 which satisfies
(33)G(gxn+1,Txn,Txn)=ΔABG∀n≥0,
and {xn} converges to g-best proximity point of T.Example 16.
LetX={0,1,2,3,…} and G:X×X×X→R+ defined by
(34)G(x,y,z)=x+y+zifx≠y≠z≠0,x+yifx=y≠z,x,y,z≠0,y+z+1ifx=0,y≠z,y,z≠0,y+2ifx=0,y=z≠0,z+1ifx=y=0,z≠0,0ifx=y=z.
It is known that X is a complete G-metric space. Let A={0,2,4} and B={1,3,5,…}. Obviously A and B are closed subsets of X and ΔABG=G(A,B,B)=G(0,1,1)=3. Take A0={0,2}. Define the mapping g:A→A by
(35)g(x)=xifx=0,2x2ifx=4.
Obviously g is continuous and A0⊆g(A0). A mapping T:A→B defined by T(x)=1 is continuous. Define α:X×X→[0,∞) by α(x,y)=x+y. Clearly
(36)α(g(0),g(2))=α(0,2)=2>1,G(g(0),T(2),T(2))=G(0,1,1)=3,G(g(2),T(0),T(0))=G(2,1,1)=3.
As α(0,2)=2>1, so T is (α,g)-proximal and admissible. Since
(37)α(g(0),g(2))G(T(0),T(2),T(2))=2G(1,1,1)=0
and Mg(0,2,2)=4, therefore T is (φ,θ,α,g)-contraction. Now
(38)G(g(0),T(2),T(2))=G(0,1,1)=3=ΔABG,G(g(2),T(0),T(0))=G(2,1,1)=3=ΔABG
imply that G(g(0),g(0),g(0))=G(0,0,0)=0=G(T(2),T(0),T(0))=G(1,1,1). Hence quadruple (A,B,T,g) has weak P-property of the first kind. Note that (0,2)∈A0×A0 with G(g(2),T(0),T(0))=3=ΔABG and α(g(0),g(2))>1. Thus T has g-best proximity point (0 and 2 are g-best proximity point of T).Lemma 17.
Letϕ∈Φ be a mapping and let {am}⊂R+ be a sequence. If am+1≤ϕ(am) and am≠0 for all m, then {am}→0.Theorem 18.
If condition (h) in Theorem 13 is replaced by the following: (h′)
φ∈Φ,θ∈Ω2 and α is (2,g)-transitive, then there exists a sequence {xn}⊆A0 which satisfies
(39)G(gxn+1,Txn,Txn)=ΔABG∀n≥0and converges to a g-best proximity point of T.Proof.
Following arguments similar to those in the proof of Theorem13, we have
(40)G(gxn+1,Txn,Txn)=ΔAB,Ggxn,gxn+1,gxn+1>0,α(gxn,gxn+1)≥1,iiiiiiiiiiiiiiixn∈A0,Ggxn+1,gxn+2,gxn+2≤φGgxn,gxn+1,gxn+2iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∀n≥0.
By Lemma 17, we have
(41){G(gxn,gxn+1,gxn+1)}⟶0.
Next, we show that {gxn} is a Cauchy sequence. Assume on the contrary that {gxn} is not a Cauchy sequence. Then, by Remark 6, there exist ϵ0>0 and two subsequences {xm(k)} and {xn(k)} such that the following hold:
(42)k≤m(k)<n(k),Ggxmk,gxnk,gxnk>ϵ0,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∀k∈N,Ggxmk,gxp,gxp≤ϵ0∀p∈mk+1,mk+2,…,nk-2,nk-1,(43)lim⁡k→∞Ggxmk-1,gxnk-1,gxnk-1=ϵ0,lim⁡k→∞Ggxmk-1,gxnk+p,gxnk+p=ϵ0,∀p≥0. Note that
(44)0≤G(gxn(k),Txn(k),Txn(k))-ΔABG≤G(gxn(k),gxn(k)+1,gxn(k)+1)+G(gxn(k)+1,Txn(k),Txn(k))-ΔABG=G(gxn(k),gxn(k)+1,gxn(k)+1).
Therefore
(45)lim⁡k→∞[G(gxn(k),Txn(k),Txn(k))-ΔABG]=0.
Similarly,
(46)lim⁡k→∞[G(gxm(k),Txm(k),Txm(k))-ΔABG]=0.
Furthermore,
(47)ϵ0<Gxmk,xnk,xnk≤Mg(xm(k),xn(k),xn(k))∀k≥0,
where
(48)Mg(xm(k),xn(k),xn(k))=max⁡Ggxmk,Txnk,Txnk+G(gxn(k),Txn(k),Txm(k))2Ggxmk,gxnk,gxnk,Ggxmk,Txmk,Txmksssssisss-ΔABG,G(gxn(k),Txn(k),Txn(k))-ΔABG,sssssisssGgxmk,Txnk,Txnk+G(gxn(k),Txn(k),Txm(k))2sssssisss-ΔABGGgxmk,Txnk,Txnk+G(gxn(k),Txn(k),Txm(k))2.(49)G(gxm(k),Txn(k),Txn(k))+G(gxn(k),Txn(k),Txm(k))2-ΔABG≤12Ggxmk,gxnk+1,gxnk+1sssiss+G(gxn(k)+1,Txn(k),Txn(k))sssiss+Ggxnk,gxmk+1,gxmk+1sssiss+Ggxmk+1,Txnk,Txmk-ΔABG=G(gxm(k),gxn(k)+1,gxn(k)+1)+ΔABssss+G(gxn(k),gxm(k)+1,gxm(k)+1)+ΔAB×2-1-ΔAB=G(gxm(k),gxn(k)+1,gxn(k)+1)sss.+Ggxnk,gxmk+1,gxmk+1×2-1.
Taking limit as k→∞ in (49) and using (45), we obtain that
(50)lim⁡k→∞G(gxm(k),Txn(k),Txn(k))+G(gxn(k),Txn(k),Txm(k))2dd-ΔABGG(gxm(k),Txn(k),Txn(k))+G(gxn(k),Txn(k),Txm(k))2≤ϵ+ϵ2=ϵ0.
Taking limit as k→∞ in (48) and using (43), (45), (46), and (50), we have
(51)lim⁡k→∞Mg(xm(k),xn(k),xn(k))=max⁡(ϵ0,0,0,ϵ0)=ϵ0.
Thus a sequence {Mg(xm(k),xn(k),xn(k))} converges to ϵ0 and terms of this sequence are strictly greater than ϵ0. ln particular, since φ∈Φ,
(52)lim⁡k→∞φ({Mg(xm(k),xn(k),xn(k))})=lim⁡t→ϵ0+φ(t)<ϵ0.
From the fact that α(gxn,gxn+1)≥1 for all n≥0 and α is (2,g)-transitive, we deduce that
(53)α(gxm(k),gxn(k))≥1∀k≥0.
As (A,B,T,g) has the weak P-property of the first kind, so, for all k≥0,
(54)xm(k),xm(k)+1,xn(k),xn(k)+1∈A0,Ggxmk+1,Txmk,Txmk=ΔABG,Ggxnk+1,Txnk,Txnk=ΔABG.
This implies that
(55)G(gxm(k)+1,gxn(k)+1,gxn(k)+1)≤G(Txm(k),Txn(k),Txn(k)).
As T is (φ,θ,α,g)-contraction, so we have
(56)Ggxmk+1,gxnk+1,gxnk+1≤Ggxmk,Txnk,Txnk≤αTxmk,Txnk,TxnkGgxmk,Txnk,Txnk≤φMg(xm(k),xn(k),xn(k))iiiiiii+θG(gxn(k),Txm(k),Txm(k))vvvvvvvv-ΔAB,Ggxmk,Txnk,Txnk-ΔAB,vvvvvvvvGgxmk,Txmk,Txmk-ΔAB,vvvvvivvGgxnk,Txnk,Txnk-ΔAB.
Using (45), the third and the fourth arguments of θ converge to zero as k→∞. Since θ∈Ω2, all the terms tend to zero as k→∞. Taking limit as k→∞ in (56), using (45) and (52), we have
(57)ϵ0=lim⁡k→∞G(gxm(k)+1,gxn(k)+1,gxn(k)+1)≤lim⁡k→∞φ(Mg(xm(k),xn(k),xn(k)))<ϵ0,
which is an absurd statement. Hence {gxn} is a Cauchy sequence. The rest follows from Theorem 13.Theorem 19.
Theorem13 also holds if contractive condition (10) is valid for all x∈A0 and y∈A; conditions (b) and (g) are replaced by the following: (b′)
quadruple(A,B,T,g) has the weak P-property of the second kind;(g′)
for a sequence{xn}⊆A0 converging to x∈A and α(gxn,gxn+1)≥1 for all n≥0, there exists a subsequence {xn(k)} of {xn} such that α(gxn(k),gx)≥1 for all k≥0.Proof.
Following similar arguments to those given in proof of Theorem13, we deduce that {gxn} and {xn} are Cauchy sequences in closed subset A of X. So we obtain an x in A such that {xn}→x and {gxn}→gx. We show that x is a g-best proximity point of T.
Given that(A,B,T,g) has the weak P-property of the second kind, for all n,m∈N,
(58)xm,xm+1,xn,xn+1∈A0,G(gxm+1,Txm,Txm)=ΔABG,G(gxn+1,Txn,Txn)=ΔABG
imply that
(59)G(gxm+1,gxn+1,gxn+1)≤G(Txm,Txn,Txn).
It follows that {Txn} is also a Cauchy sequence in B. Hence, there is z∈B such that {Txn}→z. Thus
(60)G(gxn,gx,gx)⟶0,G(Txn,z,z)⟶0.
Since G(gxn+1,Txn,Txn)=ΔABG for all n≥0, we deduce that
(61)Ggx,z,z=ΔABG;
that is, gx∈A0 and z∈B0. Using condition (g′), we conclude that there exists a subsequence {xn(k)} of {xn} such that
(62)α(gxn(k),gx)≥1∀k≥0.
Note that
(63)0≤G(gxn(k),Txn(k),Txn(k))-ΔABG≤G(gxn(k),gxn(k)+1,gxn(k)+1)+G(gxn(k)+1,Txn(k),Txn(k))-ΔABG=G(gxn(k),gxn(k)+1,gxn(k)+1).
Therefore
(64)lim⁡k→∞[G(gxnk,Txnk,Txnk)-ΔABG]=0.
The first and the second arguments of
(65)Mg(xn(k),x,x)=max⁡G(gxn(k),Tx,Tx)+G(gx,Txn(k),Txn(k))2G(gxnk,gx,gx),Ggxnk,Txnk,Txnkssssssssss-ΔABG,G(gx,Tx,Tx)-ΔABG,ssssssssssG(gxn(k),Tx,Tx)+G(gx,Txn(k),Txn(k))2-ΔABG
tend to zero, while the last argument gives
(66)lim⁡k→∞G(gxn(k),Tx,Tx)+G(gx,Txn(k),Txn(k))2-ΔABG≤lim⁡k→∞G(gxn(k),Tx,Tx)+G(gx,gxn(k)+1,gxn(k)+1)iiiiiiiiiiiii+G(gxn(k)+1,Txn(k),Txn(k))×2-1-ΔABG=lim⁡k→∞G(gxn(k),Tx,Tx)+G(gx,gxn(k)+1,gxn(k)+1)ddddddd+ΔAB×2-1-ΔABG=G(gx,Tx,Tx)+0+ΔAB2-ΔABG=Ggx,Tx,Tx+0-ΔABG2.
Therefore,
(67)lim⁡k→∞Mg(xn(k),x,x)=G(gx,Tx,Tx)-ΔABG.
Suppose that G(gx,Tx,Tx)≠ΔABG; that is,
(68)t0=G(gx,Tx,Tx)-ΔABG>0.
Since the first and the second terms in (65) tend to zero, and the fourth term tends to t0/2, there exists k0∈N such that
(69)Mg(xn(k),x,x)=G(gx,Tx,Tx)-ΔABG=t0>0∀k≥k0.
Using the contractivity condition, we have
(70)G(Txn(k),Tx,Tx)≤α(gxn(k),gx,gx)G(Txn(k),Tx,Tx)≤φMgxnk,x,x+θ-ΔABGG(gx,Txn(k),Txn(k))-ΔABG,ddddddG(gxn(k),Tx,Tx)-ΔABG,ddddddG(gxn(k),Txn(k),Txn(k))-ΔABG,Ggx,Tx,Tx-ΔABG=φGgx,Tx,Tx-ΔABG+θG(gx,Txn(k),Txn(k))-ΔABG,sssssssissssGgxn(k),Tx,Tx-ΔABG,G(gxn(k),Txn(k),Txn(k))sssssssssssssss-ΔABG,Ggx,Tx,Tx-ΔABG.
Since the third argument of θ in (70) tends to zero and θ∈Ω2, its limit as k→∞ is zero. Therefore, we have
(71)Gz,Tx,Tx=lim⁡k→∞G(Txn(k),Tx,Tx)≤φ(G(gx,Tx,Tx)-ΔABG).
As G(gx,Tx,Tx)-ΔABG>0, then φ(G(gx,Tx,Tx)-ΔAB)<G(gx,Tx,Tx)-ΔABG. Thus,
(72)Gz,Tx,Tx≤φ(G(gx,Tx,Tx)-ΔABG)<G(gx,Tx,Tx)-ΔAB≤G(gx,z,z)+G(z,Tx,Tx)-ΔABG≤ΔAB+G(gx,z,z)-ΔABG=G(z,Tx,Tx),
which is a contradiction. Hence G(gx,Tx,Tx)=ΔAB and the result follows.


### 2.1. Uniqueness ofg-Best Proximity Points

In this section, we study sufficient conditions in order to prove the uniqueness ofg-best proximity point.Definition 20.
LetT:A→B, g:A→A, and α:X×X→[0,∞) be three mappings. A mapping T is called (α,g)-regular if, for all x,y∈A0, such that α(gx,gy)<1, there exists z∈A0 such that α(gx,gz)≥1 and α(gy,gz)≥1.Theorem 21.
Under the hypothesis of Theorem13, assume that θ∈Θ and T is (α,g)-regular. Then for all g-best proximity points x and y of T in A0 we have that gx=gy. In particular, if g is injective on the set of all g-best proximity points of T in A0, then T has a unique g-best proximity point.Proof.
Letx,y∈A0 be two g-best proximity points of T in A0. Since G(gx,Tx,Tx)=G(gy,Ty,Ty)=ΔABG and T is a (α,g)-proximal and admissible, we deduce that
(73)G(gx,gy,gy)≤G(Tx,Ty,Ty).
We always have α(gx,gy)≥1 or α(gx,gy)<1. If α(gx,gy)≥1, then we obtain that
(74)G(gx,gy,gy)≤G(Tx,Ty,Ty)≤α(gx,gy)G(Tx,Ty,Ty)≤φMgx,y,y+θG(gy,Tx,Tx)-ΔABG,vvvvvvvvG(gx,Ty,Ty)-ΔABG,G(gx,Tx,Tx)-ΔABG,vvvvvvvvG(gy,Ty,Ty)-△ABG=φMgx,y,y.
The last equality holds since θ∈Θ and the last two arguments of θ are zero. Note that
(75)G(gx,Ty,Ty)+G(gy,Tx,Tx)2-ΔABG≤gx,Tx,Tx×2-1Ggx,gy,gy+Ggy,Ty,Tyv+Ggy,gx,gx+Ggx,Tx,Tx×2-1-ΔABG=G(gx,gy,gy)+ΔABG+G(gy,gx,gx)+ΔABG2-ΔABG=G(gx,gy,gy)+G(gy,gx,gx)2=G(gx,gy,gy).
Hence
(76)Mg(x,y,y)=max⁡⁡Ggx,Ty,Ty+Ggy,Tx,Tx2G(gx,gy,gy),G(gx,Tx,Tx)-ΔABG,vvvvvvvvvG(gy,Ty,Ty)-ΔABG,vvvvvvvvvGgx,Ty,Ty+Ggy,Tx,Tx2-ΔABGGgx,Ty,Ty+Ggy,Tx,Tx2.
Therefore
(77)G(gx,gy,gy)≤φ(Mg(x,y,y))=φ(G(gx,gy,gy))
gives the fact that G(gx,gy,gy)=0; that is, gx=gy.
Now, ifα(gx,gy)<1, then, by the (α,g)-regularity of T, there exists z0∈A0 such that α(gx,gz0)≥1 and α(gy,gz0)≥1. Based on z0, we define a sequence {zn} such that {gzn} converges to gx and gy which proves the uniqueness. First, we will prove that {gzn} converges to gx.
Indeed,Tz0∈TA0⊆B0 implies that s0∈A0 such that G(s0,Tz0,Tz0)=ΔABG, and, for s0∈A0⊆gA0, there is z1∈A0 verifying gz1=s0. Therefore, G(gz1,Tz0,Tz0)=ΔABG. Following the similar arguments, there exists a sequence {zn}⊆A0 such that G(gzn+1,Tzn,Tzn)=ΔABG for all n≥0. In particular, gzn+1∈A0 and Tzn∈B0. We claim that
(78)αgx,gzn≥1∀n≥0.
If n=0, α(gx,gz0)≥1 by the choice of z0. Suppose that α(gx,gzn)≥1 for some n≥0. As T is (α;g)-proximal and admissible, so we have
(79)α(gx,gzn)≥1,G(gx,Tx,Tx)=ΔABG,Ggzn+1,Tzn,Tzn=ΔABGvvvvvivvvx,zn,zn+1∈A0,
which imply that α(gx,gzn+1)≥1. Hence (78) holds. For all n≥0, we have
(80)G(gx,Tzn,Tzn)+G(gzn,Tx,Tx)2-ΔABG≤2-1G(gx,gzn+1,gzn+1)+G(gzn+1,Tzn,Tzn)vvvv+G(gzn,gx,gx)+G(gx,Tx,Tx)×2-1-ΔABG=Ggx,gzn+1,gzn+1+ΔAB+Ggzn,gx,gx+ΔAB2-ΔABG=G(gx,gzn+1,gzn+1)+G(gzn,gx,gx)2≤max⁡(G(gx,gzn,gzn),G(gx,gzn+1,gzn+1)),
which implies that
(81)Mg(x,zn,zn)=max⁡G(gx,Tzn,Tzn)+G(gzn,Tx,Tx)2G(gx,gzn,gzn),G(gx,Tx,Tx)-ΔABG,vvvvvvvG(gzn,Tzn,Tzn)-ΔABG,vvvvvvvG(gx,Tzn,Tzn)+G(gzn,Tx,Tx)2-ΔABGG(gx,Tzn,Tzn)+G(gzn,Tx,Tx)2≤max⁡(G(gx,gzn,gzn),G(gx,gzn+1,gzn+1)).
By weak P-property of the first kind,
(82)G(gx,Tx,Tx)=ΔAB,G(gzn+1,Tzn,Tzn)=ΔABGiiiiiiiiiiiiiiix,zn,zn+1∈A0,implythatG(gx,gzn+1,gzn+1)≤G(Tx,Tzn,Tzn).
For all n≥0, we have
(83)G(gx,gzn+1,gzn+1)≤G(Tx,Tzn,Tzn)≤φ(Mg(x,zn,zn))+θΔABGG(gzn,Tx,Tx)-ΔABG,G(gx,Tzn,Tzn)-ΔAB,vvvvvvvvGgx,Tx,Tx-ΔABG,Ggzn,Tzn,Tzn≤φ(Mg(x,zn,zn))≤φ(max⁡(G(gx,Tzn,Tzn),G(gx,gzn+1,gzn+1))).
Suppose that there is n0∈N such that gzn0=gx. In this case, we have
(84)G(gx,gzn+1,gzn0+1)≤φ(max⁡(G(gx,Tzn0,Tzn0),G(gx,gzn0+1,gzn0+1))),
but this is possible only when G(gx,gzn0+1,gzn0+1); that is, gzn0+1=gx. Following the similar arguments, we have gzn=gx for all n≥n0. Hence {gzn} converges to gx.
Suppose thatgzn≠gx for all n≥0; that is, G(gx,gzn,gzn)>0 for all n≥0. Suppose that
(85)max⁡(G(gx,gzn,gzn),G(gx,gzn+1,gzn+1))=G(gx,gzn+1,gzn+1)
for some n. Then (83) would yield
(86)G(gx,gzn+1,gzn+1)≤φ(max⁡(G(gx,gzn,gzn),G(gx,gzn+1,gzn+1)))=φ(G(gx,gzn+1,gzn+1))<Ggx,gzn+1,gzn+1,
which is a contradiction. Therefore, max⁡⁡(G(gx,gzn,gzn), G(gx,gzn+1,gzn+1))=G(gx,gzn,gzn); that is, for all n≥0,
(87)Ggx,gzn+1,gzn+1≤φ(Mg(x,zn,zn))=φ(G(gx,gzn,gzn)).
Recursively, for all n≥0,
(88)G(gx,gzn,gzn)≤φ(G(gx,gzn-1,gzn-1))≤φ2(G(gx,gzn-2,gzn-2))≤⋯≤φn(G(gx,gz0,gz0)).
Fix ϵ>0 arbitrary and consider t0=G(gx,gz0,gz0)>0. Since φ∈Ψ, the series ∑n≥1φn(t0) converges. In particular, there exists m0∈N such that ∑k=m0∞φn(t0)<ϵ. More precisely, φn(t0)<ϵ for all n≥m0. Therefore, if n≥m0, we have that
(89)Ggx,gzn,gzn≤φnGgx,gz0,gz0=φn(t0)<ϵ.
This means that {gxn} converges to gx. Similarly, it can be shown that {gxn} converges to gy and this completes the proof.


## 2.1. Uniqueness ofg-Best Proximity Points

In this section, we study sufficient conditions in order to prove the uniqueness ofg-best proximity point.Definition 20.
LetT:A→B, g:A→A, and α:X×X→[0,∞) be three mappings. A mapping T is called (α,g)-regular if, for all x,y∈A0, such that α(gx,gy)<1, there exists z∈A0 such that α(gx,gz)≥1 and α(gy,gz)≥1.Theorem 21.
Under the hypothesis of Theorem13, assume that θ∈Θ and T is (α,g)-regular. Then for all g-best proximity points x and y of T in A0 we have that gx=gy. In particular, if g is injective on the set of all g-best proximity points of T in A0, then T has a unique g-best proximity point.Proof.
Letx,y∈A0 be two g-best proximity points of T in A0. Since G(gx,Tx,Tx)=G(gy,Ty,Ty)=ΔABG and T is a (α,g)-proximal and admissible, we deduce that
(73)G(gx,gy,gy)≤G(Tx,Ty,Ty).
We always have α(gx,gy)≥1 or α(gx,gy)<1. If α(gx,gy)≥1, then we obtain that
(74)G(gx,gy,gy)≤G(Tx,Ty,Ty)≤α(gx,gy)G(Tx,Ty,Ty)≤φMgx,y,y+θG(gy,Tx,Tx)-ΔABG,vvvvvvvvG(gx,Ty,Ty)-ΔABG,G(gx,Tx,Tx)-ΔABG,vvvvvvvvG(gy,Ty,Ty)-△ABG=φMgx,y,y.
The last equality holds since θ∈Θ and the last two arguments of θ are zero. Note that
(75)G(gx,Ty,Ty)+G(gy,Tx,Tx)2-ΔABG≤gx,Tx,Tx×2-1Ggx,gy,gy+Ggy,Ty,Tyv+Ggy,gx,gx+Ggx,Tx,Tx×2-1-ΔABG=G(gx,gy,gy)+ΔABG+G(gy,gx,gx)+ΔABG2-ΔABG=G(gx,gy,gy)+G(gy,gx,gx)2=G(gx,gy,gy).
Hence
(76)Mg(x,y,y)=max⁡⁡Ggx,Ty,Ty+Ggy,Tx,Tx2G(gx,gy,gy),G(gx,Tx,Tx)-ΔABG,vvvvvvvvvG(gy,Ty,Ty)-ΔABG,vvvvvvvvvGgx,Ty,Ty+Ggy,Tx,Tx2-ΔABGGgx,Ty,Ty+Ggy,Tx,Tx2.
Therefore
(77)G(gx,gy,gy)≤φ(Mg(x,y,y))=φ(G(gx,gy,gy))
gives the fact that G(gx,gy,gy)=0; that is, gx=gy.
Now, ifα(gx,gy)<1, then, by the (α,g)-regularity of T, there exists z0∈A0 such that α(gx,gz0)≥1 and α(gy,gz0)≥1. Based on z0, we define a sequence {zn} such that {gzn} converges to gx and gy which proves the uniqueness. First, we will prove that {gzn} converges to gx.
Indeed,Tz0∈TA0⊆B0 implies that s0∈A0 such that G(s0,Tz0,Tz0)=ΔABG, and, for s0∈A0⊆gA0, there is z1∈A0 verifying gz1=s0. Therefore, G(gz1,Tz0,Tz0)=ΔABG. Following the similar arguments, there exists a sequence {zn}⊆A0 such that G(gzn+1,Tzn,Tzn)=ΔABG for all n≥0. In particular, gzn+1∈A0 and Tzn∈B0. We claim that
(78)αgx,gzn≥1∀n≥0.
If n=0, α(gx,gz0)≥1 by the choice of z0. Suppose that α(gx,gzn)≥1 for some n≥0. As T is (α;g)-proximal and admissible, so we have
(79)α(gx,gzn)≥1,G(gx,Tx,Tx)=ΔABG,Ggzn+1,Tzn,Tzn=ΔABGvvvvvivvvx,zn,zn+1∈A0,
which imply that α(gx,gzn+1)≥1. Hence (78) holds. For all n≥0, we have
(80)G(gx,Tzn,Tzn)+G(gzn,Tx,Tx)2-ΔABG≤2-1G(gx,gzn+1,gzn+1)+G(gzn+1,Tzn,Tzn)vvvv+G(gzn,gx,gx)+G(gx,Tx,Tx)×2-1-ΔABG=Ggx,gzn+1,gzn+1+ΔAB+Ggzn,gx,gx+ΔAB2-ΔABG=G(gx,gzn+1,gzn+1)+G(gzn,gx,gx)2≤max⁡(G(gx,gzn,gzn),G(gx,gzn+1,gzn+1)),
which implies that
(81)Mg(x,zn,zn)=max⁡G(gx,Tzn,Tzn)+G(gzn,Tx,Tx)2G(gx,gzn,gzn),G(gx,Tx,Tx)-ΔABG,vvvvvvvG(gzn,Tzn,Tzn)-ΔABG,vvvvvvvG(gx,Tzn,Tzn)+G(gzn,Tx,Tx)2-ΔABGG(gx,Tzn,Tzn)+G(gzn,Tx,Tx)2≤max⁡(G(gx,gzn,gzn),G(gx,gzn+1,gzn+1)).
By weak P-property of the first kind,
(82)G(gx,Tx,Tx)=ΔAB,G(gzn+1,Tzn,Tzn)=ΔABGiiiiiiiiiiiiiiix,zn,zn+1∈A0,implythatG(gx,gzn+1,gzn+1)≤G(Tx,Tzn,Tzn).
For all n≥0, we have
(83)G(gx,gzn+1,gzn+1)≤G(Tx,Tzn,Tzn)≤φ(Mg(x,zn,zn))+θΔABGG(gzn,Tx,Tx)-ΔABG,G(gx,Tzn,Tzn)-ΔAB,vvvvvvvvGgx,Tx,Tx-ΔABG,Ggzn,Tzn,Tzn≤φ(Mg(x,zn,zn))≤φ(max⁡(G(gx,Tzn,Tzn),G(gx,gzn+1,gzn+1))).
Suppose that there is n0∈N such that gzn0=gx. In this case, we have
(84)G(gx,gzn+1,gzn0+1)≤φ(max⁡(G(gx,Tzn0,Tzn0),G(gx,gzn0+1,gzn0+1))),
but this is possible only when G(gx,gzn0+1,gzn0+1); that is, gzn0+1=gx. Following the similar arguments, we have gzn=gx for all n≥n0. Hence {gzn} converges to gx.
Suppose thatgzn≠gx for all n≥0; that is, G(gx,gzn,gzn)>0 for all n≥0. Suppose that
(85)max⁡(G(gx,gzn,gzn),G(gx,gzn+1,gzn+1))=G(gx,gzn+1,gzn+1)
for some n. Then (83) would yield
(86)G(gx,gzn+1,gzn+1)≤φ(max⁡(G(gx,gzn,gzn),G(gx,gzn+1,gzn+1)))=φ(G(gx,gzn+1,gzn+1))<Ggx,gzn+1,gzn+1,
which is a contradiction. Therefore, max⁡⁡(G(gx,gzn,gzn), G(gx,gzn+1,gzn+1))=G(gx,gzn,gzn); that is, for all n≥0,
(87)Ggx,gzn+1,gzn+1≤φ(Mg(x,zn,zn))=φ(G(gx,gzn,gzn)).
Recursively, for all n≥0,
(88)G(gx,gzn,gzn)≤φ(G(gx,gzn-1,gzn-1))≤φ2(G(gx,gzn-2,gzn-2))≤⋯≤φn(G(gx,gz0,gz0)).
Fix ϵ>0 arbitrary and consider t0=G(gx,gz0,gz0)>0. Since φ∈Ψ, the series ∑n≥1φn(t0) converges. In particular, there exists m0∈N such that ∑k=m0∞φn(t0)<ϵ. More precisely, φn(t0)<ϵ for all n≥m0. Therefore, if n≥m0, we have that
(89)Ggx,gzn,gzn≤φnGgx,gz0,gz0=φn(t0)<ϵ.
This means that {gxn} converges to gx. Similarly, it can be shown that {gxn} converges to gy and this completes the proof.



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*Source: 243753-2015-01-05.xml*