# New Approach for Common Fixed Point Theorems viaC-Class Functions in Gp-Metric Spaces

Authors: A. H. Ansari; M. A. Barakat; H. Aydi
Journal: Journal of Function Spaces
 (2017)

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## Abstract

We prove a new approach for some common fixed point results in completeGp-metric spaces for weakly increasing self-mappings satisfying ψ,φ-contractions via the concept of C-class functions. An example is also provided.


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## Body

## 1. Introduction and Mathematical Preliminaries

In 1922, Banach [1] proved his classical theorem which asserts suitable conditions ensuring the existence and uniqueness of fixed point of the underlying mapping. Over the last several decades, this theorem has been generalized and improved in various spaces (e.g., [2–6]). In 1994, Matthews [7] introduced the notion of a partial metric space and established the Banach contraction theorem in the class of partial metric spaces. Notably, in partial metric spaces, the distance from a point to itself need not be zero. In recent years, several authors proved variant (common) fixed point theorems in partial metric spaces. For more details, see [8–17].For the sake of completeness, we recall the definition of a partial metric space (in short PMS) which runs as follows.Definition 1 (see [7]).
A partial metric on a nonempty setX is a function p:X×X→R+,  R+:=[0,∞), such that for all x,y,z∈Xp1x=y⇔p(x,x)=p(x,y)=p(y,y),p2p(x,x)≤p(x,y),p3p(x,y)=p(y,x),p4p(x,y)≤p(x,z)+p(z,y)-pz,z.A partial metric space is a pair (X,p) such that X is a nonempty set and p is a partial metric on X.On the other hand, the notation of generalized metric spaces (in shortG-metric spaces) was introduced by Mustafa and Sims [18] who presented and improved the Banach contraction principle in the class of G-metric spaces. The definition of a G-metric space is introduced as follows.Definition 2 (see [18]).
LetX be a nonempty set. Suppose that G:X×X×X→R+ satisfies(a)
G(x,y,z)=0 if x=y=z,(b)
G(x,y,z)>0,∀x,y,z∈X,x≠y,(c)
G(x,x,y)≤G(x,y,z),∀x,y,z∈X,y≠z,(d)
G(x,y,z)=G(x,z,y)=G(y,z,x)=⋯ (symmetry in all three variables),(e)
G(x,y,z)≤G(x,a,a)+G(a,y,z),∀x,y,z,a∈X.
ThenG is called a G-metric on X and (X,G) is called a G-metric space.Recently, as a unification between partial metric spaces andG-metric spaces, Zand and Nezhad [19] defined the concept of a Gp-metric space in the following way.Definition 3 (see [19]).
LetX be a nonempty set. Suppose that Gp:X×X×X→R+ satisfies(a)
x=y=z if Gp(x,x,x)=Gp(y,y,y)=Gpz,z,z∀x,y,z∈X;(b)
Gp(x,x,x)≤Gp(x,x,y)≤Gp(x,y,z),∀x,y,z∈X;(c)
Gp(x,y,z)=Gp(x,z,y)=Gp(y,z,x)=⋯ (symmetry in all three variables);(d)
Gp(x,y,z)≤Gp(x,a,a)+Gp(a,y,z)-Gp(a,a,a),∀x,y,z,a∈X.
ThenGp is called a Gp-metric on X and (X,Gp) is called a Gp-metric space.Example 4 (see [19]).
LetX=[0,∞) and define Gp(x,y,z)=max⁡x,y,z for all x,y,z∈X. Then (X,Gp) is a Gp-metric space. Note that (X,Gp) is not a G-metric space.Proposition 5 (see [19]).
Let(X,Gp) be a Gp-metric space. Then for any x,y,z∈X and a∈X, one has(i)
Gp(x,y,z)≤Gp(x,x,y)+Gp(x,x,z)-Gp(x,x,x);(ii)
Gp(x,y,y)≤2Gp(x,x,y)-Gp(x,x,x);(iii)
Gp(x,y,z)≤Gp(x,a,a)+Gp(y,a,a)+Gp(z,a,a)-2Gp(a,a,a);(iv)
Gp(x,y,z)≤Gp(x,a,z)+Gp(a,y,z)-Gp(a,a,a).Proposition 6 (see [19]).
EveryGp-metric space (X,Gp) defines a metric space (X,DGp), where (1)DGpx,y=Gpx,y,y+Gpy,x,x-Gpx,x,x-Gpy,y,y,for all x,y∈X.Definition 7 (see [19]).
Let(X,Gp) be a Gp-metric space. A sequence {xn} is Gp-convergent to x∈X if limn,m→∞Gp(x,xm,xn)=Gp(x,x,x).
We may write the above asxn→x.
Thus ifxn→x in a Gp-metric space (X,Gp), then for any ϵ>0, there exists l∈N such that |Gp(x,xn,xm)-Gp(x,x,x)|<ϵ for all n,m>l.Proposition 8 (see [19]).
Let(X,Gp) be a Gp-metric space. Take a sequence {xn} in X and a point x∈X. The following are equivalent:(i)
{xn} is Gp-convergent to x;(ii)
Gp(xn,xn,x)→Gp(x,x,x) as n→∞;(iii)
Gp(xn,x,x)→Gp(x,x,x) as n→∞.Definition 9 (see [19]).
Let(X,Gp) be a Gp-metric space.(i)
A sequence{xn} is called a Gp-Cauchy if and only if limm,n→∞Gp(xn,xm,xm) exists (and is finite).(ii)
AGp-metric space (X,Gp) is said to be Gp-complete if and only if every Gp-Cauchy sequence in X is Gp-convergent to x∈X; that is, Gp(x,x,x)=limm,n→∞Gp(xn,xm,xm).TakeΨ={ψ:[0,∞)→[0,∞) is continuous, nondecreasing, and ψ-1({0})={0}}, and Φ={φ:φ:[0,∞)→[0,∞) is lower semicontinuous, nondecreasing, and φ-1({0})={0}}.Definition 10 (see [20]).
Let(X,⪯) be a partially ordered set. Two maps f,g:X→X are said to be weakly increasing if fx⪯gfx and gx⪯fgx for all x∈X.Definition 11.
Let(X,Gp) be a Gp-metric space endowed with a partial order ⪯. Let {xn} and z be in X. (X,Gp,⪯) is said to be regular if xn→z and {xn} is nondecreasing; then xn⪯z for all n∈N.Lemma 12 (see [21]).
Let(X,Gp) be a Gp-metric space. One has the following.(i)
IfGp(x,y,z)=0, then x=y=z.(ii)
Ifx≠y, then Gp(x,y,y)>0.We rewrite the continuity of mappings inGp-metric spaces.Definition 13.
Let(X,Gp) be a Gp-metric space and let T:X→X be a given mapping. One says that T is continuous at u∈X if for every sequence {xn} converging to u in X, the sequence {Txn} converges to Tu in X. If T is continuous at each point u∈X, then one says that T is continuous on X.Ansari [22] introduced the class of C-functions which covers a large class of contractive conditions.Definition 14 (see [22]).
A mappingF:[0,∞)2→R is called a C-function if it is continuous and satisfies the following axioms:(1)
F(s,t)≤s for all s,t∈[0,∞);(2)
F(s,t)=s implies that either s=0 or t=0.Mention that anyC-function F verifies F(0,0)=0. We denote by C the set of C-class functions.Example 15 (see [22]).
The following functionsF:[0,∞)2→R are elements of C. For all s,t∈[0,∞), consider(1)
F(s,t)=s-t;(2)
F(s,t)=ms, where 0<m<1;(3)
F(s,t)=s/(1+t)r, where r∈(0,∞);(4)
F(s,t)=sβ(s), where β:[0,∞)→[0,1) is continuous;(5)
F(s,t)=s-φ(s), where φ:[0,∞)→[0,∞) is a continuous function such that φ(t)=0⇔t=0.Abbas et al. [20] proved the following result.Theorem 16 (see [20]).
LetX,⪯ be a partially ordered set and let f and g be weakly increasing self-mappings on a complete G-metric space X. Suppose that there exist ψ∈Ψ and φ∈Φ such that(2)ψGfx,gy,gy≤ψMx,y,y-φMx,y,yfor all comparable x,y∈X, where (3)Mx,y,y=a1Gx,y,y+a2Gx,fx,fx+a3Gy,gy,gy+a4Gx,gy,gy+Gy,fx,fx,where ai>0 for i={1,2,3,4} with a1+a2+a3+2a4≤1. Assume either f or g is continuous, or (X,Gp,⪯) is regular. Then f and g have a common fixed point.Very recently, Barakat and Zidan [13] extended Theorem 16 to the class of Gp-metric spaces where a general contractive condition is considered.Theorem 17 (see [13]).
Let(X,⪯) be a partially ordered set. Let f and g be weakly increasing self-mappings on a complete Gp-metric space X satisfying(4)ψGpfx,gy,gy≤ψMx,y,y-φMx,y,yfor all comparable x,y∈X, where ψ∈Ψ,  φ∈Φ, and(5)Mx,y,y=max⁡Gpx,y,y,Gpx,fx,fx,Gpy,gy,gy,12Gpx,gy,gy+Gpy,fx,fx.Assume either f or g is continuous or (X,Gp,⪯) is regular. Then f and g have a common fixed point.In this paper, we prove a common fixed point result in completeGp-metric spaces for weakly increasing self-mappings satisfying (ψ,φ)-contractions via the concept of C-class functions. Some corollaries are also presented for particular cases of the C-function. For a given C-function, Theorem 17 is reached.


## 2. Main Results

First, we introduce an auxiliary lemma as follows.Lemma 18.
Let(X,Gp) be a Gp-metric space and let {xn} be a sequence in X such that {Gp(xn,xn,xn+1)} is decreasing and(6)limn→∞Gpxn,xn,xn+1=0.If {x2n} is not a Gp-Cauchy sequence, then there exist an ϵ>0 and {mk},{nk} of positive integers such that the following sequences {Gp(x2mk,x2nk,x2nk)}, {Gp(x2mk,x2nk+1,x2nk+1)}, {Gp(x2mk-1,x2nk,x2nk)}, {Gp(x2mk-1,x2nk+1,x2nk+1)}, and {Gp(x2mk,x2nk+2,x2nk+2)} tend to ϵ when k→∞.Proof.
Assume that{x2n} is not a Gp-Cauchy sequence. So there exist ϵ>0, and {mk} and {nk} of positive integers such that (7)Gpx2mk,x2nk-2,x2nk-2<ϵ,Gpx2mk,x2nk,x2nk>ϵ,nk>mk>k for all k∈N. Then (8)ϵ≤Gpx2mk,x2nk,x2nk≤Gpx2mk,x2nk-2,x2nk-2+Gpx2nk-2,x2nk-1,x2nk-1+Gpx2nk-1,x2nk,x2nk-Gpx2nk-2,x2nk-2,x2nk-2-Gpx2nk-1,x2nk-1,x2nk-1<ϵ+Gpx2nk-2,x2nk-1,x2nk-1+Gpx2nk-1,x2nk,x2nk. By taking the limit in above inequalities and using (6), we get(9)limk→∞Gpx2mk,x2nk,x2nk=ϵ.On the other hand (10)Gpx2mk,x2nk,x2nk≤Gpx2mk,x2nk+1,x2nk+1+Gpx2nk+1,x2nk,x2nk-Gpx2nk+1,x2nk+1,x2nk+1≤Gpx2mk,x2nk+1,x2nk+1+2Gpx2nk,x2nk+1,x2nk+1-2Gpx2nk+1,x2nk+1,x2nk+1≤Gpx2mk,x2nk+1,x2nk+1+2Gpx2nk,x2nk+1,x2nk+1≤Gpx2mk,x2nk,x2nk+Gpx2nk,x2nk+1,x2nk+1+2Gpx2nk,x2nk+1,x2nk+1. Letting k→∞, again using (6) and (9), we obtain(11)limk→∞Gpx2mk,x2nk+1,x2nk+1=ϵ. Similarly, we can prove that the remaining sequences tend to ϵ as k→∞.Now, we state and prove our main result in the following way.Theorem 19.
Let(X,⪯) be a partially ordered set. Let f and g be weakly increasing self-mappings on a complete Gp-metric space X. Assume there exist ψ,φ∈Ψ and F∈C such that(12)ψGpfx,gy,gy≤FψMx,y,y,φMx,y,yfor all comparable x,y∈X, where (13)Mx,y,y=max⁡Gpx,y,y,Gpx,fx,fx,Gpy,gy,gy,Gpx,gy,gy+Gpy,fx,fx2.Suppose that one of the following two cases is satisfied:(i)
f or g is continuous;(ii)
(X,Gp,⪯) is regular.
Then the mapsf and g have a common fixed point.Proof.
Assume thatu is a fixed point of f. Taking x=y=u in (12), we have(14)ψGpu,gu,gu=ψGpfu,gu,gu≤FψMu,u,u,φMu,u,u,where (15)Mu,u,u=max⁡Gpu,u,u,Gpu,fu,fu,Gpu,gu,gu,Gpu,gu,gu+Gpu,fu,fu2=max⁡Gpu,u,u,Gpu,u,u,Gpu,gu,gu,Gpu,gu,gu+Gpu,u,u2=max⁡Gpu,u,u,Gpu,gu,gu=Gpu,gu,gu.Hence we get (16)ψGpu,gu,gu=ψGpfu,gu,gu≤FψGpu,gu,gu,φGpu,gu,gu≤ψGpu,gu,gu.We deduce F(ψGpu,gu,gu),φGpu,gu,gu≤ψ(Gpu,gu,gu). By a property of the C-class F, we get ψ(Gpu,gu,gu)=0 or φ(Gpu,gu,gu)=0. The functions ψ and φ are in Ψ, so Gpu,gu,gu=0; that is, u=gu; that is, u is a common fixed point of f and g. Now, if u is a fixed point of g, similarly we get that u is also fixed point of f. Let x0 be an arbitrary point of X. The pair (f,g) is weakly increasing, so we construct a sequence {xn} in X as follows: (17)x1=fx0⪯gfx0=gx1=x2,x2=gx1⪯fgx1=fx2=x3,⋮x2n+1=fx2n⪯x2n+2=gx2n+1.We have xn⪯xn+1 for all n≥0.
Now, suppose thatGp(x2n,x2n+1,x2n+1)=0 for some n≥0. Then x2n=x2n+1=fx2n; that is, x2n is a fixed point of f. Proceeding similarly, we get that x2n is a fixed point of g.
From now on, we suppose thatGp(x2n,x2n+1,x2n+1)>0 for every n∈N. Since x2n and x2n+1 are comparable, by (12),(18)ψGpx2n+1,x2n+2,x2n+2=ψGpfx2n,gx2n+1,gx2n+1≤FψMx2n,x2n+1,x2n+1,φMx2n,x2n+1,x2n+1,where (19)Mx2n,x2n+1,x2n+1=max⁡Gpx2n,x2n+1,x2n+1,Gpx2n,fx2n,fx2n,Gpx2n+1,gx2n+1,gx2n+1,Gpx2n,gx2n+1,gx2n+1+Gpx2n+1,fx2n,fx2n2=max⁡Gpx2n,x2n+1,x2n+1,Gpx2n,x2n+1,x2n+1,Gpx2n+1,x2n+2,x2n+2,Gpx2n,x2n+2,x2n+2+Gpx2n+1,x2n+1,x2n+12≤max⁡Gpx2n,x2n+1,x2n+1,Gpx2n+1,x2n+2,x2n+2,Gpx2n,x2n+1,x2n+1+Gpx2n+1,x2n+2,x2n+2-Gpx2n+1,x2n+1,x2n+1+Gpx2n+1,x2n+1,x2n+12≤max⁡Gpx2n,x2n+1,x2n+1,Gpx2n+1,x2n+2,x2n+2,12Gpx2n,x2n+1,x2n+1+Gpx2n+1,x2n+2,x2n+2=max⁡Gpx2n,x2n+1,x2n+1,Gpx2n+1,x2n+2,x2n+2.If Gp(x2n+1,x2n+2,x2n+2)≥Gp(x2n,x2n+1,x2n+1) for some n≥0, then M(x2n,x2n+1,x2n+1)=Gp(x2n+1,x2n+2,x2n+2). Using (18), we have (20)ψGpx2n+1,x2n+2,x2n+2≤FψGpx2n+1,x2n+2,x2n+2,φGpx2n+1,x2n+2,x2n+2.By a property of F, this implies that ψGpx2n+1,x2n+2,x2n+2 or φ(Gp(x2n+1,x2n+2,x2n+2))=0, which is a contradiction. Therefore, for all n≥0, Gp(x2n+1,x2n+2,x2n+2)<Gpx2n,x2n+1,x2n+1. Similarly, we may show that Gp(x2n,x2n+1,x2n+1)<Gpx2n-1,x2n,x2n for all n≥0. We deduce that for all n≥0(21)Gpxn+1,xn+2,xn+2<Gpxn,xn+1,xn+1.So the sequence {Gp(xn+1,xn+2,xn+2)} is decreasing. Then there exists L≥0, such that limn→∞Gpxn+1,xn+2,xn+2=L. We claim that L=0. We have(22)limn→∞Mxn,xn+1,xn+1=L.Recall that(23)ψGpxn+1,xn+2,xn+2≤FψMxn,xn+1,xn+1,ϕMxn,xn+1,xn+1. As n→∞, by continuity of F, ψ, and φ, we get (24)ψL≤FψL,φL≤ψL. By a property of F, we get ψ(L)=0 or φ(L)=0; that is, L=0. We conclude that(25)limn→∞Gpxn+1,xn+2,xn+2=0.We shall show that {xn} is a Gp-Cauchy sequence. Suppose that {x2n} is not a Gp-Cauchy sequences. By (12), (26)ψGpx2mk+1,x2nk+2,x2nk+2=ψGpfx2mk,gx2nk+1,gx2nk+1≤FψMx2mk,x2nk+1,x2nk+1,φMx2mk,x2nk+1,x2nk+1, where(27)Mx2mk,x2nk+1,x2nk+1=max⁡Gpx2mk,x2nk+1,x2nk+1,Gpx2mk,fx2mk,fx2mk,Gpx2nk+1,gx2nk+1,gx2nk+1,12Gpx2mk,gx2nk+1,gx2nk+1+Gpx2nk+1,fx2mk,gx2mk=max⁡Gpx2mk,x2nk+1,x2nk+1,Gpx2mk,x2mk+1,x2mk+1,Gpx2nk+1,x2nk+2,x2nk+2,12Gpx2mk,x2nk+2,x2nk+2+Gpx2nk+1,x2mk+1,x2mk+1. By taking the limit as k→∞, from Lemma 18, we have (28)limk→∞Mx2mk,x2nk+1,x2nk+1=max⁡ϵ,0,0,ϵ=ϵ>0.Hence ψ(ϵ)≤F(ψ(ϵ),ϕ(ϵ)). We deduce that ψ(ϵ)=0 or ϕ(ϵ)=0, which is a contradiction. This proves that {x2n} is a Gp-Cauchy sequence and hence {xn} is a Gp-Cauchy sequence. By Gp-completeness of X, there exists z∈X such that {xn} converges to z as n→∞.
Now, we will distinguish the cases (i) and (ii) of Theorem19.
(i) Without loss of generality, suppose thatg is continuous. Since x2n+1→z, we obtain that x2n+2=g(x2n+1)→gz. But as x2n+2→z (as a subsequence of {xn}), it follows that gz=z. From the beginning of the proof, we get gz=z=fz.
The case thatf is continuous is treated similarly.
(ii) Suppose that(X,Gp,⪯) is regular. We know that sequence {xn} is nondecreasing and xn→z in X; then by regularity of (X,Gp,⪯),  x2n+1⪯z for all n∈N. By (12) (29)ψGpx2n+1,gz,gz=ψGpfx2n,gz,gz≤FψMx2n,z,z,φMx2n,z,z,where (30)Mx2n,z,z=max⁡Gpx2n,z,z,Gpx2n,fx2n,fx2n,Gpz,gz,gz,Gpx2n,gz,gz+Gpz,fx2n,fx2n2=max⁡Gpx2n,z,z,Gpx2n,x2n+1,x2n+1,Gpz,gz,gz,Gpx2n,gz,gz+Gpz,x2n+1,x2n+12. By taking the limit as n→∞, we have limn→∞Mx2n,z,z=Gp(z,gz,gz). Thus (31)ψGpz,gz,gz=limn→∞sup⁡ψGpfx2n,gz,gz≤limn→∞sup⁡FψMx2n,z,z,φMx2n,z,z≤FψGpz,gz,gz,φGpz,gz,gz≤ψGpz,gz,gz. Similarly, we may get Gp(z,gz,gz)=0 and so z=fz=gz.Now, we provide some corollaries from our obtained result given by Theorem19. First, putting ψ(t)=t in Theorem 19, we obtain the following.Corollary 20.
Let(X,⪯) be a partially ordered set. Let f and g be weakly increasing self-mappings on a complete Gp-metric space X satisfying(32)Gpfx,gy,gy≤FMx,y,y,φMx,y,yfor all comparable x,y∈X, where φ∈Ψ,  F∈C, and (33)Mx,y,y=max⁡Gpx,y,y,Gpx,fx,fx,Gpy,gy,gy,Gpx,gy,gy+Gpy,fx,fx2.Assume either f or g is continuous, or (X,Gp,⪯) is regular. Then f and g have a common fixed point.Proceeding as Theorem19, we have the following.Corollary 21.
Let(X,⪯) be a partially ordered set. Let f and g be weakly increasing self-mappings on a complete Gp-metric space X. Assume that that there exist ψ,φ∈Ψ and F∈C such that(34)ψGpfx,gy,gy≤FψMx,y,y,φMx,y,yfor all comparable x,y∈X, where (35)Mx,y,y=a1Gpx,y,y+a2Gpx,fx,fx+a3Gpy,gy,gy+a4Gpx,gy,gy+Gpy,fx,fx,where ai>0 for i={1,2,3,4} with a1+a2+a3+a4≤1. Assume either f or g is continuous or (X,Gp,⪯) is regular. Then f and g have a common fixed point.The above corollary is theGp-metric space version of Theorem 16 via the C-class function F, except that φ is taken in addition to the fact that it is continuous in [0,∞) (with respect to the conditions on φ in Theorem 16). If we set ψ(t)=t in Corollary 21, we get the following.Corollary 22.
Let(X,⪯) be a partially ordered set. Let f and g be weakly increasing self-mappings on a complete Gp-metric space X satisfying(36)Gpfx,gy,gy≤FMx,y,y,φMx,y,yfor all comparable x,y∈X, where φ∈Ψ,  F∈C, and (37)Mx,y,y=a1Gpx,y,y+a2Gpx,fx,fx+a3Gpy,gy,gy+a4Gpx,gy,gy+Gpy,fx,fx,where ai>0 for i=1,2,3,4 with a1+a2+a3+2a4≤1. Assume either f or g is continuous or (X,Gp,⪯) is regular. Then f and g have a common fixed point.TakingF(s,t)=ms with 0<m<1 (ψ(t)=φ(t)=t) in Theorem 19, we state the following.Corollary 23.
Let(X,⪯) be a partially ordered set. Let f and g be weakly increasing self-mappings on a complete Gp-metric space X satisfying(38)Gpfx,gy,gy≤m·max⁡Gpx,y,y,Gpx,fx,fx,Gpy,gy,gy,Gpx,gy,gy+Gpy,fx,fx2,for all comparable x,y∈X. Assume either f or g is continuous or (X,Gp,⪯) is regular. Then f and g have a common fixed point.Proceeding similarly as Theorem19, we have the following.Corollary 24.
Let(X,⪯) be a partially ordered set. Let f and g be weakly increasing self-mappings on a complete Gp-metric space X satisfying (39)ψGpfx,gy,gy≤FψGpx,y,y,φGpx,y,yfor all comparable x,y∈X, where ψ,φ∈Ψ and F∈C.
Assume eitherf or g is continuous or (X,Gp,⪯) is regular. Then f and g have a common fixed point.TakingF(s,t)=s/(1+t) in Theorem 19, we have the following.Corollary 25.
Let(X,⪯) be a partially ordered set. Let f and g be weakly increasing self-mappings on a complete Gp-metric space X satisfying(40)Gpfx,gy,gy≤kGpx,y,y1+Gpx,y,yfor all comparable x,y∈X, where 0<k<1. Assume either f or g is continuous or (X,Gp,⪯) is regular. Then f and g have a common fixed point.We provide the following example illustrating Theorem19.Example 26.
LetF(s,t)=s/(1+t) for all s,t≥0. Let X=[0,1] be a set endowed with the partial order x⪯y⇔y≤x. Let Gp(x,y,z)=max⁡x,y,z be a Gp-metric on X, given f,g:X→X as fx=x/12 and (41)gx=x6;x∈0,12,x22;x∈12,1.It is clear that f is continuous on (X,Gp) and the pair (f,g) is weakly increasing. Take ψ(t)=t2 and φ(t)=t for all t≥0. We shall prove that, for all x,y∈[0,1], we have(42)ψGpfx,gy,gy≤ψMx,y,y1+φMx,y,y.
First, forx∈0,1/2 we have(43)ψGpfx,gy,gy=ψmax⁡fx,gy,gy=ψmax⁡x12,y6,y6.
Now, we will discuss the following two cases.
Case  1. Suppose that x/12<y/6. We have(44)ψGpfx,gy,gy=ψy6=y236.Moreover (45)Mx,y,y=max⁡Gpx,y,y,Gpx,fx,fx,Gpy,gy,gy,Gpx,gy,gy+Gpy,fx,fx2=max⁡max⁡x,y,y,max⁡x,fx,fx,max⁡y,gy,gy,12max⁡x,gy,gy+max⁡y,fx,fx=max⁡x,x,y,12x+maxy,x12=x.Thus(46)ψMx,y,y1+φMx,y,y=ψx1+φx=x21+x.Therefore(47)ψGpfx,gy,gy=y236≤x236≤x21+x=ψMx,y,y1+φMx,y,y. Thus (42) holds.
Case  2. Suppose that x/12≥y/6; hence(48)ψGpfx,gy,gy=ψx12=x2144.Moreover(49)Mx,y,y=max⁡Gpx,y,y,Gpx,fx,fx,Gpy,gy,gy,Gpx,gy,gy+Gpy,fx,fx2=max⁡max⁡x,y,y,max⁡x,fx,fx,max⁡y,gy,gy,12max⁡x,gy,gy+max⁡y,fx,fx=max⁡x,x,y,12x+max⁡y,x12=x.Therefore(50)ψMx,y,y1+φMx,y,y=ψx1+φx=x21+x.Thus(51)ψGpfx,gy,gy=x2144≤x21+x=ψMx,y,y1+φMx,y,y;that is, (42) holds.
Second, forx∈[1/2,1](52)ψGpfx,gy,gy=ψmax⁡fx,gy,gy=ψmax⁡x12,y2,y2.Let us discuss the following two cases.
Case  1. Suppose that x/12<y2/2. Hence(53)ψGpfx,gy,gy=ψy22=y44.Moreover (54)Mx,y,y=max⁡Gpx,y,y,Gpx,fx,fx,Gpy,gy,gy,Gpx,gy,gy+Gpy,fx,fx2=max⁡max⁡x,y,y,max⁡x,fx,fx,max⁡y,gy,gy,12max⁡x,gy,gy+max⁡y,fx,fx=max⁡x,x,y,12x+max⁡y,x12=x.Therefore(55)ψMx,y,y1+φMx,y,y=ψx1+φx=x21+x.We deduce(56)ψGpfx,gy,gy=y44≤x24≤x21+x=ψMx,y,y1+φMx,y,y;that is, (42) holds.
Case  2. Suppose that x/12≥y2/2. Then(57)ψGpfx,gy,gy=ψx12=x2144.Moreover(58)Mx,y,y=max⁡Gpx,y,y,Gpx,fx,fx,Gpy,gy,gy,Gpx,gy,gy+Gpy,fx,fx2=max⁡max⁡x,y,y,max⁡x,fx,fx,max⁡y,gy,gy,12max⁡x,gy,gy+max⁡y,fx,fx=max⁡x,x,y,12x+max⁡y,x12=x.Therefore(59)ψMx,y,y1+φMx,y,y=ψx1+φx=x21+x.Thus (60)ψGpfx,gy,gy=x2144≤x21+x=ψMx,y,y1+φMx,y,y.Then (42) holds. All the conditions of Theorem 19 are satisfied and 0 is the common fixed point for f and g.
On the other hand, the contractive condition of Theorem17 is not satisfied. Indeed, for x=1 and y=1/3, we have (61)ψGpfx,gy,gy=x2144=1144>0=x2-x=ψMx,y,y-φMx,y,y.



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