# Common Fixed Point Results for Four Mappings on Partial Metric Spaces

Authors: A. Duran Turkoglu; Vildan Ozturk
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We give fixed point results for four mappings which satisfy almost generalized contractive condition on partial metric space and we support the results with an example.


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## Body

## 1. Introduction and Preliminaries

Partial metric spaces, introduced by Matthews [1, 2], are a generalization of the notion of the metric space in which in definition of metric, the condition d(x,x)=0 is replaced by the condition d(x,x)≤d(x,y).In [1], Matthews discussed some properties of convergence of sequence and proved the fixed point theorems for contractive mapping on partial metric spaces: any mapping T of a complete partial metric space X into itself that satisfies, where 0≤k<1, the inequality d(Tx,Ty)≤kd(x,y) for all x,y∈X, has a unique fixed point. Recently, many authors (see [3–15]) have focused on this subject and generalized some fixed point theorems from the class of metric spaces.The definition of partial metric space is given by Matthews (see [2]) as follows.Definition 1.1.
LetX be a nonempty set and let p:X×X→ℝ0+ satisfy(PM1)x=y⇔p(x,x)=p(y,y)=p(x,y),(PM2)p(x,x)≤p(x,y),(PM3)p(x,y)=p(y,x),(PM4)p(x,y)≤p(x,z)+p(z,y)-p(z,z),for all x, y and z∈X, where ℝ0+=[0,∞). Then the pair (X,p) is called a partial metric space (in short PMS) and p is called a partial metric on X.Let(X,p) be a PMS. Then, the functions ps,pw:X×X→ℝ0+ given by
(1.1)ps(x,y)=2p(x,y)-p(x,x)-p(y,y)pw(x,y)=p(x,y)-min{p(x,x),p(y,y)}
are ordinary equivalent metrics on X. Each partial metric p on X generates a T0 topology τp on X with a base of the family of open p-balls {Bp(x,ε):x∈X,ε>0}, where Bp(x,ε)={y∈X:p(x,y)<p(x,x)+ε} for all x∈X and ε>0.Example 1.2 (see [1, 2]).
LetX={[a,b]:a,b∈ℝ,a≤b} and define
(1.2)p([a,b],[c,d])=max{b,d}-min{a,c}.
Then (X,p) is a partial metric space.We give same topological definitions on partial metric spaces.Definition 1.3 (see [1, 2, 4]).
(i)
A sequence{xn} in a PMS (X,p) converges to x∈X if and only if p(x,x)=limn→∞p(x,xn).
(ii)
A sequence{xn} in a PMS (X,p) is called a Cauchy sequence if and only if limn,m→∞p(xn,xm) exists (and finite).
(iii)
A PMS(X,p) is said to be complete if every Cauchy sequence {xn} in X converges, with respect to τp, to a point x∈X such that p(x,x)=limn,m→∞p(xn,xm).
(iv)
A mappingf:X→X is said to be continuous at x0∈X if for every ε>0, there exists δ>0 such that f(B(x0,δ))⊂B(f(x0),ε).Lemma 1.4 (see [1, 2, 4]).
(A)
A sequence{xn} is Cauchy in a PMS (X,p) if and only if {xn} is Cauchy in a metric space (X,ps).
(B)
A PMS(X,p) is complete if and only if the metric space (X,ps) is complete. Moreover,
(1.3)limn→∞ps(x,xn)=0⟺p(x,x)=limn→∞p(x,xn)=limn,m→∞p(xn,xm),where x is a limit of {xn} in (X,ps).Remark 1.5 (see [11]).
Let(X,p) be a PMS. Therefore,(A)
ifp(x,y)=0, then x=y;(B)
ifx≠y, then p(x,y)>0.Lemma 1.6 (see [10]).
Assumexn→z as n→∞ in a PMS (X,p) such that p(z,z)=0. Then limn→∞p(xn,y)=p(z,y) for every y∈X.On the other hand, Kannan [16] proved a fixed point theorem for a map satisfying a contractive condition that did not require continuity at each point. Afterward Sessa [17] introduced the notion of weakly commuting maps, which generalized the concept of commuting maps. Then Jungck generalized this idea, first to compatible mappings [18] and then to weakly compatible mappings [19].A pair(f,T) of self-mappings on X is said to be weakly compatible if they commute at their coincidence point (i.e., fTx=Tfx whenever fx=Tx). A point y∈X is called point of coincidence of a family Tj, j∈J, of self-mappings on X if there exists a point x∈X such that y=Tjx for all j∈J.The concept of almost contraction property was given to as follows by Berinde.Definition 1.7 (see [20, 21]).
Let(X,d) be a metric space. A map f:X→X is called an almost contraction if there exist a constant δ∈[0,1[ and some L≥0 such that for all x,y∈X(1.4)d(fx,fy)≤δd(x,y)+Ld(fx,y).
Berinde called this as “weak contraction” in [20], then he renamed it as “almost contraction” in [21, 22], also Berinde [21] proved some fixed point theorems for almost contraction in complete metric space. Definition 1.7 is a special case of the following definition (choose g=IX, IX is the identity map on X).Definition 1.8 (see [7]).
Let(X,d) be a metric space. A map f:X→X is called an almost contraction with respect to a mapping g:X→X if there exist a constant δ∈[0,1[ and some L≥0 such that for all x, y∈X(1.5)d(fx,fy)≤δd(gx,gy)+Ld(fx,gy).Babu et al. [23] considered the class of mappings that satisfy “condition (B).”Let(X,d) be a metric space. A map T:X→X is said to satisfy “condition (B)” if there exist a constant δ∈[0,1[ and some L≥0 such that for all x,y∈X,
(1.6)d(fx,fy)≤δd(x,y)+Lmin{p(x,fx),p(y,fy),p(x,fy),p(y,fx)}.Afterward, Berinde [21], Abbas and Ilić [24], and Ćirić et al. [7] generalized the above definition and proved some fixed point results.In recent paper, Altun and Acar [25] introduced the notion of (δ,L) weak contraction in the sense of Berinde in partial metric space.Definition 1.9 (see [25]).
Let(X,p) be a partial metric space. A map T:X→X is called (δ,L)-weak contraction if there exist a δ∈[0,1) and some L≥0 such that
(1.7)p(Tx,Ty)≤δp(x,y)+Lpw(y,Tx),
for all x,y∈X.In this paper, we give a fixed point theorem for four mappings satisfying almost generalized contractive condition in [26] on partial metric spaces.


## 2. Main Results

Theorem 2.1.
Let(X,p) be a complete partial metric space and f, g, S and T be self maps on X, with f(X)⊆T(X) and g(X)⊆S(X). If there exists δ∈[0,1[ and L≥0 with such that
(2.1)p(fx,gy)≤δM(x,y)+LN(x,y),
for any x, y∈X, where,
(2.2)M(x,y)=max{p(Sx,Ty),p(fx,Sx),p(gy,Ty),p(Sx,gy)+p(fx,Ty)2},N(x,y)=min{pw(fx,Sx),pw(gy,Ty),pw(Sx,gy),pw(fx,Ty)}.If{f,S} and {g,T} are weakly compatible and one of f(X),g(X),S(X), and T(X) is a complete subspace of X, then f, g, S, and T have a common fixed point.Proof.
Letx0 be an arbitrary point in X. Since f(X)⊆T(X), we can find x1∈X such that fx0=Tx1 and also, as gx1∈S(X), there exist x2∈X such that gx1=Sx2. In general, x2n+1∈X is chosen such that fx2n=Tx2n+1 and x2n+2∈X such that gx2n+1=Sx2n+2, we obtain a sequences {yn} in X such that
(2.3)y2n=fx2n=Tx2n+1,y2n+1=gx2n+1=Sx2n+2,∀n≥0.
Suppose y2m=y2m+1 for some m. Thus, g and T have a coincidence point. Due to (2.1), we have
(2.4)p(y2m+2,y2m+1)=p(fx2m+2,gx2m+1)≤δM(x2m+2,x2m+1)+LN(x2m+2,x2m+1),
where
(2.5)N(x2m+2,x2m+1)=min{pw(fx2m+2,Sx2m+2),pw(gx2m+1,Tx2m+1),pw(Sx2m+2,gx2m+1),pw(fx2m+2,Tx2m+1)}=min{pw(y2m+2,y2m+1),pw(y2m+1,y2m),pw(y2m+1,y2m+1),pw(y2m+2,y2m)}=0,M(x2m+2,x2m+1)=max{p(Sx2m+2,Tx2m+1),p(fx2m+2,Sx2m+2),p(gx2m+1,Tx2m+1),p(Sx2m+2,gx2m+1)+p(fx2m+2,Tx2m+1)2}=max{p(y2m+1,y2m),p(y2m+2,y2m+1),p(y2m+1,y2m),p(y2m+1,y2m+1)+p(y2m+2,y2m)2}=p(y2m+2,y2m+1).
So,
(2.6)p(y2m+2,y2m+1)≤δp(y2m+2,y2m+1).
Therefore, by δ∈[0,1[, we have p(y2m+2,y2m+1)=0, that is, y2m+1=y2m+2. So, f and S have a coincidence point.
Suppose now thatyn≠yn+1 for all n≥0. From (2.1), we obtain
(2.7)p(y2n,y2n+1)=p(fx2n,gx2n+1)≤δM(x2n,x2n+1)+LN(x2n,x2n+1),
where
(2.8)N(x2n,x2n+1)=min{pw(fx2n,Sx2n),pw(gx2n+1,Tx2n+1),pw(Sx2n,gx2n+1),pw(fx2n,Tx2n+1)}=min{pw(y2n,y2n-1),pw(y2n+1,y2n),pw(y2n-1,y2n+1),pw(y2n,y2n)}=0,M(x2n,x2n+1)=max{p(Sx2n,Tx2n+1),p(fx2n,Sx2n),p(gx2n+1,Tx2n+1),p(Sx2n,gx2n+1)+p(fx2n,Tx2n+1)2}=max{p(y2n-1,y2n),p(y2n,y2n-1),p(y2n+1,y2n),p(y2n-1,y2n+1)+p(y2n,y2n)2}.
Due to (2.7), we have
(2.9)p(y2n,y2n+1)≤δM(x2n,x2n+1).
Due to PM4, we have
(2.10)p(y2n-1,y2n+1)+p(y2n,y2n)≤p(y2n-1,y2n)+p(y2n,y2n+1).
Hence, M(x2n,x2n+1)=max{p(y2n,y2n-1),p(y2n+1,y2n)}. If M(x2n,x2n+1)=p(y2n+1,y2n), then by (2.7)
(2.11)p(y2n+1,y2n)≤δp(y2n+1,y2n).
Since δ∈[0,1[, the inequality (2.9) yields a contradiction. Hence, M(x2n,x2n+1)=p(y2n,y2n-1), then by (2.7) we have
(2.12)p(y2n+1,y2n)≤δp(y2n,y2n-1).
Thus, one can observe that
(2.13)p(yn+1,yn)≤δnp(y0,y1),∀n=0,1,2,….
Consider now
(2.14)ps(yn+2,yn+1)=2p(yn+2,yn+1)-p(yn+2,yn+2)-p(yn+1,yn+1)≤2p(yn+2,yn+1)≤δn+1p(y0,y1).
Hence, regarding (2.13), we have
(2.15)limn→∞ps(yn+2,yn+1)=0.
Moreover,
(2.16)ps(yn+1,yn+k)≤ps(yn+k-1,yn+k)+⋯+ps(yn+1,yn+2)≤2δn+k-1p(y0,y1)+⋯+2δn+1p(y0,y1).
After standard calculation, we obtain that {yn} is a Cauchy sequence in (X,ps), that is, ps(yn,ym)→0 as n, m→∞. Since (X,p) is complete, by Lemma 1.4, (X,ps) is complete and sequence {yn} is convergent in (X,ps) to say z∈X. From Lemma 1.4,
(2.17)p(z,z)=limn→∞p(yn,z)=limn,m→∞p(yn,ym).
Since {yn} is a Cauchy sequence in (X,ps), we have
(2.18)limn,m→∞ps(yn,ym)=0.
We assert that limn,m→∞p(yn,ym)=0. Without loss of generality, we assume that n>m,
(2.19)p(yn+2,yn)≤p(yn+2,yn+1)+p(yn+1,yn)-p(yn+1,yn+1)≤p(yn+2,yn+1)+p(yn+1,yn).
Similarly,
(2.20)p(yn+3,yn)≤p(yn+3,yn+2)+p(yn+2,yn)-p(yn+2,yn+2)≤p(yn+3,yn+2)+p(yn+2,yn).
Taking into account (2.20), the expression (2.19) yields
(2.21)p(yn+3,yn)≤p(yn+3,yn+2)+p(yn+2,yn+1)+p(yn+1,yn).
Inductively, we obtain
(2.22)p(ym,yn)≤p(ym,ym+1)+⋯+p(yn-2,yn-1)+p(yn-1,yn).
Due to (2.13),
(2.23)p(ym,yn)≤δmp(y0,y1)+⋯+δn-2p(y0,y1)+δn-1p(y0,y1)≤δm(1+δ+⋯+δn-m-1)p(y0,y1).
Regarding δ∈[0,1[, we can observe that limn,m→∞p(yn,ym)=0.
Sinceyn→z in X, {fx2n},  {Tx2n+1},  {gx2n+1},  {Sx2n+2} converge to z.
Now we show thatz is the fixed point for maps g and T. Assume that T(X) is complete, there exists u∈X such that z=Tu. We will show that gu=z. On the contrary, assume that gu≠z.
From, (2.1) we have
(2.24)p(fx2n,gu)≤δM(x2n,u)+LN(x2n,u),
where
(2.25)N(x2n,u)=min{pw(fx2n,Sx2n),pw(gu,Tu),pw(Sx2n,gu),pw(fx2n,Tu)}=min{pw(fx2n,Sx2n),pw(gu,z),pw(Sx2n,gu),pw(fx2n,z)},M(x2n,u)=max{p(Sx2n,Tu),p(fx2n,Sx2n),p(gu,Tu),p(Sx2n,gu)+p(fx2n,Tu)2}=max{p(Sx2n,z),p(fx2n,Sx2n),p(gu,z),p(Sx2n,gu)+p(fx2n,z)2}.
Since limn→∞M(x2n,u)=p(gu,z) and limn→∞N(x2n,u)=0. We get
(2.26)p(z,gu)≤δp(gu,z).
Since δ∈[0,1[, we get p(z,gu)=0. Therefore, gu=Tu=z. Since the maps g and T are weakly compatible, we have gz=gTu=Tgu=Tz. We will also show that gz=z. From (2.1), we have
(2.27)p(fx2n,gz)≤δM(x2n,z)+LN(x2n,z),
where
(2.28)N(x2n,z)=min{pw(fx2n,Sx2n),pw(gz,Tz),pw(Sx2n,gz),pw(fx2n,Tz)},M(x2n,z)=max{p(Sx2n,Tz),p(fx2n,Sx2n),p(gz,Tz),p(Sx2n,gz)+p(fx2n,Tz)2}=max{p(Sx2n,gz),p(fx2n,Sx2n),p(gz,gz),p(Sx2n,gz)+p(fx2n,Tz)2}.
Since limn→∞M(x2n,z)=p(z,gz) and limn→∞N(x2n,z)=0, then
(2.29)p(z,gz)=limn→∞p(fx2n,gz)≤δp(z,gz).
Since δ∈[0,1[, p(z,gz)=0. By Remark 1.5, we get z=gz.
Similarly, we show thatz is also fixed point of f and S. Hence, fz=gz=Tz=Sz=z.
The proofs for the cases in whichS(X), f(X), or g(X) is complete are similar.
Last, we showz is unique. Suppose on the contrary that there is another common fixed point t of f, g, S, and T. Then
(2.30)p(z,t)=p(fz,gt)≤δM(z,t)+LN(z,t),
where
(2.31)N(z,t)=min{pw(fz,Sz),pw(gt,Tt),pw(Sz,gt),pw(fz,Tt)}=0,M(z,t)=max{p(Sz,Tt),p(fz,Sz),p(gt,Tt),p(Sz,gt)+p(fz,Tt)2}=p(Sz,Tt)=p(z,t).
Thus,
(2.32)p(z,t)≤δp(z,t).
Therefore, p(z,t)=0 and Remark 1.5 z=t. So, z is the unique common fixed point os f, g, S, and T.Example 2.2.
LetX={0,1,2} endowed with the partial metric p given by p(x,y)=max{x,y} for all x,y∈X. It is clear that (X,p) is a complete partial metric space. Define the mappings f, g, S, T:X→X by
(2.33)f=g,S=T,f0=f2=0,f1=1T0=0,T1=2,T2=1.
We have f(X)⊆T(X)=X. For δ=1/2, L=1,
(2.34)p(f0,f1)=1≤δ.2+L.1,p(f2,f1)=1≤δ.2+L.0,p(f2,f2)=p(f0,f0)=0≤δ.0+L.1,p(f1,f1)=1≤δ.2+L.0.
Then, the contractive condition (2.1) is satisfied for every x, y∈X. Moreover, {f,T} is weakly compatible. So all conditions of Theorem 2.1 are satisfied. We deduce the existence and uniqueness of a common fixed point of f and T. Here, 0 is the unique common fixed point.Corollary 2.3.
Let(X,p) is complete PMS and f and T be self maps on X, with f(X)⊆T(X). If there exists δ∈[0,1[ and L≥0 such that
(2.35)p(fx,fy)≤δM(x,y)+LN(x,y),
where,
(2.36)M(x,y)=max{p(Tx,Ty),p(fx,Tx),p(fy,Ty),p(Tx,fy)+p(fx,Ty)2},N(x,y)=min{pw(fx,Tx),pw(fy,Ty),pw(Tx,fy),pw(fx,Ty)},
for every x, y∈X. If {f,T} is weakly compatible and one of f(X) and T(X) is a complete subspace of X, then f and T have a common fixed point.Remark 2.4.
It is easy to see that for every mapT:X→X, {T,IX} is weakly compatible, where IX is identity map on X, so by taking f=g=IX in Theorem 2.1 we have the following results.Corollary 2.5.
Let(X,p) is complete PMS and S and T be self maps on X. If there exists δ∈[0,1[ and L≥0 such that
(2.37)p(x,y)≤δM(x,y)+LN(x,y),
for every x, y∈X, where
(2.38)M(x,y)=max{p(Sx,Ty),p(x,Sx),p(y,Ty),p(Sx,y)+p(x,Ty)2},N(x,y)=min{pw(x,Sx),pw(y,Ty),pw(Sx,y),pw(x,Ty)}.
Then S and T have a common fixed point.



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*Source: 190862-2012-10-04.xml*