# Coupled Fixed Point Results in Complete Partial Metric Spaces

Authors: H. Alaeidizaji; V. Parvaneh
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We establish some coupled fixed point theorems for a mapping satisfying some contraction conditions in complete partial metric spaces. Our consequences extend the results of H. Aydi (2011).


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## Body

## 1. Introduction and Mathematical Preliminaries

The notion of a partial metric space (PMS) was introduced in 1992 by Matthews [1, 2]. Matthews proved a fixed point theorem on this spaces, analogous to the Banach's fixed point theorem. Recently, many authors have focused on partial metric spaces and their topological properties (see, e.g., [3–9]).The definition of a partial metric space is given by Matthews (see [1, 2]) as follows:Definition 1.1.
LetX be a nonempty set and let p:X×X→R+ satisfies(P1)
x=y⇔p(x,x)=p(y,y)=p(x,y), for all x,y∈X,(P2)
p(x,x)≤p(x,y), for all x,y∈X,(P3)
p(x,y)=p(y,x), for all x,y∈X,(P4)
p(x,y)≤p(x,z)+p(z,y)-p(z,z), for all x,y,z∈X.
Then the pair(X,p) is called a partial metric space and p is called a partial metric on X.
The functiondp:X×X→R+ defined by
(1.1)dp(x,y)=2p(x,y)-p(x,x)-p(y,y)
satisfies the conditions of a metric on X; therefore it is a (usual) metric on X.Remark 1.2.
ifx=y, p(x,y) may not be 0.(1)
A famous example of partial metric spaces is the pair(R+,p), where p(x,y)=max{x,y} for all x,y∈R+. In this case, dp is the Euclidian metric dp(x,y)=|x-y|.(2)
Each partial metricp on X generates a T0 topology τp on X which has a base of open p-balls Bp(x,ε), where x∈X and ε>0 (Bp(x,ε)={y∈X:p(x,y)<p(x,x)+ε}).The following concepts has been defined as follows on a partial metric space.Definition 1.3 (see e.g., [1, 2]).
(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 Cauchy if and only if limn,m→∞p(xn,xm) exists (and is 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).
The concept of coupled fixed point have been introduced in [10] by Bhaskar and Lakshmikantham as follows.Definition 1.4 (see [10]).
An element(x,y)∈X×X is called a coupled fixed point of mapping F:X×X→X if x=F(x,y) and y=F(y,x).
Aydi in [11] has obtained some coupled fixed point results for mappings satisfying different contractive conditions on complete partial metric spaces. Some of these results are the following cases.Theorem 1.5 (see [11, Theorem 2.1]).
Let(X,p) be a complete partial metric space. Suppose that the mapping F:X×X→X satisfies the following contractive condition:
(1.2)p(F(x,y),F(u,v))≤kp(x,u)+lp(y,v),
for all x,y,u,v∈X, where k,l are nonnegative constants with k+l<1. Then, F has a unique coupled fixed point.Theorem 1.6 (see [11, Theorem 2.4]).
Let (X, p) be a complete partial metric space. Suppose that the mappingF:X×X→X satisfies the following contractive condition:
(1.3)p(F(x,y),F(u,v))≤kp(F(x,y),x)+lp(F(u,v),u),
for all x,y,u,v∈X, where k,l are nonnegative constants with k+l<1. Then, F has a unique coupled fixed point.Theorem 1.7 (see [11, Theorem 2.5]).
Let (X, p) be a complete partial metric space. Suppose that the mappingF:X×X→X satisfies the following contractive condition:
(1.4)p(F(x,y),F(u,v))≤kp(F(x,y),u)+lp(F(u,v),x),
for all x,y,u,v∈X, where k,l are nonnegative constants with k+l<1. Then, F has a unique coupled fixed point.For a survey of fixed point theory, its applications, and related results in partial metric spaces we refer the reader to [4, 5, 12–20] and the references mentioned therein. Also, many researchers have obtained coupled fixed point results for mappings under various contractive conditions in the framework of partial metric spaces (see, e.g., [21, 22]).In this paper we establish some coupled fixed point results of contractive mappings in the framework of complete partial metric spaces. Our results extend and generalize the results of Aydi [11].


## 2. Main Results

We recall three easy lemmas which have an essential role in the proof of the main result. These results can be derived easily (see, e.g., [1, 2, 6]).Lemma 2.1.
(
1
) A sequence {xn} is a Cauchy sequence in the PMS (X,p) if and only if it is a Cauchy sequence in the metric space (X,dp).
(
2
) A PMS (X,p) is complete if and only if the metric space (X,dp) is complete. Moreover,
(2.1)limn→∞dp(x,xn)=0⟺p(x,x)=limn→∞p(x,xn)=limn,m→∞p(xn,xm).Lemma 2.2 (see [3]).
Assume thatxn→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.Lemma 2.3 (see, e.g., [3, 4]).
Let (X, p) be a complete PMS. Then,(a)
ifp(x,y)=0 then, x=y,(b)
ifx≠y, then p(x,y)>0.Throughout this paper, we assume that all of the constants are nonnegative. Our main result is the following. The method of the proof can be found in [11].Theorem 2.4.
Let(X,p) be a complete partial metric space and F:X×X→X be a mapping such that
(2.2)p(F(x,y),F(u,v))≤α1p(x,u)+α2p(y,v)+α3p(F(x,y),x)+α4p(F(y,x),y)+α5p(F(x,y),u)+α6p(F(y,x),v)+α7p(F(u,v),x)+α8p(F(v,u),y)+α9p(F(u,v),u)+α10p(F(v,u),v),
for every pairs (x,y),(u,v)∈X×X, where ∑i=110‍αi<1. Then, F has a unique coupled fixed point in X.Proof.
Letx0,y0∈X be arbitrary. Define x1,y1∈X such that x1=F(x0,y0) and y1=F(y0,x0) and in this way, we construct the sequences {xn} and {yn} as xn=F(xn-1,yn-1) and yn=F(yn-1,xn-1), for all n≥0.
We will complete the proof in three steps.
Step I. Letδn=p(xn-1,xn)+p(yn-1,yn). We will show that limn→∞δn=0.
Using (2.2) we obtain that
(2.3)p(xn,xn+1)=p(F(xn-1,yn-1),F(xn,yn))≤α1p(xn-1,xn)+α2p(yn-1,yn)+α3p(F(xn-1,yn-1),xn-1)+α4p(F(yn-1,xn-1),yn-1)+α5p(F(xn-1,yn-1),xn)+α6p(F(yn-1,xn-1),yn)+α7p(F(xn,yn),xn-1)+α8p(F(yn,xn),yn-1)+α9p(F(xn,yn),xn)+α10p(F(yn,xn),yn)=α1p(xn-1,xn)+α2p(yn,yn-1)+α3p(xn,xn-1)+α4p(yn,yn-1)+α5p(xn,xn)+α6p(yn,yn)+α7p(xn+1,xn-1)+α8p(yn+1,yn-1)+α9p(xn+1,xn)+α10p(yn+1,yn)≤α1p(xn-1,xn)+α2p(yn-1,yn)+α3p(xn,xn-1)+α4p(yn,yn-1)+(α5-α7)p(xn,xn)+(α6-α8)p(yn,yn)+α7[p(xn+1,xn)+p(xn,xn-1)]+α8[p(yn+1,yn)+p(yn,yn-1)]+α9p(xn+1,xn)+α10p(yn+1,yn).
Analogously, starting fromp(xn+1,xn)=p(F(xn,yn),F(xn-1,yn-1)), we have
(2.4)p(xn+1,xn)=p(F(xn,yn),F(xn-1,yn-1))≤α1p(xn,xn-1)+α2p(yn,yn-1)+α3p(F(xn,yn),xn)+α4p(F(yn,xn),yn)+α5p(F(xn,yn),xn-1)+α6p(F(yn,xn),yn-1)+α7p(F(xn-1,yn-1),xn)+α8p(F(yn-1,xn-1),yn)+α9p(F(xn-1,yn-1),xn-1)+α10p(F(yn-1,xn-1),yn-1)=α1p(xn-1,xn)+α2p(yn,yn-1)+α3p(xn+1,xn)+α4p(yn+1,yn)+α5p(xn+1,xn-1)+α6p(yn+1,yn-1)+α7p(xn,xn)+α8p(yn,yn)+α9p(xn,xn-1)+α10p(yn,yn-1)≤α1p(xn-1,xn)+α2p(yn-1,yn)+α3p(xn+1,xn)+α4p(yn+1,yn)+α5[p(xn+1,xn)+p(xn,xn-1)]+α6[p(yn+1,yn)+p(yn,yn-1)]+(α7-α5)p(xn,xn)+(α8-α6)p(yn,yn)+α9p(xn,xn-1)+α10p(yn,yn-1).
In a similar way, we have(2.5)p(yn,yn+1)=p(F(yn-1,xn-1),F(yn,xn))≤α1p(yn-1,yn)+α2p(xn-1,xn)+α3p(yn,yn-1)+α4p(xn,xn-1)+(α5-α7)p(yn,yn)+(α6-α8)p(xn,xn)+α7[p(yn+1,yn)+p(yn,yn-1)]+α8[p(xn+1,xn)+p(xn,xn-1)]+α9p(yn+1,yn)+α10p(xn+1,xn).
Analogously, starting fromp(yn+1,yn)=p(F(yn,xn),F(yn-1,xn-1)), we have
(2.6)p(yn+1,yn)=p(F(yn,xn),F(yn-1,xn-1))≤α1p(yn-1,yn)+α2p(xn-1,xn)+α3p(yn+1,yn)+α4p(xn+1,xn)+α5[p(yn+1,yn)+p(yn,yn-1)]+α6[p(xn+1,xn)+p(xn,xn-1)]+(α7-α5)p(yn,yn)+(α8-α6)p(xn,xn)+α9p(yn,yn-1)+α10p(xn,xn-1).
Adding (2.3), (2.4), (2.5), and (2.6) we obtain that
(2.7)2δn+1≤[2α1+2α2+α3+α4+α5+α6+α7+α8+α9+α10]δn+[α3+α4+α5+α6+α7+α8+α9+α10]δn+1,
or, equivalently,
(2.8)δn+1≤λδn,
where, λ=[2α1+2α2+α3+α4+α5+α6+α7+α8+α9+α10]/(2-[α3+α4+α5+α6+α7+α8+α9+α10]).
Repeating the above mentioned process, we have(2.9)δn+1≤λδn≤λ2δn-1≤⋯≤λn+1δ0,
where, from our assumption about coefficients αi, λ∈[0,1); hence,
(2.10)limn→∞δn=0.
Step II.{xn} and {yn} are Cauchy.
Ifδ0=0 then, p(x0,x1)+p(y0,y1)=0. Hence, we get x0=x1=F(x0,y0) and y0=y1=F(y0,x0); that is, (x0,y0) is a coupled fixed point of F. Now, let δ0>0. For each m≥n, we have
(2.11)p(xm,xn)+p(ym,yn)≤p(xm,xm-1)+p(ym,ym-1)+p(xm-1,xm-2)+p(ym-1,ym-2)+⋯+p(xn+1,xn)+p(yn+1,yn)=δm+δm-1+⋯+δn+1≤[λm+λm-1+⋯+λn+1]δ0≤λn+11-λδ0.
So, we havelimn,m→∞p(xn,xm)+p(yn,ym)=0. This proves that {xn} and {yn} are Cauchy sequences in (X,p) and hence {xn} and {yn} are Cauchy sequences in the metric space (X,dp). From Lemma 2.1, (X,dp) is complete, so {xn} and {yn} converge to some x,y∈X, respectively; that is, limn→∞dp(xn,x)=0 and limn→∞dp(yn,y)=0. Therefore, from Lemma 2.1 and (2.10), we have
(2.12)p(x,x)=limn→∞p(xn,x)=limn,m→∞p(xn,xm)=0,(2.13)p(y,y)=limn→∞p(yn,y)=limn,m→∞p(yn,ym)=0.
Step III. We will show thatF has a unique coupled fixed point.
From the above step,(2.14)limn→∞p(F(xn,yn),x)=limn→∞p(F(yn,xn),y)=0.
Next, we will prove thatx=F(x,y) and y=F(y,x).
We have(2.15)p(x,F(x,y))≤p(x,xn+1)+p(xn+1,F(x,y))-p(xn+1,xn+1).
Taking the limit asn→∞ in the above inequality, as xn+1=F(xn,yn) and using triangle inequality and (2.12), we have
(2.16)p(x,F(x,y))≤limn→∞p(x,xn+1)+limn→∞p(F(xn,yn),F(x,y))=limn→∞p(F(xn,yn),F(x,y)).
But, for alln≥0, from (2.2),
(2.17)p(F(xn,yn),F(x,y))≤α1p(xn,x)+α2p(yn,y)+α3p(F(xn,yn),xn)+α4p(F(yn,xn),yn)+α5p(F(xn,yn),x)+α6p(F(yn,xn),y)+α7p(F(x,y),xn)+α8p(F(y,x),yn)+α9p(F(x,y),x)+α10p(F(y,x),y)=α1p(xn,x)+α2p(yn,y)+α3p(xn+1,x)+α4p(yn+1,y)+α5p(xn+1,x)+α6p(yn+1,y)+α7p(F(x,y),xn)+α8p(F(y,x),yn)+α9p(F(x,y),x)+α10p(F(y,x),y).
In the above inequality, ifn→∞, using (2.12) and Lemma 2.2 we have
(2.18)limn→∞p(F(xn,yn),F(x,y))≤(α7+α9)p(F(x,y),x)+(α8+α10)p(F(y,x),y).
Analogously,(2.19)p(y,F(y,x))≤p(y,yn+1)+p(yn+1,F(y,x))-p(yn+1,yn+1).
Taking the limit asn→∞ in the above inequality, since yn+1=F(yn,xn) and using triangle inequality and (2.13), we have
(2.20)p(y,F(y,x))≤limn→∞p(y,yn+1)+limn→∞p(F(yn,xn),F(y,x))=limn→∞p(F(yn,xn),F(y,x)).
Similar to (2.17), we have
(2.21)limn→∞p(F(yn,xn),F(y,x))≤(α7+α9)p(F(y,x),y)+(α8+α10)p(F(x,y),x).
Adding (2.18) and (2.21) and using (2.15) and (2.19), we obtain that
(2.22)p(x,F(x,y))+p(y,F(y,x))≤(α7+α8+α9+α10)[p(x,F(x,y))+p(y,F(y,x))].
Therefore,p(x,F(x,y))+p(y,F(y,x))=0; that is, F(x,y)=x and F(y,x)=y.Remark 2.5.
(1) If in the above theorem, we assume thatαi=0, for all 3≤i≤10, then we obtain the result of Aydi in [11] which is noted here in Theorem 1.5.
(2) If in the above theorem,αi=0, for all 1≤i≤10, unless i=3,9, then we obtain the result of Aydi in [11] which is mentioned here in Theorem 1.6.
(3) If in the above theorem, we assume thatαi=0, for all 3≤i≤10, except that i≠5,7, then we obtain the result of Aydi in [11] (Theorem 1.7).
Many results can be deduced from the above theorem as follows.Corollary 2.6.
Let(X,p) be a complete partial metric space and F:X×X→X be a mapping such that
(2.23)p(F(x,y),F(u,v))≤α1p(F(x,y),x)+α2p(F(y,x),y)+α3p(F(u,v),u)+α4p(F(v,u),v),
for every pairs (x,y),(u,v)∈X×X, where ∑i=14‍αi<1. Then, F has a unique coupled fixed point in X.Corollary 2.7.
Let(X,p) be a complete partial metric space and F:X×X→X be a mapping such that
(2.24)p(F(x,y),F(u,v))≤α1p(F(x,y),u)+α2p(F(y,x),v)+α3p(F(u,v),x)+α4p(F(v,u),y),
for every pairs (x,y),(u,v)∈X×X, where ∑i=14‍αi<1. Then, F has a unique coupled fixed point in X.Corollary 2.8.
Let(X,p) be a complete partial metric space and F:X×X→X be a mapping such that
(2.25)p(F(x,y),F(u,v))≤α1p(F(x,y),x)+α2p(F(y,x),y)+α3p(F(u,v),x)+α4p(F(v,u),y),
for every pairs (x,y),(u,v)∈X×X, where α1+α2+α3+α4<1. Then, F has a unique coupled fixed point in X.Corollary 2.9.
Let(X,p) be a complete partial metric space and F:X×X→X be a mapping such that
(2.26)p(F(x,y),F(u,v))≤α1p(F(x,y),u)+α2p(F(y,x),v)+α3p(F(u,v),u)+α4p(F(v,u),v),
for every pairs (x,y),(u,v)∈X×X, where ∑i=14‍αi<1. Then, F has a unique coupled fixed point in X.Also, we have the following results, when the constants in the above corollaries are equal.Corollary 2.10.
Let(X,p) be a complete partial metric space and F:X×X→X be a mapping such that
(2.27)p(F(x,y),F(u,v))≤k2[p(x,u)+p(y,v)]+l2[p(F(x,y),x)+p(F(y,x),y)]+r2[p(F(x,y),u)+p(F(y,x),v)]+s2[p(F(u,v),x)+p(F(v,u),y)]+t2[p(F(u,v),u)+p(F(v,u),v)],
for every pairs (x,y),(u,v)∈X×X, where k+l+r+s+t<1. Then, F has a unique coupled fixed point in X.Corollary 2.11.
Let(X,p) be a complete partial metric space and F:X×X→X be a mapping such that
(2.28)p(F(x,y),F(u,v))≤k2[p(F(x,y),x)+p(F(y,x),y)]+l2[p(F(u,v),u)+p(F(v,u),v)],
for every pairs (x,y),(u,v)∈X×X, where k+l<1. Then, F has a unique coupled fixed point in X.Corollary 2.12.
Let(X,p) be a complete partial metric space and F:X×X→X be a mapping such that
(2.29)p(F(x,y),F(u,v))≤k2[p(F(x,y),u)+p(F(y,x),v)]+l2[p(F(u,v),x)+p(F(v,u),y)],
for every pairs (x,y),(u,v)∈X×X, where k+l<1. Then, F has a unique coupled fixed point in X.Corollary 2.13.
Let(X,p) be a complete partial metric space and F:X×X→X be a mapping such that
(2.30)p(F(x,y),F(u,v))≤k2[p(F(x,y),x)+p(F(y,x),y)]+l2[p(F(u,v),x)+p(F(v,u),y)],
for every pairs (x,y),(u,v)∈X×X, where k+l<1. Then, F has a unique coupled fixed point in X.Corollary 2.14.
Let(X,p) be a complete partial metric space and F:X×X→X be a mapping such that
(2.31)p(F(x,y),F(u,v))≤k2[p(x,u)+p(y,v)]+l4[p(F(x,y),x)+p(F(y,x),y)+p(F(u,v),u)+p(F(v,u),v)]+r4[p(F(x,y),u)+p(F(y,x),v)+p(F(u,v),x)+p(F(v,u),y)],
for every pairs (x,y),(u,v)∈X×X, where k+l+r<1. Then, F has a unique coupled fixed point in X.Corollary 2.15.
Let(X,p) be a complete partial metric space and F:X×X→X be a mapping such that
(2.32)p(F(x,y),F(u,v))≤k2[p(F(x,y),u)+p(F(y,x),v)]+l2[p(F(u,v),u)+p(F(v,u),v)],
for every pairs (x,y),(u,v)∈X×X, where k+l<1. Then F has a unique coupled fixed point in X.Example 2.16.
LetX=[0,∞) and p on X be given as p(a,b)=max{a,b}. Obviously, the partial metric space (X,p) is complete (see, e.g., Example  2.3 of [11]).
DefineF:X×X→X as F(x,y)=(x+y)/30, for all x,y∈X.
Now, we have(2.33)p(F(x,y),F(u,v))=130max{x+y,u+v}≤127[max{x,u}+max{y,v}]≤127[max{x,u}+max{y,v}]+127[max{x+y30,x}+max{y+x30,y}]+127[max{x+y30,u}+max{y+x30,v}]+127[max{u+v30,x}+max{v+u30,y}]+127[max{u+v30,u}+max{v+u30,v}]=α1p(x,u)+α2p(y,v)+α3p(F(x,y),x)+α4p(F(y,x),y)+α5p(F(x,y),u)+α6p(F(y,x),v)+α7p(F(u,v),x)+α8p(F(v,u),y)+α9p(F(u,v),u)+α10p(F(v,u),v).
Thus, (2.2) is satisfied with αi=1/27. Obviously, all the conditions of Theorem 2.4 are satisfied. Moreover, (0,0) is the unique coupled fixed point of F.



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*Source: 670410-2012-10-30.xml*