Some Common Fixed Point Theorems in 0σComplete MetricLike Spaces
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Abstract
In this paper, we introduce the notion of 0σcomplete metriclike space and prove some common fixed point theorems in such spaces. Our results unify and generalize several wellknown results in the literature and the recent result of AminiHarandi [Fixed Point Theory Appl. 2012:204, 2012]. Some examples are included which show that the generalization is proper.
Keywords
Common fixed point Metriclike space Partial metric spaceMathematics Subject Classification (2000)
47H10 54H251 Introduction and Preliminaries
Matthews [20] introduced the notion of partial metric space as a part of the study of denotational semantics of dataflow network. In this space, the usual metric is replaced by partial metric with an interesting property that the selfdistance of any point of the space may not be zero. Further, Matthews showed that the Banach contraction principle is valid in partial metric spaces and can be applied in program verification. Later, several authors generalized the result of Metthews (see, for example, [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]). O’Neill [22] generalized the concept of partial metric space a bit further by admitting negative distances. The partial metric defined by O’Neill is called dualistic partial metric. Heckmann [17] generalized it by omitting small selfdistance axiom. The partial metric defined by Heckmann is called weak partial metric. Romaguera [24] introduced the notion of 0Cauchy sequence, 0complete partial metric spaces and proved some characterizations of partial metric spaces in terms of completeness and 0completeness.
Recently, AminiHarandi [7] generalized the partial metric spaces by introducing the metriclike spaces and proved some fixed point theorems in such spaces. AminiHarandi defined the σcompleteness of metriclike spaces. In this paper, we introduce the notion of 0σcompleteness which generalizes the notion of the σcompleteness of [7] as well as the notion of the 0completeness of [24]. Also, we prove common fixed point results in such spaces which generalize the results of AminiHarandi and several wellknown results of metric, partial metric spaces in metriclike spaces.
First we recall some definitions and facts about partial metric and metriclike spaces.
Definition 1
 (p1)

x=y if and only if p(x,x)=p(x,y)=p(y,y);
 (p2)

p(x,x)≤p(x,y);
 (p3)

p(x,y)=p(y,x);
 (p4)

p(x,y)≤p(x,z)+p(z,y)−p(z,z).
Definition 2
[7]
 (σ1)

σ(x,y)=0 implies x=y;
 (σ2)

σ(x,y)=σ(y,x);
 (σ3)

σ(x,y)≤σ(x,z)+σ(z,y).
Example 1
[7]
Example 2
Definition 3
Let (X,σ) be a metriclike space. A sequence {x _{ n }} in X is called a 0σCauchy sequence if lim_{ n,m→∞} σ(x _{ n },x _{ m })=0. The space (X,σ) is said to be 0σcomplete if every 0σCauchy sequence in X converges with respect to τ _{ σ } to a point x∈X such that σ(x,x)=0.
It is obvious that every 0σCauchy sequence is a σCauchy sequence in (X,σ) and every σcomplete metriclike space is 0σcomplete. Also, every 0complete partial metric space is a 0σcomplete metriclike space. The following example shows that the converse assertions of these facts do not hold.
Example 3
Remark 1
It is not hard to see that, if σ(x _{ n },x)→σ(x,x)=0, then σ(x _{ n },y)→σ(x,y) for all y∈X.
For the following definition and proposition we refer to [1].
Definition 4
Let f and g be self maps of a set X. If w=fx=gx for some x∈X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g. The pair f,g of self maps is weakly compatible if they commute at their coincidence points.
Proposition 1
Let f and g be weakly compatible self maps of a set X. If f and g have a unique point of coincidence w=fx=gx, then w is the unique common fixed point of f and g.
The following lemmas will be useful in the sequel.
Lemma 1
Proof
Lemma 2
Proof
Let z∈X be the point of coincidence of f and g and u be the corresponding coincidence point, that is, gu=fu=z. Suppose to the contrary that σ(z,z)>0.
Now we can state our main results.
2 Main Results
The following theorem is a generalization and improvement of Theorem 2.4 of AminiHarandi [7].
Theorem 1
Proof
We construct a sequence {y _{ n }} in X as follows: let x _{0} be an arbitrary point in X. Choose a point x _{1}∈X such that fx _{0}=gx _{1}=y _{1} (say). This can be done, since the range of g contains the range of f. Continuing this process, having chosen x _{ n }∈X, we obtain x _{ n+1}∈X such that fx _{ n }=gx _{ n+1}=y _{ n } (say). Thus, we obtain the sequence {y _{ n }}={gx _{ n+1}} such that fx _{ n }=gx _{ n+1}=y _{ n } for all \(n\in\Bbb{N}\). Consider the two possible cases.
Suppose that y _{ n }=y _{ n+1} for some \(n\in \Bbb{N}\). Then gx _{ n }=fx _{ n }=y _{ n } is a point of coincidence and the proof is finished.
Suppose that y _{ n }≠y _{ n+1} for all n≥0. We shall show that {y _{ n }} is a 0σCauchy sequence in X.
Suppose that f and g are weakly compatible, then by Remark 1, f and g have a unique common fixed point v which is also a unique point of coincidence of f and g and σ(v,v)=0.
In the case when f(X) is a closed set in (X,σ) the proof is similar. □
Taking g=I _{ X } (the identity mapping of X) in the above theorem we obtain the following improvement of Theorem 2.4 of AminiHarandi [7] (in the sense that σcompleteness of space is replaced by 0σcompleteness as well as the uniqueness of fixed point is established).
Corollary 1
Now we give some examples which illustrate our results.
Example 4
Example 5
In the next theorem we give an improvement to Theorem 2.7 of [7].
Let Ψ={ψ∣ψ:[0,∞)→[0,∞) is continuous, nondecreasing and ψ ^{−1}({0})={0}},
Φ={φ∣φ:[0,∞)→[0,∞) is lower semicontinuous and φ ^{−1}({0})={0}}.
Theorem 2
Proof
We construct a sequence {y _{ n }} in X as follows: let x _{0} be an arbitrary point in X. Choose a point x _{1}∈X such that fx _{0}=gx _{1}=y _{1} (say). This can be done, since the range of g contains the range of f. Continuing this process, having chosen x _{ n }∈X, we obtain x _{ n+1}∈X such that fx _{ n }=gx _{ n+1}=y _{ n } (say). Thus, we obtain the sequence {y _{ n }}={gx _{ n+1}} such that fx _{ n }=gx _{ n+1}=y _{ n } for all \(n\in\Bbb{N}\). Consider the two possible cases.
Suppose that y _{ n }=y _{ n+1} for some \(n\in \Bbb{N}\). Hence y _{ n }=gx _{ n }=fx _{ n } is a point of coincidence and then the proof is finished.
We prove now that σ ^{∗}=0. Indeed, passing to the limit in (11) when n→∞, we obtain ψ(σ ^{∗})≤ψ(σ ^{∗})−φ(σ ^{∗}) and σ ^{∗}=0, by the properties of functions ψ∈Ψ,φ∈Φ. Hence, lim_{ n→∞} σ(y _{ n+1},y _{ n })=0.
This shows that {y _{ n }}={gx _{ n+1}} is a 0σCauchy sequence in (X,σ).
In the case when f(X) is a closed set in (X,σ) the proof is similar. □
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