Common Fixed Point Results for Four Mappings on Partial Metric Spaces

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.

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. Let 𝑋 be a nonempty set and let 𝑝∶𝑋×𝑋→ℝ⁺₀ 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 ℝ⁺₀=[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 𝑝^𝑠,𝑝^𝑤∶𝑋×𝑋→ℝ⁺₀ given by 𝑝^𝑠()𝑝𝑥,𝑦=2𝑝𝑥,𝑦−𝑝𝑥,𝑥−𝑝𝑦,𝑦_𝑤𝑥,𝑦=𝑝𝑥,𝑦−min𝑝(𝑥,𝑥),𝑝𝑦,𝑦 (1.1) are ordinary equivalent metrics on 𝑋. Each partial metric 𝑝 on 𝑋 generates a 𝑇₀ topology 𝜏_𝑝 on 𝑋 with a base of the family of open 𝑝-balls {𝐵_𝑝(𝑥,𝜀)∶𝑥∈𝑋,𝜀>0}, where 𝐵_𝑝(𝑥,𝜀)={𝑦∈𝑋∶𝑝(𝑥,𝑦)<𝑝(𝑥,𝑥)+𝜀} for all 𝑥∈𝑋 and 𝜀>0.

Example 1.2 (see [1, 2]). Let 𝑋={[𝑎,𝑏]∶𝑎,𝑏∈ℝ,𝑎≤𝑏} and define 𝑝([𝑎,𝑏],[𝑐,𝑑])=max{𝑏,𝑑}−min{𝑎,𝑐}. (1.2) 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 {𝑥_𝑛} in a PMS (𝑋,𝑝) converges to 𝑥∈𝑋 if and only if 𝑝(𝑥,𝑥)=lim_𝑛→∞𝑝(𝑥,𝑥_𝑛).(ii) A sequence {𝑥_𝑛} in a PMS (𝑋,𝑝) is called a Cauchy sequence if and only if lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚) exists (and finite).(iii) A PMS (𝑋,𝑝) is said to be complete if every Cauchy sequence {𝑥_𝑛} in 𝑋 converges, with respect to 𝜏_𝑝, to a point 𝑥∈𝑋 such that 𝑝(𝑥,𝑥)=lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚).(iv) A mapping 𝑓∶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 {𝑥_𝑛} is Cauchy in a PMS (𝑋,𝑝) if and only if {𝑥_𝑛} is Cauchy in a metric space (𝑋,𝑝^𝑠). (B) A PMS (𝑋,𝑝) is complete if and only if the metric space (𝑋,𝑝^𝑠) is complete. Moreover, lim_𝑛→∞𝑝^𝑠𝑥,𝑥_𝑛=0⟺𝑝(𝑥,𝑥)=lim_𝑛→∞𝑝𝑥,𝑥_𝑛=lim_{𝑛,𝑚→∞}𝑝𝑥_𝑛,𝑥_𝑚, (1.3)where 𝑥 is a limit of {𝑥_𝑛} in (𝑋,𝑝^𝑠).

Remark 1.5 (see [11]). Let (𝑋,𝑝) be a PMS. Therefore,(A) if 𝑝(𝑥,𝑦)=0, then 𝑥=𝑦;(B) if 𝑥≠𝑦, then 𝑝(𝑥,𝑦)>0.

Lemma 1.6 (see [10]). Assume 𝑥_𝑛→𝑧 as 𝑛→∞ in a PMS (𝑋,𝑝) such that 𝑝(𝑧,𝑧)=0. Then lim_𝑛→∞𝑝(𝑥_𝑛,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 𝑇_𝑗, 𝑗∈𝐽, of self-mappings on 𝑋 if there exists a point 𝑥∈𝑋 such that 𝑦=𝑇_𝑗𝑥 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∈𝑋𝑑𝑓𝑥,𝑓𝑦≤𝛿𝑑𝑥,𝑦+𝐿𝑑𝑓𝑥,𝑦. (1.4) 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=𝐼_𝑋, 𝐼_𝑋 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∈𝑋𝑑𝑓𝑥,𝑓𝑦≤𝛿𝑑𝑔𝑥,𝑔𝑦+𝐿𝑑𝑓𝑥,𝑔𝑦. (1.5)

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, 𝑑𝑓𝑥,𝑓𝑦≤𝛿𝑑𝑥,𝑦+𝐿min𝑝𝑥,𝑓𝑥,𝑝𝑦,𝑓𝑦,𝑝𝑥,𝑓𝑦,𝑝𝑦,𝑓𝑥. (1.6)

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 𝑝𝑇𝑥,𝑇𝑦≤𝛿𝑝𝑥,𝑦+𝐿𝑝^𝑤𝑦,𝑇𝑥, (1.7) 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 𝑝𝑓𝑥,𝑔𝑦≤𝛿𝑀𝑥,𝑦+𝐿𝑁𝑥,𝑦, (2.1) for any 𝑥, 𝑦∈𝑋, where, 𝑀𝑝,𝑝𝑥,𝑦=max𝑆𝑥,𝑇𝑦,𝑝𝑓𝑥,𝑆𝑥,𝑝𝑔𝑦,𝑇𝑦𝑆𝑥,𝑔𝑦+𝑝𝑓𝑥,𝑇𝑦2,𝑁𝑥,𝑦=min𝑝^𝑤𝑓𝑥,𝑆𝑥,𝑝^𝑤𝑔𝑦,𝑇𝑦,𝑝^𝑤𝑆𝑥,𝑔𝑦,𝑝^𝑤𝑓𝑥,𝑇𝑦. (2.2)

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. Let 𝑥₀ 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, 𝑥_2𝑛+1∈𝑋 is chosen such that 𝑓𝑥_2𝑛=𝑇𝑥_2𝑛+1 and 𝑥_2𝑛+2∈𝑋 such that 𝑔𝑥_2𝑛+1=𝑆𝑥_2𝑛+2, we obtain a sequences {𝑦_𝑛} in 𝑋 such that 𝑦_2𝑛=𝑓𝑥_2𝑛=𝑇𝑥_2𝑛+1,𝑦_2𝑛+1=𝑔𝑥_2𝑛+1=𝑆𝑥_2𝑛+2,∀𝑛≥0. (2.3) Suppose 𝑦_2𝑚=𝑦_2𝑚+1 for some 𝑚. Thus, 𝑔 and 𝑇 have a coincidence point. Due to (2.1), we have 𝑝𝑦_2𝑚+2,𝑦_2𝑚+1=𝑝𝑓𝑥_2𝑚+2,𝑔𝑥_2𝑚+1≤𝛿𝑀𝑥_2𝑚+2,𝑥_2𝑚+1+𝐿𝑁𝑥_2𝑚+2,𝑥_2𝑚+1, (2.4) where 𝑁𝑥_2𝑚+2,𝑥_2𝑚+1𝑝=min^𝑤𝑓𝑥_2𝑚+2,𝑆𝑥_2𝑚+2,𝑝^𝑤𝑔𝑥_2𝑚+1,𝑇𝑥_2𝑚+1,𝑝^𝑤𝑆𝑥_2𝑚+2,𝑔𝑥_2𝑚+1,𝑝^𝑤𝑓𝑥_2𝑚+2,𝑇𝑥_2𝑚+1𝑝=min^𝑤𝑦_2𝑚+2,𝑦_2𝑚+1,𝑝^𝑤𝑦_2𝑚+1,𝑦_2𝑚,𝑝^𝑤𝑦_2𝑚+1,𝑦_2𝑚+1,𝑝^𝑤𝑦_2𝑚+2,𝑦_2𝑚𝑀𝑥=0,_2𝑚+2,𝑥_2𝑚+1𝑝=max𝑆𝑥_2𝑚+2,𝑇𝑥_2𝑚+1,𝑝𝑓𝑥_2𝑚+2,𝑆𝑥_2𝑚+2,𝑝𝑔𝑥_2𝑚+1,𝑇𝑥_2𝑚+1,𝑝𝑆𝑥_2𝑚+2,𝑔𝑥_2𝑚+1+𝑝𝑓𝑥_2𝑚+2,𝑇𝑥_2𝑚+12𝑝𝑦=max^2𝑚+1,𝑦_2𝑚,𝑝𝑦_2𝑚+2,𝑦_2𝑚+1,𝑝𝑦_2𝑚+1,𝑦_2𝑚,𝑝𝑦_2𝑚+1,𝑦_2𝑚+1+𝑝𝑦_2𝑚+2,𝑦_2𝑚2=𝑝𝑦_2𝑚+2,𝑦_2𝑚+1. (2.5) So, 𝑝𝑦_2𝑚+2,𝑦_2𝑚+1≤𝛿𝑝𝑦_2𝑚+2,𝑦_2𝑚+1. (2.6) Therefore, by 𝛿∈[0,1[, we have 𝑝(𝑦_2𝑚+2,𝑦_2𝑚+1)=0, that is, 𝑦_2𝑚+1=𝑦_2𝑚+2. So, 𝑓 and 𝑆 have a coincidence point.
Suppose now that 𝑦_𝑛≠𝑦_𝑛+1 for all 𝑛≥0. From (2.1), we obtain 𝑝𝑦_2𝑛,𝑦_2𝑛+1=𝑝𝑓𝑥_2𝑛,𝑔𝑥_2𝑛+1≤𝛿𝑀𝑥_2𝑛,𝑥_2𝑛+1+𝐿𝑁𝑥_2𝑛,𝑥_2𝑛+1, (2.7) where 𝑁𝑥_2𝑛,𝑥_2𝑛+1𝑝=min^𝑤𝑓𝑥_2𝑛,𝑆𝑥_2𝑛,𝑝^𝑤𝑔𝑥_2𝑛+1,𝑇𝑥_2𝑛+1,𝑝^𝑤𝑆𝑥_2𝑛,𝑔𝑥_2𝑛+1,𝑝^𝑤𝑓𝑥_2𝑛,𝑇𝑥_2𝑛+1𝑝=min^𝑤𝑦_2𝑛,𝑦_2𝑛−1,𝑝^𝑤𝑦_2𝑛+1,𝑦_2𝑛,𝑝^𝑤𝑦_2𝑛−1,𝑦_2𝑛+1,𝑝^𝑤𝑦_2𝑛,𝑦_2𝑛𝑀𝑥=0,_2𝑛,𝑥_2𝑛+1𝑝=max𝑆𝑥_2𝑛,𝑇𝑥_2𝑛+1,𝑝𝑓𝑥_2𝑛,𝑆𝑥_2𝑛,𝑝𝑔𝑥_2𝑛+1,𝑇𝑥_2𝑛+1,𝑝𝑆𝑥_2𝑛,𝑔𝑥_2𝑛+1+𝑝𝑓𝑥_2𝑛,𝑇𝑥_2𝑛+12𝑝𝑦=max^2𝑛−1,𝑦_2𝑛,𝑝𝑦_2𝑛,𝑦_2𝑛−1,𝑝𝑦_2𝑛+1,𝑦_2𝑛,𝑝𝑦_2𝑛−1,𝑦_2𝑛+1+𝑝𝑦_2𝑛,𝑦_2𝑛2. (2.8) Due to (2.7), we have 𝑝𝑦_2𝑛,𝑦_2𝑛+1≤𝛿𝑀𝑥_2𝑛,𝑥_2𝑛+1. (2.9) Due to PM4, we have 𝑝𝑦_2𝑛−1,𝑦_2𝑛+1+𝑝𝑦_2𝑛,𝑦_2𝑛≤𝑝𝑦_2𝑛−1,𝑦_2𝑛+𝑝𝑦_2𝑛,𝑦_2𝑛+1. (2.10) Hence, 𝑀(𝑥_2𝑛,𝑥_2𝑛+1)=max{𝑝(𝑦_2𝑛,𝑦_2𝑛−1),𝑝(𝑦_2𝑛+1,𝑦_2𝑛)}. If 𝑀(𝑥_2𝑛,𝑥_2𝑛+1)=𝑝(𝑦_2𝑛+1,𝑦_2𝑛), then by (2.7) 𝑝𝑦_2𝑛+1,𝑦_2𝑛≤𝛿𝑝𝑦_2𝑛+1,𝑦_2𝑛. (2.11) Since 𝛿∈[0,1[, the inequality (2.9) yields a contradiction. Hence, 𝑀(𝑥_2𝑛,𝑥_2𝑛+1)=𝑝(𝑦_2𝑛,𝑦_2𝑛−1), then by (2.7) we have 𝑝𝑦_2𝑛+1,𝑦_2𝑛≤𝛿𝑝𝑦_2𝑛,𝑦_2𝑛−1. (2.12) Thus, one can observe that 𝑝𝑦_𝑛+1,𝑦_𝑛≤𝛿^𝑛𝑝𝑦₀,𝑦₁,∀𝑛=0,1,2,…. (2.13) Consider now 𝑝^𝑠𝑦_𝑛+2,𝑦_𝑛+1=2𝑝𝑦_𝑛+2,𝑦_𝑛+1−𝑝𝑦_𝑛+2,𝑦_𝑛+2−𝑝𝑦_𝑛+1,𝑦_𝑛+1≤2𝑝𝑦_𝑛+2,𝑦_𝑛+1≤𝛿^𝑛+1𝑝𝑦₀,𝑦₁. (2.14) Hence, regarding (2.13), we have lim_𝑛→∞𝑝^𝑠𝑦_𝑛+2,𝑦_𝑛+1=0. (2.15) Moreover, 𝑝^𝑠𝑦_𝑛+1,𝑦_𝑛+𝑘≤𝑝^𝑠𝑦_{𝑛+𝑘−1},𝑦_𝑛+𝑘+⋯+𝑝^𝑠𝑦_𝑛+1,𝑦_𝑛+2≤2𝛿^{𝑛+𝑘−1}𝑝𝑦₀,𝑦₁+⋯+2𝛿^𝑛+1𝑝𝑦₀,𝑦₁. (2.16) After standard calculation, we obtain that {𝑦_𝑛} is a Cauchy sequence in (𝑋,𝑝^𝑠), that is, 𝑝^𝑠(𝑦_𝑛,𝑦_𝑚)→0 as 𝑛, 𝑚→∞. Since (𝑋,𝑝) is complete, by Lemma 1.4, (𝑋,𝑝^𝑠) is complete and sequence {𝑦_𝑛} is convergent in (𝑋,𝑝^𝑠) to say 𝑧∈𝑋. From Lemma 1.4, 𝑝(𝑧,𝑧)=lim_𝑛→∞𝑝𝑦_𝑛,𝑧=lim_{𝑛,𝑚→∞}𝑝𝑦_𝑛,𝑦_𝑚. (2.17) Since {𝑦_𝑛} is a Cauchy sequence in (𝑋,𝑝^𝑠), we have lim_{𝑛,𝑚→∞}𝑝^𝑠𝑦_𝑛,𝑦_𝑚=0. (2.18) We assert that lim_{𝑛,𝑚→∞}𝑝(𝑦_𝑛,𝑦_𝑚)=0. Without loss of generality, we assume that 𝑛>𝑚, 𝑝𝑦_𝑛+2,𝑦_𝑛≤𝑝𝑦_𝑛+2,𝑦_𝑛+1+𝑝𝑦_𝑛+1,𝑦_𝑛−𝑝𝑦_𝑛+1,𝑦_𝑛+1≤𝑝𝑦_𝑛+2,𝑦_𝑛+1+𝑝𝑦_𝑛+1,𝑦_𝑛. (2.19) Similarly, 𝑝𝑦_𝑛+3,𝑦_𝑛≤𝑝𝑦_𝑛+3,𝑦_𝑛+2+𝑝𝑦_𝑛+2,𝑦_𝑛−𝑝𝑦_𝑛+2,𝑦_𝑛+2≤𝑝𝑦_𝑛+3,𝑦_𝑛+2+𝑝𝑦_𝑛+2,𝑦_𝑛. (2.20) Taking into account (2.20), the expression (2.19) yields 𝑝𝑦_𝑛+3,𝑦_𝑛≤𝑝𝑦_𝑛+3,𝑦_𝑛+2+𝑝𝑦_𝑛+2,𝑦_𝑛+1+𝑝𝑦_𝑛+1,𝑦_𝑛. (2.21) Inductively, we obtain 𝑝𝑦_𝑚,𝑦_𝑛≤𝑝𝑦_𝑚,𝑦_𝑚+1+⋯+𝑝𝑦_𝑛−2,𝑦_𝑛−1+𝑝𝑦_𝑛−1,𝑦_𝑛. (2.22) Due to (2.13), 𝑝𝑦_𝑚,𝑦_𝑛≤𝛿^𝑚𝑝𝑦₀,𝑦₁+⋯+𝛿^𝑛−2𝑝𝑦₀,𝑦₁+𝛿^𝑛−1𝑝𝑦₀,𝑦₁≤𝛿^𝑚1+𝛿+⋯+𝛿^{𝑛−𝑚−1}𝑝𝑦₀,𝑦₁. (2.23) Regarding 𝛿∈[0,1[, we can observe that lim_{𝑛,𝑚→∞}𝑝(𝑦_𝑛,𝑦_𝑚)=0.
Since 𝑦_𝑛→𝑧 in 𝑋, {𝑓𝑥_2𝑛},  {𝑇𝑥_2𝑛+1},  {𝑔𝑥_2𝑛+1},  {𝑆𝑥_2𝑛+2} converge to 𝑧.
Now we show that 𝑧 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 𝑝𝑓𝑥_2𝑛,𝑔𝑢≤𝛿𝑀𝑥_2𝑛,𝑢+𝐿𝑁𝑥_2𝑛,𝑢, (2.24) where 𝑁𝑥_2𝑛,𝑢=min𝑝^𝑤𝑓𝑥_2𝑛,𝑆𝑥_2𝑛,𝑝^𝑤𝑔𝑢,𝑇𝑢,𝑝^𝑤𝑆𝑥_2𝑛,𝑔𝑢,𝑝^𝑤𝑓𝑥_2𝑛𝑝,𝑇𝑢=min^𝑤𝑓𝑥_2𝑛,𝑆𝑥_2𝑛,𝑝^𝑤𝑔𝑢,𝑧,𝑝^𝑤𝑆𝑥_2𝑛,𝑔𝑢,𝑝^𝑤𝑓𝑥_2𝑛,𝑀𝑥,𝑧_2𝑛𝑝,𝑢=max𝑆𝑥_2𝑛,𝑇𝑢,𝑝𝑓𝑥_2𝑛,𝑆𝑥_2𝑛,𝑝,𝑝𝑔𝑢,𝑇𝑢𝑆𝑥_2𝑛,𝑔𝑢+𝑝𝑓𝑥_2𝑛,𝑇𝑢2𝑝=max𝑆𝑥_2𝑛,𝑧,𝑝𝑓𝑥_2𝑛,𝑆𝑥_2𝑛,𝑝,𝑝𝑔𝑢,𝑧𝑆𝑥_2𝑛,𝑔𝑢+𝑝𝑓𝑥_2𝑛,𝑧2. (2.25) Since lim_𝑛→∞𝑀(𝑥_2𝑛,𝑢)=𝑝(𝑔𝑢,𝑧) and lim_𝑛→∞𝑁(𝑥_2𝑛,𝑢)=0. We get 𝑝𝑧,𝑔𝑢≤𝛿𝑝𝑔𝑢,𝑧. (2.26) 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 𝑝𝑓𝑥_2𝑛,𝑔𝑧≤𝛿𝑀𝑥_2𝑛,𝑧+𝐿𝑁𝑥_2𝑛,𝑧, (2.27) where 𝑁𝑥_2𝑛,𝑧=min𝑝^𝑤𝑓𝑥_2𝑛,𝑆𝑥_2𝑛,𝑝^𝑤𝑔𝑧,𝑇𝑧,𝑝^𝑤𝑆𝑥_2𝑛,𝑔𝑧,𝑝^𝑤𝑓𝑥_2𝑛,𝑀𝑥,𝑇𝑧_2𝑛𝑝,𝑧=max𝑆𝑥_2𝑛,𝑇𝑧,𝑝𝑓𝑥_2𝑛,𝑆𝑥_2𝑛,𝑝,𝑝𝑔𝑧,𝑇𝑧𝑆𝑥_2𝑛,𝑔𝑧+𝑝𝑓𝑥_2𝑛,𝑇𝑧2𝑝=max𝑆𝑥_2𝑛,𝑔𝑧,𝑝𝑓𝑥_2𝑛,𝑆𝑥_2𝑛,𝑝,𝑝𝑔𝑧,𝑔𝑧𝑆𝑥_2𝑛,𝑔𝑧+𝑝𝑓𝑥_2𝑛,𝑇𝑧2. (2.28) Since lim_𝑛→∞𝑀(𝑥_2𝑛,𝑧)=𝑝(𝑧,𝑔𝑧) and lim_𝑛→∞𝑁(𝑥_2𝑛,𝑧)=0, then 𝑝𝑧,𝑔𝑧=lim_𝑛→∞𝑝𝑓𝑥_2𝑛,𝑔𝑧≤𝛿𝑝𝑧,𝑔𝑧. (2.29) Since 𝛿∈[0,1[, 𝑝(𝑧,𝑔𝑧)=0. By Remark 1.5, we get 𝑧=𝑔𝑧.
Similarly, we show that 𝑧 is also fixed point of 𝑓 and 𝑆. Hence, 𝑓𝑧=𝑔𝑧=𝑇𝑧=𝑆𝑧=𝑧.
The proofs for the cases in which 𝑆(𝑋), 𝑓(𝑋), or 𝑔(𝑋) is complete are similar.
Last, we show 𝑧 is unique. Suppose on the contrary that there is another common fixed point 𝑡 of 𝑓, 𝑔, 𝑆, and 𝑇. Then 𝑝(𝑧,𝑡)=𝑝𝑓𝑧,𝑔𝑡≤𝛿𝑀(𝑧,𝑡)+𝐿𝑁(𝑧,𝑡), (2.30) where 𝑁(𝑧,𝑡)=min𝑝^𝑤𝑓𝑧,𝑆𝑧,𝑝^𝑤𝑔𝑡,𝑇𝑡,𝑝^𝑤𝑆𝑧,𝑔𝑡,𝑝^𝑤𝑀()𝑝(),𝑝𝑓𝑧,𝑇𝑡=0,𝑧,𝑡=max𝑆𝑧,𝑇𝑡,𝑝𝑓𝑧,𝑆𝑧,𝑝𝑔𝑡,𝑇𝑡𝑆𝑧,𝑔𝑡+𝑝𝑓𝑧,𝑇𝑡2()().=𝑝𝑆𝑧,𝑇𝑡=𝑝𝑧,𝑡 (2.31) Thus, 𝑝(𝑧,𝑡)≤𝛿𝑝(𝑧,𝑡). (2.32) 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. Let 𝑋={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 𝑓=𝑔,𝑆=𝑇,𝑓0=𝑓2=0,𝑓1=1𝑇0=0,𝑇1=2,𝑇2=1. (2.33) We have 𝑓(𝑋)⊆𝑇(𝑋)=𝑋. For 𝛿=1/2, 𝐿=1, 𝑝𝑝𝑝𝑝𝑓0,𝑓1=1≤𝛿.2+𝐿.1,𝑓2,𝑓1=1≤𝛿.2+𝐿.0,𝑓2,𝑓2=𝑝𝑓0,𝑓0=0≤𝛿.0+𝐿.1,𝑓1,𝑓1=1≤𝛿.2+𝐿.0. (2.34) 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 𝑝𝑓𝑥,𝑓𝑦≤𝛿𝑀𝑥,𝑦+𝐿𝑁𝑥,𝑦, (2.35) where, 𝑀𝑝,𝑝𝑥,𝑦=max𝑇𝑥,𝑇𝑦,𝑝𝑓𝑥,𝑇𝑥,𝑝𝑓𝑦,𝑇𝑦𝑇𝑥,𝑓𝑦+𝑝𝑓𝑥,𝑇𝑦2,𝑁𝑥,𝑦=min𝑝^𝑤𝑓𝑥,𝑇𝑥,𝑝^𝑤𝑓𝑦,𝑇𝑦,𝑝^𝑤𝑇𝑥,𝑓𝑦,𝑝^𝑤𝑓𝑥,𝑇𝑦, (2.36) 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 map 𝑇∶𝑋→𝑋, {𝑇,𝐼_𝑋} is weakly compatible, where 𝐼_𝑋 is identity map on 𝑋, so by taking 𝑓=𝑔=𝐼_𝑋 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 𝑝𝑥,𝑦≤𝛿𝑀𝑥,𝑦+𝐿𝑁𝑥,𝑦, (2.37) for every 𝑥, 𝑦∈𝑋, where 𝑀𝑝(),𝑝𝑥,𝑦=max𝑆𝑥,𝑇𝑦,𝑝𝑥,𝑆𝑥,𝑝𝑦,𝑇𝑦𝑆𝑥,𝑦+𝑝𝑥,𝑇𝑦2,𝑁𝑥,𝑦=min𝑝^𝑤(𝑥,𝑆𝑥),𝑝^𝑤𝑦,𝑇𝑦,𝑝^𝑤𝑆𝑥,𝑦,𝑝^𝑤𝑥,𝑇𝑦. (2.38) Then 𝑆 and 𝑇 have a common fixed point.