A Generalization of Prešić Type Mappings in Metric-Like Spaces

Abstract

We generalize the result of Prešić in metric-like spaces by proving some common fixed point theorems for Prešić type mappings in metric-like spaces. An example is given which shows that the generalization is proper.

1. Introduction and Preliminaries

Let (𝑋,𝑑) be any metric space, and let 𝑓∶𝑋→𝑋 be any mapping; then 𝑓 is said to be a contraction on 𝑋 if there exists 𝜆∈[0,1) such that 𝑑𝑓𝑥,𝑓𝑦≤𝜆𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑋. (1)

A point 𝑥∈𝑋 is called a fixed point of 𝑓 if 𝑓𝑥=𝑥. Banach [1] proved that every contraction on a complete metric space has a unique fixed point and this result is known as the Banach contraction principle. There are several generalizations of this famous principle. One such generalization is given by Prešić [2, 3]. When studying the convergence of some particular sequences, Prešić [2, 3] proved the following theorem.

Theorem 1. Let (𝑋,𝑑) be a complete metric space, 𝑘 a positive integer, and 𝑓∶𝑋^𝑘→𝑋 a mapping satisfying the following contractive type condition: 𝑑𝑓𝑥₁,𝑥₂,…,𝑥_𝑘,𝑓𝑥₂,𝑥₃,…,𝑥_𝑘+1≤^𝑘_𝑖=1𝑞_𝑖𝑑𝑥_𝑖,𝑥_𝑖+1, (2) for every 𝑥₁,𝑥₂,…,𝑥_𝑘+1∈𝑋, where 𝑞₁,𝑞₂,…,𝑞_𝑘 are nonnegative constants such that 𝑞₁+𝑞₂+⋯+𝑞_𝑘<1. Then there exists a unique point 𝑥∈𝑋 such that 𝑓(𝑥,𝑥,…,𝑥)=𝑥. Moreover, if 𝑥₁,𝑥₂,…,𝑥_𝑘 are arbitrary points in 𝑋 and for 𝑛∈ℕ, 𝑥_𝑛+𝑘=𝑓𝑥_𝑛,𝑥_𝑛+1,…,𝑥_{𝑛+𝑘−1}, (3) then the sequence {𝑥_𝑛} is convergent and lim𝑥_𝑛=𝑓(lim𝑥_𝑛,lim𝑥_𝑛,…,lim𝑥_𝑛).

Note that the 𝑘-step iterative sequence given by (3) represents a nonlinear difference equation. In view of Prešić theorem, it is obvious that if this sequence is convergent (which is ensured by the Prešić theorem) then the limit of the sequence is a fixed point of 𝑓. The result of Prešić is generalized by several authors, and some generalizations and applications of Prešić theorem can be seen in [4–15].

On the other hand, Matthews [16] introduced the notion of a partial metric space as a part of the study of denotational semantics of a dataflow network. In this space, the usual metric is replaced by a partial metric with an interesting property that the self-distance of any point of space may not be zero. Further, Matthews showed that the Banach contraction principle is valid in a partial metric space and can be applied in program verifications. O'Neill [17] generalized the concept of a partial metric space a bit further by admitting negative distances. The partial metric defined by O'Neill is called the dualistic partial metric. Heckmann [18] generalized it by omitting the small self-distance axiom. The partial metric defined by Heckmann is called a weak partial metric.

Recently, Amini-Harandi [19] generalized the partial metric spaces by introducing the metric-like spaces. Amini-Harandi introduced the notion of a 𝜎-Cauchy sequence and completeness of metric-like spaces and proved some fixed point theorems in such spaces. In this paper, we prove some common fixed point theorems for Prešić type mappings in metric-like spaces. Our results generalize and extend the result of Prešić from complete spaces into 𝜎-complete metric-like spaces. An example is given which shows that the generalization is proper.

First we recall some definitions about partial metric and metric-like spaces.

Definition 2 (see [16]). A partial metric on a nonempty set 𝑋 is a function 𝑝∶𝑋×𝑋→ℝ⁺ (ℝ⁺ stands for nonnegative reals) such that for all 𝑥,𝑦,𝑧∈𝑋:(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).A partial metric space is a pair (𝑋,𝑝) such that 𝑋 is a nonempty set and 𝑝 is a partial metric on 𝑋. A sequence {𝑥_𝑛} in (𝑋,𝑝) converges to a point 𝑥∈𝑋 if and only if 𝑝(𝑥,𝑥)=lim_𝑛→∞𝑝(𝑥_𝑛,𝑥). A sequence {𝑥_𝑛} in (𝑋,𝑝) is called a 𝑝-Cauchy sequence if lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚) exists and is finite. (𝑋,𝑝) is said to be complete if every 𝑝-Cauchy sequence {𝑥_𝑛} in 𝑋 converges to a point 𝑥∈𝑋 such that 𝑝(𝑥,𝑥)=lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚).

Definition 3 (see [19]). A metric-like on a nonempty set 𝑋 is a function 𝜎∶𝑋×𝑋→ℝ⁺ such that for all 𝑥,𝑦,𝑧∈𝑋:(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).A metric-like space is a pair (𝑋,𝜎) such that 𝑋 is a nonempty set and 𝜎 is a metric-like on 𝑋. Note that a metric-like satisfies all the conditions of metric except that 𝜎(𝑥,𝑥) may be positive for some 𝑥∈𝑋. Each metric-like 𝜎 on 𝑋 generates a topology 𝜏_𝜎 on 𝑋 whose base is the family of open 𝜎-balls 𝐵_𝜎(𝑥,𝜖)=𝑦∈𝑋∶𝜎𝑥,𝑦−𝜎(𝑥,𝑥)<𝜖,∀𝑥∈𝑋,𝜖>0. (4)
A sequence {𝑥_𝑛} in 𝑋 converges to a point 𝑥∈𝑋 if and only if lim_𝑛→∞𝜎(𝑥_𝑛,𝑥)=𝜎(𝑥,𝑥). A sequence {𝑥_𝑛} is said to be 𝜎-Cauchy if lim_{𝑛,𝑚→∞}𝜎(𝑥_𝑛,𝑥_𝑚) exists and is finite. The metric-like space (𝑋,𝜎) is called complete if for each 𝜎-Cauchy sequence {𝑥_𝑛}, there exists 𝑥∈𝑋 such that lim_𝑛→∞𝜎𝑥_𝑛,𝑥=𝜎(𝑥,𝑥)=lim_{𝑚,𝑛→∞}𝜎𝑥_𝑛,𝑥_𝑚. (5)

Every partial metric space is a metric-like space but the converse may not be true.

Example 4 (see [19]). Let 𝑋={0,1} and 𝜎∶𝑋×𝑋→ℝ⁺ be defined by 𝜎𝑥,𝑦=2,if𝑥=𝑦=0,1,otherwise. (6)
Then (𝑋,𝜎) is a metric-like space, but it is not a partial metric space, as 𝜎(0,0)≰𝜎(0,1).

Example 5. Let 𝑋=ℝ, 𝑘≥0, and 𝜎∶𝑋×𝑋→ℝ⁺ be defined by 𝜎𝑥,𝑦=2𝑘,if𝑥=𝑦=0,𝑘,otherwise. (7)
Then (𝑋,𝜎) is a metric-like space, but for 𝑘>0, it is not a partial metric space, as 𝜎(0,0)≰𝜎(0,1).

Example 6. Let 𝑋=ℝ⁺ and 𝜎∶𝑋×𝑋→ℝ⁺ be defined by 𝜎𝑥,𝑦=2𝑥,if𝑥=𝑦,max𝑥,𝑦,otherwise. (8)
Then (𝑋,𝑝) is a metric-like space, but it is not a partial metric space, as 𝜎(1,1)=2≰𝜎(0,1)=1.

For the following definition, we refer to [10] and the references therein.

Definition 7. Let 𝑋 be a nonempty set, 𝑘 a positive integer, and 𝑓∶𝑋^𝑘→𝑋 a mapping. If 𝑓(𝑥,𝑥,…,𝑥)=𝑥, then 𝑥∈𝑋 is called a fixed point of 𝑓.

Definition 8. Let 𝑋 be a nonempty set, 𝑘 a positive integer, and 𝑓∶𝑋^𝑘→𝑋 and 𝑔∶𝑋→𝑋 mappings.(a)An element 𝑥∈𝑋 is said to be a coincidence point of 𝑓 and 𝑔 if 𝑔𝑥=𝑓(𝑥,…,𝑥).(b)If 𝑤=𝑔𝑥=𝑓(𝑥,…,𝑥), then 𝑤 is called a point of coincidence of 𝑓 and 𝑔.(c)If 𝑥=𝑔𝑥=𝑓(𝑥,…,𝑥), then 𝑥 is called a common fixed point of 𝑓 and 𝑔.(d)Mappings 𝑓 and 𝑔 are said to be commuting if 𝑔(𝑓(𝑥,…,𝑥))=𝑓(𝑔𝑥,…,𝑔𝑥) for all 𝑥∈𝑋.(e)Mappings 𝑓 and 𝑔 are said to be weakly compatible if they commute at their coincidence points.

Let (𝑋,𝜎) be a metric-like space, 𝑘 a positive integer, and 𝑓∶𝑋^𝑘→𝑋 a mapping. 𝑓 is said to be Prešić type if 𝜎𝑓𝑥₁,𝑥₂,…,𝑥_𝑘,𝑓𝑥₂,𝑥₃.…,𝑥_𝑘+1≤^𝑘_𝑖=1𝛼_𝑖𝜎𝑥_𝑖,𝑥_𝑖+1, (9) for all 𝑥₁,𝑥₂,…,𝑥_𝑘,𝑥_𝑘+1∈𝑋, where 𝛼_𝑖 are nonnegative constants such that ∑^𝑘_𝑖=1𝛼_𝑖<1.

The following lemma shows that the self-distance of a point of coincidence of two maps of a metric-like space satisfying Prešić type contractive condition is minimum (zero), and it will be used in the sequel.

Lemma 9. Let (𝑋,𝜎) be a metric-like space, 𝑘 a positive integer, and 𝑓∶𝑋^𝑘→𝑋, 𝑔∶𝑋→𝑋 two mappings. Suppose that the following condition holds: 𝜎𝑓𝑥₁,𝑥₂,…,𝑥_𝑘,𝑓𝑥₂,𝑥₃,…,𝑥_𝑘+1≤^𝑘_𝑖=1𝛼_𝑖𝜎𝑔𝑥_𝑖,𝑔𝑥_𝑖+1, (10) for all 𝑥₁,𝑥₂,…,𝑥_𝑘+1∈𝑋, where 𝛼₁,𝛼₂,…,𝛼_𝑘 are nonnegative constants such that 𝛼₁+𝛼₂+⋯+𝛼_𝑘<1. If 𝑓 and 𝑔 have a point of coincidence 𝑣∈𝑋, then 𝜎(𝑣,𝑣)=0.

Proof. Let 𝑣∈𝑋 be any point of coincidence of 𝑓 and 𝑔. Then there exists 𝑢∈𝑋 such that 𝑓(𝑢,𝑢,…,𝑢)=𝑔𝑢=𝑣.
Now suppose that 𝜎(𝑣,𝑣)≠0. Then it follows from (10) that 𝜎(𝑣,𝑣)=𝜎𝑓(𝑢,𝑢,…,𝑢),𝑓(𝑢,𝑢,…,𝑢)≤𝛼₁𝜎𝑔𝑢,𝑔𝑢+𝛼₂𝜎𝑔𝑢,𝑔𝑢+⋯+𝛼_𝑘𝜎=𝛼𝑔𝑢,𝑔𝑢₁+𝛼₂+⋯+𝛼_𝑘𝜎=𝛼𝑔𝑢,𝑔𝑢₁+𝛼₂+⋯+𝛼_𝑘𝜎()()𝑣,𝑣<𝜎𝑣,𝑣as𝛼₁+𝛼₂+⋯+𝛼_𝑘<1, (11) a contradiction. Therefore we must have 𝜎(𝑣,𝑣)=0.

Now we can state our main results.

2. Main Results

The following theorem extends and generalizes the result of Prešić in metric-like spaces.

Theorem 10. Let (𝑋,𝜎) be a metric-like space, 𝑘 a positive integer, and 𝑓∶𝑋^𝑘→𝑋, 𝑔∶𝑋→𝑋 two mappings such that 𝑓(𝑋^𝑘)⊂𝑔(𝑋) and 𝑔(𝑋) is a complete subspace of 𝑋. Suppose that the following condition holds: 𝜎𝑓𝑥₁,𝑥₂,…,𝑥_𝑘,𝑓𝑥₂,𝑥₃,…,𝑥_𝑘+1≤^𝑘_𝑖=1𝛼_𝑖𝜎𝑔𝑥_𝑖,𝑔𝑥_𝑖+1, (12) for every 𝑥₁,𝑥₂,…,𝑥_𝑘+1∈𝑋, where 𝛼₁,𝛼₂,…,𝛼_𝑘 are nonnegative constants such that 𝛼₁+𝛼₂+⋯+𝛼_𝑘<1. Then 𝑓 and 𝑔 have a unique point of coincidence 𝑣∈𝑋 and 𝜎(𝑣,𝑣)=0. Moreover, if 𝑓 and 𝑔 are weakly compatible, then 𝑣 is the unique common fixed point of 𝑓 and 𝑔.

Proof. Let 𝑥₁,𝑥₂,…,𝑥_𝑘∈𝑋 be arbitrary points. As 𝑓(𝑋^𝑘)⊂𝑔(𝑋), we can define a sequence {𝑦_𝑛} in 𝑋 by 𝑦_𝑛=𝑔𝑥_𝑛 for 𝑛=1,2,…,𝑘 and 𝑦_𝑛+𝑘=𝑔𝑥_𝑛+𝑘=𝑓(𝑥_𝑛,𝑥_𝑛+1,…,𝑥_{𝑛+𝑘−1}) for 𝑛=1,2,….
For simplicity put 𝜎_𝑛=𝜎(𝑦_𝑛,𝑦_𝑛+1) for 𝑛=1,2,… and 𝜇=max{𝜎₁/𝜃,𝜎₂/𝜃²,…,𝜎_𝑘/𝜃^𝑘}, where 𝜃=[∑^𝑘_𝑖=1𝛼_𝑖]^1/𝑘<1.
By mathematical induction, we will show that 𝜎_𝑛≤𝜇𝜃^𝑛∀𝑛∈ℕ. (13) According to the definition of 𝜇, it is clear that (13) is true for 𝑛=1,2,…,𝑘. Let the following 𝑘 inequalities: 𝜎_𝑛≤𝜇𝜃^𝑛,𝜎_𝑛+1≤𝜇𝜃^𝑛+1,…,𝜎_{𝑛+𝑘−1}≤𝜇𝜃^{𝑛+𝑘−1}, (14) be the induction hypothesis.
Now using (12), we obtain 𝜎_𝑛+𝑘=𝜎𝑦_𝑛+𝑘,𝑦_𝑛+𝑘+1=𝜎𝑓𝑥_𝑛,𝑥_𝑛+1,…,𝑥_{𝑛+𝑘−1},𝑓𝑥_𝑛+1,𝑥_𝑛+2,…,𝑥_𝑛+𝑘≤𝛼₁𝜎𝑔𝑥_𝑛,𝑔𝑥_𝑛+1+𝛼₂𝜎𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+2+⋯+𝛼_𝑘𝜎𝑔𝑥_{𝑛+𝑘−1},𝑔𝑥_𝑛+𝑘=𝛼₁𝜎𝑦_𝑛,𝑦_𝑛+1+𝛼₂𝜎𝑦_𝑛+1,𝑦_𝑛+2+⋯+𝛼_𝑘𝜎𝑦_{𝑛+𝑘−1},𝑦_𝑛+𝑘=𝛼₁𝜎_𝑛+𝛼₂𝜎_𝑛+1+⋯+𝛼_𝑘𝜎_{𝑛+𝑘−1}≤𝛼₁𝜇𝜃^𝑛+𝛼₂𝜇𝜃^𝑛+1+⋯+𝛼_𝑘𝜇𝜃^{𝑛+𝑘−1}≤𝛼₁𝜇𝜃^𝑛+𝛼₂𝜇𝜃^𝑛+⋯+𝛼_𝑘𝜇𝜃^𝑛as𝜃=^𝑘_𝑖=1𝛼_𝑖^1/𝑘=𝛼<1₁+𝛼₂+⋯+𝛼_𝑘𝜇𝜃^𝑛=^𝑘_𝑖=1𝛼_𝑖𝜇𝜃^𝑛=𝜇𝜃^𝑛+𝑘. (15)
Thus, the inductive proof of (13) is complete.
Now let 𝑛,𝑚∈ℕ with 𝑚>𝑛. Then from (13), we have 𝜎𝑦_𝑛,𝑦_𝑚≤𝜎𝑦_𝑛,𝑦_𝑛+1+𝜎𝑦_𝑛+1,𝑦_𝑛+2+⋯+𝜎𝑦_𝑚−1,𝑦_𝑚=𝜎_𝑛+𝜎_𝑛+1+⋯+𝜎_𝑚−1≤𝜇𝜃^𝑛+𝜇𝜃^𝑛+1+𝜇𝜃^𝑛+2+⋯=𝜇𝜃^𝑛1+𝜃+𝜃²=+⋯𝜇𝜃^𝑛.1−𝜃 (16) As, 𝜃<1, it follows from the previous inequality that lim_{𝑛,𝑚→∞}𝜎𝑦_𝑛,𝑦_𝑚=0. (17) Thus, {𝑦_𝑛}={𝑔𝑥_𝑛} is a 𝜎-Cauchy sequence in (𝑋,𝜎). By the completeness of 𝑔(𝑋), there exists 𝑢,𝑣∈𝑋 such that 𝑣=𝑔𝑢 and lim_𝑛→∞𝜎𝑦_𝑛,𝑣=lim_{𝑚,𝑛→∞}𝜎𝑦_𝑛,𝑦_𝑚=𝜎(𝑣,𝑣)=0. (18)
We will now show that 𝑣 is a point of coincidence of 𝑓 and 𝑔. For any 𝑛∈ℕ we have 𝜎𝑔𝑢,𝑓(𝑢,𝑢,…,𝑢)≤𝜎𝑔𝑢,𝑦_𝑛+𝑘+𝜎𝑦_𝑛+𝑘,𝑓(𝑢,𝑢,…,𝑢)≤𝜎𝑣,𝑦_𝑛+𝑘+𝜎𝑓𝑥_𝑛,…,𝑥_{𝑛+𝑘−1},𝑓(𝑢,𝑢,…,𝑢)≤𝜎𝑣,𝑦_𝑛+𝑘+𝜎𝑓𝑥_𝑛,…,𝑥_{𝑛+𝑘−1},𝑓𝑥_𝑛+1,…,𝑥_{𝑛+𝑘−1}𝑓𝑥,𝑢+𝜎_𝑛+1,…,𝑥_{𝑛+𝑘−1},𝑓𝑥,𝑢_𝑛+2,…,𝑥_{𝑛+𝑘−1}𝑓𝑥,𝑢,𝑢+⋯+𝜎_{𝑛+𝑘−1},𝑢,…,𝑢,𝑓(𝑢,𝑢,…,𝑢). (19) Using (12) and writing 𝜎_𝑛=𝜎(𝑦_𝑛,𝑦_𝑛+1)=𝜎(𝑔𝑥_𝑛,𝑔𝑥_𝑛+1), 𝑔𝑢=𝑣 in the previous inequality we obtain 𝜎𝑣,𝑓(𝑢,𝑢,…,𝑢)≤𝜎𝑣,𝑦_𝑛+𝑘+𝛼₁𝜎_𝑛+⋯+𝛼_𝑘−1𝜎_{𝑛+𝑘−2}+𝛼_𝑘𝜎𝑦_{𝑛+𝑘−1},𝑣+𝛼₁𝜎_𝑛+1+⋯+𝛼_𝑘−2𝜎_{𝑛+𝑘−2}+𝛼_𝑘−1𝜎𝑦_{𝑛+𝑘−1},𝑣+𝛼_𝑘𝜎(𝑣,𝑣)+⋯+𝛼₁𝜎𝑦_{𝑛+𝑘−1}+𝛼,𝑣₂+⋯+𝛼_𝑘𝜎(𝑣,𝑣). (20) Letting 𝑛→∞ and using (18) in the previous inequality, we obtain 𝜎𝑣,𝑓(𝑢,𝑢,…,𝑢)=0,thatis𝑓(𝑢,𝑢,…,𝑢)=𝑣=𝑔𝑢. (21) Therefore, 𝑣 is a point of coincidence of 𝑓 and 𝑔.
We will now show that it is unique. Suppose that 𝑣^′ is another point of coincidence of 𝑓 and 𝑔. Then there exists 𝑢^′∈𝑋 such that 𝑓(𝑢^′,𝑢^′,…,𝑢^′)=𝑔𝑢^′=𝑣^′. By Lemma 9, we have 𝜎(𝑣^′,𝑣^′)=0. Now it follows from (12) that 𝜎𝑣,𝑣^′=𝜎𝑓(𝑢,𝑢,…,𝑢),𝑓𝑢^′,𝑢^′,…,𝑢^′≤𝜎𝑓(𝑢,…,𝑢),𝑓𝑢,…,𝑢,𝑢^′+𝜎𝑓𝑢,…,𝑢,𝑢^′,𝑓𝑢,…,𝑢,𝑢^′,𝑢^′+⋯+𝜎𝑓𝑢,𝑢^′…,𝑢^′,𝑓𝑢^′,𝑢^′,…,𝑢^′≤𝛼₁𝜎𝑔𝑢,𝑔𝑢+⋯+𝛼_𝑘−1𝜎𝑔𝑢,𝑔𝑢+𝛼_𝑘𝜎𝑔𝑢,𝑔𝑢^′+𝛼₁𝜎𝑔𝑢,𝑔𝑢+⋯+𝜎_𝑘−2𝜎𝑔𝑢,𝑔𝑢+𝛼_𝑘−1𝜎𝑔𝑢,𝑔𝑢^′+𝛼_𝑘𝜎𝑔𝑢^′,𝑔𝑢^′+⋯+𝛼₁𝜎𝑔𝑢,𝑔𝑢^′+𝛼₂𝜎𝑔𝑢^′,𝑔𝑢^′+⋯+𝛼_𝑘𝜎𝑔𝑢^′,𝑔𝑢^′. (22) As 𝜎(𝑔𝑢,𝑔𝑢)=𝜎(𝑣,𝑣)=0=𝜎(𝑣^′,𝑣^′)=𝜎(𝑔𝑢^′,𝑔𝑢^′), it follows from previous inequality that 𝜎𝑣,𝑣^′≤𝛼₁+𝛼₂+⋯+𝛼_𝑘𝑝𝑔𝑢,𝑔𝑢^′=𝛼₁+𝛼₂+⋯+𝛼_𝑘𝑝𝑣,𝑣^′<𝑝𝑣,𝑣^′as𝛼₁+𝛼₂+⋯+𝛼_𝑘<1, (23) a contradiction. Therefore, we must have 𝜎(𝑣,𝑣^′)=0, that is, 𝑣=𝑣^′. Thus the point of coincidence of 𝑓 and 𝑔 is unique.
Now suppose that 𝑓 and 𝑔 are weakly compatible. Put 𝑤=𝑔𝑣. Then 𝑤=𝑔𝑣=𝑔𝑓(𝑢,𝑢,…,𝑢)=𝑓(𝑔𝑢,𝑔𝑢,…,𝑔𝑢)=𝑓(𝑣,𝑣,…,𝑣), so 𝑤 is a point of coincidence of 𝑓 and 𝑔. However, by the uniqueness of 𝑣, we have 𝑤=𝑣=𝑔𝑣=𝑓(𝑣,𝑣,…,𝑣). Thus, 𝑣 is the unique common fixed point of 𝑓 and 𝑔.

Taking 𝑓=𝐼_𝑋 (i.e., the identity mapping of 𝑋) we obtain following fixed point result for Prešić type mapping in a metric-like space.

Corollary 11. Let (𝑋,𝜎) be a complete metric-like space, 𝑘 a positive integer, and 𝑓∶𝑋^𝑘→𝑋 a Prešić type mapping on 𝑋. Then 𝑓 has a unique fixed point 𝑣∈𝑋 and 𝜎(𝑣,𝑣)=0.

The following example shows that Corollary 11 is a proper generalization of Theorem 1.

Example 12. Let 𝑋=[0,1], and define 𝜎∶𝑋×𝑋→ℝ⁺ by 𝜎𝑥,𝑦=2𝑥,if𝑥=𝑦,max𝑥,𝑦,otherwise. (24)
Then, (𝑋,𝜎) is a complete metric-like space. Note that (𝑋,𝜎) is not a partial metric space, as 𝜎(1,1)≰𝜎(0,1).
Define 𝑓∶𝑋²→𝑋 by 𝑓=1𝑥,𝑦16,if𝑥=𝑦=1,𝑥+𝑦8,otherwise. (25)
Note that 𝑓 is not a Prešić type mapping in the usual metric space (𝑋,𝑑), where 𝑑(𝑥,𝑦)=|𝑥−𝑦| for all 𝑥,𝑦∈𝑋. Indeed, for the points 𝑥₁=𝑥₂=1,𝑥₃=9/10 we have 𝑑𝑓𝑥₁,𝑥₂,𝑓𝑥₂,𝑥₃1=𝑑,1619=780,40 (26) and 𝛼₁𝑑(1,1)+𝛼₂𝑑(1,9/10)=(1/10)𝛼₂. Therefore, the condition (2) of Theorem 1 is not satisfied with 𝛼₁+𝛼₂<1. Thus the result of Prešić is not applicable here.
On the other hand, it is easy to see that 𝑓 satisfies all the conditions of Corollary 11, with 𝛼₁=𝛼₂∈[1/8,1) and 𝑓(0,0)=0; that is, 0 is the unique fixed point of 𝑓.