Berinde-Type Generalized Contractions on Partial Metric Spaces

Abstract

We consider generalized Berinde-type contractions in the context of partial metric spaces. Such contractions are also known as generalized almost contractions in the literature. In this paper, we extend, generalize, and enrich the results in this direction. Some examples are presented to illustrate our results.

1. Introduction and Preliminaries

Matthews [1] introduced the notion of a partial metric space as a part of the study of denotational semantics of data for networks, showing that the contraction mapping principle [2] can be generalized to the partial metric context for applications in program verifications. Later, there have been several recent extensive researchs on (common) fixed points for different contractions on partial metric spaces, see [3–28].

First, we recall some basic concepts and notations.

Definition 1. A partial metric on a nonempty set 𝑋 is a function 𝑝∶𝑋×𝑋→[0,+∞) such that for all 𝑥,𝑦,𝑧∈𝑋: (𝑝1)𝑥=𝑦⇔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.

Example 2 (see [1]). Let 𝑋=ℝ⁺ and 𝑝 defined on 𝑋 by 𝑝(𝑥,𝑦)=max{𝑥,𝑦} for all 𝑥,𝑦∈𝑋. Then (𝑋,𝑝) is a partial metric space.

Example 3 (see [20, 26]). Let (𝑋,𝑑) and (𝑋,𝑝) be a metric space and a partial metric space, respectively. Functions 𝜌_𝑖∶𝑋×𝑋→ℝ⁺ (𝑖∈{1,2,3}) given by 𝜌₁,𝜌𝑥,𝑦=𝑑𝑥,𝑦+𝑝𝑥,𝑦₂𝑢(𝑥)𝑦,𝜌𝑥,𝑦=𝑑𝑥,𝑦+max,𝑢₃𝑥,𝑦=𝑑𝑥,𝑦+𝑎, (1) define partial metrics on 𝑋, where 𝑢∶𝑋→ℝ⁺ is an arbitrary function and 𝑎≥0.

Example 4 (see [1]). Let 𝑋={[𝑎,𝑏]∶𝑎,𝑏∈ℝ,𝑎≤𝑏} and define 𝑝([𝑎,𝑏],[𝑐,𝑑])=max{𝑏,𝑑}−min{𝑎,𝑐}. Then (𝑋,𝑝) is a partial metric space.

Example 5 (see [1]). Let 𝑋=[0,1]∪[2,3] and define 𝑝∶𝑋×𝑋→ℝ⁺ by 𝑝=∩[]⊂[].𝑥,𝑦max𝑥,𝑦,if𝑥,𝑦2,3≠∅,𝑥−𝑦,if𝑥,𝑦0,1 (2) Then (𝑋,𝑝) is a partial metric space.

Remark 6. It is clear that, if 𝑝(𝑥,𝑦)=0, then from (𝑝1) and (𝑝2), we get 𝑥=𝑦. On the other hand, 𝑝(𝑥,𝑦) may not be 0 even if 𝑥=𝑦.

Each partial metric 𝑝 on 𝑋 generates a 𝑇₀ topology 𝜏_𝑝 on 𝑋 which has as a base 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.

If 𝑝 is a partial metric on 𝑋, then the functions 𝑑_𝑝, 𝑑^𝑝_𝑚∶𝑋×𝑋→ℝ₊ given by 𝑑_𝑝(),𝑑𝑥,𝑦=2𝑝𝑥,𝑦−𝑝𝑥,𝑥−𝑝𝑦,𝑦_𝑝𝑚𝑝()𝑝()𝑥,𝑦=max𝑥,𝑦−𝑝𝑥,𝑥,𝑝𝑥,𝑦−𝑝𝑦,𝑦=𝑝𝑥,𝑦−min𝑥,𝑥,𝑝𝑦,𝑦 (3) are equivalent metrics on 𝑋.

Definition 7 (see [1]). Let (𝑋,𝑝) be a partial metric space. (1) A sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑋 is called a Cauchy sequence in (𝑋,𝑝) if lim_{𝑛,𝑚→+∞}𝑝(𝑥_𝑛,𝑥_𝑚) exists and is finite.(2)(𝑋,𝑝) is called complete if every Cauchy sequence {𝑥_𝑛}_𝑛∈ℕ converges with respect to 𝜏_𝑝 to a point 𝑥∈𝑋 such that 𝑝(𝑥,𝑥)=lim_{𝑛,𝑚→+∞}𝑝(𝑥_𝑛,𝑥_𝑚).

Lemma 8 (see [1]). Let (𝑋,𝑝) be a partial metric space. (1){𝑥_𝑛}_𝑛∈ℕ is a Cauchy sequence in (𝑋,𝑝) if and only if it is a Cauchy sequence in the metric space (𝑋,𝑑_𝑝).(2) A partial metric space (𝑋,𝑝) is complete if and only if the metric space (𝑋,𝑑_𝑝) is complete. Furthermore, lim_𝑛→+∞𝑑_𝑝(𝑥_𝑛,𝑥)=0 if and only if 𝑝(𝑥,𝑥)=lim_𝑛→+∞𝑝𝑥_𝑛,𝑥=lim_{𝑛,𝑚→+∞}𝑝𝑥_𝑛,𝑥_𝑚. (4)

Lemma 9 (see [20]). Let {𝑥_𝑛}_𝑛∈ℕ be a convergent sequence in a partial metric space 𝑋 such that 𝑥_𝑛→𝑥 and 𝑥_𝑛→𝑦 with respect to 𝜏_𝑝. If lim_𝑛→+∞𝑝𝑥_𝑛,𝑥_𝑛=𝑝(𝑥,𝑥)=𝑝𝑦,𝑦, (5) then 𝑥=𝑦.

Lemma 10 (see [20]). Let {𝑥_𝑛}_𝑛∈ℕ and {𝑦_𝑛}_𝑛∈ℕ be two sequences in a partial metric space 𝑋 such that lim_𝑛→+∞𝑝𝑥_𝑛,𝑥=lim_𝑛→+∞𝑝𝑥_𝑛,𝑥_𝑛=𝑝(𝑥,𝑥),lim_𝑛→+∞𝑝𝑦_𝑛,𝑦=lim_𝑛→+∞𝑝𝑦_𝑛,𝑦_𝑛=𝑝𝑦,𝑦, (6) then lim_𝑛→+∞𝑝(𝑥_𝑛,𝑦_𝑛)=𝑝(𝑥,𝑦). In particular, lim_𝑛→+∞𝑝(𝑥_𝑛,𝑧)=𝑝(𝑥,𝑧) for every 𝑧∈𝑋.

Lemma 11 (see [3]). Let (𝑋,𝑝) be a partial metric space and 𝑥_𝑛→𝑧, with respect to 𝜏_𝑝, with 𝑝(𝑧,𝑧)=0. Then lim_𝑛→+∞𝑝(𝑥_𝑛,𝑦)=𝑝(𝑧,𝑦) for all 𝑦∈𝑋.

The concept of almost contractions was introduced by Berinde [29, 30] on metric spaces. Other results on almost contractions could be found in [31–34]. Recently, Altun and Acar [35] characterized this concept in the setting of partial metric space and proved some fixed point theorems using these concepts. Very recently, Turkoglu and Ozturk [27] established a fixed point theorem for four mappings satisfying an almost generalized contractive condition on partial metric spaces. In this paper, we generalize the results given in [27, 35] by presenting some fixed point results for self mappings involving some almost generalized contractions in the setting of partial metric spaces. Also, we give some illustrative examples making our results proper.

2. Main Results

We start to this section by defining some sets of auxiliary functions. Let ℱ denote all functions 𝑓∶[0,+∞)→[0,+∞) such that 𝑓(𝑡)=0 if and only if 𝑡=0. We denote by Ψ and Φ be subsets of ℱ such that ,.Ψ=𝜓∈ℱ∶𝜓iscontinuousandnondecreasingΦ=𝜙∈ℱ∶𝜙islowersemicontinuous (7) Let (𝑋,𝑝) a partial metric space. We consider the following expressions: 𝑀𝑝(),1𝑥,𝑦=max𝑥,𝑦,𝑝𝑥,𝑇𝑥,𝑝𝑦,𝑇𝑦2𝑝,𝑁𝑑𝑥,𝑇𝑦+𝑝𝑦,𝑇𝑥𝑥,𝑦=min^𝑝_𝑚(𝑥,𝑇𝑥),𝑑^𝑝_𝑚,𝑑𝑦,𝑇𝑦_𝑝𝑚𝑥,𝑇𝑦,𝑑^𝑝_𝑚𝑦,𝑇𝑥, (8) for all 𝑥,𝑦∈𝑋.

Our first result is the following.

Theorem 12. Let (𝑋,𝑝) be a complete partial metric space. Let 𝑇∶𝑋→𝑋 be a self mapping. Suppose there exist 𝜓∈Ψ, 𝜙∈Φ and 𝐿≥0 such that for all 𝑥,𝑦∈𝑋𝜓𝑝𝑇𝑥,𝑇𝑦≤𝜓𝑀𝑥,𝑦−𝜙𝑀𝑥,𝑦+𝐿𝑁𝑥,𝑦. (9) Then 𝑇 has a unique fixed point, say 𝑢∈𝑋. Also, one has 𝑝(𝑢,𝑢)=0.

Proof. Let 𝑥₀∈𝑋. We construct a sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑋 in a way that 𝑥_𝑛=𝑇𝑥_𝑛−1 for all 𝑛≥1. Suppose that 𝑝(𝑥_𝑛0,𝑥_𝑛0+1)=0 for some 𝑛₀≥0. So we have 𝑥_𝑛0=𝑥_𝑛0+1=𝑇𝑥_𝑛0, that is, 𝑥_𝑛0 is the fixed point of 𝑇.
From now on, assume that 𝑝(𝑥_𝑛,𝑥_𝑛+1)>0 for all 𝑛≥0. By (9), we have 𝜓𝑝𝑥_𝑛,𝑥_𝑛+1≤𝜓𝑝𝑇𝑥_𝑛−1,𝑇𝑥_𝑛≤𝜓𝑀𝑥_𝑛−1,𝑥_𝑛−𝜙𝑀𝑥_𝑛−1,𝑥_𝑛+𝐿𝑁𝑥_𝑛−1,𝑥_𝑛, (10) where 𝑁𝑥_𝑛−1,𝑥_𝑛=min𝑑^𝑝_𝑚𝑥_𝑛−1,𝑥_𝑛,𝑑^𝑝_𝑚𝑥_𝑛,𝑥_𝑛+1,𝑑^𝑝_𝑚𝑥_𝑛−1,𝑥_𝑛+1,𝑑^𝑝_𝑚𝑥_𝑛,𝑥_𝑛. (11) Since 𝑑^𝑝_𝑚(𝑥_𝑛,𝑥_𝑛)=0, we get 𝑁(𝑥_𝑛−1,𝑥_𝑛)=0. Hence, it follows that 𝜓𝑝𝑥_𝑛,𝑥_𝑛+1≤𝜓𝑀𝑥_𝑛−1,𝑥_𝑛−𝜙𝑀𝑥_𝑛−1,𝑥_𝑛, (12) which yields that 𝜓𝑝𝑥_𝑛,𝑥_𝑛+1≤𝜓𝑀𝑥_𝑛−1,𝑥_𝑛. (13) Since 𝜓 is nondecreasing, then 𝑝𝑥_𝑛,𝑥_𝑛+1≤𝑀𝑥_𝑛−1,𝑥_𝑛, (14) where 𝑀𝑥_𝑛−1,𝑥_𝑛=max𝑝𝑥_𝑛−1,𝑥_𝑛,𝑝𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑝𝑥_𝑛,𝑇𝑥_𝑛,12𝑝𝑥_𝑛−1,𝑇𝑥_𝑛+𝑝𝑥_𝑛,𝑇𝑥_𝑛−1=max𝑝𝑥_𝑛−1,𝑥_𝑛,𝑝𝑥_𝑛−1,𝑥_𝑛,𝑝𝑥_𝑛,𝑥_𝑛+1,12𝑝𝑥_𝑛−1,𝑥_𝑛+1+𝑝𝑥_𝑛,𝑥_𝑛. (15) Due to (𝑝4), we have 𝑝𝑥_𝑛−1,𝑥_𝑛+1+𝑝𝑥_𝑛,𝑥_𝑛≤𝑝𝑥_𝑛−1,𝑥_𝑛+𝑝𝑥_𝑛,𝑥_𝑛+1. (16) Hence, the expression (15) turns into 𝑀𝑥_𝑛−1,𝑥_𝑛=max𝑝𝑥_𝑛−1,𝑥_𝑛,𝑝𝑥_𝑛,𝑥_𝑛+1. (17)
If for some 𝑛, max𝑝𝑥_𝑛−1,𝑥_𝑛,𝑝𝑥_𝑛,𝑥_𝑛+1=𝑝𝑥_𝑛,𝑥_𝑛+1 (18) then by (12) 𝜓𝑝𝑥_𝑛,𝑥_𝑛+1≤𝜓𝑝𝑥_𝑛,𝑥_𝑛+1−𝜙𝑝𝑥_𝑛,𝑥_𝑛+1, (19) so 𝜙(𝑝(𝑥_𝑛,𝑥_𝑛+1))=0. By (𝜙₂), we get 𝑝(𝑥_𝑛,𝑥_𝑛+1)=0, which is a contradiction with respect to 𝑝(𝑥_𝑛,𝑥_𝑛+1)>0 for all 𝑛≥0. Thus 𝑀𝑥_𝑛−1,𝑥_𝑛=𝑝𝑥_𝑛−1,𝑥_𝑛,foreach𝑛≥1, (20) so from (14) 𝑝𝑥_𝑛,𝑥_𝑛+1≤𝑝𝑥_𝑛−1,𝑥_𝑛,foreach𝑛≥1. (21) Thus, the sequence {𝑝(𝑥_𝑛,𝑥_𝑛+1)} is non-increasing and so there exists 𝛿≥0 such that lim_𝑛→+∞𝑝𝑥_𝑛,𝑥_𝑛+1=𝛿. (22) Suppose that 𝛿>0. Taking limsup_𝑛→+∞ in inequality (12) and using (20), we get limsup_𝑛→+∞𝜓𝑝𝑥_𝑛,𝑥_𝑛+1≤limsup_𝑛→+∞𝜓𝑝𝑥_𝑛−1,𝑥_𝑛−liminf_𝑛→+∞𝜙𝑝𝑥_𝑛−1,𝑥_𝑛. (23) By continuity of 𝜓 and lower semicontinuity of 𝜙, we get 𝜓(𝛿)≤𝜓(𝛿)−𝜙(𝛿), so 𝜙(𝛿)=0, that is, 𝛿=0, a contradiction. We conclude that lim_𝑛→+∞𝑝𝑥_𝑛,𝑥_𝑛+1=0. (24) We will show that {𝑥_𝑛}_𝑛∈ℕ is a Cauchy sequence in the partial metric space (𝑋,𝑝). From Lemma 8, we need to prove that {𝑥_𝑛}_𝑛∈ℕ is a Cauchy sequence in the metric space (𝑋,𝑑_𝑝). Suppose to the contrary that {𝑥_𝑛}_𝑛∈ℕ is not a Cauchy sequence in the metric space (𝑋,𝑑_𝑝). Then, there is a 𝜀>0 such that for an integer 𝑘 there exist integers 𝑚(𝑘)>𝑛(𝑘)>𝑘 such that 𝑑_𝑝𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)>𝜀. (25) By definition of 𝑑_𝑝, we have 𝑑_𝑝(𝑥,𝑦)≤2𝑝(𝑥,𝑦) for each 𝑥,𝑦∈𝑋, so (25) gives us 𝑝𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)>𝜀2. (26) For every integer 𝑘, let 𝑚(𝑘) be the least positive integer exceeding 𝑛(𝑘) satisfying (26) then 𝑝𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1}≤𝜀2. (27) Now, using (26), (27), and the triangular inequality (which still holds for the partial metric 𝑝), we obtain 𝜀2<𝑝𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤𝑝𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1}+𝑝𝑥_{𝑚(𝑘)−1},𝑥_𝑚(𝑘)−𝑝𝑥_{𝑚(𝑘)−1},𝑥_{𝑚(𝑘)−1}≤𝑝𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1}+𝑝𝑥_{𝑚(𝑘)−1},𝑥_𝑚(𝑘)≤𝜀2+𝑝𝑥_{𝑚(𝑘)−1},𝑥_𝑚(𝑘). (28) Then by (24) it follows that lim_𝑘→+∞𝑝𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)=𝜀2. (29) Also, by the triangle inequality, we have 𝑝𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1}−𝑝𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤𝑝𝑥_{𝑚(𝑘)−1},𝑥_𝑚(𝑘). (30) From (24) and (29) we get lim_𝑘→+∞𝑝𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1}=𝜀2. (31) Similarly, by triangle inequality 𝑝𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤𝑝𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1+𝑝𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)≤𝑝𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1+𝑝𝑥_𝑛(𝑘)+1,𝑥_{𝑚(𝑘)−1}+𝑝𝑥_{𝑚(𝑘)−1},𝑥_𝑚(𝑘)≤2𝑝𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1+𝑝𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1}+𝑝𝑥_{𝑚(𝑘)−1},𝑥_𝑚(𝑘) (32) and from (24), (29), and (31) we get lim_𝑘→+∞𝑝𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)=𝜀2,lim_𝑘→+∞𝑝𝑥_𝑛(𝑘)+1,𝑥_{𝑚(𝑘)−1}=𝜀2. (33)(34) Having 𝑑^𝑝_𝑚𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1=𝑝𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1−min𝑝𝑥_𝑛(𝑘),𝑥_𝑛(𝑘),𝑝𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1≤𝑝𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1, (35) so referring to (24), we get lim_𝑘→+∞𝑑^𝑝_𝑚𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1=0. (36) Moreover 𝑀𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1}=max𝑝𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1},𝑝𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘),𝑝𝑥_{𝑚(𝑘)−1},𝑇𝑥_{𝑚(𝑘)−1},12𝑝𝑥_𝑛(𝑘),𝑇𝑥_{𝑚(𝑘)−1}+𝑝𝑥_{𝑚(𝑘)−1},𝑇𝑥_𝑛(𝑘)=max𝑝𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1},𝑝𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑝𝑥_{𝑚(𝑘)−1},𝑥_𝑚(𝑘),12𝑝𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+𝑝𝑥_{𝑚(𝑘)−1},𝑥_𝑛(𝑘)+1. (37) Thus, from (24), (29), (31), and (34), we get lim_𝑘→+∞𝑀𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1}𝜀=max2𝜀,0,0,2=𝜀2. (38) From (9), we have 𝜓𝑝𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)≤𝜓𝑝𝑇𝑥_𝑛(𝑘),𝑇𝑥_{𝑚(𝑘)−1}≤𝜓𝑀𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1}−𝜙𝑀𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1}+𝐿𝑁𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1}, (39) where 𝑁𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1}=min𝑑^𝑝_𝑚𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘),𝑑^𝑝_𝑚𝑥_{𝑚(𝑘)−1},𝑇𝑥_{𝑚(𝑘)−1},𝑑^𝑝_𝑚𝑥_𝑛(𝑘),𝑇𝑥_{𝑚(𝑘)−1},𝑑^𝑝_𝑚𝑥_{𝑚(𝑘)−1},𝑇𝑥_𝑛(𝑘)=min𝑑^𝑝_𝑚𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑑^𝑝_𝑚𝑥_{𝑚(𝑘)−1},𝑥_𝑚(𝑘),𝑑^𝑝_𝑚𝑥_𝑛(𝑘),𝑥_𝑚(𝑘),𝑑^𝑝_𝑚𝑥_{𝑚(𝑘)−1},𝑥_𝑛(𝑘)+1. (40)
By (36), we get lim_𝑘→+∞𝑁𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−1}=0 (41) and referring to (33), (38) and letting 𝑘→+∞, we get 𝜓𝜀2𝜀≤𝜓2𝜀−𝜙2, (42) so 𝜙(𝜀/2)=0, which is a contradiction with respect to 𝜀>0. Thus we proved that {𝑥_𝑛}_𝑛∈ℕ is a Cauchy sequence in the metric space (𝑋,𝑑_𝑝).
Since (𝑋,𝑝) is complete, then from Lemma 8, (𝑋,𝑑_𝑝) is a complete metric space. Therefore, the sequence {𝑥_𝑛}_𝑛∈ℕ converges to some 𝑢∈𝑋 in (𝑋,𝑑_𝑝), that is, lim_𝑛→+∞𝑑_𝑝𝑥_𝑛,𝑢=0. (43) Again, from Lemma 8, 𝑝(𝑢,𝑢)=lim_𝑛→+∞𝑝𝑥_𝑛,𝑢=lim_𝑛→+∞𝑝𝑥_𝑛,𝑥_𝑛. (44) On the other hand, thanks to (24) and the condition (𝑝2) from Definition 1, lim_𝑛→+∞𝑝𝑥_𝑛,𝑥_𝑛=0, (45) so it follows that 𝑝(𝑢,𝑢)=lim_𝑛→+∞𝑝𝑥_𝑛,𝑢=lim_𝑛→+∞𝑝𝑥_𝑛,𝑥_𝑛=0. (46) Now, we show that 𝑝(𝑢,𝑇𝑢)=0. Assume this is not true, then from (9) we obtain 𝜓𝑝𝑥_𝑛+1,𝑇𝑢=𝜓𝑝𝑇𝑥_𝑛𝑀𝑥,𝑇𝑢≤𝜓_𝑛,𝑢−𝜙𝑀𝑥_𝑛𝑑,𝑢+𝐿min^𝑝_𝑚𝑥_𝑛,𝑇𝑥_𝑛,𝑑^𝑝_𝑚(),𝑑𝑢,𝑇𝑢_𝑝𝑚𝑢,𝑇𝑥_𝑛,𝑑^𝑝_𝑚𝑥_𝑛,𝑇𝑢, (47) where 𝑀𝑥_𝑛,𝑢=max𝑝𝑥_𝑛,𝑢,𝑝𝑥_𝑛,𝑇𝑥_𝑛(),1,𝑝𝑢,𝑇𝑢2𝑝𝑥_𝑛,𝑇𝑢+𝑝𝑢,𝑇𝑥_𝑛=max𝑝𝑥_𝑛,𝑢,𝑝𝑥_𝑛,𝑥_𝑛+1(),1,𝑝𝑢,𝑇𝑢2𝑝𝑥_𝑛,𝑇𝑢+𝑝𝑢,𝑥_𝑛+1. (48) Thanks to (46), it is obvious that lim_𝑛→+∞𝑝(𝑥_𝑛,𝑇𝑢)=𝑝(𝑢,𝑇𝑢). Therefore, using (24) and again (46), we deduce that lim_𝑛→+∞𝑀𝑥_𝑛(),1,𝑢=max0,0,𝑝𝑢,𝑇𝑢2𝑝(().𝑢,𝑇𝑢)=𝑝𝑢,𝑇𝑢 (49) Also lim_𝑛→+∞𝑁𝑥_𝑛,𝑢=0 (50) because (24) and (45) give lim_𝑛→+∞𝑑^𝑝_𝑚(𝑥_𝑛,𝑇𝑥_𝑛)=0. Now, taking the upper limit as 𝑛→+∞, we obtain using the properties of 𝜓 and 𝜙𝜓𝑝(𝑢,𝑇𝑢)≤𝜓𝑝(𝑢,𝑇𝑢)−𝜙𝑝(𝑢,𝑇𝑢), (51) so 𝜙(𝑝(𝑢,𝑇𝑢))=0, that is, 𝑝(𝑢,𝑇𝑢)=0, so 𝑇𝑢=𝑢. We conclude that 𝑇 has a fixed point 𝑢∈𝑋 and 𝑝(𝑢,𝑢)=0.
Now if 𝑣≠𝑢 (so 𝑝(𝑢,𝑣)≠0) is another fixed point of 𝑇 (with 𝑝(𝑣,𝑣)=0), then by (46), 𝑁(𝑢,𝑣)=min𝑑^𝑝_𝑚(𝑢,𝑇𝑢),𝑑^𝑝_𝑚(),𝑑𝑣,𝑇𝑣_𝑝𝑚(𝑢,𝑇𝑣),𝑑^𝑝_𝑚(𝑑𝑣,𝑇𝑢)=min^𝑝_𝑚(𝑢,𝑢),𝑑^𝑝_𝑚(𝑣,𝑣),𝑑^𝑝_𝑚(𝑢,𝑣),𝑑^𝑝_𝑚(𝑀()𝑝()()(),1𝑣,𝑢)=0,𝑢,𝑣=max𝑢,𝑣,𝑝𝑢,𝑇𝑢,𝑝𝑣,𝑇𝑣2𝑝()(𝑝()()(),1𝑢,𝑇𝑣+𝑝𝑣,𝑇𝑢)=max𝑢,𝑣,𝑝𝑢,𝑢,𝑝𝑣,𝑣2𝑝()(𝑝()1𝑢,𝑣+𝑝𝑣,𝑢)=max𝑢,𝑣,0,0,2𝑝()(().𝑢,𝑣+𝑝𝑣,𝑢)=𝑝𝑢,𝑣 (52) Hence, using (9) we obtain 𝜓𝑝()𝑝()(𝑀((𝑀(()𝑝()𝑝(),𝑢,𝑣=𝜓𝑇𝑢,𝑇𝑣≤𝜓𝑢,𝑣))−𝜙𝑢,𝑣))+𝐿𝑁𝑣,𝑢=𝜓𝑢,𝑣−𝜙𝑢,𝑣 (53) that is, 𝑝(𝑢,𝑣)=0, which is a contradiction. The proof of Theorem 12 is completed.

As a consequence of Theorem 12, we may state the following corollaries.

First, taking 𝐿=0 in Theorem 12, we have the following.

Corollary 13. Let (𝑋,𝑝) be a complete partial metric space. Let 𝑇∶𝑋→𝑋 be a self mapping. Suppose there exist 𝜓∈Ψ and 𝜙∈Φ such that for all 𝑥,𝑦∈𝑋𝜓𝑝𝑇𝑥,𝑇𝑦≤𝜓𝑀𝑥,𝑦−𝜓𝑀𝑥,𝑦. (54) Then 𝑇 has a unique fixed point, say 𝑢∈𝑋. Also, one has 𝑝(𝑢,𝑢)=0.

Corollary 14. Let (𝑋,𝑝) be a complete partial metric space. Let 𝑇∶𝑋→𝑋 be a self mapping. Suppose there exist 𝑘∈[0,1) and 𝐿≥0 such that for all 𝑥,𝑦∈𝑋𝑝𝑑𝑇𝑥,𝑇𝑦≤𝑘𝑀𝑥,𝑦+𝐿min^𝑝_𝑚(𝑥,𝑇𝑥),𝑑^𝑝_𝑚,𝑑𝑦,𝑇𝑦_𝑝𝑚𝑥,𝑇𝑦,𝑑^𝑝_𝑚𝑦,𝑇𝑥. (55) Then 𝑇 has a unique fixed point, say 𝑢∈𝑋. Also, one has 𝑝(𝑢,𝑢)=0.

Proof. It follows by taking 𝜓(𝑡)=𝑡 and 𝜙(𝑡)=(1−𝑘)(𝑡) in Theorem 12.

Denote by Λ the set of functions 𝜆∶[0,+∞)→[0,+∞) satisfying the following hypotheses: (1)𝜆 is a Lebesgue-integrable mapping on each compact subset of [0,+∞), (2) for every 𝜖>0, we have ∫^𝜖₀𝜆(𝑠)𝑑𝑠>0.

We have the following result.

Corollary 15. Let (𝑋,𝑝) be a complete partial metric space. Let 𝑇∶𝑋→𝑋 be a self mapping. Suppose there exist 𝛼,𝛽∈Λ and 𝐿≥0 such that for all 𝑥,𝑦∈𝑋₀^{𝑝(𝑇𝑥,𝑇𝑦)}𝛼(𝑠)𝑑𝑠≤₀^{𝑝(𝑇𝑥,𝑇𝑦)}𝛼(𝑠)𝑑𝑠−₀^{𝑀(𝑥,𝑦)}𝛽(𝑠)𝑑𝑑𝑠+𝐿min^𝑝_𝑚(𝑥,𝑇𝑥),𝑑^𝑝_𝑚,𝑑𝑦,𝑇𝑦_𝑝𝑚𝑥,𝑇𝑦,𝑑^𝑝_𝑚𝑦,𝑇𝑥. (56) Then 𝑇 has a unique fixed point, say 𝑢∈𝑋. Also, one has 𝑝(𝑢,𝑢)=0.

Proof. It follows from Theorem 12 by taking 𝜓(𝑡)=^𝑡₀𝛼(𝑠)𝜙(𝑡)=𝑑𝑠,_𝑡0𝛽(𝑠)𝑑𝑠. (57)

Taking 𝐿=0 in Corollary 15, we obtain the following result.

Corollary 16. Let (𝑋,𝑝) be a complete partial metric space. Let 𝑇∶𝑋→𝑋 be a self mapping. Suppose there exist 𝛼,𝛽∈Λ such that for all 𝑥,𝑦∈𝑋₀^{𝑝(𝑇𝑥,𝑇𝑦)}𝛼(𝑠)𝑑𝑠≤₀^{𝑝(𝑇𝑥,𝑇𝑦)}𝛼(𝑠)𝑑𝑠−₀^{𝑀(𝑥,𝑦)}𝛽(𝑠)𝑑𝑠. (58) Then 𝑇 has a unique fixed point, say 𝑢∈𝑋. Also, one has 𝑝(𝑢,𝑢)=0.

Now, let ℱ be the set of functions 𝜑∶[0,+∞)→[0,+∞) satisfying the following hypotheses:(𝜑₁)𝜑 is nondecreasing(𝜑₂)∑^+∞_𝑛=0𝜑^𝑛(𝑡) converges for all 𝑡>0.

Note that if 𝜑∈ℱ, 𝜑 is said a (𝐶)-comparison function. It is easily proved that if 𝜑 is a (𝐶)-comparison function, then 𝜑(𝑡)<𝑡 for any 𝑡>0. Our second main result is as follows.

Theorem 17. Let (𝑋,𝑝) be a complete partial metric space. Let 𝑇∶𝑋→𝑋 be a mapping such that there exist 𝜑∈ℱ and 𝐿≥0 such that for all 𝑥,𝑦∈𝑋𝑝𝑀𝑑𝑇𝑥,𝑇𝑦≤𝜑𝑥,𝑦+𝐿min^𝑝_𝑚(𝑥,𝑇𝑥),𝑑^𝑝_𝑚,𝑑𝑦,𝑇𝑦_𝑝𝑚𝑥,𝑇𝑦,𝑑^𝑝_𝑚𝑦,𝑇𝑥. (59) Then 𝑇 has a unique fixed point, say 𝑢∈𝑋. Also, one has 𝑝(𝑢,𝑢)=0.

Proof. Let 𝑥₀∈𝑋. Let {𝑥_𝑛}_𝑛∈ℕ in 𝑋 such that 𝑥_𝑛=𝑇𝑥_𝑛−1 for all 𝑛≥1.
If for some 𝑛∈ℕ,𝑝(𝑥_𝑛,𝑥_𝑛+1)=0, the proof is completed. Assume that 𝑝(𝑥_𝑛,𝑥_𝑛+1)≠0 for all 𝑛≥0.
From (59) 𝑝𝑥_𝑛,𝑥_𝑛+1=𝑝𝑇𝑥_𝑛−1,𝑇𝑥_𝑛≤𝜑𝑀𝑥_𝑛−1,𝑥_𝑛=𝐿min𝑑^𝑝_𝑚𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑑^𝑝_𝑚𝑥_𝑛,𝑇𝑥_𝑛,𝑑^𝑝_𝑚𝑥_𝑛−1,𝑇𝑥_𝑛,𝑑^𝑝_𝑚𝑥_𝑛,𝑇𝑥_𝑛−1. (60) As explained in the proof of Theorem 12, we may get min𝑑^𝑝_𝑚𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑑^𝑝_𝑚𝑥_𝑛,𝑇𝑥_𝑛,𝑑^𝑝_𝑚𝑥_𝑛−1,𝑇𝑥_𝑛,𝑑^𝑝_𝑚𝑥_𝑛,𝑇𝑥_𝑛−1𝑀𝑥=0,_𝑛−1,𝑥_𝑛=max𝑝𝑥_𝑛−1,𝑥_𝑛,𝑝𝑥_𝑛,𝑥_𝑛+1. (61) Therefore 𝑝𝑥_𝑛,𝑥_𝑛+1≤𝜑max𝑝𝑥_𝑛−1,𝑥_𝑛,𝑝𝑥_𝑛,𝑥_𝑛+1. (62) If for some 𝑛≥1, we have 𝑝(𝑥_𝑛−1,𝑥_𝑛)≤𝑝(𝑥_𝑛,𝑥_𝑛+1). So from (62), we obtain that 𝑝𝑥_𝑛,𝑥_𝑛+1≤𝜑𝑝𝑥_𝑛,𝑥_𝑛+1<𝑝𝑥_𝑛,𝑥_𝑛+1, (63) a contradiction. Thus, for all 𝑛≥1, we have 𝑀𝑥_𝑛−1,𝑥_𝑛=max𝑝𝑥_𝑛−1,𝑥_𝑛,𝑝𝑥_𝑛,𝑥_𝑛+1=𝑝𝑥_𝑛−1,𝑥_𝑛. (64) Using (62) and (64), we get that 𝑝𝑥_𝑛,𝑥_𝑛+1≤𝜑𝑝𝑥_𝑛−1,𝑥_𝑛∀𝑛≥1. (65) By induction, we get 𝑝𝑥_𝑛,𝑥_𝑛+1≤𝜑^𝑛𝑝𝑥₀,𝑥₁ (66) for all 𝑛≥0. By triangle inequality, we have for 𝑚>𝑛𝑝𝑥_𝑛,𝑥_𝑚≤^{𝑘=𝑚−1}_𝑘=𝑛𝑝𝑥_𝑘,𝑥_𝑘+1−^{𝑘=𝑚−1}_𝑘=𝑛+1𝑝𝑥_𝑘,𝑥_𝑘≤^{𝑘=𝑚−1}_𝑘=𝑛𝑝𝑥_𝑘,𝑥_𝑘+1≤^𝑘=+∞_𝑘=𝑛𝑝𝑥_𝑘,𝑥_𝑘+1≤^𝑘=+∞_𝑘=𝑛𝜑^𝑘𝑝𝑥₀,𝑥₁. (67) Keeping in mind that 𝜑 is a (𝐶)-comparison function, then lim_𝑛→+∞∑^𝑘=+∞_𝑘=𝑛𝜑^𝑘(𝑝(𝑥₀,𝑥₁))=0 and so {𝑥_𝑛}_𝑛∈ℕ is a Cauchy sequence in (𝑋,𝑝) with lim_{𝑛,𝑚→+∞}𝑝(𝑥_𝑛,𝑥_𝑚)=0. Since (𝑋,𝑝) is complete then {𝑥_𝑛}_𝑛∈ℕ converges, with respect to 𝜏_𝑝, to a point 𝑢∈𝑋 such that 𝑝(𝑢,𝑢)=lim_𝑛→+∞𝑝𝑥_𝑛,𝑢=lim_{𝑛,𝑚→+∞}𝑝𝑥_𝑛,𝑥_𝑚=0. (68) Now we claim that 𝑝(𝑢,𝑇𝑢)=0. Suppose the contrary, then 𝑝(𝑢,𝑇𝑢)>0. By (59), we have 𝑝(𝑢,𝑇𝑢)≤𝑝𝑢,𝑥_𝑛+1+𝑝𝑇𝑥_𝑛,𝑇𝑢≤𝑝𝑢,𝑥_𝑛+1+𝜑𝑀𝑥_𝑛𝑑,𝑢+𝐿min^𝑝_𝑚𝑥_𝑛,𝑇𝑥_𝑛,𝑑^𝑝_𝑚(),𝑑𝑢,𝑇𝑢_𝑝𝑚𝑢,𝑇𝑥_𝑛,𝑑^𝑝_𝑚𝑥_𝑛,𝑇𝑢, (69) where 𝑀𝑥_𝑛,𝑢=max𝑝𝑥_𝑛,𝑢,𝑝𝑥_𝑛,𝑇𝑥_𝑛(),1,𝑝𝑢,𝑇𝑢2𝑝𝑥_𝑛,𝑇𝑢+𝑝𝑢,𝑇𝑥_𝑛=max𝑝𝑥_𝑛,𝑢,𝑝𝑥_𝑛,𝑥_𝑛+1(),1,𝑝𝑢,𝑇𝑢2𝑝𝑥_𝑛,𝑇𝑢+𝑝𝑢,𝑥_𝑛+1. (70) By (68), we have lim_𝑛→+∞min𝑑^𝑝_𝑚𝑥_𝑛,𝑇𝑥_𝑛,𝑑^𝑝_𝑚(),𝑑𝑢,𝑇𝑢_𝑝𝑚𝑢,𝑇𝑥_𝑛,𝑑^𝑝_𝑚𝑥_𝑛,𝑇𝑢=0,lim_𝑛→+∞𝑀𝑥_𝑛,𝑢=𝑝(𝑢,𝑇𝑢). (71) Therefore 𝑝(𝑢,𝑇𝑢)≤𝜑𝑝(𝑢,𝑇𝑢)<𝑝(𝑢,𝑇𝑢), (72) which is a contradiction. That is 𝑝(𝑢,𝑇𝑢)=0. Thus, we obtained that 𝑢 is a fixed point for 𝑇 and 𝑝(𝑢,𝑢)=0.
Now if 𝑣≠𝑢 (so 𝑝(𝑢,𝑣)≠0) is another fixed point of 𝑇, then by (68), min𝑑^𝑝_𝑚(𝑢,𝑇𝑢),𝑑^𝑝_𝑚(𝑣,𝑇𝑣),𝑑^𝑝_𝑚(𝑢,𝑇𝑣),𝑑^𝑝_𝑚(𝑑𝑣,𝑇𝑢)=min^𝑝_𝑚(𝑢,𝑢),𝑑^𝑝_𝑚(𝑣,𝑣),𝑑^𝑝_𝑚(𝑢,𝑣),𝑑^𝑝_𝑚(𝑀()𝑝()()(),1𝑣,𝑢)=0,𝑢,𝑣=max𝑢,𝑣,𝑝𝑢,𝑇𝑢,𝑝𝑣,𝑇𝑣2𝑝()(𝑝()()(),1𝑢,𝑇𝑣+𝑝𝑣,𝑇𝑢)=max𝑢,𝑣,𝑝𝑢,𝑢,𝑝𝑣,𝑣2𝑝()(().𝑢,𝑣+𝑝𝑣,𝑢)=𝑝𝑢,𝑣 (73) Hence, using (59) we obtain 𝑝()()(𝑀(𝑑𝑢,𝑣=𝑝𝑇𝑢,𝑇𝑣≤𝜑𝑢,𝑣))+𝐿min^𝑝_𝑚(𝑢,𝑇𝑢),𝑑^𝑝_𝑚(),𝑑𝑣,𝑇𝑣_𝑝𝑚(𝑢,𝑇𝑣),𝑑^𝑝_𝑚(𝑝()()𝑣,𝑇𝑢)≤𝜑𝑢,𝑣<𝑝𝑢,𝑣 (74) which is a contradiction. Thus 𝑢=𝑣 and the proof of Theorem 17 is completed.

Taking 𝐿=0 in Theorem 17, we have the following.

Corollary 18. Let (𝑋,𝑝) be a complete partial metric space. Let 𝑇∶𝑋→𝑋 be a mapping such that there exists 𝜑∈ℱ such that for all 𝑥,𝑦∈𝑋𝑝𝑇𝑥,𝑇𝑦≤𝜑𝑀𝑥,𝑦. (75) Then 𝑇 has a unique fixed point, say 𝑢∈𝑋. Also, one has 𝑝(𝑢,𝑢)=0.

Taking 𝜑(𝑡)=ℎ𝑡 where 0≤ℎ<1 in Corollary 18, we obtain the Ćirić fixed point theorem [36] in the setting of metric spaces (by considering 𝑝=𝑑 is a metric).

Corollary 19. Let (𝑋,𝑑) be a complete metric space. Let 𝑇∶𝑋→𝑋 be a mapping such that there exists ℎ∈[0,1) such that for all 𝑥,𝑦∈𝑋𝑑𝑑(),1𝑇𝑥,𝑇𝑦≤ℎmax𝑥,𝑦,𝑑𝑥,𝑇𝑥,𝑑𝑦,𝑇𝑦2𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥. (76) Then 𝑇 has a unique fixed point.

Remark 20. Corollary 14 generalizes Theorem 10 (with 𝑓=𝑔=𝑇=𝑆) of Turkoglu and Ozturk [27]. Corollary 18 improves Theorem 1 of Altun et al. [4] by assuming that 𝜑 is not continuous.

3. Examples

We give in this section some examples making effective our obtained results.

Example 21. Let 𝑋=[0,1] and 𝑝(𝑥,𝑦)=max{𝑥,𝑦} for all 𝑥,𝑦∈𝑋. Then (𝑋,𝑝) is a complete partial metric space. Consider 𝑇∶𝑋→𝑋 defined by 𝑥𝑇𝑥=².1+𝑥 (77) Take 𝜓(𝑡)=𝑡 and 𝜙(𝑡)=𝑡/(1+𝑡) for all 𝑡≥0. Note that 𝜓∈Ψ and 𝜙∈Φ. Take 𝑥≤𝑦, then 𝜓𝑝=𝑦𝑇𝑥,𝑇𝑦₂𝑦1+𝑦=𝑦−𝑀𝑀𝑀𝑀𝑑1+𝑦=𝜓𝑥,𝑦−𝜙𝑥,𝑦since𝑀𝑥,𝑦=𝑦≤𝜓𝑥,𝑦−𝜙𝑥,𝑦+𝐿min^𝑝_𝑚(𝑥,𝑇𝑥),𝑑^𝑝_𝑚,𝑑𝑦,𝑇𝑦_𝑝𝑚𝑥,𝑇𝑦,𝑑^𝑝_𝑚𝑦,𝑇𝑥 (78) for all 𝐿≥0. Thus, (9) holds. Applying Theorem 12, 𝑇 has a unique fixed point, which is 𝑢=0.

Example 22. Let 𝑋={0,1,2,3,4} and 𝑝(𝑥,𝑦)=max{𝑥,𝑦}. Let 𝑇∶𝑋→𝑋 be defined as follows: 𝑇0=0,𝑇1=𝑇3=2,𝑇4=𝑇2=1. (79) By simple calculation, we get that 𝑝()()()()()𝑝()𝑝()()()()()𝑝()()()()𝑇2,𝑇2=𝑝𝑇4,𝑇4=𝑝𝑇2,𝑇0=𝑝𝑇4,𝑇0=𝑝𝑇2,𝑇4=1,𝑇0,𝑇0=0,𝑇1,𝑇1=𝑝𝑇3,𝑇3=𝑝𝑇1,𝑇0=𝑝𝑇3,𝑇0=𝑝𝑇1,𝑇3=2,𝑇4,𝑇1=𝑝𝑇4,𝑇3=𝑝𝑇2,𝑇1=𝑝𝑇2,𝑇3=2. (80) Hence, we derive that 𝑀()()()()()𝑀()()()()𝑀()𝑀()()()()()𝑁()()()()()()()𝑁()()()()()()𝑁()()𝑇1,𝑇4=𝑀𝑇2,𝑇4=𝑀𝑇3,𝑇4=𝑀𝑇4,𝑇4=𝑀𝑇0,𝑇4=4,𝑇1,𝑇3=𝑀𝑇2,𝑇3=𝑀𝑇0,𝑇3=𝑀𝑇3,𝑇3=3,𝑇0,𝑇0=0,𝑇1,𝑇2=𝑀𝑇2,𝑇0=𝑀𝑇1,𝑇1=𝑀𝑇0,𝑇1=𝑀𝑇2,𝑇2=2,𝑇1,𝑇4=𝑁𝑇2,𝑇1=𝑁𝑇1,𝑇0=𝑁𝑇2,𝑇0=𝑁𝑇3,𝑇0=𝑁𝑇4,𝑇0=𝑁𝑇0,𝑇0=0,𝑇1,𝑇1=𝑁𝑇1,𝑇3=𝑁𝑇2,𝑇3=𝑁𝑇3,𝑇3=𝑁𝑇2,𝑇2=𝑁𝑇2,𝑇4=1,𝑇4,𝑇3=2,𝑁𝑇4,𝑇4=3. (81) For 𝜓(𝑡)=𝑡/3, 𝜙(𝑡)=𝑡/6 and 𝐿≥1/5 all conditions of Theorem 12 are satisfied. Notice that 0 is the unique fixed point of 𝑇.

Example 23. Let 𝑋=[0,2] and 𝑝∶𝑋×𝑋→ℝ⁺ be defined by 𝑝(𝑥,𝑦)=max{𝑥,𝑦}. Define 𝑇∶𝑋→𝑋 by 𝑇(𝑥)=𝑥²[[,[[,4𝑥+1,if𝑥∈0,10,if𝑥∈1,23if𝑥=2 (82) and let 𝜑∶[0,+∞[→[0,+∞[ defined by 𝜑(𝑡)=𝑡².𝑡+1 (83) By induction, we have 𝜑^𝑛(𝑡)≤𝑡(𝑡/(1+𝑡))^𝑛 for all 𝑛≥1, so it is clear that 𝜑 is a (𝐶)-comparison function. Now we show that (59) is satisfied for all 𝑥,𝑦∈𝑋. It suffices to prove it for 𝑥≤𝑦. Consider the following six cases.
Case 1. Let 𝑥,𝑦∈[0,1[, then 𝑝=𝑦𝑇𝑥,𝑇𝑦²𝑝𝑀.𝑦+1=𝜑𝑥,𝑦≤𝜑𝑥,𝑦 (84)
Case 2. Let 𝑥,𝑦∈[1,2[, then 𝑝()𝑀.𝑇𝑥,𝑇𝑦=𝑝0,0=0≤𝜑𝑥,𝑦 (85)
Case 3. Let 𝑥=𝑦=2, then 𝑝4𝑇𝑥,𝑇𝑦=𝑝3,43=43(2)𝑀.=𝜑≤𝜑𝑥,𝑦 (86)
Case 4. Let 𝑥∈[0,1[ and 𝑦∈[1,2[ then 𝑝𝑥𝑇𝑥,𝑇𝑦=𝑝²=𝑥𝑥+1,0²≤𝑦𝑥+1₂𝑝𝑀.𝑦+1=𝜑𝑥,𝑦≤𝜑𝑥,𝑦 (87)
Case 5. Let 𝑥∈[0,1[ and 𝑦=2, then 𝑝𝑥𝑇𝑥,𝑇𝑦=𝑝²,4𝑥+13=43(2)𝑝𝑀.=𝜑=𝜑𝑥,𝑦≤𝜑𝑥,𝑦 (88)
Case 6. Let 𝑥∈[1,2[ and 𝑦=2 then 𝑝4𝑇𝑥,𝑇𝑦=𝑝0,3=43(2)𝑝𝑀.=𝜑=𝜑𝑥,𝑦≤𝜑𝑥,𝑦 (89) Since, for all 𝑥,𝑦∈𝑋𝐿min𝑑^𝑝_𝑚(𝑥,𝑇𝑥),𝑑^𝑝_𝑚𝑦,𝑇𝑦,𝑑^𝑝_𝑚𝑥,𝑇𝑦,𝑑^𝑝_𝑚𝑦,𝑇𝑥≥0 (90) then (59) is verified. Applying Theorem 17, 𝑇 has a unique fixed point, which is 𝑢=0.

All presented theorems involve generalized almost contractive mappings which have a unique fixed point. But, one of the main features of Berinde contractions is the fact that they do possess more that one fixed point. In this direction, Altun and Acar [35] proved the following result.

Theorem 24. Let (𝑋,𝑝) a complete partial metric space. Given 𝑇∶𝑋→𝑋 satisfying 𝑡ℎ𝑒𝑟𝑒𝑒𝑥𝑖𝑠𝑡𝑘∈[0,1),𝐿≥0𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑝𝑇𝑥,𝑇𝑦≤𝑘𝑝𝑥,𝑦+𝐿𝑑^𝑝_𝑚𝑥,𝑇𝑦, (91) for all 𝑥,𝑦∈𝑋. Then, 𝑇 has a fixed point.

The following example illustrates Theorem 24 where we have two fixed points.

Example 25. Let 𝑋={0,1,2}. A partial metric 𝑝∶𝑋×𝑋→ℝ⁺ is defined by 𝑝()()()=10,0=𝑝1,1=0,𝑝2,24,𝑝()()=10,1=𝑝1,03,𝑝(0,2)=𝑝(2,0)=11,𝑝()()=1241,2=𝑝2,12. (92) Define the mapping 𝑇∶𝑋→𝑋 by 𝑇0=𝑇2=0,𝑇1=1. (93) It is easy to show that (91) is satisfied. Applying Theorem 24, 𝑇 has a fixed point. Note that 𝑇 has two fixed points which are 0 and 1.