Further Remarks on Fixed-Point Theorems in the Context of Partial Metric Spaces

Abstract

New fixed-point theorems on metric spaces are established, and analogous results on partial metric spaces are deduced. This work can be considered as a continuation of the paper Samet et al. (2013).

1. Introduction

In 1994, Matthews [1] introduced the concept of partial metric spaces as a part of the study of denotational semantics of dataflow networks and showed that the Banach’s contraction principle can be generalized to the partial metric context for applications in program verification. Later on, many authors studied fixed-point theorems on partial metric spaces (see, e.g., [2–9] and references therein).

We start by recalling some basic definitions and properties of partial metric spaces (see [1, 5] for more details).

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, we have (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.

It is clear that, if 𝑝(𝑥,𝑦)=0, then from (P1) and (P2), 𝑥=𝑦; but if 𝑥=𝑦, 𝑝(𝑥,𝑦) may not be 0. A basic example of a partial metric space is the pair ([0,+∞),𝑝), where 𝑝(𝑥,𝑦)=max{𝑥,𝑦} for all 𝑥,𝑦∈[0,+∞). Other examples of partial metric spaces which are interesting from a computational point of view may be found in [1].

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∶𝑝𝑥,𝑦<𝑝(𝑥,𝑥)+𝜀, (1) for all 𝑥∈𝑋 and 𝜀>0.

Definition 2. Let (𝑋,𝑝) be a partial metric space and {𝑥_𝑛} a sequence in 𝑋. Then {𝑥_𝑛} converges to a point 𝑥∈𝑋 if and only if 𝑝(𝑥,𝑥)=lim_𝑛→+∞𝑝(𝑥,𝑥_𝑛). We may write this as 𝑥_𝑛→𝑥; {𝑥_𝑛} is called a Cauchy sequence if lim_{𝑛,𝑚→+∞}𝑝(𝑥_𝑛,𝑥_𝑚) exists and is finite; (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).
If 𝑝 is a partial metric on 𝑋, then the function 𝑝^𝑠∶𝑋×𝑋→[0,+∞) given by 𝑝^𝑠𝑥,𝑦=2𝑝𝑥,𝑦−𝑝(𝑥,𝑥)−𝑝𝑦,𝑦 (2) is a metric on 𝑋.

Lemma 3. Let (𝑋,𝑝) be a partial metric space. Then(a){𝑥_𝑛} is a Cauchy sequence in (𝑋,𝑝) if and only if it is a Cauchy sequence in the metric space (𝑋,𝑝^𝑠);(b)a partial metric space (𝑋,𝑝) is complete if and only if the metric space (𝑋,𝑝^𝑠) is complete; furthermore, lim_𝑛→+∞𝑝^𝑠(𝑥_𝑛,𝑥)=0 if and only if 𝑝(𝑥,𝑥)=lim_𝑛→+∞𝑝𝑥_𝑛,𝑥=lim_{𝑛,𝑚→+∞}𝑝𝑥_𝑛,𝑥_𝑚. (3)

In [1], Matthews extended the Banach contraction principle to the setting of partial metric spaces.

Theorem 4 (see [1]). Let (𝑋,𝑝) be a complete partial metric space and 𝑇∶𝑋→𝑋 a given mapping. Suppose that there exists a constant 𝑘∈(0,1) such that 𝑝𝑇𝑥,𝑇𝑦≤𝑘𝑝𝑥,𝑦,∀𝑥,𝑦∈𝑋. (4) Then 𝑇 has a unique fixed point 𝑧∈𝑋. Moreover, we have 𝑝(𝑧,𝑧)=0.

Very recently, in [10, 11], the authors proved that a large class of fixed-point theorems in partial metric spaces, including Matthews result, are immediate consequences of fixed-point theorems in metric spaces. More precisely, in [11], the authors established the following result.

Theorem 5 (see [11]). Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝑋, and 𝜑∶𝑋→[0,+∞) a lower semicontinuous function. Suppose that there exists 𝑘∈(0,1) such that 𝑑()𝑑(𝑥)𝑦,𝑇𝑥,𝑇𝑦+𝜑𝑇𝑥+𝜑𝑇𝑦≤𝑘𝑥,𝑦+𝜑+𝜑 (5) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point 𝑧∈𝑋. Moreover, one has 𝜑(𝑧)=0.

Now, taking 𝜑(𝑥)=𝑝(𝑥,𝑥) in Theorem 4, we obtain that 𝑝^𝑠()𝑝𝑇𝑥,𝑇𝑦+𝜑𝑇𝑥+𝜑𝑇𝑦≤𝑘^𝑠𝑥,𝑦+𝜑(𝑥)+𝜑𝑦, (6) for all 𝑥,𝑦∈𝑋. Applying Theorem 5, we obtain immediately the result of Matthews. Another technique suggested by Haghi et al. [10] let 𝐷 be a metric on 𝑋 defined by 𝐷𝑝𝑥,𝑦∶=𝑥,𝑦if𝑥≠𝑦,0if𝑥=𝑦. (7) It is not difficult to show that if (𝑋,𝑝) is complete, then (𝑋,𝐷) is complete. Now, Matthews’ contraction is equivalent to 𝐷𝑇𝑥,𝑇𝑦≤𝑘𝐷𝑥,𝑦, (8) for all 𝑥,𝑦∈𝑋. So, the result of Matthews can be deduced immediately from the Banach contraction principle. Observe that in Theorem 5, if we consider the mapping 𝐷_𝜑∶𝑋×𝑋→[0,+∞) defined by 𝐷_𝜑𝑑(𝑥)𝑦𝑥,𝑦∶=𝑥,𝑦+𝜑+𝜑if𝑥≠𝑦,0if𝑥=𝑦, (9) then 𝐷_𝜑 is a metric on 𝑋 and (𝑋,𝐷_𝜑) is a complete metric space. Moreover, the contractive condition in Theorem 5 is equivalent to 𝐷_𝜑𝑇𝑥,𝑇𝑦≤𝑘𝐷_𝜑𝑥,𝑦, (10) for all 𝑥,𝑦∈𝑋. So, Theorem 5 can also be deduced from the Banach contraction principle.

Define the function 𝐹∶[0,+∞)³→[0,+∞) by 𝐹(𝑎,𝑏,𝑐)∶=𝑎+𝑏+𝑐,∀𝑎,𝑏,𝑐≥0. (11) Then, the contractive condition in Theorem 5 is equivalent to 𝐹𝑑𝑇𝑥,𝑇𝑦,𝜑(𝑇𝑥),𝜑𝑇𝑦≤𝑘𝐹𝑑𝑥,𝑦,𝜑(𝑥),𝜑𝑦, (12) for all 𝑥,𝑦∈𝑋. In this paper, we establish new fixed-point theorems in metric spaces involving a function 𝐹∶[0,+∞)³→[0,+∞), where the mapping 𝐷_𝜑𝐹𝑑(𝑥)𝑦𝑥,𝑦∶=𝑥,𝑦,𝜑,𝜑if𝑥≠𝑦,0if𝑥=𝑦 (13) is not a metric on 𝑋. Some fixed-point theorems in partial metric spaces are deduced from our main results in metric spaces.

2. Main Result

We denote by ℱ the set of functions 𝐹∶[0,+∞)³→[0,+∞) satisfying the following conditions:(i)max{𝑎,𝑏}≤𝐹(𝑎,𝑏,𝑐) for all 𝑎,𝑏,𝑐∈[0,+∞),(ii)𝐹(0,0,0)=0,(iii)𝐹 is continuous.

As examples, the following functions belong to ℱ:(a)𝐹(𝑎,𝑏,𝑐)=(𝑎+𝑏)(𝑐+1),(b)𝐹(𝑎,𝑏,𝑐)=𝑎(ln(𝑐+1)+1)+𝑏(𝑒^𝑐+1),(c)𝐹(𝑎,𝑏,𝑐)=𝑎+𝑏+𝑐,(d)𝐹(𝑎,𝑏,𝑐)=max{𝑎,𝑏,𝑐},(e)𝐹(𝑎,𝑏,𝑐)=𝑎+max{𝑎,𝑏,𝑐}.

We denote by Ψ the set of functions 𝜓∶[0,+∞)→[0,+∞) satisfying the following conditions:(j)𝜓 is nondecreasing,(jj)∑_𝑛≥1𝜓^𝑛(𝑡)<∞ for each 𝑡>0, where 𝜓^𝑛 is the 𝑛th iterate of 𝜓.

It is not difficult to show that if 𝜓∈Ψ, then 𝜓(𝑡)<𝑡 for every 𝑡>0. These functions are known in the literature as (c)-comparison functions.

Definition 6. Let (𝑋,𝑑) be a metric space, 𝑇∶𝑋→𝑋, 𝜑∶𝑋→[0,+∞), and 𝐹∈ℱ. We say that the pair (𝑇,𝜑) satisfies an 𝐹-generalized 𝜓-contraction if there exists 𝜓∈Ψ such that 𝐹𝑑()𝐹𝑑(𝑥)𝑦,𝐹𝑑()()(𝑥𝑑𝑦𝑇𝑥,𝑇𝑦,𝜑𝑇𝑥,𝜑𝑇𝑦≤𝜓max𝑥,𝑦,𝜑,𝜑𝑥,𝑇𝑥,𝜑𝑇𝑥,𝜑)+𝐹𝑦,𝑇𝑦,𝜑𝑇𝑦,𝜑×2⁻¹, (14) for every 𝑥,𝑦∈𝑋.

Our main result is giving by the following theorem.

Theorem 7. Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝑋, and 𝜑∶𝑋→[0,+∞). Suppose that the following conditions hold: (1)𝜑 is lower semicontinuous,(2)there exist 𝜓∈Ψ and 𝐹∈ℱ such that the pair (𝑇,𝜑) satisfies an 𝐹-generalized 𝜓-contraction.
Then 𝑇 has a unique fixed point 𝑧∈𝑋. Moreover, one has 𝜑(𝑧)=0.

Proof. Let 𝑥₀∈𝑋. Define the sequence {𝑥_𝑛}⊂𝑋 by 𝑥_𝑛+1=𝑇𝑥_𝑛 for all positive integer 𝑛. Let 𝑛 be a positive integer such that 𝑛≥1. Using condition (2) with 𝑥=𝑥_𝑛 and 𝑦=𝑥_𝑛−1, we obtain that 𝐹𝑑𝑥_𝑛+1,𝑥_𝑛,𝜑𝑥_𝑛+1,𝜑𝑥_𝑛≤𝜓max𝐹𝑑𝑥_𝑛,𝑥_𝑛−1,𝜑𝑥_𝑛,𝜑𝑥_𝑛−1,𝐹𝑑𝑥_𝑛,𝑥_𝑛+1,𝜑𝑥_𝑛+1,𝜑𝑥_𝑛. (15) Without restriction of the generality, we can suppose that 𝑥_𝑚+1≠𝑥_𝑚 for every 𝑚. Suppose that max𝐹𝑑𝑥_𝑛,𝑥_𝑛−1,𝜑𝑥_𝑛,𝜑𝑥_𝑛−1,𝐹𝑑𝑥_𝑛,𝑥_𝑛+1,𝜑𝑥_𝑛+1,𝜑𝑥_𝑛=𝐹𝑑𝑥_𝑛,𝑥_𝑛+1,𝜑𝑥_𝑛+1,𝜑𝑥_𝑛. (16) In this case, we have 𝐹𝑑𝑥_𝑛+1,𝑥_𝑛,𝜑𝑥_𝑛+1,𝜑𝑥_𝑛≤𝜓𝐹𝑑𝑥_𝑛,𝑥_𝑛+1,𝜑𝑥_𝑛+1,𝜑𝑥_𝑛. (17) Note that 𝐹(𝑑(𝑥_𝑛,𝑥_𝑛+1),𝜑(𝑥_𝑛+1),𝜑(𝑥_𝑛))>0. Indeed, if 𝐹(𝑑(𝑥_𝑛,𝑥_𝑛+1),𝜑(𝑥_𝑛+1),𝜑(𝑥_𝑛))=0, from condition (i), we have 𝑑(𝑥_𝑛,𝑥_𝑛+1)≤𝐹(𝑑(𝑥_𝑛,𝑥_𝑛+1),𝜑(𝑥_𝑛+1),𝜑(𝑥_𝑛))=0, which implies that 𝑥_𝑛=𝑥_𝑛+1, that is a contradiction with the assumption 𝑥_𝑚+1≠𝑥_𝑚 for every 𝑚. Since 𝜓(𝑡)<𝑡 for every 𝑡>0, we get that 𝐹𝑑𝑥_𝑛+1,𝑥_𝑛,𝜑𝑥_𝑛+1,𝜑𝑥_𝑛<𝐹𝑑𝑥_𝑛+1,𝑥_𝑛,𝜑𝑥_𝑛+1,𝜑𝑥_𝑛, (18) which is a contradiction. Then, we deduce that 𝐹𝑑𝑥_𝑛+1,𝑥_𝑛,𝜑𝑥_𝑛+1,𝜑𝑥_𝑛≤𝜓𝐹𝑑𝑥_𝑛,𝑥_𝑛−1,𝜑𝑥_𝑛,𝜑𝑥_𝑛−1, (19) for every 𝑛≥1. Using the property (i) of 𝐹 and the monotony property of 𝜓, we obtain that max𝜑𝑥_𝑛+1,𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜓^𝑛𝐹𝑑𝑥₁,𝑥₀,𝜑𝑥₁,𝜑𝑥₀, (20) for every 𝑛≥1. Fix 𝜀>0, and let ℎ=ℎ(𝜀) be a positive integer (given by (jj)) such that _𝑛≥ℎ𝜓^𝑛𝐹𝑑𝑥₁,𝑥₀,𝜑𝑥₁,𝜑𝑥₀<𝜀. (21) Let 𝑚>𝑛>ℎ; using the triangular inequality and (20), we obtain 𝑑𝑥_𝑛,𝑥_𝑚≤^𝑚−1_𝑘=𝑛𝑑𝑥_𝑘,𝑥_𝑘+1≤_𝑛≥ℎ𝜓^𝑛𝐹𝑑𝑥₁,𝑥₀,𝜑𝑥₁,𝜑𝑥₀<𝜀. (22) Thus we proved that {𝑥_𝑛} is a Cauchy sequence in the metric space (𝑋,𝑑). Since (𝑋,𝑑) is complete, there exists some 𝑧∈𝑋 such that 𝑥_𝑛→𝑧 as 𝑛→∞. We shall prove that 𝑧 is a fixed point of 𝑇. At first, observe that from condition (jj) and (20), we have lim_𝑛→+∞𝜑𝑥_𝑛=0, (23) which implies (since 𝜑 is lower semicontinuous) that 𝜑(𝑧)=0. (24) Using condition (2) with 𝑥=𝑧 and 𝑦=𝑥_𝑛, we get that 𝐹𝑑𝑇𝑧,𝑥_𝑛+1,𝜑(𝑇𝑧),𝜑𝑥_𝑛+1≤𝜓max𝐹𝑑𝑧,𝑥_𝑛,𝜑(𝑧),𝜑𝑥_𝑛,𝐹𝑑()()(𝑧𝑑𝑥𝑧,𝑇𝑧,𝜑𝑇𝑧,𝜑)+𝐹_𝑛,𝑥_𝑛+1,𝜑𝑥_𝑛+1,𝜑𝑥_𝑛×2⁻¹, (25) for every 𝑛. Suppose that 𝑧≠𝑇𝑧. Using the continuity of 𝐹, (23), and (24), we have lim_𝑛→+∞max𝐹𝑑𝑧,𝑥_𝑛,𝜑(𝑧),𝜑𝑥_𝑛,𝐹𝑑()()(𝑧)𝑑𝑥𝑧,𝑇𝑧,𝜑𝑇𝑧,𝜑+𝐹_𝑛,𝑥_𝑛+1,𝜑𝑥_𝑛+1,𝜑𝑥_𝑛×2⁻¹=𝐹𝑑(𝑧,𝑇𝑧),𝜑(𝑇𝑧),02>0. (26) Then, there exists a positive integer 𝑁 such that max𝐹𝑑𝑧,𝑥_𝑛,𝜑(𝑧),𝜑𝑥_𝑛,𝐹𝑑()()(𝑧)𝑑𝑥𝑧,𝑇𝑧,𝜑𝑇𝑧,𝜑+𝐹_𝑛,𝑥_𝑛+1,𝜑𝑥_𝑛+1,𝜑𝑥_𝑛×2⁻¹>0, (27) for every 𝑛≥𝑁. Thus, we have 𝐹𝑑𝑇𝑧,𝑥_𝑛+1,𝜑(𝑇𝑧),𝜑𝑥_𝑛+1<max𝐹𝑑𝑧,𝑥_𝑛,𝜑(𝑧),𝜑𝑥_𝑛,𝐹𝑑()()(𝑧𝑑𝑥𝑧,𝑇𝑧,𝜑𝑇𝑧,𝜑)+𝐹_𝑛,𝑥_𝑛+1,𝜑𝑥_𝑛+1,𝜑𝑥_𝑛×2⁻¹. (28) Letting 𝑛→+∞ in the above inequality, we get that 𝐹𝑑()()≤𝐹𝑑()()𝑇𝑧,𝑧,𝜑𝑇𝑧,0𝑧,𝑇𝑧,𝜑𝑇𝑧,02, (29) which is a contradiction. Thus, we proved that 𝑇𝑧=𝑧. Now, suppose that 𝑧^′∈𝑋 is another fixed point of 𝑇. We can observe that 𝜑𝑧^′=0. (30) Indeed, applying (14) with 𝑥=𝑦=𝑧^′, we get that 𝐹0,𝜑𝑧^′,𝜑𝑧^′≤𝜓𝐹0,𝜑𝑧^′,𝜑𝑧^′, (31) which implies (30). Now, applying (14) with 𝑥=𝑧 and 𝑦=𝑧^′, we obtain 𝐹𝑑𝑧,𝑧^′,𝜑(𝑧),𝜑𝑧^′≤𝜓max𝐹𝑑𝑧,𝑧^′,𝜑(𝑧),𝜑𝑧^′,𝐹0,𝜑(𝑧),𝜑(𝑧)+𝐹0,𝜑𝑧^′,𝜑𝑧^′2. (32) Using (24) and (30), we obtain 𝐹𝑑𝑧,𝑧^′,0,0≤𝜓𝐹𝑑𝑧,𝑧^′,0,0, (33) which implies that 𝑑(𝑧,𝑧^′)=0; that is, 𝑧=𝑧^′. Thus, we proved that 𝑇 has a unique fixed point 𝑧∈𝑋 with 𝜑(𝑧)=0.

Example 8. Let 𝑇∶[0,1]→[0,1] be the mapping defined by 𝑥𝑇𝑥=14if0≤𝑥<1,2if𝑥=1. (34) We endow 𝑋=[0,1] with the standard metric 𝑑(𝑥,𝑦)=|𝑥−𝑦|. Let 𝜑(𝑥)=𝑥 for all 𝑥∈𝑋, 𝜓(𝑡)=(3/4)𝑡 for all 𝑡≥0, and 𝐹(𝑎,𝑏,𝑐)=𝑎+𝑏+𝑐 for all 𝑎,𝑏,𝑐≥0. Clearly, 𝜑 is continuous, 𝜓∈Ψ, and 𝐹∈ℱ. Moreover, we have 𝐹𝑑()𝐹𝑑(𝑥)𝑦,𝑇𝑥,𝑇𝑦,𝜑𝑇𝑥,𝜑𝑇𝑦≤𝜓𝑥,𝑦,𝜑,𝜑 (35) for all 𝑥,𝑦∈𝑋. By Theorem 7, 𝑇 has a unique fixed point 𝑧∈𝑋 (𝑧=0) with 𝜑(𝑧)=0. Note that in this example, the Banach contraction principle cannot be used since the mapping 𝑇 is not continuous.

3. Particular Cases

In this section, new fixed point results are deduced from our main result given by Theorem 7.

Corollary 9. Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝑋, and 𝜑∶𝑋→[0,+∞). Suppose that the following conditions hold: (1)𝜑 is lower semicontinuous,(2) there exists 𝜓∈Ψ such that 𝑑≤𝜓𝑑(𝑥)𝜑𝑦()𝜑×𝜑𝑇𝑥,𝑇𝑦𝑥,𝑦+𝜑+1−𝜑𝑇𝑥𝑇𝑦+1𝑇𝑦+1⁻¹, (36) for all 𝑥,𝑦∈𝑋.

Then 𝑇 has a unique fixed point 𝑧∈𝑋. Moreover, we have 𝜑(𝑧)=0.

Proof. It follows from Theorem 7 with 𝐹(𝑎,𝑏,𝑐)=(𝑎+𝑏)(𝑐+1).

Corollary 10. Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝑋, and 𝜑∶𝑋→[0,+∞). Suppose that the following conditions hold:(1)𝜑 is lower semicontinuous,(2) there exists 𝜓∈Ψ such that 𝑑≤𝜓𝑑()()𝜑(𝑥)𝑇𝑥,𝑇𝑦𝑥,𝑇𝑥+𝜑𝑇𝑥+12()𝜑×−𝜑𝑇𝑥𝑇𝑦+1𝜑(𝑇𝑦)+1⁻¹, (37) for all 𝑥,𝑦∈𝑋.

Then 𝑇 has a unique fixed point 𝑧∈𝑋. Moreover, we have 𝜑(𝑧)=0.

Proof. It follows from Theorem 7 with 𝐹(𝑎,𝑏,𝑐)=(𝑎+𝑏)(𝑐+1).

Corollary 11. Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝑋, and 𝜑∶𝑋→[0,+∞). Suppose that the following conditions hold:(1)𝜑 is lower semicontinuous,(2) there exists 𝜓∈Ψ such that 𝑑≤𝜓𝑑𝜑𝑦𝑇𝑥,𝑇𝑦𝑦,𝑇𝑦+𝜑𝑇𝑦+12()𝜑×−𝜑𝑇𝑥𝑇𝑦+1𝜑(𝑇𝑦)+1⁻¹, (38) for all 𝑥,𝑦∈𝑋.

Then 𝑇 has a unique fixed point 𝑧∈𝑋. Moreover, we have 𝜑(𝑧)=0.

Proof. It follows from Theorem 7 with 𝐹(𝑎,𝑏,𝑐)=(𝑎+𝑏)(𝑐+1).

Corollary 12. Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝑋, and 𝜑∶𝑋→[0,+∞). Suppose that the following conditions hold:(1)𝜑 is lower semicontinuous,(2) there exists 𝜓∈Ψ such that 𝑑≤𝜓𝑑𝜑𝑦(𝑥)𝑒𝑇𝑥,𝑇𝑦𝑥,𝑦ln+1+1+𝜑^𝜑(𝑦)(𝑒+1−𝜑𝑇𝑥)^{𝜑(𝑇𝑦)}×+1ln𝜑(𝑇𝑦)+1+1⁻¹, (39) for all 𝑥,𝑦∈𝑋.

Then 𝑇 has a unique fixed point 𝑧∈𝑋. Moreover, we have 𝜑(𝑧)=0.

Proof. It follows from Theorem 7 with 𝐹(𝑎,𝑏,𝑐)=𝑎(ln(𝑐+1)+1)+𝑏(𝑒^𝑐+1).

Corollary 13. Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝑋, and 𝜑∶𝑋→[0,+∞). Suppose that the following conditions hold:(1)𝜑 is lower semicontinuous,(2) there exists 𝜓∈Ψ such that 𝑑≤𝜓𝑑(𝜑(𝑥)()𝑒𝑇𝑥,𝑇𝑦𝑥,𝑇𝑥)ln+1+1+𝜑𝑇𝑥^𝜑(𝑥)+12−𝜑(𝑇𝑥)𝑒^{𝜑(𝑇𝑦)}×+1ln𝜑(𝑇𝑦)+1+1⁻¹, (40) for all 𝑥,𝑦∈𝑋.

Then 𝑇 has a unique fixed point 𝑧∈𝑋. Moreover, we have 𝜑(𝑧)=0.

Proof. It follows from Theorem 7 with 𝐹(𝑎,𝑏,𝑐)=𝑎(ln(𝑐+1)+1)+𝑏(𝑒^𝑐+1).

Corollary 14. Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝑋, and 𝜑∶𝑋→[0,+∞). Suppose that the following conditions hold: (1)𝜑 is lower semicontinuous,(2) there exists 𝜓∈Ψ such that 𝑑≤𝜓𝑑𝜑𝑦𝑒𝑇𝑥,𝑇𝑦𝑦,𝑇𝑦ln+1+1+𝜑𝑇𝑦^𝜑(𝑦)+12−𝜑(𝑇𝑥)𝑒^{𝜑(𝑇𝑦)}×+1ln𝜑(𝑇𝑦)+1+1⁻¹, (41) for all 𝑥,𝑦∈𝑋.

Then 𝑇 has a unique fixed point 𝑧∈𝑋. Moreover, we have 𝜑(𝑧)=0.

Proof. It follows from Theorem 7 with 𝐹(𝑎,𝑏,𝑐)=𝑎(ln(𝑐+1)+1)+𝑏(𝑒^𝑐+1).

Many other results can be deduced from Theorem 7 by considering different choices of 𝐹.

4. Applications to Partial Metric Spaces

We will show that the following fixed-point theorem in partial metric spaces can be deduced from Theorem 7.

Corollary 15. Let (𝑋,𝑝) be a complete partial metric space, and let 𝑇∶𝑋→𝑋. Suppose that there exists 𝜓∈Ψ such that 𝑝𝑝,𝑝()𝑇𝑥,𝑇𝑦≤𝜓max𝑥,𝑦𝑥,𝑇𝑥+𝑝𝑦,𝑇𝑦2, (42) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point 𝑧∈𝑋. Moreover, one has 𝑝(𝑧,𝑧)=0.

Proof. Let 𝑑(𝑥,𝑦)∶=𝑝^𝑠(𝑥,𝑦)/2 for all 𝑥,𝑦∈𝑋 and 𝜑(𝑥)=𝑝(𝑥,𝑥)/2 for all 𝑥∈𝑋. We have 𝑝𝑥,𝑦=𝐹𝑑𝑥,𝑦,𝜑(𝑥),𝜑𝑦,∀𝑥,𝑦∈𝑋, (43) where 𝐹(𝑎,𝑏,𝑐)=𝑎+𝑏+𝑐,∀𝑎,𝑏,𝑐≥0. (44) Now, inequality (42) is equivalent to 𝐹𝑑()𝐹𝑑(𝑥)𝑦,𝐹𝑑()()(𝑥𝑑𝑦𝑇𝑥,𝑇𝑦,𝜑𝑇𝑥,𝜑𝑇𝑦≤𝜓max𝑥,𝑦,𝜑,𝜑𝑥,𝑇𝑥,𝜑𝑇𝑥,𝜑)+𝐹𝑦,𝑇𝑦,𝜑𝑇𝑦,𝜑×2⁻¹, (45) for every 𝑥,𝑦∈𝑋. Applying Theorem 7, we obtain that 𝑇 has a unique fixed point 𝑧∈𝑋 with 𝜑(𝑧)=0; that is, 𝑝(𝑧,𝑧)=0.

The following results follow immediately from Corollary 15.

Corollary 16. Let (𝑋,𝑝) be a complete partial metric space, and let 𝑇∶𝑋→𝑋. Suppose that there exists 𝜓∈Ψ such that 𝑝𝑇𝑥,𝑇𝑦≤𝜓𝑝𝑥,𝑦, (46) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point 𝑧∈𝑋. Moreover, one has 𝑝(𝑧,𝑧)=0.

Corollary 17. Let (𝑋,𝑝) be a complete partial metric space, and let 𝑇∶𝑋→𝑋. Suppose that there exists 𝜓∈Ψ such that 𝑝𝑝()𝑇𝑥,𝑇𝑦≤𝜓𝑥,𝑇𝑥+𝑝𝑦,𝑇𝑦2, (47) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point 𝑧∈𝑋. Moreover, one has 𝑝(𝑧,𝑧)=0.

Note that Matthews result (see Theorem 4) follows from Corollary 16 by taking 𝜓(𝑡)=𝑘𝑡, where 𝑘∈(0,1).

Acknowledgment

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project no. RGP-VPP-237.