𝜃-Metric Space: A Generalization

Abstract

We introduce the notion of 𝜃-metric as a generalization of a metric by replacing the triangle inequality with a more generalized inequality. We investigate the topology of the spaces induced by a 𝜃-metric and present some essential properties of it. Further, we give characterization of well-known fixed point theorems, such as the Banach and Caristi types in the context of such spaces.

1. Introduction

Celebrated Banach contraction mapping principle [1] can be considered as a revolution in fixed point theory and hence in nonlinear functional analysis. The statement of this well-known principle is simple, but the consequences are so strong: every contraction mapping in a complete metric space has a unique fixed point. Since fixed point theory has been applied to different sciences and also distinct branches of mathematics, this pioneer result of Banach has been generalized, extended, and improved in various ways in several abstract spaces. After that, many authors stated many types, generalizations, and applications of fixed point theory until now (see also [2, 3]).

In 1976, Caristi [4] defined an order relation in a metric space by using a functional under certain conditions and proved a fixed point theorem for such an ordered metric space. Denote ℝ and ℕ as the sets of all real and natural numbers, respectively.

The order relation is defined as follows.

Lemma 1. Let (𝑋,𝑑) be a metric space and 𝜙∶𝑋→ℝ a functional. Define the relation “⪯” on 𝑋 by 𝑥⪯𝑦⟺𝑑𝑥,𝑦≤𝜙(𝑥)−𝜙𝑦. (1) Then, “≤” is a partial order relation on 𝑋 introduced by 𝜙, and (𝑋,𝑑) is called an ordered metric space introduced by 𝜙. Apparently, if 𝑥⪯𝑦, then 𝜙(𝑥)⪰𝜙(𝑦).

Caristi’s fixed point theorem states that a mapping 𝑇∶𝑋→𝑋 has a fixed point provided that (𝑋,𝑑) is a complete metric space and there exists a lower semicontinuous map 𝜙∶𝑋→ℝ such that 𝑑(𝑥,𝑇𝑥)≤𝜙(𝑥)−𝜙(𝑇𝑥),forevery𝑥∈𝑋. (2) This general fixed point theorem has found many applications in nonlinear analysis.

Many authors generalized Caristi’s fixed point theorem and stated many types of it in complete metric spaces (see [5–8]). In particular, in 2010, Amini-Harandi [6] extended Caristi’s fixed point and Takahashi’s minimization theorems in complete metric space via the extension of partial ordered relation which is introduced in Lemma 1 and introduced some applications of such results.

One of the interesting generalizations of the notion of a metric is the concept of a fuzzy metric, given by Kramosil and Michálek [9], and Grabiec [10], independently. Later, George and Veeramani [11] investigated fuzzy metric structure and observed some important topological properties of such spaces. Furthermore, the authors [9–11] announced existence and uniqueness of a fixed point of certain mappings in the framework of such spaces.

Definition 2 (see [11]). A binary operation ∗∶[0,1]×[0,1]→[0,1] is a continuous 𝑡-norm if it is a topological monoid with unit 1 such that 𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑 (𝑎,𝑏,𝑐,𝑑∈[0,1]).

Definition 3 (see [9, 10]). The 3-tuple (𝑋,𝑀,∗) is said to be a fuzzy metric space if 𝑋 is an arbitrary set, ∗ is a continuous 𝑡-norm, and 𝑀∶𝑋×𝑋×[0,+∞)→[0,1] is a fuzzy set satisfying the following conditions: (1)𝑀(𝑥,𝑦,0)=0, (2)𝑀(𝑥,𝑦,𝑡)=1 for all 𝑡>0 if and only if 𝑥=𝑦, (3)𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡), (4)𝑀(𝑥,𝑦,𝑡)∗𝑀(𝑦,𝑧,𝑡)≤𝑀(𝑥,𝑧,𝑡+𝑠), (5)𝑀(𝑥,𝑦,⋅)∶[0,+∞)→[0,1] is left continuous, where 𝑥,𝑦,𝑧∈𝑋 and 𝑡,𝑠>0.

In this paper, inspired from the definition of fuzzy metric spaces, we will introduce 𝜃-metric as an extension of metric spaces which is obtained by replacing the triangle inequality with a more generalized inequality. We also investigate the topology of the 𝜃-metric space and observe some fundamental properties of it. Furthermore, we give the characterization of the Banach and Caristi type fixed point theorems in the context of 𝜃-metric space.

First, we give the following definition.

Definition 4. Let 𝜃∶[0,+∞)×[0,+∞)→[0,+∞) be a continuous mapping with respect to each variable. Let Im(𝜃)={𝜃(𝑠,𝑡)∶𝑠≥0,𝑡≥0}. A mapping 𝜃 is called an 𝐵-action if and only if it satisfies the following conditions: (I)𝜃(0,0)=0 and 𝜃(𝑡,𝑠)=𝜃(𝑠,𝑡) for all 𝑡,𝑠≥0,(II)𝜃(𝑠,𝑡)<𝜃(𝑢,𝑣)ifeither𝑠<𝑢,𝑡≤𝑣,or𝑠≤𝑢,𝑡<𝑣, (3)(III)for each 𝑟∈Im(𝜃) and for each 𝑠∈[0,𝑟], there exists 𝑡∈[0,𝑟] such that 𝜃(𝑡,𝑠)=𝑟, (IV)𝜃(𝑠,0)≤𝑠, for all 𝑠>0.

We denote by Υ the set of all 𝐵-actions.

Example 5. The following functions are examples of 𝐵-action: (𝜃₁) 𝜃(𝑡,𝑠)=𝑘(𝑡+𝑠), where 𝑘∈(0,1], (𝜃₂) 𝜃(𝑡,𝑠)=𝑘(𝑡+𝑠+𝑡𝑠), where 𝑘∈(0,1], (𝜃₃) 𝜃(𝑡,𝑠)=𝑡𝑠/(1+𝑡𝑠), (𝜃₄) 𝜃(𝑠,𝑡)=√𝑠²+𝑡², (𝜃₅) 𝜃(𝑡,𝑠)=𝑡+𝑠+𝑡𝑠, (𝜃₆) 𝜃(𝑡,𝑠)=𝑡+𝑠+√𝑡𝑠,(𝜃₇) 𝜃(𝑡,𝑠)=(𝑡+𝑠)(1+𝑡𝑠).

Example 5 shows that the category of 𝐵-actions is uncountable. Next, we derive some lemmas which play a crucial role in our main results.

Lemma 6. Let [)⟶[)(0)𝑓(𝑡)Ψ=.𝑓𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠,𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑓∶0,+∞0,+∞∶𝑓=0<𝑡∀𝑡>0. (4) Then, there exists a correspondence between Υ and Ψ. In other words, Υ is an infinite set.

Proof. For each (𝑡,𝑠)∈[0,+∞)×[0,+∞), define 𝜃_𝑓(𝑡,𝑠)=𝜆𝑓(𝑡+𝑠), where 𝜆∈[0,1) and 𝑓∈Ψ. It is previous that 𝜃_𝑓∈Υ. Now, define 𝐻𝑓𝐻∶Ψ⟶Υ=𝜃_𝑓. (5)𝐻 is well-defined and injective function which completes the proof.

Lemma 7. Let 𝜃 be a 𝐵-action. For each 𝑟∈Im(𝜃) and 𝑠∈𝐵=[0,𝑟], there exist 𝑡∈[0,𝑟] and a function 𝜂∶[0,+∞)×[0,+∞)→[0,+∞) such that 𝜂(𝑟,𝑠)=𝑡. Then, one derives the following. (𝑎₁)𝜂(0,0)=0. (𝑎₂)𝜃(𝜂(𝑟,𝑠),𝑠)=𝑟 and 𝜃(𝑟,𝜂(𝑠,𝑟))=𝑠. (𝑎₃)𝜂 is continuous with respect to the first variable. (𝑎₄)If 𝜂(𝑟,𝑠)≥0, then 0≤𝑠≤𝑟.

Proof. By (III) of Definition 4, for each 𝑟∈Im(𝜃) and for each 𝑠∈[0,𝑟], there exists 𝑡≥0 such that 𝜃(𝑡,𝑠)=𝑟. Now, define 𝜂(𝑟,𝑠)=𝑡. Let 𝑡 and 𝑡^′ be two values such that 𝜂(𝑟,𝑠)=𝑡 and 𝜂(𝑟,𝑠)=𝑡^′. If 𝑡≠𝑡^′, then 𝑡<𝑡^′ or 𝑡>𝑡^′. If 𝑡<𝑡^′, then 𝑟=𝜃(𝑡,𝑠)<𝜃(𝑡^′,𝑠)=𝑟 and this is a contradiction. For 𝑡>𝑡^′, we have the same argument. Thus, 𝜂 is well defined.
On the other hand, (𝑎₁), (𝑎₂), and (𝑎₄) are straightforward from (III) of Definition 4. Also, (𝑎₃) holds since if 𝑟∈Im(𝜃) and {𝑟_𝑛} is a sequence in Im(𝜃) such that 𝑟_𝑛→𝑟, then 𝜃(𝜂(𝑟_𝑛,⋅),⋅)=𝑟_𝑛. Thus, 𝜃lim_𝑛→∞𝜂𝑟_𝑛,⋅,⋅=lim_𝑛→∞𝜃𝜂𝑟_𝑛𝜂().,⋅,⋅=𝑟=𝜃𝑟,⋅,⋅ (6) If lim_𝑛→∞𝜂(𝑟_𝑛,⋅)>𝜂(𝑟,⋅) or lim_𝑛→∞𝜂(𝑟_𝑛,⋅)<𝜂(𝑟,⋅), then by (II) of Definition 4 and (6), we conclude a contradiction. So, 𝜂 is continuous with respect to the first variable.

The following definition arises from Lemma 7.

Definition 8. The function 𝜂, mentioned in Lemma 7, is called 𝐵-inverse action of 𝜃. One says that 𝜃 is regular if 𝜂 satisfies 𝜂(𝑟,𝑟)=0, for each 𝑟>0.

The set of all regular 𝐵-inverse actions will be denoted by Υ_𝑟.

Example 9. Let 𝜃₁(𝑡,𝑠)=𝑡+𝑠 and 𝜃₂(𝑡,𝑠)=√^𝑛𝑡^𝑛+𝑠^𝑛. It is evident that 𝜂₁(𝑡,𝑠)=𝑡−𝑠 and 𝜂(𝑡,𝑠)=√^𝑛𝑡^𝑛−𝑠^𝑛. Furthermore, 𝜂₁, 𝜂₂ satisfy all the conditions of Lemma 7. Note that 𝜃₁, 𝜃₂ are regular.

Lemma 10. If 𝑎∈𝑋, 𝑐∈Im(𝜃) and 𝑏∈[0,𝑐], then 𝜃(𝑎,𝑏)≤𝑐 implies that 𝑎≤𝜂(𝑐,𝑏).

Proof. Suppose, on the contrary, that 𝜂(𝑐,𝑏)<𝑎. Then, we have 𝑐=𝜃𝜂(𝑐,𝑏),𝑏<𝜃(𝑎,𝑏), (7) a contradiction.

2. Main Results

In this section, we introduce the notion 𝜃-metric and discuss the induced topology generated by 𝑑_𝜃 on a nonempty subset 𝑋. In particular, we will show that 𝜏_𝑑𝜃 is Hausdorff and first countable. Further, we derive that (𝑋,𝑑_𝜃) is a metrizable topological space.

Definition 11. Let 𝑋 be a nonempty set. A mapping 𝑑_𝜃∶𝑋×𝑋→[0,+∞) is called a 𝜃-metric on 𝑋 with respect to 𝐵-action 𝜃∈Υ if 𝑑_𝜃 satisfies the following:(𝐀𝟏)𝑑_𝜃(𝑥,𝑦)=0 if and only if 𝑥=𝑦, (𝐀𝟐)𝑑_𝜃(𝑥,𝑦)=𝑑_𝜃(𝑦,𝑥), for all 𝑥,𝑦∈𝑋, (𝐀𝟑)𝑑_𝜃(𝑥,𝑦)≤𝜃(𝑑_𝜃(𝑥,𝑧),𝑑_𝜃(𝑧,𝑦)), for all 𝑥,𝑦,𝑧∈𝑋. A pair (𝑋,𝑑_𝜃) is called a 𝜃-metric space.

Remark 12. If (𝑋,𝑑_𝜃) is a 𝜃-metric space, 𝜃(𝑠,𝑡)=𝑘(𝑠+𝑡), and 𝑘∈(0,1], then (𝑋,𝑑_𝜃) is a metric space.
Conversely, for 𝜃(𝑠,𝑡)=𝑘(𝑠+𝑡), 𝑘∈(0,1), we have that there exists metric space (𝑋,𝑑) which is not 𝜃-metric space. For example, if 𝑋={1,2,3} and 𝑑∶𝑋×𝑋→[0,+∞) defined by 𝑑()()()𝑑()()()11,2=1,𝑑1,3=1,𝑑2,3=2,1,1=0,𝑑2,2=0,𝑑3,3=0,𝑘=2, (8)𝑑 is a metric, but 𝑑 is not a 𝜃 metric.

Example 13. Let 𝑋={𝑥,𝑦,𝑧} and 𝑑_𝜃∶𝑋×𝑋→[0,+∞) defined by 𝑑_𝜃𝑥,𝑦=2,𝑑_𝜃(𝑥,𝑧)=6,𝑑_𝜃𝑑𝑧,𝑦=10,_𝜃(𝑥,𝑥)=0,𝑑_𝜃𝑦,𝑦=0,𝑑_𝜃()𝑑𝑧,𝑧=0,_𝜃𝑥,𝑦=𝑑_𝜃𝑦,𝑥,𝑑_𝜃(𝑥,𝑧)=𝑑_𝜃(),𝑑𝑧,𝑥_𝜃𝑧,𝑦=𝑑_𝜃(𝑧,𝑥). (9) For 𝜃(𝑠,𝑡)=𝑠+𝑡+𝑠𝑡, the function 𝑑_𝜃 forms a metric, and hence, the pair (𝑋,𝑑_𝜃) is a 𝜃-metric space.

Remark 14. Notice that 𝑑_𝜃, in Example 13, is not a metric on 𝑋, since we have 𝑑_𝜃(𝑧,𝑦)>𝑑_𝜃(𝑥,𝑦)+𝑑_𝜃(𝑥,𝑧).

Definition 15. Let (𝑋,𝑑_𝜃) be a 𝜃-metric space. An open ball 𝐵_𝑑𝜃(𝑥,𝑟) at a center 𝑥∈𝑋 with a radius 𝑟∈Im(𝜃) is defined as follows: 𝐵_𝑑𝜃(𝑥,𝑟)=𝑦∈𝑋∶𝑑_𝜃𝑥,𝑦<𝑟. (10)

Lemma 16. Every open ball is an open set.

Proof. We show that, for each 𝑥∈𝑋 and 𝑟>0 and for each 𝑦∈𝐵_𝑑𝜃(𝑥,𝑟), there exists 𝛿>0 such that 𝐵_𝑑𝜃𝑦,𝛿⊂𝐵_𝑑𝜃(𝑥,𝑟). (11) By (III) of Definition 4, we can choose 𝛿>0 such that 𝜃(𝛿,𝑑_𝜃(𝑥,𝑦))=𝑟. Now, if 𝑧∈𝐵_𝑑𝜃(𝑦,𝛿), then we have 𝑑_𝜃(𝑧,𝑥)≤𝜃𝑑_𝜃𝑧,𝑦,𝑑_𝜃𝑦,𝑥<𝜃𝛿,𝑑_𝜃𝑦,𝑥=𝑟. (12) It means that 𝑧∈𝐵_𝑑𝜃(𝑥,𝑟) and (11) is proved.

Lemma 17. If (𝑋,𝑑_𝜃) is a 𝜃-metric space, then the collection of open sets forms a topology, denoted by 𝜏_𝑑𝜃. A pair (𝑋,𝜏_𝜃) is called topological space induced by a 𝜃-metric.

Lemma 18. The set {𝐵_𝑑𝜃(𝑥,1/𝑛)∶𝑛∈ℕ} is a local base at 𝑥, and the above topology is first countable.

Proof. For each 𝑥∈𝑋 and 𝑟>0, we can find 𝑛₀∈ℕ such that 1/𝑛₀<𝑟. Thus, 𝐵_𝑑𝜃(𝑥,1/𝑛)⊂𝐵_𝑑𝜃(𝑥,𝑟). This means that {𝐵_𝑑𝜃(𝑥,1/𝑛)∶𝑛∈ℕ} is a local base at 𝑥 and the above topology is first countable.

Theorem 19. A topological space (𝑋,𝜏_𝑑𝜃) is Hausdorff.

Proof. Let 𝑥, 𝑦 be two distinct points of 𝑋. Suppose that 0<𝛼<𝑑_𝜃(𝑥,𝑦) is arbitrary. By Definition 4, we conclude that 𝛼∈Im(𝜃). Therefore, there exist 𝑟,𝑠>0 such that 𝜃(𝑟,𝑠)=𝛼. It is clear that 𝐵_𝑑𝜃(𝑥,𝑟)∩𝐵_𝑑𝜃(𝑥,𝑠)=∅. For if there exists 𝑧∈𝐵_𝑑𝜃(𝑥,𝑟)∩𝐵_𝑑𝜃(𝑥,𝑠), then 𝑑_𝜃𝑥,𝑦≤𝜃𝑑_𝜃(𝑥,𝑧),𝑑_𝜃()𝑧,𝑦<𝜃𝑟,𝑠=𝛼<𝑑_𝜃𝑥,𝑦, (13) a contradiction.

Theorem 20. Let (𝑋,𝑑_𝜃) be a 𝜃-metric space and 𝜏_𝑑𝜃 the topology induced by the 𝜃-metric. Then, for a sequence {𝑥_𝑛} in 𝑋, 𝑥_𝑛→𝑥 if and only if 𝑑_𝜃(𝑥_𝑛,𝑥)→0 as 𝑛→∞.

Proof. Suppose that 𝑥_𝑛→𝑥. Then, for each 𝜖>0, there exists 𝑛₀∈ℕ such that 𝑥_𝑛∈𝐵_𝑑𝜃(𝑥,𝜖), for all 𝑛≥𝑛₀. Thus, 𝑑_𝜃(𝑥_𝑛,𝑥)<𝜖; that is, 𝑑_𝜃(𝑥_𝑛,𝑥)→0 as 𝑛→∞. The converse is verified easily.

Theorem 21. Let (𝑋,𝑑_𝜃) be a 𝜃-metric space and 𝑥_𝑛→𝑥, 𝑦_𝑛→𝑦.Then, 𝑑_𝜃𝑥_𝑛,𝑦_𝑛⟶𝑑_𝜃𝑥,𝑦. (14)

Proof. For each 𝑛∈ℕ, there exists 𝐾>0 such that, for all 𝑛≥𝐾, 𝑑_𝜃𝑥_𝑛<1,𝑥𝑛,𝑑_𝜃𝑦_𝑛<1,𝑦𝑛. (15) Thus, by the continuity of 𝜃 with respect to each variable, we have 𝑑_𝜃𝑥,𝑦≤𝜃𝑑_𝜃𝑥,𝑥_𝑛,𝜃𝑑_𝜃𝑥_𝑛,𝑦_𝑛,𝑑_𝜃𝑦_𝑛1,𝑦<𝜃𝑛,𝜃𝑑_𝜃𝑥_𝑛,𝑦_𝑛,1𝑛. (16) Therefore, 𝑑_𝜃𝑥,𝑦≤lim_𝑛→∞𝜃1𝑛,𝜃𝑑_𝜃𝑥_𝑛,𝑦_𝑛,1𝑛=𝜃0,𝜃lim_𝑛→∞𝑑_𝜃𝑥_𝑛,𝑦_𝑛,0≤lim_𝑛→∞𝑑_𝜃𝑥_𝑛,𝑦_𝑛≤lim_𝑛→∞𝜃𝑑_𝜃𝑥,𝑥_𝑛,𝜃𝑑_𝜃𝑥,𝑦,𝑑_𝜃𝑦_𝑛,𝑦≤lim_𝑛→∞𝜃1𝑛,𝜃𝑑_𝜃,1𝑥,𝑦𝑛≤𝑑_𝜃𝑥,𝑦. (17) Thus, 𝑑_𝜃(𝑥_𝑛,𝑦_𝑛)→𝑑_𝜃(𝑥,𝑦).

Lemma 22. Let (𝑋,𝑑_𝜃) be a 𝜃-metric space. Let {𝑥_𝑛} be a sequence in 𝑋 and 𝑥_𝑛→𝑥. Then, 𝑥 is unique.

Proof. Suppose that 𝑥_𝑛→𝑥 and 𝑥_𝑛→𝑦. We show that 𝑥=𝑦. For each 𝑛∈ℕ, there exists 𝑁>0 such that 𝑑_𝜃(𝑥_𝑛,𝑥)<1/𝑛 and 𝑑_𝜃(𝑥_𝑛,𝑦)<1/𝑛. By the continuity of 𝜃, we have 0≤𝑑_𝜃𝑥,𝑦≤𝜃𝑑_𝜃𝑥_𝑛,𝑥,𝑑_𝜃𝑥_𝑛1,𝑦<𝜃𝑛,1𝑛⟶0,(𝑛⟶∞). (18) Hence, we have 𝑥=𝑦.

Definition 23. Let (𝑋,𝑑_𝜃) be a 𝜃-metric space. Then, for a sequence {𝑥_𝑛} in 𝑋, one says that {𝑥_𝑛} is a Cauchy sequence if for each 𝜖>0, there exists 𝑁>0 such that, for all 𝑚≥𝑛≥𝑁, 𝑑_𝜃(𝑥_𝑛,𝑥_𝑚)<𝜖.

Definition 24. Let (𝑋,𝑑_𝜃) be a 𝜃-metric space. One says that (𝑋,𝑑_𝜃) is complete 𝜃-metric space if every Cauchy sequence {𝑥_𝑛} is convergent in 𝑋.

Lemma 25 (see [12]). A Hausdorff topological space (𝑋,𝜏_𝑑𝜃) is metrizable if and only if it admits a compatible uniformity with a countable base.

In the following theorems we apply the previous lemma and the concept of uniformity (see [12] for more information) to prove the metrizability of a topological space (𝑋,𝜏_𝑑𝜃).

Theorem 26. Let (𝑋,𝑑_𝜃) be a 𝜃-metric space. Then, (𝑋,𝜏_𝑑𝜃) is a metrizable topological space.

Proof. For each 𝑛∈ℕ, define 𝒰_𝑛=𝑥,𝑦∈𝑋×𝑋∶𝑑_𝜃<1𝑥,𝑦𝑛. (19) We will prove that {𝒰_𝑛∶𝑛∈ℕ} is a base for a uniformity 𝒰 on 𝑋 whose induced topology coincides with 𝜏_𝑑𝜃.
We first note that for each 𝑛∈ℕ, {(𝑥,𝑥)∶𝑥∈𝑋}⊆𝒰_𝑛,𝒰_𝑛+1⊆𝒰_𝑛,𝒰_𝑛=𝒰_𝑛⁻¹. (20) On the other hand, for each 𝑛∈ℕ, there is, by the continuity of 𝜃, an 𝑚∈ℕ such that 1𝑚>2𝑛,𝜃𝑚,1𝑚<1𝑛. (21) Then, 𝒰_𝑚∘𝒰_𝑚⊆𝒰_𝑛: indeed, let (𝑥,𝑦)∈𝒰_𝑚 and (𝑦,𝑧)∈𝒰_𝑚. Thus, 𝑑_𝜃(𝑥,𝑧)≤𝜃𝑑_𝜃𝑥,𝑦,𝑑_𝜃1𝑦,𝑧<𝜃𝑚,1𝑚<1𝑛. (22) Therefore, (𝑥,𝑧)∈𝒰_𝑛. Hence, {𝒰_𝑛∶𝑛∈ℕ} is a base for a uniformity 𝒰 on 𝑋. Since for each 𝑥∈𝑋 and each 𝑛∈ℕ, 𝒰_𝑛(𝑥)={𝑦∈𝑋∶𝑑_𝜃(𝑥,𝑦)<1/𝑛}. We deduce from Lemma 25 that (𝑋,𝜏_𝑑𝜃) is a metrizable topological space.

Let us recall that a metrizable topological space (𝑋,𝜏) is said to be completely metrizable if it admits a complete metric [13].

Theorem 27. Let (𝑋,𝑑_𝜃) be a complete 𝜃-metric space. Then, (𝑋,𝜏_𝑑𝜃) is completely metrizable.

Proof. It follows from the proof of Theorem 26 that {𝒰_𝑛∶𝑛∈ℕ} is a base for a uniformity 𝒰 on 𝑋 compatible with 𝜏_𝑑𝜃, where 𝒰_𝑛={(𝑥,𝑦)∈𝑋×𝑋∶𝑑_𝜃(𝑥,𝑦)<1/𝑛}, for every 𝑛∈ℕ. Then, there exists a metric 𝑑 on 𝑋 whose induced uniformity coincides with 𝒰. We want to show that the metric 𝑑 is complete. Indeed, given a Cauchy sequence {𝑥_𝑛} in (𝑋,𝑑), we will prove that {𝑥_𝑛} is a Cauchy sequence in (𝑋,𝑑). To this end, fix 𝜖>0. Choose 𝑘∈ℕ such that 1/𝑘<𝜖. Then, there exists 𝑛₀∈ℕ such that (𝑥_𝑚,𝑥_𝑛)∈𝒰_𝑘 for every 𝑛,𝑚≥𝑛₀. Consequently, for each 𝑛,𝑚≥𝑛₀, 𝑑_𝜃(𝑥_𝑛,𝑥_𝑚)≤1/𝑘<𝜖. We have shown that {𝑥_𝑛} is a Cauchy sequence in the complete 𝜃-metric space (𝑋,𝑑) and so is convergent with respect to (𝑋,𝑑). Thus, (𝑋,𝑑) is a complete metric space.

3. Two Fixed Point Theorems

In this section, we introduce two fixed point theorems in 𝜃-metric spaces. First, we introduce the Banach fixed point and Caristi’s fixed point theorems in such spaces.

3.1. Banach Fixed Point Theorem

Theorem 28. Let (𝑋,𝑑_𝜃) be a complete 𝜃-metric space and 𝑓∶𝑋→𝑋 a mapping that satisfies the following: 𝑑_𝜃𝑓𝑥,𝑓𝑦≤𝛼𝑑_𝜃𝑥,𝑦, (23) for each 𝑥,𝑦∈𝑋, where 𝛼∈[0,1). Then, 𝑓 has a unique fixed point.

Proof. Let 𝑥₀∈𝑋 and 𝑥_𝑛+1=𝑓𝑥_𝑛. We divide our proof into four steps.
Step  1. We claim that 𝑑_𝜃(𝑥_𝑛,𝑥_𝑛+1)→0. Indeed, we have 𝑑_𝜃𝑥_𝑛+1,𝑥_𝑛≤𝛼𝑑_𝜃𝑥_𝑛,𝑥_𝑛−1≤𝛼²𝑑_𝜃𝑥_𝑛−1,𝑥_𝑛−2⋮≤𝛼^𝑛𝑑_𝜃𝑥₁,𝑥₀⟶0,as(𝑛⟶∞). (24) Thus, we have 𝑑_𝜃(𝑥_𝑛,𝑥_𝑛+1)→0.
Step  2. We assert that the sequence {𝑥_𝑛} is bounded. Suppose, on the contrary, that {𝑥_𝑛} is an unbounded sequence. Thus, there exists subsequence {𝑛(𝑘)} such that 𝑛(1)=1 and for each 𝑘∈ℕ, 𝑛(𝑘+1) is minimal in the sense that the relation 𝑑_𝜃𝑥_𝑛(𝑘+1),𝑥_𝑛(𝑘)>1 (25) does not hold and 𝑑_𝜃𝑥_𝑚,𝑥_𝑛(𝑘)≤1 (26) holds for all 𝑚∈{𝑛(𝑘)+1,𝑛(𝑘)+2,…,𝑛(𝑘+1)−1}. Hence, by using the triangle inequality, we derive that 1<𝑑_𝜃𝑥_𝑛(𝑘+1),𝑥_𝑛(𝑘)≤𝜃𝑑_𝜃𝑥_𝑛(𝑘+1),𝑥_{𝑛(𝑘+1)−1},𝑑_𝜃𝑥_{𝑛(𝑘+1)−1},𝑥_𝑛(𝑘)≤𝜃𝑑_𝜃𝑥_𝑛(𝑘+1),𝑥_{𝑛(𝑘+1)−1},1. (27) By taking the limit from two sides of (27) and using (II) of Definition 4, we derive that 𝑑_𝜃𝑥_𝑛(𝑘+1),𝑥_𝑛(𝑘)⟶1⁺as(𝑘⟶+∞). (28) Also, we have 1<𝑑_𝜃𝑥_𝑛(𝑘+1),𝑥_𝑛(𝑘)≤𝑑_𝜃𝑥_{𝑛(𝑘+1)−1},𝑥_{𝑛(𝑘)−1}≤𝜃𝑑_𝜃𝑥_{𝑛(𝑘+1)−1},𝑥_𝑛(𝑘),𝑑_𝜃𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−1}≤𝜃1,𝑑_𝜃𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−1}, (29) which implies that 𝑑_𝜃𝑥_{𝑛(𝑘+1)−1},𝑥_{𝑛(𝑘)−1}⟶1⁺as(𝑘⟶+∞). (30) Since 𝑑_𝜃(𝑥_𝑛(𝑘+1),𝑥_𝑛(𝑘))≤𝛼𝑑_𝜃(𝑥_{𝑛(𝑘+1)−1},𝑥_{𝑛(𝑘)−1}), we have 1≤𝛼1, a contradiction. Thus, the sequence {𝑥_𝑛} is bounded.
Step  3. We will show that {𝑥_𝑛} is a Cauchy sequence. Let 𝑚,𝑛∈ℕ with 𝑚>𝑛𝑑_𝜃𝑥_𝑚,𝑥_𝑛=𝑑_𝜃𝑓𝑥_𝑚−1,𝑓𝑥_𝑛−1≤𝛼𝑑_𝜃𝑓𝑥_𝑚−2,𝑓𝑥_𝑛−2⋮≤𝛼^𝑛𝑑_𝜃𝑓𝑥_𝑚−𝑛,𝑥₀. (31) Since {𝑥_𝑛} is a bounded sequence, therefore, lim_{𝑛,𝑚→∞}𝑑_𝜃(𝑥_𝑚,𝑥_𝑛)=0; that is, {𝑥_𝑛} is a Cauchy sequence. Thus, there exists 𝑥∈𝑋 such that 𝑥_𝑛→𝑥. Further, we derive that 𝑑_𝜃𝑥_𝑛+1,𝑓𝑥=𝑑_𝜃𝑓𝑥_𝑛,𝑓𝑥≤𝛼𝑑_𝜃𝑥_𝑛,𝑥⟶0,(𝑛⟶∞). (32) It means that 𝑥_𝑛+1→𝑓𝑥; that is, 𝑓𝑥=𝑥.
Step  4. In the last step, we prove that the 𝑥 is the unique fixed point of 𝑇. Suppose, on the contrary, that 𝑥, 𝑦 are two distinct fixed points of 𝑓. So, we get that 𝑑_𝜃𝑦,𝑥=𝑑_𝜃𝑓𝑦,𝑓𝑥≤𝛼𝑑_𝜃𝑦,𝑥 (33) is a contradiction. This completes the proof.

3.2. Caristi-Type Fixed Point Theorem

Definition 29. Suppose that (𝑋,𝑑_𝜃) be a complete 𝜃-metric space and 𝒫 the class of all maps 𝜓∶𝑋×𝑋→[0,+∞) which satisfies the following conditions: (𝐸₁)there exists 𝑥∈𝑋 such that 𝜓(𝑥,⋅) is bounded below and lower semicontinuous, and 𝜓(⋅,𝑦) is upper semicontinuous for each 𝑦∈𝑋, (𝐸₂)𝜓(𝑥,𝑥)=0, for each 𝑥∈𝑋, (𝐸₃)𝜃(𝜓(𝑥,𝑦),𝜓(𝑦,𝑧))≤𝜓(𝑥,𝑧), for each 𝑥,𝑦,𝑧∈𝑋.

Lemma 30. By Definition 29, one has 𝜓𝑥,𝑦≤𝜂𝜓(𝑥,𝑧),𝜓𝑦,𝑧, (34) for each 𝑥,𝑦,𝑧∈𝑋.

Proof. By Lemma 10, we obtain the desired result.

Example 31. Let 𝜃(𝑡,𝑠)=𝑡𝑠/(1+𝑡𝑠); thus, Im(𝜃)=[0,1). Now, let 𝜙∶𝑋→ℝ be a lower bounded, lower semicontinuous function and 𝜓=𝑒𝑥,𝑦^{𝜙(𝑦)−𝜙(𝑥)}𝑥≠𝑦0𝑥=𝑦. (35) Then, 𝜓 satisfies all conditions of Definition 29.

Example 32. Let 𝜃(𝑡,𝑠)=√^2𝑛+1𝑡^2𝑛+1+𝑠^2𝑛+1; thus, Im(𝜃)=[0,+∞). Now, let 𝜙∶𝑋→ℝ be a lower bounded, lower semicontinuous function and 𝜓𝑥,𝑦=^2𝑛+1𝜙𝑦−𝜙(𝑥). (36) Then, 𝜓 satisfies all conditions of Definition 29. Also, 𝜂(𝑡,𝑠)=√^2𝑛+1𝑡^2𝑛+1−𝑠^2𝑛+1, and 𝜃 is regular.

From now to end, we assume that 𝜃 is regular (see Definition 8).

Lemma 33. Let (𝑋,𝑑_𝜃) be a complete 𝜃-metric space and 𝜓∈𝒫. Let 𝛾∶[0,+∞)→[0,+∞) be 𝜃-subadditive; that is, 𝛾(𝜃(𝑥,𝑦))≤𝜃(𝛾(𝑥),𝛾(𝑦)), for each 𝑥,𝑦∈[0,+∞), nondecreasing continuous map such that 𝛾⁻¹{0}={0}. Define the order ≺ on 𝑋 by 𝑥≺𝑦⟺𝛾𝑑_𝜃𝑥,𝑦≤𝜓𝑥,𝑦, (37) for any 𝑥,𝑦∈𝑋. Then, (𝑋,≺) is a partial order set which has minimal elements.

Proof. At first, we show that (𝑋,≺) is a partial ordered set. For each 𝑥∈𝑋, we have 0=𝛾(0)=𝛾(𝑑_𝜃(𝑥,𝑥))≤𝜓(𝑥,𝑥)=0. Thus, 𝑥≺𝑥. If 𝑥≺𝑦 and 𝑦≺𝑥, then 𝛾(𝑑_𝜃(𝑥,𝑦))≤𝜓(𝑥,𝑦) and 𝛾(𝑑_𝜃(𝑥,𝑦))≤𝜓(𝑦,𝑥). Thus, we give 𝛾𝜃𝑑_𝜃𝑥,𝑦,𝑑_𝜃𝑥,𝑦≤𝜃𝛾𝑑_𝜃𝑥,𝑦,𝛾𝑑_𝜃𝜓()𝑥,𝑦≤𝜃𝑥,𝑦,𝜓𝑦,𝑥≤𝜓𝑥,𝑥=0. (38) It means that 𝑥=𝑦. Finally, if 𝑥≺𝑦 and 𝑦≺𝑧, then 𝛾(𝑑_𝜃(𝑥,𝑦))≤𝜓(𝑥,𝑦) and 𝛾(𝑑_𝜃(𝑦,𝑧))≤𝜓(𝑦,𝑧). Thus, we give 𝛾𝑑_𝜃(𝑥,𝑧)≤𝛾𝜃𝑑_𝜃𝑥,𝑦,𝑑_𝜃𝛾𝑑𝑦,𝑧≤𝜃_𝜃𝑥,𝑦,𝛾𝑑_𝜃𝜓().𝑦,𝑧≤𝜃𝑥,𝑦,𝜓𝑦,𝑧≤𝜓𝑥,𝑧 (39) It means that 𝑥≺𝑧. Thus, (𝑋,≺) is a partial ordered set.
To show that (𝑋,≺) has minimal elements, we show that any decreasing chain has a lower bound. Indeed, let {𝑥_𝛼}_𝛼∈Γ be a decreasing chain; then we have 0≤𝛾𝑑_𝜃𝑥_𝛼,𝑥_𝛽≤𝜓𝑥_𝛼,𝑥_𝛽≤𝜂𝜓𝑥,𝑥_𝛽,𝜓𝑥,𝑥_𝛼; (40) by definition of 𝜂, we have 𝜓(𝑥,𝑥_𝛼)≤𝜓(𝑥,𝑥_𝛽). Thus, {𝜓(𝑥,𝑥_𝛼)}_𝛼∈Γ is decreasing net of reals which is bounded below. Let {𝛼_𝑛} be an increasing sequence of elements from Γ such that lim_𝑛→∞𝜓𝑥,𝑥_𝛼𝑛=inf𝜓𝑥,𝑥_𝛼∶𝛼∈Γ=𝜌. (41) Then, for each 𝑚≥𝑛, we infer that 𝛾𝑑_𝜃𝑥_𝛼𝑛,𝑥_𝛼𝑚≤𝜓𝑥_𝛼𝑛,𝑥_𝛼𝑚≤𝜂𝜓𝑥,𝑥_𝛼𝑛,𝜓𝑥,𝑥_𝛼𝑚. (42) By taking limit from two sides of (42), the regularity of 𝜃, and continuity of 𝜂, we give limsup_{𝑛,𝑚→∞}𝛾𝑑_𝜃𝑥_𝛼𝑛,𝑥_𝛼𝑚≤𝜓𝑥_𝛼𝑛,𝑥_𝛼𝑚≤limsup_{𝑛,𝑚→∞}𝜂𝜓𝑥,𝑥_𝛼𝑛,𝜓𝑥,𝑥_𝛼𝑚≤𝜂𝜌,𝜌=0. (43) Then, our assumption on 𝛾 implies that {𝑥_𝛼𝑛} is a Cauchy sequence and therefore converges to some 𝑥∈𝑋. Since 𝛾 is continuous and 𝜓(⋅,𝑥_𝛼𝑛) is upper semicontinuous, then we have 𝛾𝑑_𝜃𝑥,𝑥_𝛼𝑛=limsup_𝑚→∞𝛾𝑑_𝜃𝑥_𝛼𝑚,𝑥_𝛼𝑛≤limsup_𝑚→∞𝜓𝑥_𝛼𝑚,𝑥_𝛼𝑛≤𝜓𝑥,𝑥_𝛼𝑛. (44) This shows that 𝑥≺𝑥_𝛼𝑛 for all 𝑛≥1, which means that 𝑥 is lower bound for {𝑥_𝛼𝑛}. In order to see that 𝑥 is also a lower bound for {𝑥_𝛼}_𝛼∈Γ, let 𝛽∈Γ be such that 𝑥_𝛽≺𝑥_𝛼𝑛 for all 𝑛≥1. Then, for each 𝑛∈ℕ, we have 0≤𝛾𝑑_𝜃𝑥_𝛽,𝑥_𝛼𝑛≤𝜓𝑥_𝛽,𝑥_𝛼𝑛≤𝜂𝜓𝑥,𝑥_𝛼𝑛,𝜓𝑥,𝑥_𝛽. (45) Hence, for all 𝑛≥1, 𝜓𝑥,𝑥_𝛽≤𝜓𝑥,𝑥_𝛼𝑛 (46) which implies that 𝜓𝑥,𝑥_𝛽=inf𝜓𝑥,𝑥_𝛼∶𝛼∈Γ=lim_𝑛→∞𝜓𝑥,𝑥_𝛼𝑛. (47) Thus, from (46), we get lim_𝑛→∞𝑥_𝛼𝑛=𝑥_𝛽, which implies that 𝑥_𝛽=𝑥. Therefore, for any 𝛼∈Γ, there exists 𝑛∈ℕ such that 𝑥_𝛼𝑛≺𝑥_𝛼; that is, 𝑥 is a lower bound of {𝑥_𝛼}. Zorn’s lemma will therefore imply that (𝑋,≺) has minimal elements.

Theorem 34. Let (𝑋,𝑑_𝜃) be a complete 𝜃-metric space and 𝜓∈𝒫. Let 𝛾∶[0,+∞)→[0,+∞) be as in Lemma 33. Let 𝑇∶𝑋→𝑋 be a map satisfying the following: 𝛾𝑑_𝜃(𝑥,𝑇𝑥)≤𝜓(𝑇𝑥,𝑥), (48) for any 𝑥∈𝑋. Then, 𝑇 has a fixed point.

Proof. By Lemma 33, (𝑋,≺) has a minimal element say 𝑥. Thus, 𝑇𝑥≺𝑥. It means that 𝑇𝑥=𝑥.

Corollary 35. Let (𝑋,𝑑_𝜃) be a complete 𝜃-metric space and 𝜓∈𝒫. Let 𝛾∶[0,+∞)→[0,+∞) be as in Lemma 33. Let 𝑇∶𝑋→2^𝑋 be a multivalued mapping satisfying the following: 𝛾𝑑_𝜃𝑥,𝑦≤𝜓𝑦,𝑥,∀𝑦∈𝑇𝑥. (49) Then, 𝑇 has an endpoint; that is, there exists 𝑥∈𝑋 such that 𝑇𝑥={𝑥}.

In Corollary 35, we can introduce many types of Caristi’s fixed point theorem as follows.

If we set 𝜓 as in Example 32, then (48) has the following form: 𝛾𝑑_𝜃(𝑥,𝑇𝑥)≤^2𝑛+1𝜙(𝑇𝑥)−𝜙(𝑥). (50)