The Generalized Weaker (𝜶-𝝓-𝝋)-Contractive Mappings and Related Fixed Point Results in Complete Generalized Metric Spaces

Abstract

We introduce the notion of generalized weaker (𝛼-𝜙-𝜑)-contractive mappings in the context of generalized metric space. We investigate the existence and uniqueness of fixed point of such mappings. Some consequences on existing fixed point theorems are also derived. The presented results generalize, unify, and improve several results in the literature.

1. Introduction and Preliminaries

In [1], Branciari introduced the notion of generalized metric space by weakening the triangular inequality of metric assumption with quadrilateral inequality. The author [1] characterized and proved the analog of famous Banach fixed point theorem in the setting of generalized metric space. Although the theorem of Branciari [1] is correct, the proofs had gaps [2] since the topology of generalized metric space is not strong enough as the topology of metric space. The disadvantages of generalized metric space can be listed as follows:(𝑤1)generalized metric need not be continuous;(𝑤2)a convergent sequence in generalized metric space need not be Cauchy;(𝑤3)generalized metric space need not be Hausdorff, and hence the uniqueness of limits cannot be guaranteed.Despite the weakness of the topology of generalized metric space, in [3, 4], the authors suggested some techniques to get a (unique) fixed point in such spaces.

On the other hand, Samet et al. [5] introduced the notion of 𝛼-𝜓 contraction mappings and proved the existence and uniqueness of such mappings in complete metric space. The results of this paper are very impressive since several existing results derived from the main theorem of Samet et al. [5] quiet easily. Later, a number of authors have appreciated these results and have used this technique to get further generalization via 𝛼-𝜓 contraction mappings; see, for example, [6–10].

In this paper, we introduce the generalized weaker 𝛼-𝜓 contraction mappings in the setting of generalized metric spaces. Consequently, we investigate the existence and uniqueness of fixed point by caring the problems (𝑤1)--(𝑤3) mentioned above.

Let us recall basic definitions and notations and interesting results that will be in the sequel.

Let Ψ be the family of functions 𝜓∶[0,∞)→[0,∞) satisfying the following conditions:(i)𝜓 is nondecreasing;(ii)there exist 𝑘₀∈ℕ and 𝑎∈(0,1) and a convergent series of nonnegative terms ∑^∞_𝑘=1𝑣_𝑘 such that 𝜓^𝑘+1(𝑡)≤𝑎𝜓^𝑘(𝑡)+𝑣_𝑘, (1) for 𝑘≥𝑘₀ and any 𝑡∈ℝ⁺.In the literature such functions are called either Bianchini-Grandolfi gauge functions (see, e.g., [11–13]) or (𝑐)-comparison functions (see, e.g., [14]).

Lemma 1 (see, e.g., [14]). If 𝜓∈Ψ, then the following hold:(i)(𝜓^𝑛(𝑡))_𝑛∈ℕ converges to 0 as 𝑛→∞ for all 𝑡∈ℝ⁺;(ii)𝜓(𝑡)<𝑡, for any 𝑡∈ℝ⁺;(iii)𝜓 is continuous at 0;(iv)the series ∑^∞_𝑘=1𝜓^𝑘(𝑡) converges for any 𝑡∈ℝ⁺.

In the following, we recall the notion of generalized metric spaces.

Definition 2 (see [1]). Let 𝑋 be a nonempty set and let 𝑑∶𝑋×𝑋→[0,∞] satisfy the following conditions for all 𝑥,𝑦∈𝑋 and all distinct 𝑢,𝑣∈𝑋 each of which is different from 𝑥 and 𝑦: (GMS1)𝑑𝑥,𝑦=0iff(𝑥=𝑦,GMS2)𝑑,(𝑥,𝑦=𝑑𝑦,𝑥GMS3)𝑑𝑥,𝑦≤𝑑(𝑥,𝑢)+𝑑(𝑢,𝑣)+𝑑𝑣,𝑦. (2) Then, the map 𝑑 is called generalized metric. Here, the pair (𝑋,𝑑) is called a generalized metric space and abbreviated as GMS.

In the above definition, if 𝑑 satisfies only (GMS1) and (GMS2), then it is called semimetric (see, e.g., [15]).

The concepts of convergence, Cauchy sequence, and completeness in a GMS are defined as follows.

Definition 3. (1) A sequence {𝑥_𝑛} in a GMS (𝑋,𝑑) is GMS convergent to a limit 𝑥 if and only if 𝑑(𝑥_𝑛,𝑥)→0 as 𝑛→∞.
(2) A sequence {𝑥_𝑛} in a GMS (𝑋,𝑑) is GMS Cauchy if and only if for every 𝜀>0 there exists positive integer 𝑁(𝜀) such that 𝑑(𝑥_𝑛,𝑥_𝑚)<𝜀 for all 𝑛>𝑚>𝑁(𝜀).
(3) A GMS (𝑋,𝑑) is called complete if every GMS Cauchy sequence in 𝑋 is GMS convergent.

The following assumption was suggested by Wilson [15] to replace the triangle inequality with the weakened condition.

(𝑊) For each pair of (distinct) points 𝑢,𝑣 there is a number 𝑟_𝑢,𝑣>0 such that, for every 𝑧∈𝑋, 𝑟_𝑢,𝑣<𝑑(𝑢,𝑧)+𝑑(𝑧,𝑣). (3)

Proposition 4 (see [3]). In a semimetric space, the assumption (𝑊) is equivalent to the assertion that limits are unique.

Proposition 5 (see [3]). Suppose that {𝑥_𝑛} is a Cauchy sequence in a GMS (𝑋,𝑑) with lim_𝑛→∞𝑑(𝑥_𝑛,𝑢)=0, where 𝑢∈𝑋. Then lim_𝑛→∞𝑑(𝑥_𝑛,𝑧)=𝑑(𝑢,𝑧) for all 𝑧∈𝑋. In particular, the sequence {𝑥_𝑛} does not converge to 𝑧 if 𝑧≠𝑢.

A function 𝜙∶[0,∞)→[0,∞) is said to be a Meir-Keeler function [16] if, for each 𝜂>0, there exists 𝛿>0 such that for 𝑡∈[0,∞) with 𝜂≤𝑡<𝜂+𝛿, we have 𝜙(𝑡)<𝜂. Such mapping has been improved and used by several authors [17, 18]. In what follows we recall the notion of weaker Meir-Keeler function.

Definition 6 (see, e.g., [19]). We call 𝜙∶[0,∞)→[0,∞) a weaker Meir-Keeler function if for each 𝜂>0, there exists 𝛿>0 such that for 𝑡∈[0,∞) with 𝜂≤𝑡<𝜂+𝛿, there exists 𝑛₀∈ℕ such that 𝜙^𝑛0(𝑡)<𝜂.

Let Φ be the class of all nondecreasing function 𝜙∶[0,∞)→[0,∞) satisfying the following conditions:(𝜙₁)𝜙∶[0,∞)→[0,∞) is a weaker Meir-Keeler function;(𝜙₂)0<𝜙(𝑡)<𝑡 for all 𝑡>0, 𝜙(0)=0;(𝜙₃)for all 𝑡∈(0,∞), {𝜙^𝑛(𝑡)}_𝑛∈ℕ is decreasing;(𝜙₄)if lim_𝑛→∞𝑡_𝑛=𝛾, then lim_𝑛→∞𝜙(𝑡_𝑛)≤𝛾.Let Θ be the class of functions 𝜑∶[0,∞)→[0,∞) satisfying the following conditions:(𝜑₁)𝜑 is continuous;(𝜑₂)𝜑(𝑡)>0 for 𝑡>0 and 𝜑(0)=0.By using the auxiliary functions, defined above, Chen and Sun [19] proved the following theorem.

Theorem 7. Let (𝑋,𝑑) be a Hausdorff and complete generalized metric space, and let 𝑓∶𝑋→𝑋 be a function satisfying 𝑑𝑓𝑥,𝑓𝑦≤𝜙𝑑𝑥,𝑦−𝜑𝑑𝑥,𝑦 (4) for all 𝑥,𝑦∈𝑋 and 𝜙∈Φ, 𝜑∈Θ. Then 𝑓 has a periodic point 𝜇 in 𝑋; that is, there exists 𝜇∈𝑋 such that 𝜇=𝑓^𝑝𝜇 for some 𝑝∈ℕ.

Another interesting auxiliary function, 𝛼-admissible, was defined by Samet et al. [5].

Definition 8 (see [5]). For a nonempty set 𝑋, let 𝑇∶𝑋→𝑋 and 𝛼∶𝑋×𝑋→[0,∞) be mappings. We say that 𝑇 is 𝛼-admissible if 𝛼𝑥,𝑦≥1⟹𝛼𝑇𝑥,𝑇𝑦≥1, (5) for all 𝑥,𝑦∈𝑋.

Example 9. Let 𝑋=[2,∞) and 𝑇∶𝑋→𝑋 by 𝑇𝑥=(𝑥+1)/(𝑥−1). Define 𝛼(𝑥,𝑦)∶𝑋×𝑋→[0,∞) and 𝛼=𝑒𝑥,𝑦^𝑥+1if0𝑥≥𝑦,ifotherwise. (6) Then 𝑇 is 𝛼-admissible.

Example 10. Let 𝑋=ℝ and 𝑇∶𝑋→𝑋. Define 𝛼(𝑥,𝑦)∶𝑋×𝑋→[0,∞) by 𝑇𝑥=𝑒^𝑥+1 and 𝛼=𝑥𝑥,𝑦²if0𝑥≥𝑦,ifotherwise. (7) Then 𝑇 is 𝛼-admissible.

Some interesting examples of such mappings were given in [5].

The notion of an 𝛼-𝜓 contractive mapping is defined in the following way.

Definition 11 (see [5]). Let (𝑋,𝑑) be a metric space and let 𝑇∶𝑋→𝑋 be a given mapping. We say that 𝑇 is an 𝛼-𝜓 contractive mapping if there exist two functions 𝛼∶𝑋×𝑋→[0,∞) and 𝜓∈Ψ such that 𝛼𝑥,𝑦𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑋. (8)

Clearly, any contractive mapping, that is, a mapping satisfying the Banach contraction, is an 𝛼-𝜓 contractive mapping with 𝛼(𝑥,𝑦)=1 for all 𝑥,𝑦∈𝑋 and 𝜓(𝑡)=𝑘𝑡, 𝑘∈(0,1).

Very recently, Karapınar [20] gave the analog of the notion of an 𝛼-𝜓 contractive mapping, in the context of generalized metric spaces as follows.

Definition 12. Let (𝑋,𝑑) be a generalized metric space and let 𝑇∶𝑋→𝑋 be a given mapping. We say that 𝑇 is an 𝛼-𝜓 contractive mapping if there exist two functions 𝛼∶𝑋×𝑋→[0,∞) and 𝜓∈Ψ such that 𝛼𝑥,𝑦𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑋. (9)

Karapınar [20] also stated the following fixed point theorems.

Theorem 13. Let (𝑋,𝑑) be a complete generalized metric space and let 𝑇∶𝑋→𝑋 be an 𝛼-𝜓 contractive mapping. Suppose that (i)𝑇 is 𝛼-admissible;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1 and 𝛼(𝑥₀,𝑇²𝑥₀)≥1;(iii)𝑇 is continuous.Then there exists a 𝑢∈𝑋 such that 𝑇𝑢=𝑢.

Theorem 14. Let (𝑋,𝑑) be a complete generalized metric space and let 𝑇∶𝑋→𝑋 be an 𝛼-𝜓 contractive mapping. Suppose that (i)𝑇 is 𝛼-admissible;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1 and 𝛼(𝑥₀,𝑇²𝑥₀)≥1;(iii)if {𝑥_𝑛} is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑥∈𝑋 as 𝑛→∞, then 𝛼(𝑥_𝑛,𝑥)≥1 for all 𝑛.Then there exists a 𝑢∈𝑋 such that 𝑇𝑢=𝑢.

For the uniqueness, Karapınar [20] (see also [21]) added the following additional conditions.(𝑈)For all 𝑥,𝑦∈Fix(𝑇), we have 𝛼(𝑥,𝑦)≥1, where Fix(𝑇) denotes the set of fixed points of 𝑇.(𝐻)For all 𝑥,𝑦∈Fix(𝑇), there exists 𝑧∈𝑋 such that 𝛼(𝑥,𝑧)≥1 and 𝛼(𝑦,𝑧)≥1.

Theorem 15. Adding condition (𝑈) to the hypotheses of Theorem 13 (resp., Theorem 14), one obtains that 𝑢 is the unique fixed point of 𝑇.

Theorem 16. Adding conditions (𝐻) and (𝑊) to the hypotheses of Theorem 13 (resp., Theorem 14), one obtains that 𝑢 is the unique fixed point of 𝑇.

Corollary 17. Adding condition (𝐻) to the hypotheses of Theorem 13 (resp., Theorem 14) and assuming that (𝑋,𝑑) is Hausdorff, one obtains that 𝑢 is the unique fixed point of 𝑇.

In this paper, we define the notion of weaker generalized 𝛼-𝜓 contractive mappings and prove some fixed point results in the setting of generalized metric spaces by using such mappings. We state some examples to illustrate the validity of the main results of this paper.

2. Main Results

In this section, we will state and prove our main results.

We give an extension of the notion of 𝛼-𝜓 contractive mappings, in the context of generalized metric space as follows.

Definition 18. Let (𝑋,𝑑) be a generalized metric space and let 𝑇∶𝑋→𝑋 be a given mapping. We say that 𝑇 is a (𝛼-𝜙-𝜑)-contractive mapping of type I if there exist functions 𝛼∶𝑋×𝑋→[0,∞), 𝜑∈Θ, and 𝜙∈Φ such that 𝛼𝑥,𝑦𝑑𝑇𝑥,𝑇𝑦≤𝜙𝑀𝑥,𝑦−𝜑𝑀𝑥,𝑦 (10) for all 𝑥,𝑦∈𝑋, where 𝑀𝑥,𝑦=max𝑑𝑥,𝑦,𝑑(𝑥,𝑇𝑥),𝑑𝑦,𝑇𝑦. (11)

Definition 19. Let (𝑋,𝑑) be a generalized metric space and let 𝑇∶𝑋→𝑋 be a given mapping. We say that 𝑇 is a (𝛼-𝜙-𝜑)-contractive mapping of type II if there exist functions 𝛼∶𝑋×𝑋→[0,∞), 𝜑∈Θ, and 𝜙∈Φ such that 𝛼𝑥,𝑦𝑑𝑇𝑥,𝑇𝑦≤𝜙𝑁𝑥,𝑦−𝜑𝑁𝑥,𝑦 (12) for all 𝑥,𝑦∈𝑋, where 𝑁𝑑,𝑑()𝑥,𝑦=max𝑥,𝑦𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑦2. (13)

Next, we introduce the notion of triangular 𝛼-admissible as follows.

Definition 20. Let 𝑇∶𝑋→𝑋 and 𝛼∶𝑋×𝑋→[0,∞). The mapping 𝑇 is said to be weak triangular 𝛼-admissible if for all 𝑥∈𝑋, one has 𝛼(𝑥,𝑇𝑥)≥1,𝛼𝑇𝑥,𝑇²𝑥≥1⟹𝛼𝑥,𝑇²𝑥≥1. (14)

Now, we state the first fixed point theorem.

Theorem 21. Let (𝑋,𝑑) be a complete generalized metric space and let 𝑇∶𝑋→𝑋 be a (𝛼-𝜙-𝜑)-contractive mapping of type I. Suppose that (i)𝑇 is weak triangular 𝛼-admissible;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1;(iii)𝑇 is continuous.Then, 𝑇 has a fixed point 𝑢∈𝑋; that is 𝑇𝑢=𝑢.

Proof. Due to statement (ii) of the theorem, there exists a point 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1 and 𝛼(𝑥₀,𝑇𝑥₀)≥1. First, we define a sequence {𝑥_𝑛} in 𝑋 by 𝑥_𝑛+1=𝑇𝑥_𝑛=𝑇^𝑛+1𝑥₀ for all 𝑛≥0. Notice that if 𝑥_𝑛0=𝑥_𝑛0+1 for some 𝑛₀, then the proof is completed. Indeed, we have 𝑢=𝑥_𝑛0=𝑥_𝑛0+1=𝑇𝑥_𝑛0=𝑇𝑢. Thus, for the rest of the proof, we assume that 𝑥_𝑛≠𝑥_𝑛+1∀𝑛. (15) Owing to the fact that 𝑇 is 𝛼-admissible, we derive that 𝛼𝑥₀,𝑥₁=𝛼𝑥₀,𝑇𝑥₀≥1⟹𝛼𝑇𝑥₀,𝑇𝑥₁=𝛼𝑥₁,𝑥₂≥1. (16) Utilizing the expression above, we find that 𝛼𝑥_𝑛,𝑥_𝑛+1≥1,∀𝑛=0,1,…. (17) Since 𝑇 is a weak triangular 𝛼-admissible mapping, we obtain that 𝛼𝑥₀,𝑥₂≥1. (18) Since 𝛼(𝑥₀,𝑇^𝑥₀)≥1 and 𝛼(𝑇𝑥₀,𝑇²𝑥₀)=𝛼(𝑥₀,𝑥₂), iteratively, we conclude that 𝛼𝑥_𝑛,𝑥_𝑛+𝑘≥1,∀𝑛,𝑘=0,1,…. (19) Taking (10) and (17) into account, we observe that 𝑑𝑥_𝑛+1,𝑥_𝑛=𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑛−1≤𝛼𝑥_𝑛,𝑥_𝑛−1𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑛−1≤𝜙𝑀𝑥_𝑛,𝑥_𝑛−1−𝜑𝑀𝑥_𝑛,𝑥_𝑛−1, (20) for all 𝑛≥1, where 𝑀𝑥_𝑛,𝑥_𝑛−1=max𝑑𝑥_𝑛,𝑥_𝑛−1,𝑑𝑥_𝑛,𝑇𝑥_𝑛,𝑑𝑥_𝑛−1,𝑇𝑥_𝑛−1=max𝑑𝑥_𝑛,𝑥_𝑛−1,𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛−1,𝑥_𝑛=max𝑑𝑥_𝑛,𝑥_𝑛−1,𝑑𝑥_𝑛,𝑥_𝑛+1. (21) If 𝑀(𝑥_𝑛,𝑥_𝑛−1)=𝑑(𝑥_𝑛,𝑥_𝑛+1), then by (15) and property of the function 𝜑, inequality (20) turns into 𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜙𝑀𝑥_𝑛,𝑥_𝑛−1−𝜑𝑀𝑥_𝑛,𝑥_𝑛−1=𝜙𝑑𝑥_𝑛,𝑥_𝑛+1−𝜑𝑑𝑥_𝑛,𝑥_𝑛+1<𝜙𝑑𝑥_𝑛,𝑥_𝑛+1. (22) Since {𝜙^𝑛(𝑡)} is decreasing, the inequality above yields a contradiction. Hence, we conclude that 𝑀(𝑥_𝑛,𝑥_𝑛−1)=𝑑(𝑥_𝑛,𝑥_𝑛−1) and (20) becomes 𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜙𝑑𝑥_𝑛,𝑥_𝑛−1, (23) for all 𝑛≥1. Recursively, we derive that 𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜙^𝑛𝑑𝑥₁,𝑥₀,∀𝑛≥1. (24)
Owing to the fact that the sequence {𝜙^𝑛(𝑑(𝑥₀,𝑥₁))}_𝑛∈ℕ is decreasing, it converges to some 𝜂≥0. We will show that 𝜂=0. Suppose, on the contrary, that 𝜂>0. Taking the definition of weaker Meir-Keeler function 𝜙 into account, there exists 𝛿>0 such that for 𝑥₀,𝑥₁∈𝑋 with 𝜂≤𝑑(𝑥₀,𝑥₁)<𝛿+𝜂, and there exists 𝑛₀∈ℕ such that 𝜙^𝑛0(𝑑(𝑥₀,𝑥₁))<𝜂. Regarding lim_𝑛→∞𝜙^𝑛(𝑑(𝑥₀,𝑥₁))=𝜂, there exists 𝑝₀∈ℕ such that 𝜂≤𝜙^𝑝(𝑑(𝑥₀,𝑥₁))<𝛿+𝜂, for all 𝑝≥𝑝₀. Hence, we deduce that 𝜙^𝑝0+𝑛0(𝑑(𝑥₀,𝑥₁))<𝜂, which is a contradiction. Thus, lim_𝑛→∞𝜙^𝑛(𝑑(𝑥₀,𝑥₁))=0, and hence lim_𝑛→∞𝑑𝑥_𝑛+1,𝑥_𝑛=0. (25) Regarding (10) and (19), we deduce that 𝑑𝑥_𝑛+2,𝑥_𝑛=𝑑𝑇𝑥_𝑛+1,𝑇𝑥_𝑛−1≤𝛼𝑥_𝑛+1,𝑥_𝑛−1𝑑𝑇𝑥_𝑛+1,𝑇𝑥_𝑛−1≤𝜙𝑀𝑥_𝑛+1,𝑥_𝑛−1−𝜑𝑀𝑥_𝑛+1,𝑥_𝑛−1, (26) for all 𝑛≥1, where 𝑀𝑥_𝑛+1,𝑥_𝑛−1=max𝑑𝑥_𝑛+1,𝑥_𝑛−1,𝑑𝑥_𝑛+1,𝑇𝑥_𝑛+1,𝑑𝑥_𝑛−1,𝑇𝑥_𝑛−1=max𝑑𝑥_𝑛+1,𝑥_𝑛−1,𝑑𝑥_𝑛+1,𝑥_𝑛+2,𝑑𝑥_𝑛−1,𝑥_𝑛. (27)
If 𝑀(𝑥_𝑛,𝑥_𝑛−1)=𝑑(𝑥_𝑛−1,𝑥_𝑛+1) then inequality (26) turns into 𝑑𝑥_𝑛+2,𝑥_𝑛≤𝜙𝑑𝑥_𝑛+1,𝑥_𝑛−1−𝜑𝑑𝑥_𝑛+1,𝑥_𝑛−1≤𝜙𝑑𝑥_𝑛+1,𝑥_𝑛−1 (28) for all 𝑛≥1. By repeating the same argument, inequality (15) implies that 𝑑𝑥_𝑛+2,𝑥_𝑛≤𝜙^𝑛𝑑𝑥₂,𝑥₀,∀𝑛≥1. (29) Due to the fact that the sequence {𝜙^𝑛(𝑑(𝑥₀,𝑥₂))}_𝑛∈ℕ is decreasing, we conclude that lim_𝑛→∞𝑑𝑥_𝑛+2,𝑥_𝑛=0, (30) by following the lines at the proof of (25).
If either 𝑀(𝑥_𝑛,𝑥_𝑛−1)=𝑑(𝑥_𝑛−1,𝑥_𝑛) or 𝑀(𝑥_𝑛+1,𝑥_𝑛+2)=𝑑(𝑥_𝑛+1,𝑥_𝑛+2), then inequality (26) becomes either 𝑑𝑥_𝑛+2,𝑥_𝑛≤𝜙𝑑𝑥_𝑛−1,𝑥_𝑛−𝜑𝑑𝑥_𝑛−1,𝑥_𝑛<𝜙𝑑𝑥_𝑛−1,𝑥_𝑛 (31) or 𝑑𝑥_𝑛+2,𝑥_𝑛≤𝜙𝑑𝑥_𝑛+1,𝑥_𝑛+2−𝜑𝑑𝑥_𝑛+1,𝑥_𝑛+2<𝜙𝑑𝑥_𝑛+1,𝑥_𝑛+2, (32) for all 𝑛≥1. Letting 𝑛→∞ in any of the cases, (31) or (32), together with (25), we have lim_𝑛→∞𝑑𝑥_𝑛+2,𝑥_𝑛≤lim_𝑛→∞𝜙𝑑𝑥_𝑛−1,𝑥_𝑛≤0orlim_𝑛→∞𝑑𝑥_𝑛+2,𝑥_𝑛≤lim_𝑛→∞𝜙𝑑𝑥_𝑛+1,𝑥_𝑛+2≤0. (33)
Let 𝑥_𝑛=𝑥_𝑚 for some 𝑚,𝑛∈ℕ with 𝑚≠𝑛. Without loss of generality, assume that 𝑚>𝑛. Thus, 𝑥_𝑚=𝑇^𝑚−𝑛(𝑇^𝑛𝑥₀)=𝑇^𝑛𝑥₀=𝑥_𝑛. Regarding (15), we consider now𝑑𝑥_𝑛+1,𝑥_𝑛=𝑑𝑇𝑥_𝑛,𝑥_𝑛=𝑑𝑇𝑥_𝑚,𝑥_𝑚=𝑑𝑇𝑥_𝑚,𝑇𝑥_𝑚−1≤𝛼𝑥_𝑚,𝑥_𝑚−1𝑑𝑇𝑥_𝑚,𝑇𝑥_𝑚−1≤𝜙𝑀𝑥_𝑚,𝑥_𝑚−1−𝜑𝑀𝑥_𝑚,𝑥_𝑚−1, (34) where 𝑀𝑥_𝑚,𝑥_𝑚−1=max𝑑𝑥_𝑚,𝑥_𝑚−1,𝑑𝑥_𝑚,𝑇𝑥_𝑚,𝑑𝑥_𝑚−1,𝑇𝑥_𝑚−1=max𝑑𝑥_𝑚,𝑥_𝑚−1,𝑑𝑥_𝑚,𝑥_𝑚+1,𝑑𝑥_𝑚−1,𝑥_𝑚=max𝑑𝑥_𝑚−1,𝑥_𝑚,𝑑𝑥_𝑚,𝑥_𝑚+1. (35) If 𝑀(𝑥_𝑚,𝑥_𝑚−1)=𝑑(𝑥_𝑚−1,𝑥_𝑚), then from (34) and (23) we get that 𝑑𝑥_𝑛+1,𝑥_𝑛=𝑑𝑇𝑥_𝑛,𝑥_𝑛=𝑑𝑇𝑥_𝑚,𝑥_𝑚=𝑑𝑇𝑥_𝑚,𝑇𝑥_𝑚−1≤𝛼𝑥_𝑚,𝑥_𝑚−1𝑑𝑇𝑥_𝑚,𝑇𝑥_𝑚−1≤𝜙𝑑𝑥_𝑚,𝑥_𝑚−1−𝜑𝑑𝑥_𝑚,𝑥_𝑚−1≤𝜙𝑑𝑥_𝑚,𝑥_𝑚−1≤𝜙^𝑚−𝑛𝑑𝑥_𝑛+1,𝑥_𝑛. (36) If 𝑀(𝑥_𝑚,𝑥_𝑚−1)=𝑑(𝑥_𝑚,𝑥_𝑚+1), inequalities (34) and (23) become 𝑑𝑥_𝑛+1,𝑥_𝑛=𝑑𝑇𝑥_𝑛,𝑥_𝑛=𝑑𝑇𝑥_𝑚,𝑥_𝑚=𝑑𝑇𝑥_𝑚,𝑇𝑥_𝑚−1≤𝛼𝑥_𝑚,𝑥_𝑚−1𝑑𝑇𝑥_𝑚,𝑇𝑥_𝑚−1≤𝜙𝑑𝑥_𝑚,𝑥_𝑚+1−𝜑𝑑𝑥_𝑚,𝑥_𝑚+1≤𝜙𝑑𝑥_𝑚,𝑥_𝑚+1≤𝜙^{𝑚−𝑛+1}𝑑𝑥_𝑛+1,𝑥_𝑛. (37) Due to (𝜙₂), inequalities (36) and (37) yield that 𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜙^𝑚−𝑛𝑑𝑥_𝑛+1,𝑥_𝑛<𝑑𝑥_𝑛+1,𝑥_𝑛,𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜙^{𝑚−𝑛+1}𝑑𝑥_𝑛+1,𝑥_𝑛<𝑑𝑥_𝑛+1,𝑥_𝑛, (38) which is a contradiction. Hence {𝑥_𝑛} has no periodic point.
In what follows we will prove that the sequence {𝑥_𝑛} is Cauchy by standard technique. Suppose, on the contrary, that there exists 𝜀>0 such that for any 𝑘∈ℕ, there are 𝑚(𝑘),𝑛(𝑘)∈ℕ with 𝑛(𝑘)>𝑚(𝑘)>𝑘 satisfying 𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)≥𝜀. (39) Furthermore, corresponding to 𝑚(𝑘), one can choose 𝑛(𝑘) in a way that it is the smallest integer 𝑛(𝑘)>𝑚(𝑘) with 𝑑(𝑥_𝑚(𝑘),𝑥_𝑛(𝑘))≥𝜀. Consequently, we have 𝑑(𝑥_𝑚(𝑘),𝑥_{𝑛(𝑘)−1})<𝜀. Consider 𝜀≤𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤𝑑𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−2}+𝑑𝑥_{𝑛(𝑘)−2},𝑥_{𝑛(𝑘)−1}+𝑑𝑥_{𝑛(𝑘)−1},𝑥_𝑚(𝑘)≤𝑑𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−2}+𝑑𝑥_{𝑛(𝑘)−2},𝑥_{𝑛(𝑘)−1}+𝜀. (40) Letting 𝑘→∞, we get that 𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)⟶𝜀. (41)
On the other hand, again by using the quadrilateral inequality, we find 𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤𝑑𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−1}+𝑑𝑥_{𝑛(𝑘)−1},𝑥_{𝑚(𝑘)−1}+𝑑𝑥_{𝑚(𝑘)−1},𝑥_𝑚(𝑘),𝑑𝑥_{𝑛(𝑘)−1},𝑥_{𝑚(𝑘)−1}≤𝑑𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘)+𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+𝑑𝑥_𝑚(𝑘),𝑥_{𝑚(𝑘)−1}. (42) Letting 𝑛→∞, in the inequalities above, we get that 𝑑𝑥_{𝑛(𝑘)−1},𝑥_{𝑚(𝑘)−1}⟶𝜀. (43) On account of (10), we have 𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)=𝑑𝑇𝑥_{𝑛(𝑘)−1},𝑇𝑥_{𝑚(𝑘)−1}≤𝛼𝑥_{𝑛(𝑘)−1},𝑥_{𝑚(𝑘)−1}𝑑𝑇𝑥_{𝑛(𝑘)−1},𝑇𝑥_{𝑚(𝑘)−1}≤𝜙𝑀𝑥_{𝑛(𝑘)−1},𝑥_{𝑚(𝑘)−1}−𝜑𝑀𝑥_{𝑛(𝑘)−1},𝑥_{𝑚(𝑘)−1}, (44) where 𝑀𝑥_{𝑛(𝑘)−1},𝑥_{𝑚(𝑘)−1}=max𝑑𝑥_{𝑛(𝑘)−1},𝑥_{𝑚(𝑘)−1},𝑑𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑑𝑥_{𝑚(𝑘)−1},𝑥_𝑚(𝑘). (45) Letting 𝑛→∞, in (44), and regarding definitions of auxiliary functions 𝜙,𝜑 and (45), we conclude that 𝜀≤𝜀−𝜑(𝜀), (46) which yields that 𝜑(𝜀)=0. By definition of 𝜑, we derive that 𝜀=0, which is a contradiction. Hence, we conclude that {𝑥_𝑛} is a Cauchy sequence in (𝑋,𝑑). Since (𝑋,𝑑) is complete, there exists 𝑢∈𝑋 such that lim_𝑛→∞𝑑𝑥_𝑛,𝑢=0. (47)
Since 𝑇 is continuous, we obtain from (47) that lim_𝑛→∞𝑑𝑥_𝑛+1,𝑇𝑢=lim_𝑛→∞𝑑𝑇𝑥_𝑛,𝑇𝑢=0. (48) From (47) and (48) we get immediately that lim_𝑛→∞𝑇^𝑛𝑥₀=lim_𝑛→∞𝑇𝑥_𝑛=𝑇𝑢. Taking Proposition 5 into account, we conclude that 𝑇𝑢=𝑢.

The following result is deduced from the obvious inequality 𝑁(𝑥,𝑦)≤𝑀(𝑥,𝑦).

Theorem 22. Let (𝑋,𝑑) be a complete generalized metric space and let 𝑇∶𝑋→𝑋 be a (𝛼-𝜙-𝜑)-contractive mapping of type II. Suppose that (i)𝑇 is a weak triangular 𝛼-admissible;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1;(iii)𝑇 is continuous.Then there exists a 𝑢∈𝑋 such that 𝑇𝑢=𝑢.

Theorem 23. Let (𝑋,𝑑) be a complete generalized metric space and let 𝑇∶𝑋→𝑋 be a (𝛼-𝜙-𝜑)-contractive mapping of type I. Suppose that (i)𝑇 is a weak triangular 𝛼-admissible and 𝜙 is upper semicontinuous function;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1;(iii)if {𝑥_𝑛} is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑥∈𝑋 as 𝑛→∞, then 𝛼(𝑥_𝑛,𝑥)≥1 for all 𝑛.Then there exists a 𝑢∈𝑋 such that 𝑇𝑢=𝑢.

Proof. Following the proof of Theorem 21, we know that the sequence {𝑥_𝑛} defined by 𝑥_𝑛+1=𝑇𝑥_𝑛 for all 𝑛≥0 converges for some 𝑢∈𝑋. We will show that 𝑇𝑢=𝑢. Suppose, on the contrary, that 𝑇𝑢≠𝑢. From (17) and condition (iii), there exists a subsequence {𝑥_𝑛(𝑘)} of {𝑥_𝑛} such that 𝛼(𝑥_𝑛(𝑘),𝑢)≥1 for all 𝑘. By applying the quadrilateral inequality together with (10) and (15), for all 𝑘, we get that 𝑑(𝑢,𝑇𝑢)≤𝑑𝑢,𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+2,𝑥_𝑛(𝑘)+1+𝑑𝑥_𝑛(𝑘)+1,𝑇𝑢≤𝑑𝑢,𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+2,𝑥_𝑛(𝑘)+1+𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑢≤𝑑𝑢,𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+2,𝑥_𝑛(𝑘)+1+𝛼𝑥_𝑛(𝑘),𝑢𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑢≤𝑑𝑢,𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+2,𝑥_𝑛(𝑘)+1+𝜙𝑀𝑥_𝑛(𝑘),𝑢, (49) where 𝑀𝑥_𝑛(𝑘),𝑢=max𝑑𝑥_𝑛(𝑘),𝑢,𝑑𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘),𝑑(𝑢,𝑇𝑢). (50)
Letting 𝑘→∞ in the above equality and regarding that the 𝜙 is an upper semicontinuous mapping, we find that 𝑑(𝑢,𝑇𝑢)≤𝜙(𝑑(𝑢,𝑇𝑢)). (51) It implies that from (𝜙₂) 𝑑(𝑢,𝑇𝑢)≤𝜙(𝑑(𝑢,𝑇𝑢))<𝑑(𝑢,𝑇𝑢), (52) which is a contradiction. Hence, we obtain that 𝑢 is a fixed point of 𝑇; that is, 𝑇𝑢=𝑢.

In the following theorem, we remove the semicontinuity of 𝜙 by weakening the contractive mapping type.

Theorem 24. Let (𝑋,𝑑) be a complete generalized metric space and 𝑇∶𝑋→𝑋 be a (𝛼-𝜙-𝜑)-contractive mapping of type II. Suppose that (i)𝑇 is a weak triangular 𝛼-admissible;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1;(iii)if {𝑥_𝑛} is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑥∈𝑋 as 𝑛→∞, then 𝛼(𝑥_𝑛,𝑥)≥1 for all 𝑛.Then there exists a 𝑢∈𝑋 such that 𝑇𝑢=𝑢.

Proof. Following the proof of Theorem 21, we know that the sequence {𝑥_𝑛} defined by 𝑥_𝑛+1=𝑇𝑥_𝑛 for all 𝑛≥0 converges for some 𝑢∈𝑋. We will show that 𝑇𝑢=𝑢. Suppose, on the contrary, that 𝑇𝑢≠𝑢. From (17) and condition (iii), there exists a subsequence {𝑥_𝑛(𝑘)} of {𝑥_𝑛} such that 𝛼(𝑥_𝑛(𝑘),𝑢)≥1 for all 𝑘. By applying the quadrilateral inequality together with (10) and (15), for all 𝑘, we get that 𝑑(𝑢,𝑇𝑢)≤𝑑𝑢,𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+2,𝑥_𝑛(𝑘)+1+𝑑𝑥_𝑛(𝑘)+1,𝑇𝑢≤𝑑𝑢,𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+2,𝑥_𝑛(𝑘)+1+𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑢≤𝑑𝑢,𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+2,𝑥_𝑛(𝑘)+1+𝛼𝑥_𝑛(𝑘),𝑢𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑢≤𝑑𝑢,𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+2,𝑥_𝑛(𝑘)+1+𝜙𝑁𝑥_𝑛(𝑘),𝑢, (53) where 𝑁𝑥_𝑛(𝑘)𝑑𝑥,𝑢=max_𝑛(𝑘),𝑑𝑥,𝑢^𝑛(𝑘),𝑇𝑥_𝑛(𝑘)+𝑑(𝑢,𝑇𝑢)2. (54) Letting 𝑘→∞ in the above equality and regarding (𝜙₄), we find that 𝑑()𝑑()𝑢,𝑇𝑢≤𝜙𝑢,𝑇𝑢2≤𝑑(𝑢,𝑇𝑢)2 (55) which is a contradiction. Hence, we obtain that 𝑢 is a fixed point of 𝑇; that is, 𝑇𝑢=𝑢.

Theorem 25. Adding condition (𝑈) to the hypotheses of Theorem 21 (resp., Theorem 23), one obtains that 𝑢 is the unique fixed point of 𝑇.

Proof. In what follows we will show that 𝑢 is a unique fixed point of 𝑇. We will use the reductio ad absurdum. Let 𝑣 be another fixed point of 𝑇 with 𝑣≠𝑢. It is evident that 𝛼(𝑢,𝑣)=𝛼(𝑇𝑢,𝑇𝑣).
Now, due to (10) and (𝜙₂), we have 𝑑()()𝑑()()𝑑()(𝑀((𝑀((𝑑((𝑑((𝑑(()𝑢,𝑣≤𝛼𝑢,𝑣𝑢,𝑣=𝛼𝑢,𝑣𝑇𝑢,𝑇𝑣≤𝜙𝑢,𝑣))−𝜑𝑢,𝑣))=𝜙𝑢,𝑣))−𝜑𝑢,𝑣))≤𝜙𝑢,𝑣))<𝑑𝑢,𝑣 (56) which is a contradiction, where 𝑀(𝑢,𝑣)=max{𝑑(𝑢,𝑣),𝑑(𝑢,𝑇𝑢),𝑑(𝑣,𝑇𝑣)}=𝑑(𝑢,𝑣). (57) Hence, 𝑢=𝑣.

Theorem 26. Adding condition (𝑈) to the hypotheses of Theorem 22 (resp., Theorem 24), one obtains that 𝑢 is the unique fixed point of 𝑇.

Proof. The proof is analog of the proof of Theorem 25 which will be concluded by using the reductio ad absurdum. Suppose, on the contrary, that 𝑣 is another fixed point of 𝑇 with 𝑣≠𝑢. It is evident that 𝛼(𝑢,𝑣)=𝛼(𝑇𝑢,𝑇𝑣).
Now, due to (12) and (𝜙₂), we have 𝑑()()𝑑()()𝑑()(𝑁((𝑁((𝑑((𝑑((𝑑(()𝑢,𝑣≤𝛼𝑢,𝑣𝑢,𝑣=𝛼𝑢,𝑣𝑇𝑢,𝑇𝑣≤𝜙𝑢,𝑣))−𝜑𝑢,𝑣))=𝜙𝑢,𝑣))−𝜑𝑢,𝑣))=𝜙𝑢,𝑣))<𝑑𝑢,𝑣 (58) which is a contradiction, where 𝑁()𝑑(),𝑑()()𝑢,𝑣=max𝑢,𝑣𝑢,𝑇𝑢+𝑑𝑣,𝑇𝑣2=𝑑(𝑢,𝑣). (59) Hence, 𝑢=𝑣.

For the uniqueness, we can also consider the following condition.(𝐻^∗)For all 𝑥,𝑦∈Fix(𝑇), there exists 𝑧∈𝑋 such that 𝛼(𝑥,𝑧)≥1 and 𝛼(𝑦,𝑧)≥1. Further, lim_𝑛→∞𝑑(𝑧_𝑛,𝑧_𝑛+1)=0, where 𝑧₁=𝑧 and 𝑧_𝑛+1=𝑇𝑧_𝑛 for 𝑛=1,2,3,….

Theorem 27. Adding conditions (𝐻^∗) and (𝑊) to the hypotheses of Theorem 21 (resp., Theorem 23), one obtains that 𝑢 is the unique fixed point of 𝑇.

Proof. Suppose that 𝑣 is another fixed point of 𝑇. From (𝐻^∗), there exists 𝑧∈𝑋 such that 𝛼(𝑢,𝑧)≥1,𝛼(𝑣,𝑧)≥1. (60) Since 𝑇 is 𝛼-admissible, from (60), we have 𝛼𝑢,𝑇^𝑛𝑧≥1,𝛼𝑣,𝑇^𝑛𝑧≥1,∀𝑛. (61) Define the sequence {𝑧_𝑛} in 𝑋 by 𝑧_𝑛+1=𝑇𝑧_𝑛 for all 𝑛≥0 and 𝑧₀=𝑧. From (61), for all 𝑛, we have 𝑑𝑢,𝑧_𝑛+1=𝑑𝑇𝑢,𝑇𝑧_𝑛≤𝛼𝑢,𝑧_𝑛𝑑𝑇𝑢,𝑇𝑧_𝑛≤𝜙𝑀𝑢,𝑧_𝑛−𝜑𝑀𝑢,𝑧_𝑛, (62) where 𝑀𝑢,𝑧_𝑛=max𝑑𝑢,𝑧_𝑛,𝑑(𝑢,𝑇𝑢),𝑑𝑧_𝑛,𝑇𝑧_𝑛=max𝑑𝑢,𝑧_𝑛,𝑑𝑧_𝑛,𝑇𝑧_𝑛. (63) If 𝑀(𝑢,𝑧_𝑛)=𝑑(𝑧_𝑛,𝑇𝑧_𝑛) then by letting 𝑛→∞ in (62) we get that lim_𝑛→∞𝑑𝑧_𝑛,𝑢=0, (64) due to the continuity of 𝜑, (𝜙₄) and the fact that lim_𝑛→∞𝑑(𝑧_𝑛,𝑧_𝑛+1)=0. If 𝑀(𝑢,𝑧_𝑛)=𝑑(𝑢,𝑧_𝑛) then (62) turns into 𝑑𝑢,𝑧_𝑛+1≤𝜙𝑑𝑢,𝑧_𝑛−𝜑𝑑𝑢,𝑧_𝑛≤𝜙𝑑𝑢,𝑧_𝑛. (65) Iteratively, by using inequality (62), we get that 𝑑𝑢,𝑧_𝑛+1≤𝜙^𝑛𝑑𝑢,𝑧₀, (66) for all 𝑛. Letting 𝑛→∞ in the above inequality, we obtain lim_𝑛→∞𝑑𝑧_𝑛,𝑢=0. (67) Similarly, one can show that lim_𝑛→∞𝑑𝑧_𝑛,𝑣=0. (68) Regarding (𝑊) together with (67) and (68), it follows that 𝑢=𝑣. Thus we proved that 𝑢 is the unique fixed point of 𝑇.

Theorem 28. Adding conditions (𝐻^∗) and (𝑊) to the hypotheses of Theorem 22 (resp., Theorem 24), one obtains that 𝑢 is the unique fixed point of 𝑇.

The proof is the analog of the proof of Theorem 27; hence we omit it.

Corollary 29. Adding condition (𝐻^∗) to the hypotheses of Theorem 21 (resp., Theorems 23, 22, and 24) and assuming that (𝑋,𝑑) is Hausdorff, one obtains that 𝑢 is the unique fixed point of 𝑇.

The proof is clear, and hence it is omitted. Indeed, Hausdorffness implies the uniqueness of the limit. Thus, the theorem above yields the conclusions.

3. Consequences

Now, we will show that many existing results in the literature can be deduced easily from Theorems 13 and 14.

Definition 30. Let (𝑋,𝑑) be a generalized metric space and let 𝑇∶𝑋→𝑋 be a given mapping. We say that 𝑇 is a (𝛼-𝜙-𝜑)-contractive mapping of type III if there exist functions 𝛼∶𝑋×𝑋→[0,∞), 𝜑∈Θ, and 𝜙∈Φ such that 𝛼𝑥,𝑦𝑑𝑇𝑥,𝑇𝑦≤𝜙𝑑𝑥,𝑦−𝜑𝑑𝑥,𝑦 (69) for all 𝑥,𝑦∈𝑋.

Now, we state the first fixed point theorem.

Theorem 31. Let (𝑋,𝑑) be a complete generalized metric space and let 𝑇∶𝑋→𝑋 be a (𝛼-𝜙-𝜑)-contractive mapping of type III. Suppose that (i)𝑇 is 𝛼-admissible;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1 and 𝛼(𝑥₀,𝑇²𝑥₀)≥1;(iii)𝑇 is continuous.Then, 𝑇 has a fixed point 𝑢∈𝑋; that is, 𝑇𝑢=𝑢.

We omit the proof of Theorem 31, since it can be derived easily by following the lines in the proof of Theorem 21, analogously.

Theorem 32. Let (𝑋,𝑑) be a complete generalized metric space and let 𝑇∶𝑋→𝑋 be a (𝛼-𝜙-𝜑)-contractive mapping of type III. Suppose that (i)𝑇 is 𝛼-admissible;(ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1 and 𝛼(𝑥₀,𝑇²𝑥₀)≥1;(iii)if {𝑥_𝑛} is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑥∈𝑋 as 𝑛→∞, then 𝛼(𝑥_𝑛,𝑥)≥1 for all 𝑛.Then there exists a 𝑢∈𝑋 such that 𝑇𝑢=𝑢.

Proof. Following the proof of Theorem 21 (resp., Theorem 31), we know that the sequence {𝑥_𝑛} defined by 𝑥_𝑛+1=𝑇𝑥_𝑛 for all 𝑛≥0 converges for some 𝑢∈𝑋. We will show that 𝑇𝑢=𝑢. Suppose, on the contrary, that 𝑇𝑢≠𝑢. From (17) and condition (iii), there exists a subsequence {𝑥_𝑛(𝑘)} of {𝑥_𝑛} such that 𝛼(𝑥_𝑛(𝑘),𝑢)≥1 for all 𝑘. By applying the quadrilateral inequality together with (10) and (15), for all 𝑘, we get that 𝑑(𝑢,𝑇𝑢)≤𝑑𝑢,𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+2,𝑥_𝑛(𝑘)+1+𝑑𝑥_𝑛(𝑘)+1,𝑇𝑢≤𝑑𝑢,𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+2,𝑥_𝑛(𝑘)+1+𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑢≤𝑑𝑢,𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+2,𝑥_𝑛(𝑘)+1+𝛼𝑥_𝑛(𝑘),𝑢𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑢≤𝑑𝑢,𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+2,𝑥_𝑛(𝑘)+1+𝜙𝑑𝑥_𝑛(𝑘),𝑢. (70)
Letting 𝑘→∞ in the above equality and regarding (𝜙₄), we find that 𝑑(𝑢,𝑇𝑢)≤lim_𝑛→∞𝜙𝑑𝑥_𝑛(𝑘),𝑢≤0, (71) which is a contradiction. Hence, we obtain that 𝑢 is a fixed point of 𝑇; that is, 𝑇𝑢=𝑢.

Theorem 33. Adding condition (𝑈) to the hypotheses of Theorem 31 (resp., Theorem 32), one obtains that 𝑢 is the unique fixed point of 𝑇.

Proof. In what follows we will show that 𝑢 is a unique fixed point of 𝑇. We will use the reductio ad absurdum. Let 𝑣 be another fixed point of 𝑇 with 𝑣≠𝑢. It is evident that 𝛼(𝑢,𝑣)=𝛼(𝑇𝑢,𝑇𝑣).
Now, due to (10) and (𝜙₂), we have 𝑑()()𝑑()()𝑑()(𝑑((𝑑((𝑑(()𝑢,𝑣≤𝛼𝑢,𝑣𝑢,𝑣=𝛼𝑇𝑢,𝑇𝑣𝑇𝑢,𝑇𝑣≤𝜙𝑢,𝑣))−𝜑𝑢,𝑣))≤𝜙𝑢,𝑣))<𝑑𝑢,𝑣 (72) which is a contradiction.

Theorem 34. Adding conditions (𝐻) and (𝑊) to the hypotheses of Theorem 31 (resp., Theorem 32), one obtains that 𝑢 is the unique fixed point of 𝑇.

Corollary 35. Let (𝑋,𝑑) be a complete generalized metric space and let 𝑇∶𝑋→𝑋 be a given mapping. Suppose that there exist 𝜙∈Φ and 𝜑∈Θ such that 𝑑𝑇𝑥,𝑇𝑦≤𝜙𝑀𝑥,𝑦−𝜑𝑀𝑥,𝑦, (73) for all 𝑥,𝑦∈𝑋, where 𝑀𝑥,𝑦=max𝑑𝑥,𝑦,𝑑(𝑥,𝑇𝑥),𝑑𝑦,𝑇𝑦. (74) Then 𝑇 has a unique fixed point.

Proof. Let 𝛼∶𝑋×𝑋→[0,∞) be the mapping defined by 𝛼(𝑥,𝑦)=1, for all 𝑥,𝑦∈𝑋. Then 𝑇 is a (𝛼-𝜙-𝜑)-contractive mapping of type I. It is evident that all conditions of Theorem 21 are satisfied. Hence, 𝑇 has a unique fixed point.

Corollary 36. Let (𝑋,𝑑) be a complete generalized metric space and let 𝑇∶𝑋→𝑋 be a given mapping. Suppose that there exist 𝜙∈Φ and 𝜑∈Θ such that 𝑑𝑇𝑥,𝑇𝑦≤𝜙𝑁𝑥,𝑦−𝜑𝑁𝑥,𝑦, (75) for all 𝑥,𝑦∈𝑋, where 𝑁𝑑,𝑑()𝑥,𝑦=max𝑥,𝑦𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑦2. (76) Then 𝑇 has a unique fixed point.

The following Corollary is stronger than the main result of [19]. Notice that we do not need the Hausdorffness condition although it was required in [19].

Corollary 37. Let (𝑋,𝑑) be a complete generalized metric space and let 𝑇∶𝑋→𝑋 be a given mapping. Suppose that there exist 𝜙∈Φ and 𝜑∈Θ such that 𝑑𝑇𝑥,𝑇𝑦≤𝜙𝑑𝑥,𝑦−𝜑𝑑𝑥,𝑦, (77) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors' Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.