An Extension of Generalized (𝜓,𝜑)-Weak Contractions

Abstract

We prove a fixed-point theorem for a class of maps that satisfy generalized (𝜓,𝜑)-weak contractions depending on a given function. An example is given to illustrate our extensions.

1. Introduction

Because fixed-point theory has a wide array of applications in many areas such as economics, computer science, and engineering, it plays evidently a crucial role in nonlinear analysis. One of the cornerstones of this theory is the Banach fixed-point theorem, also known as the Banach contraction mapping theorem [1], which can be stated as follows.

Let 𝑇∶𝑋→𝑋 be a contraction on a compete metric space (𝑋;𝑑); that is, there is a nonnegative real number 𝑘<1 such that 𝑑(𝑇(𝑥),𝑇(𝑦))≤𝑘𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋. Then the map 𝑇 admits one and only one point 𝑥^∗∈𝑋 such that 𝑇𝑥^∗=𝑥^∗. Moreover, this fixed point is the limit of the iterative sequence 𝑥_𝑛+1=𝑇(𝑥_𝑛) for 𝑛=0,1,2,…, where 𝑥₀ is an arbitrary starting point in 𝑋. This theorem attracted a lot of attention because of its importance in the field. Many authors have started studying on fixed-point theory to explore some new contraction mappings to generalize the Banach contraction mapping theorem. In particular, Boyd and Wong [2] introduced the notion of Φ-contractions. In 1997 Alber and Guerre-Delabriere [3] defined the 𝜑-weak contraction which is a generalization of Φ-contractions (see also [4–8]).

On the other hand, the notion of 𝑇-contractions introduced and studied by the authors of the interesting papers in [9–11]. Following this trend, we explore in this paper another extension of (𝜓,𝜑)-weak contractions in the context of 𝑇-contractions.

2. Preliminaries

Let (𝑋,𝑑) be a metric space. Boyd and Wong [2] introduced the notion of Φ-contraction as follows. A map 𝑇∶𝑋→𝑋 is called a Φ-contraction if there exists an upper semicontinuous function Φ∶[0,+∞)→[0,+∞) such that 𝑑𝑇𝑥,𝑇𝑦≤Φ𝑑𝑥,𝑦 (2.1) for all 𝑥,𝑦∈𝑋. The concept of the 𝜑-weak contraction was defined by Alber and Guerre-Delabriere [3] as a generalization of Φ-contraction under the setting of Hilbert spaces and obtained fixed-point results. A map 𝑇∶𝑋→𝑋 is a 𝜑-weak contraction, if there exists a function 𝜑∶[0,+∞)→[0,+∞) such that 𝑑𝑇𝑥,𝑇𝑦≤𝑑𝑥,𝑦−𝜑𝑑𝑥,𝑦 (2.2) for all 𝑥,𝑦∈𝑋 provided that the function 𝜑 satisfies the following condition: 𝜑(𝑡)=0iff𝑡=0. (2.3) Later Rhoades [7] proved analogs of the result in [3] in the context of metric spaces.

Theorem 2.1. Let (𝑋,𝑑) be a complete metric space. Let 𝜑∶[0,+∞)→[0,+∞) be a continuous and nondecreasing function such that 𝜑(𝑡)=0 if and only if 𝑡=0. If 𝑇∶𝑋→𝑋 is a 𝜑weak contraction, then 𝑇 has a unique fixed point.

In [12], Dutta and Choudhury proved an extension of Rhoades.

Theorem 2.2. Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝑋 be a self-mapping satisfying 𝜓𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑑𝑥,𝑦−𝜑𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑋, (2.4) where 𝜓,𝜑∶[0,+∞)→[0,+∞) are continuous and nondecreasing functions with 𝜑(𝑡)=𝜓(𝑡)=0 if and only if 𝑡=0. Then 𝑇 has a unique fixed point.

Zhang and Song [8] improved Theorem 2.1 and gave the following result which states the existence of common fixed points of certain maps in metric spaces.

Theorem 2.3. Let (𝑋,𝑑) be a complete metric space, and let 𝑓,𝑔∶𝑋→𝑋 be self-mappings satisfying 𝑑𝑓𝑥,𝑔𝑦≤𝑀𝑥,𝑦−𝜑𝑀𝑥,𝑦,∀𝑥,𝑦∈𝑋, (2.5) where 𝑀𝑑,𝑑𝑥,𝑦=max𝑥,𝑦,𝑑𝑥,𝑓𝑥,𝑑𝑦,𝑔𝑦𝑥,𝑓𝑦+𝑑𝑦,𝑔𝑥2 (2.6) and 𝜑∶[0,+∞)→[0,+∞) are lower semicontinuous functions with 𝜑(𝑡)=0 if and only if 𝑡=0. Then 𝑓,𝑔 have a unique common fixed point.

Combining the theorems above with the results of Dutta and Choudhury [12], Đoricorić [13] obtained the following theorem.

Theorem 2.4. Let (𝑋,𝑑) be a complete metric space, and let 𝑇,𝑆∶𝑋→𝑋 be self-mappings satisfying 𝜓𝑑𝑓𝑥,𝑔𝑦≤𝜓𝑀𝑥,𝑦−𝜑𝑀𝑥,𝑦,∀𝑥,𝑦∈𝑋, (2.7) where 𝑀𝑑,𝑑𝑥,𝑦=max𝑥,𝑦,𝑑𝑥,𝑓𝑥,𝑑𝑦,𝑔𝑦𝑥,𝑔𝑦+𝑑𝑦,𝑓𝑥2, (2.8)𝜓∶[0,+∞)→[0,+∞) is a continuous and nondecreasing function with 𝜓(𝑡)=0 if and only if 𝑡=0, and 𝜑∶[0,+∞)→[0,+∞) is a lower semicontinuous function with 𝜑(𝑡)=0 if and only if 𝑡=0. Then 𝑓,𝑔 have a unique common fixed point.

The notion of the 𝑇-contraction is defined in ([10, 11]) as follows.

Definition 2.5. Let 𝑇 and 𝑆 be two self-mappings on a metric space (𝑋,𝑑). The mapping 𝑆 is said to be a 𝑇-contraction if there exists 𝑘∈(0,1) such that 𝑑𝑇𝑆𝑥,𝑇𝑆𝑦≤𝑘𝑑𝑇𝑥,𝑇𝑦,∀𝑥,𝑦∈𝑋. (2.9)

It can be easily seen that if 𝑇 is the identity map, then the 𝑇-contraction coincides with the usual contraction.

Example 2.6. Let 𝑋=(0,∞) with the usual metric 𝑑(𝑥,𝑦)=|𝑥−𝑦| induced by (ℝ,𝑑). Consider the following self-mappings 𝑇(𝑥)=1/𝑥 and 𝑆𝑥=3𝑥 on 𝑋. It is clear that 𝑆 is not a contraction. On the contrary, 𝑑=1𝑇𝑆𝑥,𝑇𝑆𝑦−13𝑥=13𝑦31𝑦−1𝑥≤13𝑑𝑇𝑥,𝑇𝑦,∀𝑥,𝑦∈𝑋. (2.10)

Definition 2.7 (see, e.g., [9, 11]). Let (𝑋,𝑑) be a metric space. If {𝑦_𝑛} is a convergent sequence whenever {𝑇𝑦_𝑛} is convergent, then 𝑇∶𝑋→𝑋 is called sequentially convergent.

The aim of this work is to give a proper extension of Đoricorić’s result of using the concept of 𝑇-contraction, that is, the contraction depending on a given function. We will show the existence of a common fixed point for a class of certain maps.

3. Main Results

We start this section by recalling the following two classes of functions.

Let Ψ denote the set of all functions 𝜓∶[0,∞)→[0,∞) which satisfy (i)𝜓is continuous and nondecreasing, (ii)𝜓(𝑡)=0 if and only if 𝑡=0.

Similarly Φ denotes the set of all functions 𝜑∶[0,∞)→[0,∞) which satisfy (i)𝜑 is lower semi continuous, (ii)𝜑(𝑡)=0 if and only if 𝑡=0.

It is easy to see that 𝜓₁(𝑡)=𝑡,𝜓₂(𝑡)=𝑡/(𝑡+1),𝜓₃(𝑡)=𝑡² belong to Ψ and 𝜑₁(𝑡)=min{𝑡,1},  𝜑₂(𝑡)=ln(1+𝑡) belong to Φ.

We are ready to state our main theorem that is a proper extension of Theorem 2.4.

Theorem 3.1. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 an injective, continuous, and sequentially convergent mapping. Let 𝑓,𝑔∶𝑋→𝑋 be self-mappings. If there exist 𝜓∈Ψ and 𝜑∈Φ such that 𝜓𝑑𝑇𝑓𝑥,𝑇𝑔𝑦≤𝜓𝑀𝑇𝑥,𝑇𝑦−𝜑𝑀𝑇𝑥,𝑇𝑦, (3.1) for all 𝑥,𝑦∈𝑋, where 𝑀𝑑,𝑑𝑇𝑥,𝑇𝑦=max𝑇𝑥,𝑇𝑦,𝑑𝑇𝑥,𝑇𝑓𝑥,𝑑𝑇𝑦,𝑇𝑔𝑦𝑇𝑥,𝑇𝑔𝑦+𝑑𝑇𝑦,𝑇𝑓𝑥2, (3.2) then 𝑓,𝑔 have a unique common fixed point.

Proof. We will follow the lines in the proof of the main result in [13]. By injection of 𝑇, we easily check that 𝑀(𝑇𝑥,𝑇𝑦)=0 if and only if 𝑥=𝑦 is a common fixed point of 𝑓 and 𝑔. Let 𝑥₀∈𝑋. We define two iterative sequences {𝑥_𝑛} and {𝑦_𝑛} in the following way: 𝑥_2𝑛+2=𝑓𝑥_2𝑛+1,𝑥_2𝑛+1=𝑔𝑥_2𝑛,𝑦_𝑛=𝑇𝑥_𝑛,∀𝑛=0,1,2,…. (3.3) We prove {𝑦_𝑛} is a Cauchy sequence. For this purpose, we first claim that lim_𝑛→∞𝑑(𝑦_𝑛+1,𝑦_𝑛)=0. It follows from property of 𝜑 that if 𝑛 is odd 𝜓𝑑𝑦_𝑛+1,𝑦_𝑛=𝜓𝑑𝑇𝑥_𝑛+1,𝑇𝑥_𝑛=𝜓𝑑𝑇𝑓𝑥_𝑛,𝑇𝑔𝑥_𝑛−1≤𝜓𝑀𝑇𝑥_𝑛,𝑇𝑥_𝑛−1−𝜑𝑀𝑇𝑥_𝑛,𝑇𝑥_𝑛−1≤𝜓𝑀𝑇𝑥_𝑛,𝑇𝑥_𝑛−1, (3.4) where 𝑀𝑇𝑥_𝑛,𝑇𝑥_𝑛−1=max𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑛−1,𝑑𝑇𝑓𝑥_𝑛,𝑇𝑥_𝑛,𝑑𝑇𝑔𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑑𝑇𝑔𝑥_𝑛−1,𝑇𝑥_𝑛+𝑑𝑇𝑓𝑥_𝑛,𝑇𝑥_𝑛−12=max𝑑𝑦_𝑛,𝑦_𝑛−1,𝑑𝑦_𝑛+1,𝑦_𝑛,𝑑𝑦_𝑛,𝑦_𝑛−1,𝑑𝑦_𝑛−1,𝑦_𝑛+12≤max𝑑𝑦_𝑛,𝑦_𝑛−1,𝑑𝑦_𝑛+1,𝑦_𝑛,𝑑𝑦_𝑛−1,𝑦_𝑛+𝑑𝑦_𝑛,𝑦_𝑛+12. (3.5) Hence, we have 𝜓𝑑𝑦_𝑛+1,𝑦_𝑛≤𝜓max𝑑𝑦_𝑛,𝑦_𝑛−1,𝑑𝑦_𝑛+1,𝑦_𝑛,𝑑𝑦_𝑛−1,𝑦_𝑛+𝑑𝑦_𝑛,𝑦_𝑛+12. (3.6) If 𝑑(𝑦_𝑛,𝑦_𝑛+1)>𝑑(𝑦_𝑛−1,𝑦_𝑛)≥0 then 𝑀(𝑇𝑥_𝑛,𝑇𝑥_𝑛−1)=𝑑(𝑦_𝑛,𝑦_𝑛+1), hence 𝜓𝑑𝑦_𝑛,𝑦_𝑛+1≤𝜓𝑑𝑦_𝑛,𝑦_𝑛+1−𝜑𝑑𝑦_𝑛,𝑦_𝑛+1 (3.7) and which contradicts with 𝑑(𝑦_𝑛,𝑦_𝑛+1)>0 and the property of 𝜑. Thus, it follows from (3.5) that 𝑑𝑦_𝑛+1,𝑦_𝑛≤𝑀𝑇𝑥_𝑛,𝑇𝑥_𝑛−1=𝑑𝑦_𝑛,𝑦_𝑛−1. (3.8) If 𝑛 is even then by the same argument above, we obtain 𝑑𝑦_𝑛+1,𝑦_𝑛≤𝑀𝑇𝑥_𝑛−1,𝑇𝑥_𝑛=𝑑𝑦_𝑛,𝑦_𝑛−1. (3.9) Therefore, 𝑑𝑦_𝑛+1,𝑦_𝑛≤𝑀𝑇𝑥_𝑛,𝑇𝑥_𝑛−1=𝑑𝑦_𝑛,𝑦_𝑛−1 (3.10) for all 𝑛 and {𝑑(𝑦_𝑛,𝑦_𝑛+1)} is a nonincreasing sequence of nonnegative real numbers. Hence, there exists 𝑟≥0 such that lim_𝑛→∞𝑑𝑦_𝑛,𝑦_𝑛+1=lim_𝑛→∞𝑀𝑇𝑥_𝑛,𝑇𝑥_𝑛−1=𝑟. (3.11) By the lower semicontinuity of 𝜑, we have 𝜑(𝑟)≤lim_𝑛→∞inf𝜑𝑀𝑇𝑥_𝑛,𝑇𝑥_𝑛−1. (3.12) Taking the upper limits as 𝑛→∞ on either side of 𝜓𝑑𝑦_𝑛,𝑦_𝑛+1≤𝜓𝑀𝑇𝑥_𝑛,𝑇𝑥_𝑛−1−𝜑𝑀𝑇𝑥_𝑛,𝑇𝑥_𝑛−1, (3.13) we get 𝜓(𝑟)≤𝜓(𝑟)−lim_𝑛→∞inf𝜑𝑀𝑇𝑥_𝑛,𝑇𝑥_𝑛−1≤𝜓(𝑟)−𝜑(𝑟), (3.14) that is, 𝜑(𝑟)≤0. By the property of 𝜑, this implies that 𝜑(𝑟)=0. It follows that 𝑟=0 and lim_𝑛→∞𝑑𝑦_𝑛,𝑦_𝑛+1=0. (3.15) It is implied from (3.10) that lim_𝑛→∞𝑀𝑇𝑥_𝑛,𝑇𝑥_𝑛−1=0. (3.16) Now, we claim that {𝑦_𝑛} is a Cauchy sequence. Since lim_𝑛→∞𝑑(𝑦_𝑛,𝑦_𝑛+1)=0, it is sufficient to prove that {𝑦_2𝑛} is a Cauchy sequence. Suppose on the contrary that {𝑦_2𝑛} is not a Cauchy sequence. Then, there exist 𝜀>0 and subsequences {𝑦_2𝑛(𝑘)} and {𝑦_2𝑚(𝑘)} of {𝑦_2𝑛} such that 𝑛(𝑘) is the smallest index for which 𝑛(𝑘)>𝑚(𝑘)>𝑘,𝑑𝑦_2𝑚(𝑘),𝑦_2𝑛(𝑘)>𝜀. (3.17) This means that 𝑑𝑦_2𝑚(𝑘),𝑦_{2𝑛(𝑘)−2}<𝜀. (3.18) From (3.18) and the triangle inequality, we get 𝜀≤𝑑𝑦_2𝑚(𝑘),𝑦_2𝑛(𝑘)≤𝑑𝑦_2𝑚(𝑘),𝑦_{2𝑛(𝑘)−2}+𝑑𝑦_{2𝑛(𝑘)−2},𝑦_{2𝑛(𝑘)−1}+𝑑𝑦_{2𝑛(𝑘)−1},𝑦_2𝑛(𝑘)<𝜀+𝑑𝑦_{2𝑛(𝑘)−2},𝑦_{2𝑛(𝑘)−1}+𝑑𝑦_{2𝑛(𝑘)−1},𝑦_2𝑛(𝑘). (3.19) Letting 𝑘→∞ and using (3.15), we get lim_𝑘→∞𝑑𝑦_2𝑚(𝑘),𝑦_2𝑛(𝑘)=𝜀. (3.20) By the fact 𝑑𝑦_2𝑚(𝑘),𝑦_{2𝑛(𝑘)+1}−𝑑𝑦_2𝑚(𝑘),𝑦_2𝑛(𝑘)≤𝑑𝑦_2𝑛(𝑘),𝑦_{2𝑛(𝑘)+1}𝑑𝑦_{2𝑚(𝑘)−1},𝑦_2𝑛(𝑘)−𝑑𝑦_2𝑚(𝑘),𝑦_2𝑛(𝑘)≤𝑑𝑦_{2𝑚(𝑘)−1},𝑦_2𝑚(𝑘) (3.21) and using (3.15) and (3.20), we obtain lim_𝑘→∞𝑑𝑦_{2𝑚(𝑘)−1},𝑦_2𝑛(𝑘)=lim_𝑘→∞𝑑𝑦_2𝑚(𝑘),𝑦_{2𝑛(𝑘)+1}=𝜀. (3.22) Moreover, from 𝑑𝑦_{2𝑚(𝑘)−1},𝑦_{2𝑛(𝑘)+1}−𝑑𝑦_{2𝑚(𝑘)−1},𝑦_2𝑛(𝑘)≤𝑑𝑦_2𝑛(𝑘),𝑦_{2𝑛(𝑘)+1} (3.23) and combining with (3.15) and (3.22), we conclude that lim_𝑘→∞𝑑𝑦_{2𝑚(𝑘)−1},𝑦_{2𝑛(𝑘)+1}=𝜀. (3.24) Now, by the definition of 𝑀(𝑇𝑥,𝑇𝑦) and from (3.10), (3.15), and (3.20)–(3.24), we can deduce that lim_𝑘→∞𝑀𝑇𝑥_{2𝑚(𝑘)−1},𝑇𝑥_2𝑛(𝑘)=𝜀. (3.25) Due to (3.1), we have 𝜓𝑑𝑦_2𝑚(𝑘),𝑦_{2𝑛(𝑘)+1}=𝜓𝑑𝑇𝑥_2𝑚(𝑘),𝑇𝑥_{2𝑛(𝑘)+1}=𝜓𝑑𝑇𝑓𝑥_{2𝑚(𝑘)−1},𝑇𝑔𝑥_2𝑛(𝑘)≤𝜓𝑀𝑇𝑥_{2𝑚(𝑘)−1},𝑇𝑥_2𝑛(𝑘)−𝜑𝑀𝑇𝑥_{2𝑚(𝑘)−1},𝑇𝑥_2𝑛(𝑘). (3.26) Letting 𝑘→∞ and using (3.22) and (3.25), we have 𝜓(𝜀)≤𝜓(𝜀)−𝜑(𝜀). (3.27) It is a contradiction to 𝜑(𝑡)>0 for every 𝑡>0. This proves that {𝑦_𝑛} is a Cauchy sequence.
Since 𝑋 is a complete metric space, there exists 𝑢∈𝑋 such that lim_𝑛→∞𝑦_𝑛=𝑢. Since 𝑇 is sequentially convergent, we can deduce that {𝑥_𝑛} converges to 𝑣∈𝑋. By the continuity of 𝑇, we infer that 𝑢=lim_𝑛→∞𝑦_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=𝑇𝑣. (3.28)
We will show that 𝑣=𝑓𝑣=𝑔𝑣. Indeed, suppose that 𝑣≠𝑓𝑣, since 𝑇 is injective, we have 𝑢=𝑇𝑣≠𝑇𝑓𝑣. Hence, 𝑑(𝑇𝑣,𝑇𝑓𝑣)>0. Since lim_𝑛→∞𝑦_2𝑛+1=lim_𝑛→∞𝑦_2𝑛=𝑢,lim_𝑛→∞𝑑𝑦_2𝑛,𝑦_2𝑛+1=0, (3.29) we can seek 𝑁₀∈ℕ such that for any 𝑛≥𝑁₀𝑑𝑦_2𝑛+1<𝑑,𝑢𝑇𝑣,𝑇𝑓𝑣4,𝑑𝑦_2𝑛<𝑑(,𝑢𝑇𝑣,𝑇f𝑣)4,𝑑𝑦_2𝑛,𝑦_2𝑛+1<𝑑𝑇𝑣,𝑇𝑓𝑣4. (3.30) Then, we have 𝑑𝑇𝑣,𝑇𝑓𝑣≤𝑀𝑇𝑣,𝑇𝑥_2𝑛=max𝑑𝑇𝑣,𝑇𝑥_2𝑛,𝑑𝑇𝑣,𝑇𝑓𝑣,𝑑𝑇𝑥_2𝑛,𝑇𝑔𝑥_2𝑛,𝑑𝑇𝑣,𝑇𝑔𝑥_2𝑛+𝑑𝑇𝑥_2𝑛,𝑇𝑓𝑣2=max𝑑𝑢,𝑦_2𝑛,𝑑𝑇𝑣,𝑇𝑓𝑣,𝑑𝑦_2𝑛,𝑦_2𝑛+1,𝑑𝑢,𝑦_2𝑛+1+𝑑𝑦_2𝑛,𝑇𝑓𝑣2≤max𝑑𝑢,𝑦_2𝑛,𝑑𝑇𝑣,𝑇𝑓𝑣,𝑑𝑦_2𝑛,𝑦_2𝑛+1,𝑑𝑢,𝑦_2𝑛+1+𝑑𝑦_2𝑛,𝑇𝑣+𝑑𝑇𝑣,𝑇𝑓𝑣2𝑑≤max𝑇𝑣,𝑇𝑓𝑣4,𝑑,𝑑𝑇𝑣,𝑇𝑓𝑣𝑇𝑣,𝑇𝑓𝑣4,𝑑𝑇𝑣,𝑇𝑓𝑣/4+𝑑𝑇𝑣,𝑇𝑓𝑣/4+𝑑𝑇𝑣,𝑇𝑓𝑣2𝑑,3≤max𝑇𝑣,𝑇𝑓𝑣4𝑑𝑇𝑣,𝑇𝑓𝑣=𝑑𝑇𝑣,𝑇𝑓𝑣. (3.31) Therefore, 𝑀(𝑇𝑣,𝑇𝑥_2𝑛)=𝑑(𝑇𝑣,𝑇𝑓𝑣) for every 𝑛≥𝑁₀. Since 𝜓𝑑𝑇𝑓𝑣,𝑦_2𝑛+1=𝜓𝑑𝑇𝑓𝑣,𝑇𝑥_2𝑛+1=𝜓𝑑𝑇𝑓𝑣,𝑇𝑔𝑥_2𝑛≤𝜓𝑀𝑇𝑣,𝑇𝑥_2𝑛−𝜑𝑀𝑇𝑣,𝑇𝑥_2𝑛=𝜓𝑑𝑇𝑣,𝑇𝑓𝑣−𝜑𝑑𝑇𝑣,𝑇𝑓𝑣 (3.32) and letting 𝑛→∞, we arrive at 𝜓𝑑𝑇𝑓𝑣,𝑇𝑣≤𝜓𝑑𝑇𝑣,𝑇𝑓𝑣−𝜑𝑑𝑇𝑣,𝑇𝑓𝑣. (3.33) We get a contradiction. Hence, 𝑣=𝑓𝑣. By the same argument, we get 𝑣=𝑔𝑣.
Let 𝑤∈𝑋 such that 𝑤=𝑓𝑤=𝑔𝑤. Then, we have 𝑀()𝑑(),𝑑𝑇𝑣,𝑇𝑤=max𝑇𝑣,𝑇𝑤,𝑑𝑇𝑣,𝑇𝑓𝑣,𝑑𝑇𝑤,𝑇𝑔𝑤𝑇𝑣,𝑇𝑔𝑤+𝑑𝑇𝑓𝑣,𝑇𝑤2𝑑(),𝑑()()=max𝑇𝑣,𝑇𝑤𝑇𝑣,𝑇𝑤+𝑑𝑇𝑣,𝑇𝑤2=𝑑(𝑇𝑤,𝑇𝑣). (3.34) Thus 𝜓(𝑑(𝑑(𝑀((𝑀((𝑑((𝑑(.𝑇𝑣,𝑇𝑤))=𝜓𝑇𝑓𝑣,𝑇𝑔𝑤≤𝜓𝑇𝑣,𝑇𝑤))−𝜑𝑇𝑣,𝑇𝑤))=𝜓𝑇𝑣,𝑇𝑤))−𝜑𝑇𝑣,𝑇𝑤)) (3.35) This implies that 𝑑(𝑇𝑣,𝑇𝑤)=0, or 𝑇𝑣=𝑇𝑤. Since 𝑇 is injective, we have 𝑤=𝑣. The theorem is proved.

Remark 3.2. (1) In Theorem 3.1, if we choose 𝑇𝑥=𝑥 for all 𝑥∈𝑋, then we get Theorem 2.4.
(2) In Theorem 3.1, if we fix 𝜓(𝑡)=𝑡 for all 𝑡, then we obtain another extension of Theorem 2.3.
(3) In Theorem 3.1, if we choose 𝑓=𝑔, then we get the uniqueness and existence of fixed point of generalized 𝜑-weak 𝑇-contractions.
The following example shows that Theorem 3.1 is a proper extension of Theorem 2.4.

Example 3.3. Let 𝑋=[1,+∞) and 𝑑 be the usual metric in 𝑋. Consider the maps 𝑓(𝑥)=𝑔(𝑥)=4√𝑥. It is easy to see that 16 is the unique fixed point of 𝑓 and 𝑔. We claim that 𝑓 and 𝑔 are not generalized 𝜑-weak contraction. Indeed, if there exist lower semicontinuous functions 𝜓,𝜑∶[0,∞)→[0,∞) with 𝜓(𝑡)>0,𝜑(𝑡)>0 for 𝑡∈(0,∞) and 𝜑(0)=𝜓(0)=0, such that 𝜓𝑑𝑓𝑥,𝑔𝑦≤𝜓𝑀𝑥,𝑦−𝜑𝑀𝑥,𝑦,∀𝑥,𝑦∈𝑋, (3.36) then 𝜓4√𝑥−√𝑦≤𝜓𝑀𝑥,𝑦−𝜑𝑀𝑥,𝑦,∀𝑥,𝑦∈𝑋, (3.37) where 𝑀(𝑥,𝑦)=max{𝑑(𝑥,𝑦),𝑑(𝑓𝑥,𝑥),𝑑(𝑔𝑦,𝑦),(1/2)[𝑑(𝑔𝑥,𝑦)+𝑑(𝑓𝑦,𝑥)]}. For 𝑥=4 and 𝑦=1, we obtain 𝑀7𝑥,𝑦=max3,4,3,2=4. (3.38) It follows from (3.37) that 𝜓(4)≤𝜓(4)−𝜑(4). (3.39) Hence, 𝜑(4)≤0. We arrive at a contradiction with 𝜑(𝑡)>0 for 𝑡∈(0,∞).
Consider the map 𝑇𝑥=ln𝑥+1, for all  𝑥∈𝑋. It is easy to see that 𝑇 is injective, continuous, and sequentially convergent. Let 𝜓(𝑡)=𝑡 and 𝜑(𝑡)=𝑡/3, for all  𝑡∈[0,+∞). Now, we show that 𝑓 and 𝑔 are generalized 𝜑-weak 𝑇-contractions. It reduces to check the following inequality: ln4√𝑥−ln4√𝑦≤23𝑀𝑇𝑥,𝑇𝑦,∀𝑥,𝑦∈[1,+∞). (3.40) We have ln4√𝑥−ln4√𝑦=12𝑥ln𝑦𝑀𝑇𝑥,𝑇𝑦=maxln𝑥−ln𝑦,ln4√𝑥−ln𝑥,ln4√√𝑦−ln𝑦ln4𝑦−ln𝑥+ln4√𝑥−ln𝑦2≥=𝑥ln𝑥−ln𝑦ln𝑦. (3.41)(3.42) It follows from (3.41) and (3.42) that ln4√𝑥−ln4√𝑦≤12𝑀𝑇𝑥,𝑇𝑦 (3.43) for every 𝑥,𝑦∈𝑋. This proves that (3.40) is true.

By the same method used in the proof of Theorem 3.1, we get the following theorem.

Theorem 3.4. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 an injective, continuous, and sequentially convergent mapping. Let 𝑓,𝑔∶𝑋→𝑋 be self-mappings. If there exist 𝜓∈Ψ and 𝜑∈Φ such that 𝜓𝑑𝑓𝑇𝑥,𝑔𝑇𝑦≤𝜓𝑑𝑇𝑥,𝑇𝑦−𝜑𝑑𝑇𝑥,𝑇𝑦 (3.44) for all 𝑥,𝑦∈𝑋, then 𝑓,𝑔 have a unique common fixed point.

Proof. It follows from the proof of Theorem 3.1 with necessary modifications.