Some Generalizations of Jungck's Fixed Point Theorem

Abstract

We are going to generalize the Jungck's fixed point theorem for commuting mappings by mean of the concepts of altering distance functions and compatible pair of mappings, as well as, by using contractive inequalities of integral type and contractive inequalities depending on another function.

1. Introduction and Preliminary Facts

In 1922, Banach introduced his famous result in the metric fixed point theory, the Banach contraction Principle (BCP), as follows.

Theorem 1.1 (see [1]). Let (𝑀,𝑑) be a complete metric space and let 𝑆∶𝑀→𝑀 be a self-map that satisfies the following condition: there is 𝑎∈[0,1) such that 𝑑𝑆x,𝑆𝑦≤𝑎𝑑𝑥,𝑦 (𝐵𝐶) for all 𝑥,𝑦∈𝑀. Then, 𝑆 has a unique fixed point 𝑧₀∈𝑀 such that for each 𝑥∈𝑀,lim_𝑛→∞𝑆^𝑛𝑥=𝑧₀. One says that a mapping 𝑆 belongs to the class 𝐵𝐶 if it satisfies the condition (𝐵𝐶).

Since then, several generalizations of the BCP have been appeared, some of them can be found, for instance, in [2–7] and into the references therein. In this paper we will focus our attention on an extension of the BCP introduced in 1976 by Jungck. More precisely, we are going to improve and generalize the following extension of the BCP.

Theorem 1.2 (see [8]). Let 𝑆₁,𝑆₂ be two self-maps on a complete metric space, (𝑀,𝑑) such that  (𝐽₁) (𝑆₁,𝑆₂) is a commuting pair, (𝐽₂) 𝑆₂ is continuous, (𝐽₃) 𝑆₁(𝑀)⊂𝑆₂(𝑀),  (𝐽₄) there is 𝑎∈[0,1) such that 𝑑𝑆₁𝑥,𝑆₁𝑦≤𝑎𝑑𝑆₂𝑥,𝑆₂𝑦 (𝐽𝐶)
for all 𝑥,𝑦∈𝑀.
Then, 𝑆₁ and 𝑆₂ have a unique common fixed point 𝑧₀∈𝑀.

The class of all pair of mappings satisfying condition(𝐽𝐶) will be denoted by 𝐽𝐶.

Notice that by taking 𝑆₂𝑥=𝑥, for all 𝑥∈𝑀, in (𝐽𝐶) we obtain the condition (𝐵𝐶). That is, the Theorem 1.1 is obtained from Theorem 1.2.

In order to attain our goals, we are going to use the notions of altering distance functions [9] and compatible pair of mappings [10], also we will use some contractive inequalities of integral type [11] and contractive inequalities depending on another function [12].

We would like to start recalling that in 1984 Khan et al. [9] introduced the concept of altering distance function as follows. A function 𝜓∶ℝ₊→ℝ₊ is called an altering distance function if the following conditions are satisfied:(𝜓₁)𝜓(0)=0, (𝜓₂)𝜓 is monotonically nondecreasing,(𝜓₃)𝜓 is continuous. We will denote by Ψ the set of all altering distance functions. Using this notion the same authors introduce the following class of mappings and studied the conditions for the existence of fixed points.

Definition 1.3 (see [9]). Let (𝑀,𝑑) be a metric space, 𝜓∈Ψ and 𝑆∶𝑀→𝑀 be a self-map. The map 𝑆 is called a 𝜓-Banach contraction, 𝜓-𝐵𝐶, if for each 𝑥,𝑦∈𝑀 there exists 𝑎∈[0,1) such that 𝜓𝑑𝑆𝑥,𝑆𝑦≤𝑎𝜓𝑑𝑥,𝑦. 𝜓-𝐵𝐶 The class of all mappings satisfying condition 𝜓-𝐵𝐶 will be called the 𝜓-𝐵𝐶 class. Notice that letting 𝜓(𝑥)=𝑖_𝑀𝑥=𝑥 in the inequality 𝜓-𝐵𝐶 we obtain(𝐵𝐶), so the 𝜓-𝐵𝐶 class generalizes the 𝐵𝐶 ones.

In 2002, Branciari [12] introduced a new generalization of Banach contraction mappings, which satisfies a general contractive condition of integral type. For that generalization first let us consider the set Φ={𝜑/𝜑∶ℝ₊→ℝ₊} where 𝜑 satisfies the following conditions:(𝜑₁)𝜑 is nonnegative,(𝜑₂)𝜑 is Lebesgue integrable mapping which is summable on each compact subset of ℝ₊,(𝜑₃) for each 𝜀>0,∫^𝜀₀𝜑(𝑡)𝑑𝑡>0.

Definition 1.4 (see [12]). Let (𝑀,𝑑) be a metric space. A mapping 𝑆₁∶𝑀→𝑀 is said to be a ∫𝜑-Banach contraction, (∫𝜑-𝐵𝐶), if there is 𝑎∈[0,1) such that for all 𝑥,𝑦∈𝑀, one has ^{𝑑(𝑆1𝑥,𝑆1}₀^𝑦)𝜑(𝑡)𝑑𝑡≤𝑎₀^{𝑑(𝑥,𝑦)}𝜑(𝑡)𝑑𝑡, ∫𝜑-𝐵𝐶 where 𝜑∈Φ.

The class of all the mappings satisfying the definition above will be denoted by∫𝜑-𝐵𝐶.

Notice that the Definition 1.4 is a generalization of the BCP. In fact, letting 𝜑(𝑡)=1 for each 𝑡>0 in (∫𝜑-𝐵𝐶), we have ^{𝑑(𝑆1𝑥,𝑦𝑆1)}₀𝜑(𝑡)𝑑𝑡=𝑑𝑆₁𝑥,𝑆₁𝑦≤𝑎𝑑𝑥,𝑦=𝑎₀^{𝑑(𝑥,𝑦)}𝜑(𝑡)𝑑𝑡. (1.1) Thus, a Banach contraction mapping also satisfies the condition (∫𝜑-𝐵𝐶). The converse is not true as we will see in [12, Example 3.6].

Theorem 1.5 (see [12]). Let (𝑀,𝑑) be a complete metric space and let 𝑆₁∶𝑀→𝑀 be a ∫𝜑-𝐵𝐶. Then 𝑆₁ has a unique fixed point 𝑧₀∈𝑀 such that for each 𝑥∈𝑀,lim_𝑛→∞𝑆^𝑛₁𝑥=𝑧₀.

General contractive inequalities of integral type are becoming popular for extend classes of contractive mappings with fixed points. See for example, [13–20] and references therein.

In 2009, Beiranvand et al. [11] introduced a new class of contraction mappings, by generalizing the contraction condition (𝐵𝐶) in terms of another function.

Definition 1.6 (see [11]). Let (𝑀,𝑑) be a metric space and 𝑇,𝑆₁∶𝑀→𝑀 be two mappings. The mapping 𝑆₁ is said to be a 𝑇-Banach contraction, (or belongs to the 𝑇-𝐵𝐶 class). If there is 𝑎∈[0,1) such that 𝑑𝑇𝑆₁𝑥,𝑇𝑆₁𝑦≤𝑎𝑑𝑇𝑥,𝑇𝑦 (𝑇-𝐵𝐶) for all 𝑥,𝑦∈𝑀.

By taking 𝑇𝑥=𝑥, for all 𝑥∈𝑀, we get that the (𝑇-𝐵𝐶) and (𝐵𝐶) are equivalent conditions.

Contraction conditions depending on another function have been used in order to generalize other well-known contraction type maps as the Contractive, Chatterjea, Ćirić, Hardy-Rogers, Kannan, Reich and Rhoades mappings [11, 21–25].

Definition 1.7 (see [11]). Let (𝑀,𝑑) be a metric space. A mapping 𝑇∶𝑀→𝑀 is said sequentially convergent if one has, for every sequence (𝑦_𝑛), if 𝑇(𝑦_𝑛) is convergent, then (𝑦_𝑛) also is convergent. 𝑇 is said subsequentially convergent if one has, for every sequence (𝑦_𝑛), if (𝑇(𝑦_𝑛)) is convergent, then (𝑦_𝑛) has a convergent subsequence.

The conditions for the existence of a unique fixed point for mappings in the class 𝑇-𝐵𝐶 are given in the following result.

Theorem 1.8 (see [11]). Let (𝑀,𝑑) be a complete metric space and 𝑇∶𝑀→𝑀 be a one to one, continuous, and subsequentially convergent mapping. Then, every 𝑇-𝐵𝐶 continuous mapping 𝑆₁∶𝑀→𝑀, has a unique fixed point 𝑧₀∈𝑀. Moreover, if 𝑇 is sequentially convergent, then for each 𝑥∈𝑀,lim_𝑛→∞𝑆^𝑛𝑥=𝑧₀.

2. A Version of the Jungck's Fixed Point Theorem Using Altering Distance Functions

In this section we are going to generalize the Jungck’s fixed point Theorem 1.2 by using the altering distance function and the 𝐽𝐶 class. More precisely, we will introduce the class of 𝜓-𝐽-contraction mappings which generalize the 𝐽𝐶 class, and therefore the 𝐵𝐶 class of mappings.

Definition 2.1. Let (𝑀,𝑑) be a metric space, 𝑆,𝑇∶𝑀→𝑀 be self-mappings and 𝜓∈Ψ. The pair (𝑆,𝑇) is called a 𝜓-Jungck contraction, (𝜓-𝐽-Contraction) if for each 𝑥,𝑦∈𝑀 there exists 𝑎∈[0,1) such that 𝜓𝑑𝑆𝑥,𝑆𝑦≤𝑎𝜓𝑑𝑇𝑥,𝑇𝑦. 𝜓-𝐽𝐶

It is clear that letting 𝑇𝑥=𝑥 in 𝜓-𝐽𝐶 we obtain 𝜓-𝐵𝐶.

Example 2.2. Let 𝑀=[0,1]⊂ℝ endowed with the Euclidean metric. We define 𝑆𝑥=𝑥², 𝑇𝑥=(2/√𝑎)𝑥, with 0<𝑎<1 and 𝜓(𝑥)=𝑥². Then, 𝜓𝑑𝑆𝑥,𝑆𝑦=𝑥²−𝑦²²,𝜓𝑑=4𝑇𝑥,𝑇𝑦𝑎𝑥−𝑦². (2.1) From here we get that the pair (𝑆,𝑇) belongs to the class of mappings 𝜓-𝐽𝐶. On the other hand, 𝜓𝑑𝑆𝑥,𝑆𝑦=𝑥²−𝑦²²,𝜓𝑑𝑥,𝑦=𝑥−𝑦². (2.2) Hence, the pair (𝑆,𝑇) does not belong to the class 𝜓-𝐵𝐶.

In order to prove the fixed point theorem for pair of mappings belonging to the class 𝜓-𝐽𝐶 we will need the following notions.

Definition 2.3 (see [26]). Let (𝑀,𝑑) be a metric space and let 𝑆₁,𝑆₂∶𝑀→𝑀 be two mappings. Suppose that 𝑆₁(𝑀)⊂𝑆₂(𝑀) and for every 𝑥₀∈𝑀 one considers the sequence (𝑥_𝑛)⊂𝑀 defined by, 𝑆₁(𝑥_𝑛−1)=𝑆₂(𝑥_𝑛), for all 𝑛∈ℕ, one says that (𝑆₂(𝑥_𝑛)) is a (𝑆₁,𝑆₂)-sequence of initial point 𝑥₀.

On the other hand, recall that a pair of mappings (𝑇,𝑆) is said to be compatible [10] if and only if 𝑑(𝑇𝑆𝑥_𝑛,𝑆𝑇𝑥_𝑛)=0, whenever (𝑥_𝑛)⊂𝑀 is such that lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=𝑡 (2.3) for some 𝑡∈𝑀.

Theorem 2.4. Let (𝑀,𝑑) be a complete metric space and let 𝑆,𝑇∶𝑀→𝑀 be self-mappings such that(a)𝑇 is a continuous function,(b)𝑆(𝑀)⊂𝑇(𝑀),(c)(𝑆,𝑇) is compatible,(d)the pair (𝑆,𝑇) belongs to the class of mappings 𝜓-𝐽𝐶. Then, 𝑆 and 𝑇 have a unique common fixed point 𝑧₀∈𝑀.

Proof. Let 𝑥₀∈𝑀 be an arbitrary point, we will prove that the (𝑆,𝑇)-sequence (𝑇(𝑥_𝑛)) of initial point 𝑥₀ is a Cauchy sequence in 𝑀. For each 𝑛∈ℕ from inequality contraction 𝜓-𝐽𝐶 we have 𝜓𝑑𝑇𝑥_𝑛+1,𝑇𝑥_𝑛=𝜓𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛−1≤𝑎𝜓𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑛−1≤⋯≤𝑎^𝑛𝜓𝑑𝑇𝑥₁,𝑇𝑥₀ (2.4) since 0≤𝑎<1, it follows that lim_𝑛→∞𝜓𝑑𝑇𝑥_𝑛+1,𝑇𝑥_𝑛=0. (2.5) From conditions (𝜓₁) and (𝜓₃) of the altering distance function 𝜓 we obtain lim_𝑛→∞𝑑𝑇𝑥_𝑛+1,𝑇𝑥_𝑛=0,lim_𝑛→∞𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛−1=0. (2.6) Now, we want to prove that (𝑇𝑥_𝑛)⊂𝑀 is a Cauchy sequence in 𝑀. Suppose that (𝑇𝑥_𝑛) is not a Cauchy sequence in 𝑀. So, there exists 𝜀₀>0 for which we can find subsequences (𝑇𝑥_𝑚(𝑘)) and (𝑇𝑥_𝑛(𝑘)) of (𝑇𝑥_𝑛) with 𝑘<𝑚(k)<𝑛(𝑘) such that 𝑑𝑇𝑥_𝑚(𝑘),𝑇𝑥_𝑛(𝑘)≥𝜀₀ (2.7) and for this 𝑚(𝑘) we can choose 𝑛(𝑘) in such a way that it is the smallest integer with 𝑛(𝑘)>𝑚(𝑘) satisfying (2.7). Then 𝑑𝑇𝑥_𝑚(𝑘),𝑇𝑥_{𝑛(𝑘)−1}<𝜀₀. (2.8) Thus, we have 𝜀₀≤𝑑𝑇𝑥_𝑚(𝑘),𝑇𝑥_𝑛(𝑘)≤𝑑𝑇𝑥_𝑚(𝑘),𝑇𝑥_{𝑛(𝑘)−1}+𝑑𝑇𝑥_{𝑛(𝑘)−1},𝑇𝑥_𝑛(𝑘)<𝜀₀+𝑑𝑇𝑥_{𝑛(𝑘)−1},𝑇𝑥_𝑛(𝑘). (2.9) From (2.6), (2.7), and (2.9), it follows that lim_𝑘→∞𝑑𝑇𝑥_𝑚(𝑘),𝑇𝑆𝑥_𝑛(𝑘)=𝜀₀. (2.10) In a similar way we obtain lim_𝑘→∞𝑑𝑇𝑥_{𝑛(𝑘)−1},𝑇𝑥_{𝑚(𝑘)−1}=𝜀₀. (2.11) Now, using the definition of 𝜓-𝐽𝐶 mappings for 𝑥=𝑇𝑥_𝑚(𝑘) and 𝑦=𝑇𝑥_𝑛(𝑘) we have 𝜓𝜀₀≤𝜓𝑑𝑇𝑥_𝑚(𝑘),𝑇𝑥_𝑛(𝑘)=𝜓𝑑𝑆𝑥_{𝑚(𝑘)−1},𝑆𝑥_{𝑛(𝑘)−1}≤𝑎𝜓𝑑𝑇𝑥_{𝑚(𝑘)−1},𝑇𝑥_{𝑛(𝑘)−1}. (2.12) Letting 𝑘→∞ and using (2.11) we have 𝜓𝜀₀≤𝑎𝜓𝜀₀, (2.13) which is a contraction if 𝜀₀>0. This shows that (𝑇𝑥_𝑛) is a Cauchy sequence in 𝑀 and hence it is convergent in the complete metric space 𝑀. So there is 𝑧₀∈𝑀 such that lim_𝑛→∞𝑇𝑥_𝑛+1=lim_𝑛→∞𝑆𝑥_𝑛=𝑧₀. (2.14) Since 𝑇 is a continuous function lim_𝑛→∞𝑇²𝑥_𝑛+1=lim_𝑛→∞𝑇𝑆𝑥_𝑛=𝑇𝑧₀, (2.15) which proves that the pair (𝑆,𝑇) is compatible. Thus lim_𝑛→∞𝑑𝑆𝑇𝑥_𝑛,𝑇𝑆𝑥_𝑛=0. (2.16) Now 𝜓𝑑𝑇𝑆𝑥_𝑛,𝑆𝑥_𝑛≤𝑎𝜓𝑑𝑇²𝑥_𝑛,T𝑥_𝑛 (2.17) letting 𝑛→∞ we have 𝜓𝑑𝑇𝑧₀,𝑧₀≤𝑎𝜓𝑑𝑇𝑧₀,𝑧₀ (2.18) then, it follows that 𝜓𝑑𝑇𝑧₀,𝑧₀=0, (2.19) which implies that 𝑑(𝑇𝑧₀,𝑧₀)=0, so 𝑇𝑧₀=𝑧₀. Again, 𝜓𝑑𝑆𝑇𝑥_𝑛,𝑆𝑧₀≤𝑎𝜓𝑑𝑇²𝑥_𝑛,𝑇𝑧₀ (2.20) letting 𝑛→∞𝜓𝑑𝑇𝑧₀,𝑆𝑧₀≤𝑎𝜓𝑑𝑇𝑧₀,𝑇𝑧₀=0 (2.21) we have that 𝜓[𝑑(𝑇𝑧₀,𝑆𝑧₀)]=0, so it follows that 𝑑(𝑇𝑧₀,𝑆𝑧₀)=0. Then 𝑧₀=𝑇𝑧₀=𝑆𝑧₀.
Therefore, 𝑧₀ is a common fixed point of 𝑆 and 𝑇. Now, we are going to suppose that 𝑧₁ is another common fixed point of 𝑆 and 𝑇. Then 𝜓𝑑𝑧₀,𝑧₁=𝜓𝑑𝑆𝑧₀,𝑆𝑧₁≤𝑎𝜓𝑑𝑧₀,𝑧₁ (2.22) thus we conclude that 𝜓[𝑑(𝑧₀,𝑧₁)]=0, therefore 𝑑(𝑧₀,𝑧₁)=0, or equivalently, 𝑧₀=𝑧₁.

Notice that letting 𝜓=𝑖_𝑀, in the definition of the 𝜓-𝐽𝐶 class, we obtain a natural generalization of Jungck’s fixed point Theorem 1.2 to compatible pair of mappings, by replacing condition (𝐽₁) for the following:

(𝐽^′₁) (𝑆₁,𝑆₂) is compatible.

Also, notice that the fixed point criterion for the class of mappings introduced by Khan et al. in [9] is obtained now as a consequence of Theorem 2.4 by letting 𝑇𝑥=𝑥.

3. The Jungck's Fixed Point Theorem with Contractive Inequality Depending on Another Function

In this section our main goal is to generalize the Theorem 1.2 by considering its contractive condition depending on another function. Now, using the ideas of Beiranvand et al. given in [11], we introduce the following class of pair of mappings.

Definition 3.1. Let (𝑀,𝑑) be a metric space and let 𝑇,𝑆₁,𝑆₂∶𝑀→𝑀 be mappings. The pair (𝑆₁,𝑆₂) is said to be a 𝑇-𝐽-contraction, (𝑇-𝐽𝐶) if there is 𝑎∈[0,1) such that 𝑑𝑇𝑆₁𝑥,𝑇𝑆₁𝑦≤𝑎𝑑𝑇𝑆₂𝑥,𝑇𝑆₂𝑦 (𝑇-𝐽𝐶) for all 𝑥,𝑦∈𝑀.

Notice that taking 𝑇𝑥=𝑥 in the inequality above, we get that (𝑇-𝐽𝐶) and (𝐽𝐶) are equivalent conditions. The next example shows that the class 𝑇-𝐽𝐶 is more general than the 𝐽𝐶 ones.

Example 3.2. Let 𝑀=[1,+∞)⊂ℝ endowed with the Euclidean metric. We define 𝑆₁𝑥=4√𝑥, 𝑆₂𝑥=𝑥, and 𝑇𝑥=1+ln𝑥 for all 𝑥,𝑦∈𝑀. Then, 𝑑𝑆₁𝑥,𝑆₁𝑦=𝑆₁𝑥−𝑆₁𝑦=4√𝑥−4√𝑦≤2𝑥−𝑦=2𝑑𝑆₂𝑥,𝑆₂𝑦. (3.1) Thus, the pair (𝑆₁,𝑆₂) does not belong to 𝐽𝐶. However, 𝑑𝑇𝑆₁𝑥,𝑇𝑆₁𝑦=𝑇𝑆₁𝑥−𝑇𝑆₁𝑦=𝑇4√𝑥−𝑇4√𝑦=12=1ln𝑥−ln𝑦2𝑑𝑇𝑆₂𝑥,𝑇𝑆₂𝑦. (3.2) Hence, the pair (𝑆₁,𝑆₂) belongs to the class 𝑇-𝐽𝐶.

Theorem 3.3. Let (𝑀,𝑑) be a complete metric space and let 𝑇,𝑆₁,𝑆₂∶𝑀→𝑀 be mappings such that(a)𝑇 is one to one, continuous, and sequentially convergent,(b)𝑆₂ is continuous,(c)𝑆₁(𝑀)⊂𝑆₂(𝑀),(d)𝑆₁ and 𝑆₂ are commuting mappings,(e)the pair (𝑆₁,𝑆₂) is a member of the class 𝑇-𝐽𝐶. Then, 𝑆₁ and 𝑆₂ have a unique common fixed point 𝑧₀∈𝑀.

Proof. Let 𝑥₀∈𝑀 be an arbitrary point. We will prove that the (𝑆₁,𝑆₂)-sequence (𝑆₂(𝑥_𝑛)) of initial point 𝑥₀ is a Cauchy sequence in 𝑀. For each 𝑛∈ℕ from the (𝑇-𝐽𝐶) condition we get 𝑑𝑇𝑆₂𝑥_𝑛+1,𝑇𝑆₂𝑥_𝑛=𝑑𝑇𝑆₁𝑥_𝑛,𝑇𝑆₁𝑥_𝑛−1≤𝑎𝑑𝑇𝑆₂𝑥_𝑛,𝑇𝑆₂𝑥_𝑛−1 (3.3) consequently, by repeating the argument above we can conclude that 𝑑𝑇𝑆₂𝑥_𝑛+1,𝑇S₂𝑥_𝑛≤𝑎^𝑛𝑑𝑇𝑆₂𝑥₁,𝑇𝑆₂𝑥₀. (3.4) Taking limits in the last inequality we have that lim_𝑛→∞𝑑𝑇𝑆₂𝑥_𝑛+1,𝑇𝑆₂𝑥_𝑛=lim_𝑛→∞𝑑𝑇𝑆₁𝑥_𝑛,𝑇𝑆₁𝑥_𝑛−1=0. (3.5) From (3.3), we have that (𝑇𝑆₂𝑥_𝑛) is a Cauchy sequence in 𝑀. Since 𝑀 is a complete metric space, then (𝑇𝑆₂𝑥_𝑛) is convergent in 𝑀, and due to the fact that 𝑇 is sequentially convergent, then (𝑆₂𝑥_𝑛) is convergent in 𝑀. So, there exists 𝑧₀∈𝑀 such that lim_𝑛→∞𝑆₂𝑥_𝑛=𝑧₀,lim_𝑛→∞𝑆₁𝑥_𝑛=𝑧₀. (3.6) Since 𝑇 and 𝑆₂ are continuous maps, then 𝑇𝑆₂ is a continuous mapping, and using the contractive inequality (𝑇-𝐽𝐶) we conclude then that 𝑇𝑆₁ is continuous. Thus lim_𝑛→∞𝑇𝑆₁𝑆₂𝑥_𝑛=𝑇𝑆₁𝑧₀,lim_𝑛→∞𝑇𝑆₁𝑆₁𝑥_𝑛=𝑇𝑆₁𝑧₀. (3.7) Using that 𝑆₁ and 𝑆₂ are commuting mappings we obtain lim_𝑛→∞𝑇𝑆₁𝑆₂𝑥_𝑛=𝑇𝑆₁𝑧₀,lim_𝑛→∞𝑇𝑆₂𝑆₁𝑥_𝑛=𝑇𝑆₂𝑧₀. (3.8) Therefore 𝑇𝑆₁𝑧₀=𝑇𝑆₂𝑧₀,𝑇𝑆²₁𝑧₀=𝑇𝑆₂𝑆₁𝑧₀. (3.9)
Now 𝑑𝑇𝑆²₁𝑧₀,𝑇𝑆₁𝑧₀≤𝑎𝑑𝑇𝑆₂𝑆₁𝑧₀,𝑇𝑆₂𝑧₀=𝑎𝑑𝑇𝑆²₁𝑧₀,𝑇𝑆₁𝑧₀. (3.10) Then, it follows that 𝑑𝑇𝑆²₁𝑧₀,𝑇𝑆₁𝑧₀=0 (3.11) so, 𝑇𝑆²₁𝑧₀=𝑇𝑆₁𝑧₀. Since 𝑇 is one to one, we obtain that 𝑆₁(𝑆₁𝑧₀)=𝑆₁𝑧₀, which implies that 𝑆₁𝑧₀ is a fixed point of 𝑆₁ and also that 𝑆₁𝑧₀ is a fixed point of 𝑆₂. Hence, 𝑆₁𝑧₀ is the common fixed point of 𝑆₁ and 𝑆₂.
Now, we will prove that 𝑧₀∈𝑀 is a common fixed point of 𝑆₁ and 𝑆₂. Using the (𝑇-𝐽𝐶) condition, we have 𝑑𝑇𝑧₀,𝑇𝑆₁𝑧₀≤𝑑𝑇𝑧₀,𝑇𝑆₁𝑥_𝑛+𝑑𝑇S₁𝑥_𝑛,𝑇𝑆₁𝑧₀≤𝑑𝑇𝑧₀,𝑇𝑆₁𝑥_𝑛+𝑎𝑑𝑇𝑆₂𝑥_𝑛,𝑇𝑆₂𝑧₀ (3.12) taking the limit as 𝑛→∞ we get 𝑑𝑇𝑧₀,𝑇𝑆₁𝑧₀≤𝑑𝑇𝑧₀,𝑇𝑧₀+𝑎𝑑𝑇𝑧₀,𝑇𝑆₂𝑧₀. (3.13) That is, 𝑑𝑇𝑧₀,𝑇𝑆₁𝑧₀≤𝑎𝑑𝑇𝑧₀,𝑇𝑆₁𝑧₀. (3.14) Therefore, 𝑑(𝑇𝑧₀,𝑇𝑆₁𝑧₀)=0, then 𝑇𝑧₀=𝑇𝑆₁𝑧₀. Since 𝑇 is one to one, we conclude that 𝑧₀=𝑆₁𝑧₀=𝑆₂𝑧₀. Finally, we suppose that 𝑦₀=𝑆₁𝑦₀=𝑆₂𝑦₀ and 𝑧₀≠𝑦₀. Then 𝑑𝑇𝑧₀,𝑇𝑦₀=𝑑𝑇𝑆₁𝑧₀,𝑇𝑆₁𝑧₀≤𝑎𝑑𝑇𝑆₂𝑧₀,𝑇𝑆₂𝑦₀=𝑎𝑑𝑇𝑧₀,𝑇𝑦₀, (3.15) which implies that 𝑇𝑧₀=𝑇𝑦₀, or equivalently, 𝑧₀=𝑦₀. Therefore, the common fixed point of 𝑆₁ and 𝑆₂ is unique.

Example 3.4. As in Example 3.2, let 𝑀=[1,+∞)⊂ℝ be endowed with the euclidean metric and we define 𝑆₁(𝑥)=4√𝑥, 𝑆₂𝑥=𝑥, and 𝑇𝑥=1+ln𝑥, for all 𝑥∈𝑀. Then,(a)𝑇 is one to one, continuous and sequentially convergent,(b)𝑆₂ is continuous,(c)𝑆₁(𝑀)⊂𝑆₂(𝑀), (d)𝑆₁ and 𝑆₂ are commuting mappings,(e)In the Example 3.2, we proved that the pair (𝑆₁,𝑆₂) does not belong to 𝐽𝐶, thus we cannot apply the Theorem 1.2.(f)In the Example 3.2 we have proved that the pair (𝑆₁,𝑆₂) is a 𝑇-𝐽𝐶, so we apply the Theorem 3.3,(g)It is clear that 𝑧₀=16 is the unique common fixed point of 𝑆₁ and 𝑆₂.

Notice that if in the Theorem 3.3 we take 𝑇𝑥=𝑥, for all 𝑥∈𝑀, then we obtain Theorem 1.2.

4. Contractive Conditions of Integral Type for the Jungck's Fixed Point Theorem

This section is devoted to generalize the Theorem 1.2 using the ideas of Branciari [12]. More precisely, for a pair of mappings we introduce contractive conditions of integral type depending on another function. Afterwards, we will give (common) fixed point results for this new class of mappings.

Definition 4.1. Let (𝑀,𝑑) be a metric space and let 𝑇,𝑆₁,𝑆₂∶𝑀→𝑀 be mappings. The pair (𝑆₁,𝑆₂) is called 𝑇-∫𝜑-𝐽𝐶 if there is 𝑎∈[0,1) such that for all 𝑥,𝑦∈𝑀, we have ^{𝑑(𝑇𝑆1𝑥,𝑇𝑆1}₀^𝑦)𝜑(𝑡)𝑑𝑡≤𝑎^{𝑑(𝑇𝑆2𝑥,𝑇𝑆2}₀^𝑦)𝜑(𝑡)𝑑𝑡, 𝑇-∫𝜑-𝐽𝐶 where 𝜑∈Φ.

The class of pair of mappings satisfying the definition above will be denoted by 𝑇-∫𝜑-𝐽𝐶.

Example 4.2. Let us consider 𝑇,𝑆₁,𝑆₂ and (𝑀,‖⋅‖) as in Example 3.2 and consider the function 𝜑∈Φ defined by 𝜑(𝑡)=𝑡. So, the pair (𝑆₁,𝑆₂) belongs to the class 𝑇-∫𝜑-𝐽𝐶 if we have that ^{𝑑(𝑇𝑆1𝑥,𝑇𝑆1}₀^𝑦)𝑡𝑑𝑡≤𝑎^{𝑑(𝑇𝑆2𝑥,𝑇𝑆2}₀^𝑦)𝑡𝑑𝑡,0≤𝑎<1. (4.1) Or, equivalently 12𝑑𝑇𝑆₁𝑥,𝑇𝑆₁𝑦²≤𝑎2𝑑𝑇𝑆₂𝑥,𝑇𝑆₂𝑦², (4.2) which is hold if and only if 𝑑𝑇𝑆₁𝑥,𝑇𝑆₁𝑦≤√𝑎𝑑𝑇𝑆₂𝑥,𝑇𝑆₂𝑦. (4.3) On the other hand, from Example 3.2 we have that the inequality above is valid if we take 𝑎=1/4. Therefore, the pair (𝑆₁,𝑆₂) belongs to 𝑇-∫𝜑-𝐽𝐶 with contractive constant 𝑎=1/4.

Theorem 4.3. Let (𝑀,𝑑) be a complete metric space and let 𝑇,𝑆₁,𝑆₂∶𝑀→𝑀 be mappings such that(a)𝑇 is one to one, continuous, and sequentially convergent,(b)𝑆₁ and 𝑆₂ are continuous mappings,(c)𝑆₁(𝑀)⊂𝑆₂(𝑀), (d)𝑆₁ and 𝑆₂ are commuting mappings,(e)The pair (𝑆₁,𝑆₂) is a 𝑇-∫𝜑-𝐽𝐶. Then, 𝑆₁ and 𝑆₂ have a unique common fixed point.

Proof. Let 𝑥₀∈𝑀 be an arbitrary point. We must prove that the (𝑆₁,𝑆₂)-sequence (𝑆₁𝑥_𝑛−1)=(𝑆₂𝑥_𝑛) of initial point 𝑥₀ is a Cauchy sequence in 𝑀. For each 𝑛∈ℕ from the condition 𝑇-∫𝜑-𝐽𝐶 we have ^{𝑑(𝑇𝑆1𝑥𝑛−1,𝑇𝑆1𝑥𝑛)}₀𝜑(𝑡)𝑑𝑡≤𝑎^{𝑑(𝑇𝑆2𝑥𝑛−1,𝑇𝑆2𝑥𝑛)}₀𝜑(𝑡)𝑑𝑡=𝑎^{𝑑(𝑇𝑆1𝑥𝑛−2,𝑇𝑆1𝑥𝑛−1)}₀𝜑(𝑡)𝑑𝑡. (4.4) Consequently ^{𝑑(𝑇𝑆1𝑥𝑛−1,𝑇𝑆1𝑥𝑛)}₀𝜑(𝑡)𝑑𝑡≤𝑎^{𝑛𝑑(𝑇𝑆2𝑥0,𝑇𝑆2𝑥1)}₀𝜑(𝑡)𝑑𝑡 (4.5) since 0≤𝑎<1, it follows from (4.5) that lim_𝑛→∞^{𝑑(𝑇𝑆1𝑥𝑛−1,𝑇𝑆1𝑥𝑛)}₀𝜑(𝑡)𝑑𝑡≤0 (4.6) using that ∫^𝜀₀𝜑(𝑡)𝑑𝑡>0 for each 𝜀>0, we conclude that lim_𝑛→∞𝑑𝑇𝑆₁𝑥_𝑛−1,𝑇𝑆₁𝑥_𝑛=0. (4.7) Now, we can prove that (𝑇𝑆₁𝑥_𝑛−1) is a Cauchy sequence in 𝑀. We are going to follow the proof of the Theorem 2.1 of [27].
In first place, we prove that for all 𝜀>0, 𝑑(𝑇𝑆₁𝑥_𝑚(𝑝),𝑇𝑆₁𝑥_{𝑛(𝑝)−1})≤𝜀. Suppose that it is not true. Then, there exist 𝜀>0 and subsequences (𝑚_𝑝(𝑘)) and (𝑛_𝑝(𝑘)) such that for each positive integer 𝑘, 𝑛_𝑝(𝑘) is minimal in the sense that 𝑑𝑇𝑆₁𝑥_{𝑚𝑝(𝑘)},𝑇𝑆₁𝑥_{𝑛𝑝(𝑘)}𝑑≥𝜀,𝑇𝑆₁𝑥_{𝑚𝑝(𝑘)},𝑇𝑆₁𝑥_{𝑛𝑝(𝑘)−1}<𝜀. (4.8) Now, 𝜀<𝑑𝑇𝑆₁𝑥_{𝑚𝑝(𝑘)−1},𝑇𝑆₁𝑥_{𝑛𝑝(𝑘)−1}≤𝑑𝑇𝑆₁𝑥_{𝑚𝑝(𝑘)−1},𝑇𝑆₁𝑥_{𝑚𝑝(𝑘)}+𝑑𝑇𝑆₁𝑥_{𝑚𝑝(𝑘)},𝑇𝑆₁𝑥_{𝑛𝑝(𝑘)−1}<𝜀+𝑑𝑇𝑆₁𝑥_{𝑚𝑝(𝑘)−1},𝑇𝑆₁𝑥_{𝑚𝑝(𝑘)}. (4.9) From (4.7) and (4.8), letting 𝑘→∞, we have lim_𝑘→∞𝑑𝑇𝑆₁𝑥_{𝑚𝑝(𝑘)−1},𝑇𝑆₁𝑥_{𝑛𝑝(𝑘)−1}=𝜀,_𝜀0𝜑(𝑡)𝑑𝑡≤^{𝑑(𝑇𝑆1𝑥𝑚𝑝(𝑘),𝑇𝑆1𝑥𝑛𝑝(𝑘))}₀𝜑(𝑡)𝑑𝑡≤𝑎^{𝑑(𝑇𝑆2𝑥𝑚𝑝(𝑘),𝑇𝑆2𝑥𝑛𝑝(𝑘))}₀𝜑(𝑡)𝑑𝑡=𝑎^{𝑑(𝑇𝑆1𝑥𝑚𝑝(𝑘)−1,𝑇𝑆1𝑥𝑛𝑝(𝑘)−1)}₀𝜑(𝑡)𝑑𝑡 (4.10) letting 𝑘→∞, we have the contradiction ^𝜀₀𝜑(𝑡)𝑑𝑡≤𝑎^𝜀₀𝜑(𝑡)𝑑𝑡. (4.11) Thus 𝑑𝑇𝑆₁𝑥_{𝑚𝑝(𝑘)−1},𝑇𝑆₁𝑥_{𝑛𝑝(𝑘)−1}≤𝜀. (4.12) Now, we prove that (𝑇𝑆₁𝑥_𝑛) is a Cauchy sequence in 𝑀. From (4.7)–(4.12), ^𝜀₀𝜑(𝑡)𝑑𝑡≤^{𝑑(𝑇𝑆1𝑥𝑚𝑝(𝑘),𝑇𝑆1𝑥𝑛𝑝(𝑘))}₀𝜑(𝑡)𝑑𝑡≤𝑎^{𝑑(𝑇𝑆2𝑥𝑚𝑝(𝑘),𝑇𝑆2𝑥𝑛𝑝(𝑘))}_𝑎=𝑎^{𝑑(𝑇𝑆1𝑥𝑚𝑝(𝑘)−1,𝑇𝑆1𝑥𝑛𝑝(𝑘)−1)}₀𝜑(𝑡)𝑑𝑡≤𝑎^𝜀₀𝜑(𝑡)𝑑𝑡, (4.13) which is a contradiction. Therefore (𝑇𝑆₁𝑥_𝑛) is a Cauchy sequence in 𝑀, and since 𝑀 is a complete metric space, then (𝑇𝑆₁𝑥_𝑛) is a convergent sequence in 𝑀. Now, from the fact that 𝑇 is sequentially convergent, we get that (𝑆₁𝑥_𝑛) is a convergent sequence in 𝑀. Thus, there exists 𝑧₀∈𝑀 satisfying lim_𝑛→∞𝑆₁𝑥_𝑛=𝑧₀,lim_𝑛→∞𝑆₂𝑥_𝑛=𝑧₀ (4.14) using that 𝑇, 𝑆₁ and 𝑆₂ are continuous mappings, we conclude that 𝑇𝑆₁ and 𝑇𝑆₂ are continuous, hence from (4.14) we get lim_𝑛→∞𝑇𝑆₂𝑆₁𝑥_𝑛=𝑇𝑆₂𝑧₀,lim_𝑛→∞𝑇𝑆₁𝑆₂𝑥_𝑛=𝑇𝑆₁𝑧₀. (4.15) On the other hand, by the commutative property of 𝑆₁ and 𝑆₂, it follows that lim_𝑛→∞𝑇𝑆₁𝑆₂𝑥_𝑛=𝑇𝑆₂𝑧₀,lim_𝑛→∞𝑇𝑆₁𝑆₂𝑥_𝑛=𝑇𝑆₁𝑧₀ (4.16) therefore, 𝑇𝑆₁𝑧₀=𝑇𝑆₂𝑧₀ and because 𝑇 is injective, we have 𝑆₁𝑧₀=𝑆₂𝑧₀. (4.17) It follows for the commuting of the pair (𝑆₁,𝑆₂) that 𝑇𝑆₁𝑆₂𝑧₀=𝑇𝑆₂𝑆₁𝑧₀=𝑇𝑆₂𝑆₂𝑧₀. (4.18) Now, suppose that 𝑇𝑆₁𝑧₀≠𝑇𝑆₁(𝑆₁𝑧₀). Using the condition 𝑇-∫𝜑-𝐽𝐶 we have ^{𝑑(𝑇𝑆1𝑧0,𝑇𝑆1𝑆1𝑧0)}₀𝜑(𝑡)𝑑𝑡≤𝑎^{𝑑(𝑇𝑆2𝑧0,𝑇𝑆2𝑆1𝑧0)}₀𝜑(𝑡)𝑑𝑡=𝑎^{𝑑(𝑇𝑆1𝑧0,𝑇𝑆1S1𝑧0)}₀𝜑(𝑡)𝑑𝑡. (4.19) That is, (1−𝑎)^{𝑑(𝑇𝑆1𝑧0,𝑇𝑆1𝑆1𝑧0)}₀𝜑(𝑡)𝑑𝑡≤0, (4.20) or equivalently ^{𝑑(𝑇𝑆1𝑧0,𝑇𝑆1𝑆1𝑧0)}₀𝜑(𝑡)𝑑𝑡≤0, (4.21) which is a contradiction. Therefore, ^{𝑑(𝑇𝑆1𝑧0,𝑇𝑆1𝑆1𝑧0)}₀𝜑(𝑡)𝑑𝑡=0. (4.22) Since ∫^𝜀₀𝜑(𝑡)𝑑𝑡>0, for each 𝜀>0, then 𝑑(𝑇𝑆₁𝑧₀,𝑇𝑆₁𝑆₁𝑧₀)=0 which implies that 𝑇𝑆₁𝑧₀=𝑇𝑆₁𝑆₁𝑧₀ (4.23) again, since 𝑇 is one to one, we have that 𝑆₁𝑧₀=𝑆₁(𝑆₁𝑧₀). Thus, 𝑆₁𝑧₀ is a fixed point of 𝑆₁. Now, due to the equalities 𝑇𝑆₂𝑆₁𝑧₀=𝑇𝑆₁𝑆₁𝑧₀=𝑇𝑆₁𝑧₀ and because 𝑇 is one to one, we get that 𝑆₂𝑆₁𝑧₀=𝑆₁𝑧₀ (4.24) therefore, 𝑆₁𝑧₀ is a fixed point of 𝑆₂, hence 𝑆₁𝑧₀ is a common fixed point of 𝑆₁ and 𝑆₂.
Now we are going to prove that 𝑧₀∈𝑀 is a common fixed point of 𝑆₁ and 𝑆₂. Using the inequality condition 𝑇-∫𝜑-𝐽𝐶, we have ^{𝑑(𝑇𝑧0,𝑇𝑆1𝑧0)}₀𝜑(𝑡)𝑑𝑡≤^{𝑑(𝑇𝑧0,𝑇𝑆1𝑥𝑛)}₀𝜑(𝑡)𝑑𝑡+^{𝑑(𝑇𝑆1𝑥𝑛,𝑇𝑆1𝑧0)}₀𝜑(𝑡)≤𝑑𝑡_{𝑑(𝑇𝑧0,𝑇𝑆1𝑥𝑛)0}𝜑(𝑡)𝑑𝑡+𝑎^{𝑑(𝑇𝑆2𝑥𝑛,𝑇𝑆2𝑧0)}₀𝜑(𝑡)𝑑𝑡. (4.25) Taking the limit as 𝑛→∞ we obtain ^{𝑑(𝑇𝑧0,𝑇𝑆1𝑧0)}₀𝜑(𝑡)𝑑𝑡≤^{𝑑(𝑇𝑧0,𝑇𝑧0)}₀𝜑(𝑡)𝑑𝑡+𝑎^{𝑑(𝑇𝑧0,𝑇𝑆2𝑧0)}₀𝜑(𝑡)𝑑𝑡=𝑎^{𝑑(𝑇𝑧0,𝑇𝑆1𝑧0)}₀𝜑(𝑡)𝑑𝑡 (4.26) thus we have (1−𝑎)^{𝑑(𝑇𝑧0,𝑇𝑆1𝑧0)}₀𝜑(𝑡)𝑑𝑡≤0 (4.27) since 0≤𝑎<1, then we get that ^{𝑑(𝑇𝑧0,𝑇𝑆1𝑧0)}₀𝜑(𝑡)𝑑𝑡≤0, (4.28) which is a contradiction, therefore ^{𝑑(𝑇𝑧0,𝑇𝑆1𝑧0)}₀𝜑(𝑡)𝑑𝑡=0. (4.29) From here we conclude that 𝑑𝑇𝑧₀,𝑇𝑆₁𝑧₀=0 (4.30) therefore, 𝑇𝑧₀=𝑇𝑆₁𝑧₀, or equivalently, 𝑧₀=𝑆₁𝑧₀=𝑆₂𝑧₀. Now, we will prove the uniqueness of the common fixed point. Let us suppose that 𝑦₀=𝑆₁𝑦₀=𝑆₂𝑦₀ with 𝑦₀≠𝑧₀. Then ^{𝑑(𝑇𝑧0,𝑇𝑦0)}₀𝜑(𝑡)𝑑𝑡=^{𝑑(𝑇𝑆1𝑧0,𝑇𝑆1𝑦0)}₀𝜑(𝑡)𝑑𝑡≤𝑎^{𝑑(𝑇𝑆2𝑧0,𝑇𝑆2𝑦0)}₀𝜑(𝑡)𝑑𝑡=𝑎^{𝑑(𝑇𝑧0,𝑇𝑦0)}₀𝜑(𝑡)𝑑𝑡 (4.31) then (1−𝑎)^{𝑑(𝑇𝑧0,𝑇𝑦0)}₀𝜑(𝑡)𝑑𝑡≤0, (4.32) which implies that ^{𝑑(𝑇𝑧0,𝑇𝑦0)}₀𝜑(𝑡)𝑑𝑡=0 (4.33) thus, 𝑑(𝑇𝑧₀,𝑇𝑦₀)=0. That is, 𝑇𝑧₀=𝑇𝑦₀ and since 𝑇 is one to one, then 𝑧₀=𝑦₀. Therefore, we have proved that 𝑧₀ is the unique common fixed point of 𝑆₁ and 𝑆₂.

5. Further Generalizations

Using contractive conditions of integral type and altering distance functions we can introduce new classes of mapping that generalize the 𝐽𝐶 class.

Definition 5.1. Let (𝑀,𝑑) be a metric space, 𝜓∈Ψ and 𝑆,𝑇∶𝑀→𝑀 be self-mappings. The pair (𝑆,𝑇) is called a ∫𝜑-𝜓-Jungck contraction, ∫𝜑-𝜓-𝐽𝐶 if for each 𝑥,𝑦∈𝑀 there exists 𝑎∈[0,1) such that ₀^{𝜓[𝑑(𝑆𝑥,𝑆𝑦)]}𝜑(𝑡)𝑑𝑡≤𝑎₀^{𝜓[𝑑(𝑇𝑥,𝑇𝑦)]}𝜑(𝑡)𝑑𝑡, ∫𝜑-𝜓-𝐽𝐶 where 𝜑∈Φ.

The class of pairs of mappings fulfilling inequality above will be denoted by ∫𝜑-𝜓-𝐽𝐶.

Similarly, we introduce the following class of mappings.

Definition 5.2. Let (𝑀,𝑑) be a metric space, 𝜓∈Ψ and 𝑆,𝑇∶𝑀→𝑀 be self-mappings. The pair (𝑆,𝑇) is called a 𝜓-∫𝜑-Jungck contraction, 𝜓-∫𝜑-𝐽𝐶 if for each 𝑥,𝑦∈𝑀 there exists 𝑎∈[0,1) such that 𝜓₀^{𝑑(𝑆𝑥,𝑆𝑦)}𝜑(𝑡)𝑑𝑡≤𝑎𝜓₀^{𝑑(𝑇𝑥,𝑇𝑦)}𝜑(𝑡)𝑑𝑡, 𝜓-∫𝜑-𝐽𝐶 where 𝜑∈Φ.

By 𝜓-∫𝜑-𝐽𝐶 we mean the class of mappings given by definition above.

The existence and uniqueness of the common fixed point of pair of mappings in these new classes of self-maps is a consequence of the Theorem 2.4.

Proposition 5.3. Let (𝑀,𝑑) be a complete metric space and 𝑆,𝑇∶𝑀→𝑀 be self-mappings satisfying the following conditions: (a)𝑇 is a continuous function,(b)𝑆(𝑀)⊂𝑇(𝑀),(c)The pair (𝑆,𝑇) is compatible,(d)The pair (𝑆,𝑇) belongs to ∫𝜑-𝜓-𝐽𝐶. Then, 𝑆 and 𝑇 have a unique common fixed point 𝑧₀∈𝑀.

Proof. Let 𝜓₀∶ℝ₊→ℝ₊ be a function defined by 𝜓₀(𝑆)=^𝑆₀𝜑(𝑡)𝑑𝑡, (5.1) where 𝜑∈Φ. It is clear that 𝜓₀∈Ψ. Moreover, 𝜓₀𝜓𝑑𝑆𝑥,𝑆𝑦=₀^{𝜓(𝑑(𝑆𝑥,𝑆𝑦))}𝜑(𝑡)𝑑𝑡≤𝑎₀^{𝜓(𝑑(𝑇𝑥,𝑇𝑦))}𝜑(𝑡)𝑑𝑡=𝑎𝜓₀𝜓𝑑𝑇𝑥,𝑇𝑦 (5.2) it follows that 𝜓₀∘𝜓𝑑𝑆𝑥,𝑆𝑦≤𝑎𝜓₀∘𝜓𝑑𝑇𝑥,𝑇𝑦 (5.3) for all 𝑥,𝑦∈𝑀 and 𝜓₀∘𝜓∈Ψ. Hence by the Theorem 2.4, the mappings 𝑆 and 𝑇 have a unique common fixed point 𝑧₀∈𝑀.

In similar form we can prove the following result.

Proposition 5.4. Let (𝑀,𝑑) be a complete metric space and 𝑆,𝑇∶𝑀→𝑀 be self-mappings satisfying the conditions:(a)𝑇 is a continuous mapping,(b)𝑆(𝑀)⊂𝑇(𝑀),(c)The pair (𝑆,𝑇) is compatible,(d)The pair (𝑇,𝑆) belongs to 𝜓-∫𝜑-𝐽𝐶. Then, 𝑆 and 𝑇 have a unique common fixed point 𝑧₀∈𝑀.

Notice that taking 𝜓=𝑖_𝑀 in (∫𝜑-𝜓-𝐽𝐶) or 𝜓-∫𝜑-𝐽𝐶 we obtain the results given by Kumar et al. in [28].

Acknowledgments

The authors sincerely thank the referees for their valuable comments and suggestions which improved the presentation of the paper. E. M. Rojas is sponsored by the Pontificia Universidad Javeriana under Grant no. 004805.