Invariant Approximation Results via Common Fixed Point Theorems for Generalized Weak Contraction Maps

Abstract

A common fixed point theorem for generalized 𝜑,𝜙,𝐿_𝑓,𝑔-weak contraction in a metric space is established. As an application, some common fixed point results in normed linear spaces are obtained. We also study some results on best approximation via common fixed point theorems. Our result improves some results from the existing literature. Some illustrative examples to highlight the realized improvements are also furnished.

1. Introduction

Let (𝑋,𝑑) be a metric space. A mapping 𝑇∶𝑋→𝑋 is said to be contraction if there exists 0<𝑘<1 such that for all 𝑥,𝑦∈𝑋𝑑𝑇𝑥,𝑇𝑦≤𝑘𝑑𝑥,𝑦. (1) If the metric space (𝑋,𝑑) is complete, then the mapping satisfying (1) has a unique fixed point. This is known as the Banach contraction principle and is one of the significant results in nonlinear analysis. Inequality (1) also implies the continuity of 𝑇.

Due to its importance and usefulness, generalizations of the above contraction principle have been a very active field of research for the last four decades (see, e.g., [1–27]).

In 1997, Alber and Guerre-Delabriere [1] introduced the concept of weakly contractive mappings and proved the existence of fixed points for weakly contractive mappings in Hilbert spaces. Thereafter, Rhoades [2] assumed 𝜙-weakly contractive mappings 𝑇∶𝑋→𝑋 which satisfies the condition 𝑑𝑇𝑥,𝑇𝑦≤𝑑𝑥,𝑦−𝜙𝑑𝑥,𝑦, (2) where 𝑥,𝑦∈𝑋 and 𝜙∶[0,∞)→[0,∞) is a continuous and nondecreasing function such that 𝜙(𝑡)=0 if and only if 𝑡=0 and shows that most of the results of Alber and Guerre-Delabriere [1] are still true in Banach spaces. If one takes 𝜙(𝑡)=𝑘𝑡, where 0<𝑘<1, then (2) reduces to (1).

Recently, Zhang and Song [3] used the generalized 𝜙-weak contraction and proved the following result.

Theorem 1 (see [3]). Let (𝑋,𝑑) be a complete metric space and let 𝑇∶𝑋→𝑋 be a map such that for all 𝑥,𝑦∈𝑋𝑑𝑇𝑥,𝑇𝑦≤𝑀𝑥,𝑦−𝜙𝑀𝑥,𝑦, (3) where 𝜙∶[0,∞)→[0,∞) is a lower semicontinuous function with 𝜙(𝑡)>0 for 𝑡∈(0,∞) and 𝜙(0)=0, 𝑀𝑑(),1𝑥,𝑦=max𝑥,𝑦,𝑑𝑇𝑥,𝑥,𝑑𝑇𝑦,𝑦2𝑑𝑦,𝑇𝑥+𝑑𝑥,𝑇𝑦. (4) Then there exists the unique point 𝑢∈𝑋 such that 𝑢=𝑇𝑢.

Further, using the control function defined by Khan et al. [4], the above result has been generalized by many authors (see [5–7]). On the other hand, Berinde [8] introduced the notion of (𝑘,𝐿)-weak contraction and proved that many well-known contractive conditions do imply (𝑘,𝐿)-weak contraction. The concept of (𝑘,𝐿)-weak contraction does not ask 𝑘+𝐿 to be less than 1 as happens in condition (1). Afterward, many authors study this new class of weak contraction and obtained some significant result (see [8–12]).

Definition 2. A map 𝑇∶𝑋→𝑋 is called generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction, if for each 𝑥,𝑦∈𝑋𝜑𝑑𝑇𝑥,𝑇𝑦≤𝜑𝑚𝑥,𝑦−𝜙𝑚𝑥,𝑦+𝐿𝜑𝑛𝑥,𝑦, (5) where 𝑓,𝑔∶𝑋→𝑋,𝐿≥0, 𝑚𝑑,1𝑥,𝑦=max𝑓𝑥,𝑔𝑦,𝑑𝑇𝑥,𝑓𝑥,𝑑𝑇𝑦,𝑔𝑦2𝑑,𝑛𝑑,𝑑,𝑔𝑦,𝑇𝑥+𝑑𝑓𝑥,𝑇𝑦𝑥,𝑦=min𝑇𝑥,𝑓𝑥+𝑑𝑇𝑦,𝑔𝑦𝑔𝑦,𝑇𝑥,𝑑𝑓𝑥,𝑇𝑦 (6) and 𝜑∶[0,∞)→[0,∞) is a continuous monotone nondecreasing function with 𝜑(𝑡)=0 if and only if 𝑡=0 and 𝜙∶[0,∞)→[0,∞) is a lower semicontinuous function from right such that 𝜙 is positive on (0,∞) and 𝜙(0)=0.
If 𝐿=0, then 𝑇 is said to be generalized (𝜑,𝜙)_𝑓,𝑔-weak contraction. If 𝑔=𝑓, then 𝑇 is called generalized (𝜑,𝜙)_𝑓-weak contraction. If 𝑔=𝑓= identity map, that is, 𝑚(𝑥,𝑦) coincides with 𝑀(𝑥,𝑦), then 𝑇 is called generalized (𝜑,𝜙)-weak contraction which is exactly the maps studied by Doric [5]. Again, if 𝐿=0 and 𝜑(𝑡)=𝑡, then 𝑇 is called generalized 𝜙_𝑓,𝑔-weak contraction which is the same as generalized (𝑓,𝑔)-weak contraction investigated by Akbar et al. (see [22]) and if 𝜙(𝑡)=(1−𝑘)𝑡 for a constant 𝑘 with 0<𝑘<1, then 𝑇 is called generalized (𝑓,𝑔)-contraction which has been introduced by Song [21]. In Equation (5), if 𝑓=𝑔=𝐼 (identity map), then 𝑇 is called generalized (𝜑,𝜙,𝐿)-weak contraction.

Remark 3. It is obvious that the class of generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction contains the class of generalized (𝜑,𝜙)_𝑓,𝑔-weak contraction and hence contains the class of generalized 𝜙_𝑓,𝑔-weak contraction, but the converse is not true, as shown by Remark 30.
In this paper, we prove that there is a unique common fixed point for generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contractive mappings in a metric space. As an application, some common fixed point results in normed linear space are obtained. We apply these theorems to obtain some results on invariant approximation. Our results generalize and extend the corresponding results of [5, 11, 12, 16–23, 28] to the class of generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contractive maps.

2. Preliminaries

We need the following known definitions and standard notations in the sequel.

Let 𝑀 be a nonempty subset of a normed space (𝑋,‖⋅‖). The set 𝐵_𝑀(𝑝)={𝑥∈𝑀∶‖𝑥−𝑝‖=dist(𝑝,𝑀)} is called the set of best approximants to 𝑝∈𝑋 out of 𝑀, where dist(𝑝,𝑀)=inf{‖𝑦−𝑝‖∶𝑦∈𝑀}. We denote ℕ and cl(𝑀) (resp., wcl(𝑀)) by the set of positive integers and the closure (resp., weak closure) of a set 𝑀 in 𝑋. Let 𝑓,𝑇∶𝑀→𝑀 be mappings. The set of fixed point of 𝑇 is denoted by 𝐹(𝑇). A point 𝑥∈𝑀 is a coincidence point (common fixed point) of 𝑓 and 𝑇 if 𝑓𝑥=𝑇𝑥(𝑓𝑥=𝑇𝑥=𝑥). The set of coincidence point of 𝑓and 𝑇 is denoted by 𝐶(𝑓,𝑇). The pair {𝑓,𝑇} is called(1)commuting if 𝑓𝑇𝑥=𝑇𝑓𝑥 for all 𝑥∈𝑀,(2)compatible [13] if lim_𝑛→∞‖𝑇𝑓𝑥_𝑛−𝑓𝑇𝑥_𝑛‖=0 whenever {𝑥_𝑛} is a sequence in 𝑀 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=𝑡 for some 𝑡∈𝑀,(3)weakly compatible [14] if 𝑓𝑇𝑥=𝑇𝑓𝑥 for all 𝑥∈𝐶(𝑓,𝑇),(4)Banach operator pair [16] if the set 𝐹(𝑓) is 𝑇-invariant; that is, 𝑇(𝐹(𝑓))⊆𝐹(𝑓).Apparently, a commuting pair (𝑓,𝑇) is a Banach operator pair but not conversely. If (𝑓,𝑇) is a Banach operator pair, then (𝑇,𝑓) need not be Banach operator pair (see [16]).

The set 𝑀 is called 𝑞-star shaped with 𝑞∈𝑀 if the segment [𝑞,𝑥]={(1−𝑘)𝑞+𝑘𝑥∶0≤𝑘≤1} joining 𝑞 to 𝑥 is contained in 𝑀 for all 𝑥∈𝑀. The map 𝑓 defined on 𝑀 is said to be (5) affine [18] if 𝑀 is convex and 𝑓(𝑘𝑥+(1−𝑘)𝑦)=𝑘𝑓𝑥+(1−𝑘)𝑓𝑦 for all 𝑥,𝑦∈𝑀 and 𝑘∈[0,1] and (6) 𝑞-affine [18] if 𝑀 is 𝑞-star shaped and 𝑓(𝑘𝑥+(1−𝑘)𝑞)=𝑘𝑓𝑥+(1−𝑘)𝑞 for all 𝑥∈𝑀 and 𝑘∈[0,1].

Suppose that 𝑀 is 𝑞-star shaped with 𝑞∈𝐹(𝑓) and is both 𝑇- and 𝑓-invariant. Then 𝑇 and 𝑓 are called(1)𝐶_𝑞-commuting [18] if 𝑓𝑇𝑥=𝑇𝑓𝑥 for all 𝑥∈𝐶_𝑞(𝑓,𝑇), where 𝐶_𝑞(𝑓,𝑇)=∪{𝐶(𝑓,𝑇_𝑘)∶0≤𝑘≤1}, where 𝑇_𝑘(𝑥)=(1−𝑘)𝑞+𝑘𝑇𝑥,(2)𝑅-subweakly commuting on 𝑀 [15] if, for all 𝑥∈𝑀, there exists a real number 𝑅>0 such that ‖𝑇𝑓𝑥−𝑓𝑇𝑥‖≤𝑅dist(𝑓𝑥,[𝑞,𝑇𝑥]).

Definition 4. A map 𝑇∶𝑀→𝑋 is said to be demiclosed at 0∈𝑋 if whenever {𝑥_𝑛} is a sequence in 𝑀 such that {𝑥_𝑛} converges weakly to 𝑥∈𝑀 and {𝑇𝑥_𝑛} converges strongly to 0; then 𝑇𝑥=0.

Definition 5. A Banach space 𝑋 is said to satisfy Opial’s condition if whenever {𝑥_𝑛} is a sequence in 𝑋 such that {𝑥_𝑛} converges weakly to 𝑥∈𝑋; the inequality liminf_𝑛→∞𝑥_𝑛−𝑥<liminf_𝑛→∞𝑥_𝑛−𝑦 (7) holds for all 𝑦≠𝑥. Every Hilbert space and the space 𝑙_𝑝(1<𝑝<∞) satisfy Opial’s condition.

Lemma 6. If  lim_𝑛→∞𝑑(𝑦_𝑛+1,𝑦_𝑛)=0 and {𝑦_2𝑛} is a Cauchy sequence, then {𝑦_𝑛} is also a Cauchy sequence.

Proof. Since {𝑦_2𝑛} is a Cauchy sequence, we have that, for a given 𝜀>0, there is a 𝑛₀>0 such that for every 𝑛,𝑚>𝑛₀ we have 𝑑(𝑦_2𝑛,𝑦_2𝑚)≤𝜀/3. Also, since lim_𝑛→∞𝑑(𝑦_𝑛+1,𝑦_𝑛)=0 we have that, given 𝜀>0, there is a 𝑛₁>0 such that, for every 𝑛,𝑚>𝑛₁, 𝑑(𝑦_𝑛+1,𝑦_𝑛)<𝜀/3. Now, suppose 𝑚,𝑛>𝑚₀=max{𝑛₀,𝑛₁}; then we have three possibilities.(a)If 𝑚 and 𝑛 are both even numbers, let 𝑚=2𝑘, 𝑛=2𝑠; then 𝑑(𝑦_𝑚,𝑦_𝑛)=𝑑(𝑦_2𝑘,𝑦_2𝑠)≤𝜀/3<𝜀.(b)If 𝑚 is even and 𝑛 is odd, let 𝑚=2𝑘, 𝑛=2𝑠+1; then by the triangle inequality 𝑑(𝑦_𝑚,𝑦_𝑛)≤𝑑(𝑦_𝑚,𝑦_𝑛+1)+𝑑(𝑦_𝑛+1,𝑦_𝑛)=𝑑(𝑦_2𝑘,𝑦_2𝑠+2)+𝑑(𝑦_2𝑠+2,𝑦_2𝑠+1)≤𝜀/3+𝜀/3=2𝜀/3<𝜀.(c)If 𝑚,𝑛 are both odd numbers, let 𝑚=2𝑘+1, 𝑛=2𝑠+1; then using the previous estimate 𝑑(𝑦_𝑚,𝑦_𝑛)=𝑑(𝑦_2𝑘+1,𝑦_2𝑠+1)≤𝑑(𝑦_2𝑘+1,𝑦_2𝑠+2)+𝑑(𝑦_2𝑠+2,𝑦_2𝑠+1)≤2𝜀/3+𝜀/3=𝜀.Therefore {𝑦_𝑛} is Cauchy.

3. Main Results

The following result extends and improves Theorem 2.2 of [5], Theorem 2.4 of [11], Theorem 2.1 of [12], and Theorem 2.1 of [18].

Theorem 7. Let 𝑀 be a nonempty subset of a metric space (𝑋,𝑑) and 𝑓,𝑔, and 𝑇 are self-mappings of 𝑀 such that 𝑐𝑙(𝑇(𝑀))⊆𝑓(𝑀)∩𝑔(𝑀). Assume that 𝑐𝑙(𝑇(𝑀)) is complete and 𝑇 is generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction. Then the pairs (𝑇,𝑓) and (𝑇,𝑔) have a unique point of coincidence in 𝑀. Also, if the pairs (𝑇,𝑓) and (𝑇,𝑔) are weakly compatible, then 𝑀∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔) is singleton.

Proof. Let 𝑥₀∈𝑀 be an arbitrary point. Since cl(𝑇(𝑀))⊆𝑓(𝑀)∩𝑔(𝑀), we can choose a point 𝑥₁∈𝑀 such that 𝑦₀=𝑇𝑥₀=𝑓𝑥₁. Similarly a point 𝑥₂∈𝑀 can be chosen such that 𝑦₁=𝑇𝑥₁=𝑔𝑥₂. Continuing this process, we obtain a sequence {𝑦_𝑛} in 𝑀 such that 𝑦_2𝑛=𝑇𝑥_2𝑛=𝑓𝑥_2𝑛+1 and 𝑦_2𝑛+1=𝑇𝑥_2𝑛+1=𝑔𝑥_2𝑛+2 for every 𝑛≥0. If, for some 𝑛, 𝑦_2𝑛+1= 𝑦_2𝑛, then {𝑦_𝑛} turns out to be a constant sequence and hence it is Cauchy. Now suppose that 𝑦_2𝑛+1≠𝑦_2𝑛.
Using the fact that 𝑇 is generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction, for each 𝑛≥0, we have 𝜑𝑑𝑦_2𝑛+1,𝑦_2𝑛=𝜑𝑑𝑇𝑥_2𝑛+1,𝑇𝑥_2𝑛≤𝜑𝑚𝑥_2𝑛+1,𝑥_2𝑛−𝜙𝑚𝑥_2𝑛+1,𝑥_2𝑛+𝐿𝜑𝑛𝑥_2𝑛+1,𝑥_2𝑛, (8) where 𝑚𝑥_2𝑛+1,𝑥_2𝑛=max𝑑𝑓𝑥_2𝑛+1,𝑔𝑥_2𝑛,𝑑𝑇𝑥_2𝑛+1,𝑓𝑥_2𝑛+1,𝑑𝑇𝑥_2𝑛,𝑔𝑥_2𝑛,12𝑑𝑔𝑥_2𝑛,𝑇𝑥_2𝑛+1+𝑑𝑓𝑥_2𝑛+1,𝑇𝑥_2𝑛,𝑛𝑥_2𝑛+1,𝑥_2𝑛=min𝑑𝑇𝑥_2𝑛+1,𝑓𝑥_2𝑛+1+𝑑𝑇𝑥_2𝑛,𝑔𝑥_2𝑛,𝑑𝑔𝑥_2𝑛,𝑇𝑥_2𝑛+1,𝑑𝑓𝑥_2𝑛+1,𝑇𝑥_2𝑛=min𝑑𝑦_2𝑛+1,𝑦_2𝑛+𝑑𝑦_2𝑛,𝑦_2𝑛−1,𝑑𝑦_2𝑛−1,𝑦_2𝑛+1,0=0. (9) Since 𝜑(0)=0, therefore we have 𝜑𝑑𝑦_2𝑛+1,𝑦_2𝑛≤𝜑𝑚𝑥_2𝑛+1,𝑥_2𝑛−𝜙𝑚𝑥_2𝑛+1,𝑥_2𝑛. (10) This implies 𝜑𝑑𝑦_2𝑛+1,𝑦_2𝑛≤𝜑𝑚𝑥_2𝑛+1,𝑥_2𝑛. (11) As 𝜑 is a nondecreasing function, therefore for each 𝑛≥0 we have 𝑑𝑦_2𝑛+1,𝑦_2𝑛≤𝑚𝑥_2𝑛+1,𝑥_2𝑛. (12) Now from the triangle inequality for 𝑑 we have 𝑚𝑥_2𝑛+1,𝑥_2𝑛=max𝑑𝑓𝑥_2𝑛+1,𝑔𝑥_2𝑛,𝑑𝑇𝑥_2𝑛+1,𝑓𝑥_2𝑛+1,𝑑𝑇𝑥_2𝑛,𝑔𝑥_2𝑛,12𝑑𝑔𝑥_2𝑛,𝑇𝑥_2𝑛+1+𝑑𝑓𝑥_2𝑛+1,𝑇𝑥_2𝑛=max𝑑𝑦_2𝑛,𝑦_2𝑛−1,𝑑𝑦_2𝑛+1,𝑦_2𝑛,𝑑𝑦_2𝑛,𝑦_2𝑛−1,12𝑑𝑦_2𝑛−1,𝑦_2𝑛+1+𝑑𝑦_2𝑛,𝑦_2𝑛≤max𝑑𝑦_2𝑛,𝑦_2𝑛−1,𝑑𝑦_2𝑛+1,𝑦_2𝑛,12𝑑𝑦_2𝑛−1,𝑦_2𝑛+𝑑𝑦_2𝑛,𝑦_2𝑛+1. (13) If 𝑑(𝑦_2𝑛+1,𝑦_2𝑛)>𝑑(𝑦_2𝑛,𝑦_2𝑛−1), then 𝑚(𝑥_2𝑛+1,𝑥_2𝑛)=𝑑(𝑦_2𝑛+1,𝑦_2𝑛) > 0. It furthermore implies that 𝜑𝑑𝑦_2𝑛+1,𝑦_2𝑛≤𝜑𝑑𝑦_2𝑛+1,𝑦_2𝑛−𝜙𝑑𝑦_2𝑛+1,𝑦_2𝑛. (14) This is a contradiction; therefore we have 𝑑𝑦_2𝑛+1,𝑦_2𝑛≤𝑚𝑥_2𝑛+1,𝑥_2𝑛≤𝑑𝑦_2𝑛,𝑦_2𝑛−1. (15) Similarly, it can be shown that 𝑑𝑦_2𝑛,𝑦_2𝑛−1≤𝑚𝑥_2𝑛,𝑥_2𝑛−1≤𝑑𝑦_2𝑛−1,𝑦_2𝑛−2. (16) Therefore, for each 𝑛≥0, we have 𝑑𝑦_𝑛+1,𝑦_𝑛≤𝑚𝑥_𝑛+1,𝑥_𝑛≤𝑑𝑦_𝑛,𝑦_𝑛−1. (17) Thus, the sequence {𝑑(𝑦_𝑛+1,𝑦_𝑛)} is monotone nonincreasing and bounded. So there exist 𝑟≥0 such that lim_𝑛→∞𝑑𝑦_𝑛+1,𝑦_𝑛=lim_𝑛→∞𝑚𝑥_𝑛+1,𝑥_𝑛=𝑟. (18) After letting 𝑛→∞ in (10), we obtain 𝜑(𝑟) ≤ 𝜑(𝑟)−𝜙(𝑟), which is a contradiction unless 𝑟=0.
Hence lim_𝑛→∞𝑑𝑦_𝑛+1,𝑦_𝑛=0. (19) Because of (19) and Lemma 6, to show {𝑦_𝑛} to be a Cauchy sequence in 𝑀, it is sufficient to show that {𝑦_2𝑛} is a Cauchy in 𝑀.
Suppose not, then there exists 𝜖>0 for which we can find subsequences {𝑦_2𝑚(𝑘)} and {𝑦_2𝑛(𝑘)} of {𝑦_2𝑛} with 𝑛(𝑘)>𝑚(𝑘)>𝑘 such that, for every 𝑘, 𝑑(𝑦_2𝑚(𝑘),𝑦_2𝑛(𝑘))≥𝜖 and 𝑑(𝑦_2𝑚(𝑘),𝑦_{2𝑛(𝑘)−2})<𝜖. So we have 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). (20) Now using (19), we have lim_𝑘→∞𝑑𝑦_2𝑚(𝑘),𝑦_2𝑛(𝑘)=𝜖. (21) Moreover, using the known relation |𝑑(𝑥,𝑧)−𝑑(𝑥,𝑦)|≤𝑑(𝑦,𝑧), we obtain 𝑑𝑦_2𝑚(𝑘),𝑦_{2𝑛(𝑘)−1}−𝑑𝑦_2𝑚(𝑘),𝑦_2𝑛(𝑘)≤𝑑𝑦_2𝑛(𝑘),𝑦_{2𝑛(𝑘)−1}. (22) Then by using (19) and (21), we get lim_𝑘→∞𝑑𝑦_2𝑚(𝑘),𝑦_{2𝑛(𝑘)−1}=𝜖. (23) Again from the relation 𝑑𝑦_{2𝑛(𝑘)−1},𝑦_{2𝑚(𝑘)+1}−𝑑𝑦_{2𝑛(𝑘)−1},𝑦_2𝑚(𝑘)≤𝑑𝑦_{2𝑚(𝑘)+1},𝑦_2𝑚(𝑘), (24) using (19) and (23), we get lim_𝑘→∞𝑑𝑦_{2𝑛(𝑘)−1},𝑦_{2𝑚(𝑘)+1}=𝜖. (25) Now from 𝑑𝑦_{2𝑚(𝑘)+1},𝑦_2𝑛(𝑘)−𝑑𝑦_{2𝑚(𝑘)+1},𝑦_{2𝑛(𝑘)−1}≤𝑑𝑦_2𝑛(𝑘),𝑦_{2𝑛(𝑘)−1}, (26) using (19) and (25), we get lim_𝑘→∞𝑑𝑦_{2𝑚(𝑘)+1},𝑦_2𝑛(𝑘)=𝜖. (27) Again, using the fact that 𝑇 is generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction, we get 𝜑𝑑𝑦_{2𝑚(𝑘)+1},𝑦_2𝑛(𝑘)=𝜑𝑑𝑇𝑥_{2𝑚(𝑘)+1},𝑇𝑥_2𝑛(𝑘)≤𝜑𝑚𝑥_{2𝑚(𝑘)+1},𝑥_2𝑛(𝑘)−𝜙𝑚𝑥_{2𝑚(𝑘)+1},𝑥_2𝑛(𝑘)+𝐿𝜑𝑛𝑥_{2𝑚(𝑘)+1},𝑥_2𝑛(𝑘), (28) where 𝑚𝑥_{2𝑚(𝑘)+1},𝑥_2𝑛(𝑘)=max𝑑𝑓𝑥_{2𝑚(𝑘)+1},𝑔𝑥_2𝑛(𝑘),𝑑𝑇𝑥_{2𝑚(𝑘)+1},𝑓𝑥_{2𝑚(𝑘)+1},𝑑𝑇𝑥_2𝑛(𝑘),𝑔𝑥_2𝑛(𝑘),12𝑑𝑔𝑥_2𝑛(𝑘),𝑇𝑥_{2𝑚(𝑘)+1}+𝑑𝑓𝑥_{2𝑚(𝑘)+1},𝑇𝑥_2𝑛(𝑘)=max𝑑𝑦_2𝑚(𝑘),𝑦_{2𝑛(𝑘)−1},𝑑𝑦_{2𝑚(𝑘)+1},𝑦_2𝑚(𝑘),𝑑𝑦_2𝑛(𝑘),𝑦_{2𝑛(𝑘)−1},12𝑑𝑦_{2𝑛(𝑘)−1},𝑦_{2𝑚(𝑘)+1}+𝑑𝑦_2𝑚(𝑘),𝑦_2𝑛(𝑘)1⟶max𝜖,0,0,2((),(),(),(𝜖+𝜖)=𝜖as𝑘⟶∞.Using19212325) (29) And 𝑛𝑥_{2𝑚(𝑘)+1},𝑥_2𝑛(𝑘)=min𝑑𝑇𝑥_{2𝑚(𝑘)+1},𝑓𝑥_{2𝑚(𝑘)+1}+𝑑𝑇𝑥_2𝑛(𝑘),𝑔𝑥_2𝑛(𝑘),𝑑𝑔𝑥_2𝑛(𝑘),𝑇𝑥_{2𝑚(𝑘)+1},𝑑𝑓𝑥_{2𝑚(𝑘)+1},𝑇𝑥_2𝑛(𝑘)=min𝑑𝑦_{2𝑚(𝑘)+1},𝑦_2𝑚(𝑘)+𝑑𝑦_2𝑛(𝑘),𝑦_{2𝑛(𝑘)−1},𝑑𝑦_{2𝑛(𝑘)−1},𝑦_{2𝑚(𝑘)+1},𝑑𝑦_2𝑚(𝑘),𝑦_2𝑛(𝑘)⟶0as𝑘⟶∞.Using(19),(21),(25) (30) Letting 𝑘→∞ in (28), we get 𝜑(𝜖)≤𝜑(𝜖)−𝜙(𝜖), (31) which is a contradiction with 𝜖>0. Thus {𝑦_2𝑛} is a Cauchy sequence and hence {𝑦_𝑛} is a Cauchy sequence; therefore by the completeness of cl(𝑇(𝑀))there is some 𝑧∈cl(𝑇(𝑀)) such that lim_𝑛→∞𝑦_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=lim_𝑛→∞𝑓𝑥_2𝑛+1=𝑧=lim_𝑛→∞𝑔𝑥_2𝑛+2. (32) Further, cl(𝑇(𝑀))⊆𝑓(𝑀)∩𝑔(𝑀); therefore there exist 𝑢,𝑣∈𝑀 such that 𝑓𝑢=𝑧=𝑔𝑣. (33) Since 𝑇 is generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction, therefore 𝜑𝑑𝑇𝑢,𝑓𝑥_2𝑛+1=𝜑𝑑𝑇𝑢,𝑇𝑥_2𝑛≤𝜑𝑚𝑢,𝑥_2𝑛−𝜙𝑚𝑢,𝑥_2𝑛+𝐿𝜑𝑛𝑢,𝑥_2𝑛, (34) where 𝑚𝑢,𝑥_2𝑛=max𝑑𝑓𝑢,𝑔𝑥_2𝑛,𝑑𝑇𝑢,𝑓𝑢,𝑑𝑇𝑥_2𝑛,𝑔𝑥_2𝑛,12𝑑𝑔𝑥_2𝑛,𝑇𝑢+𝑑𝑓𝑢,𝑇𝑥_2𝑛,𝑛𝑢,𝑥_2𝑛=min𝑑𝑇𝑢,𝑓𝑢+𝑑𝑇𝑥_2𝑛,𝑔𝑥_2𝑛,𝑑𝑔𝑥_2𝑛,𝑇𝑢,𝑑𝑓𝑢,𝑇𝑥_2𝑛. (35) Now using (32) and (33), we can write 𝑚𝑢,𝑥_2𝑛⟶𝑑(𝑇𝑢,𝑧),𝑛𝑢,𝑥_2𝑛⟶0as𝑛⟶∞. (36) Therefore, letting 𝑛→∞ in (34), we get 𝜑(𝑑(𝑇𝑢,𝑧))≤𝜑(𝑑(𝑇𝑢,𝑧))−𝜙(𝑑(𝑇𝑢,𝑧)). (37) This is true only if 𝑑(𝑇𝑢,𝑧)=0; that is, 𝑇𝑢=𝑧=𝑓𝑢 and 𝑢 is coincidence point of 𝑇 and 𝑓.
Since 𝑇 is generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction, therefore 𝜑𝑑𝑔𝑥_2𝑛+2𝑑,𝑇𝑣=𝜑𝑇𝑥_2𝑛+1𝑚𝑥,𝑇𝑣≤𝜑_2𝑛+1,𝑣−𝜙𝑚𝑥_2𝑛+1,𝑣+𝐿𝜑𝑛𝑥_2𝑛+1,𝑣, (38) where 𝑚𝑥_2𝑛+1𝑑,𝑣=max𝑓𝑥_2𝑛+1,𝑔𝑣,𝑑𝑇𝑥_2𝑛+1,𝑓𝑥_2𝑛+1,1,𝑑𝑇𝑣,𝑔𝑣2𝑑𝑔𝑣,𝑇𝑥_2𝑛+1+𝑑𝑓𝑥_2𝑛+1,𝑛𝑥,𝑇𝑣_2𝑛+1𝑑,𝑣=min𝑇𝑥_2𝑛+1,𝑓𝑥_2𝑛+1,𝑑+𝑑𝑇𝑣,𝑔𝑣𝑔𝑣,𝑇𝑥_2𝑛+1,𝑑𝑓𝑥_2𝑛+1,𝑇𝑣. (39) Therefore, (32) and (33) imply 𝑚𝑥_2𝑛+1,𝑣⟶𝑑(𝑇𝑣,𝑧),𝑛𝑥_2𝑛+1,𝑣⟶0as𝑛⟶∞. (40) Then, letting 𝑛→∞ in (38), we obtain 𝜑(𝑑(𝑧,𝑇𝑣))≤𝜑(𝑑(𝑧,𝑇𝑣))−𝜙(𝑑(𝑧,𝑇𝑣)). (41) This is true only if 𝑑(𝑇𝑣,𝑧)=0; that is, 𝑇𝑣=𝑧=𝑔𝑣 and 𝑣 is coincidence point of 𝑇 and 𝑔. Thus the pairs (𝑇,𝑓) and (𝑇,𝑔) have a common point of coincidence in 𝑀.
If the pairs (𝑇,𝑓) and (𝑇,𝑔) are weakly compatible, then 𝑇𝑧=𝑇𝑓𝑢=𝑓𝑇𝑢=𝑓𝑧 and 𝑇𝑧=𝑇𝑔𝑣=𝑔𝑇𝑣=𝑔𝑧 and hence 𝑇𝑧=𝑓𝑧=𝑔𝑧. Now, we have to show that 𝑇𝑧=𝑧.
Further, using the fact that 𝑇 is generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction, we have 𝜑(𝑑((𝑑((𝑚((𝑚((𝑛(,𝑇𝑧,𝑧))=𝜑𝑇𝑧,𝑇𝑣))≤𝜑𝑧,𝑣))−𝜙𝑧,𝑣))+𝐿𝜑𝑧,𝑣)) (42) where 𝑚()𝑑,1𝑧,𝑣=max𝑓𝑧,𝑔𝑣,𝑑𝑇𝑧,𝑓𝑧,𝑑𝑇𝑣,𝑔𝑣2𝑑𝑑()1𝑔𝑣,𝑇𝑧+𝑑𝑓𝑧,𝑇𝑣=max𝑇𝑧,𝑧,0,0,2[𝑑()((),𝑛()𝑑,𝑑𝑧,𝑇𝑧+𝑑𝑇𝑧,𝑧)]=𝑑𝑇𝑧,𝑧𝑧,𝑣=min𝑇𝑧,𝑓𝑧+𝑑𝑇𝑣,𝑔𝑣𝑔𝑣,𝑇𝑧,𝑑𝑓𝑧,𝑇𝑣=0. (43) Then from (42), we get 𝜑(𝑑(𝑇𝑧,𝑧))≤𝜑(𝑑(𝑇𝑧,𝑧))−𝜙(𝑑(𝑇𝑧,𝑧)). (44) This is true only if 𝑇𝑧=𝑧 which implies 𝑇𝑧=𝑓𝑧=𝑔𝑧=𝑧.
Moreover, it can be easily shown that this 𝑧 is unique and hence 𝑀∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔) is singleton.
If we take 𝑓=𝑔 = identity mapping in Theorem 7, then we have the following result.

Corollary 8. Let 𝑀 be a nonempty subset of a metric space (𝑋,𝑑) and let 𝑇 be a self-mapping of 𝑀 such that 𝑐𝑙(𝑇(𝑀))⊆𝑀. Assume that 𝑐𝑙(𝑇(𝑀))is complete and 𝑇 is generalized (𝜑,𝜙,𝐿)-weak contraction. Then 𝑀∩𝐹(𝑇) is singleton.

Corollary 9 (see [5], Theorem 2.2). Let (𝑋,𝑑) be a complete metric space and let 𝑇 be a self-mapping of 𝑋. If 𝑇 is generalized (𝜑,𝜙)-weak contractions, then there is a unique fixed point of 𝑇.

In Corollary 8, if 𝜑(𝑡)=𝑡 and 𝐿=0, then Theorem 1 can be obtained as a particular case of the following result.

Corollary 10. Let 𝑀 be a subset of a metric space (𝑋,𝑑) and let 𝑇 be a self-mapping of 𝑀 such that 𝑐𝑙(𝑇(𝑀))⊆𝑀. Assume that 𝑐𝑙(𝑇(𝑀)) is complete and 𝑇 is generalized 𝜙-weak contractions. Then 𝑀∩𝐹(𝑇) is singleton.

Remark 11. (1) In Theorem 7, if 𝐿=0,𝜑(𝑡)=𝑡 and 𝜙(𝑡)=(1−𝑘)𝑡 for a constant 𝑘 with 0<𝑘<1, then we obtain Theorem 2.1 of Song [21] as a particular case of Theorem 7.
(2) In Theorem 7, if 𝐿=0,𝜑(𝑡)=𝑡 and 𝜙(𝑡)=(1−𝑘)𝑡 for a constant 𝑘 with 0<𝑘<1, then for 𝑔=𝑓 = identity map (resp., 𝑔=𝑓) we obtain Theorem 2.4 of Berinde [11] (resp., Theorem 2.1 of Abbas and Ilić [12]) as a particular case of Theorem 7.

As an application of Corollary 8, we obtain the following general common fixed point result.

Theorem 12. Let 𝑀 be a nonempty subset of a metric space (𝑋,𝑑) and 𝑇,𝑓, and 𝑔 are self-maps of 𝑀. Assume that 𝑐𝑙(𝑇(𝑀)) is complete, 𝑇 is generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction, 𝐹(𝑓)∩𝐹(𝑔) is nonempty, and 𝑐𝑙(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔). Then 𝑀∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔) is singleton.

Proof. cl(𝑇(𝐹(𝑓)∩𝐹(𝑔))) is complete by the completeness of cl(𝑇(𝑀)). Also, for all 𝑥,𝑦∈𝐹(𝑓)∩𝐹(𝑔), we have by generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contractiveness of 𝑇: 𝜑𝑑𝑚𝑚𝑛𝑀𝑀𝑁,𝑇𝑥,𝑇𝑦≤𝜑𝑥,𝑦−𝜙𝑥,𝑦+𝐿𝜑𝑥,𝑦=𝜑𝑥,𝑦−𝜙𝑥,𝑦+𝐿𝜑𝑥,𝑦 (45) where 𝑁(𝑥,𝑦)=min{[𝑑(𝑇𝑥,𝑥)+𝑑(𝑇𝑦,𝑦)], 𝑑(𝑦,𝑇𝑥),𝑑(𝑥,𝑇𝑦)}.
Hence 𝑇 is generalized (𝜑,𝜙,𝐿)-weak contraction mapping on 𝐹(𝑓)∩𝐹(𝑔). As cl(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔), thus, by Corollary 8, 𝑇 has a unique fixed point 𝑧 in 𝐹(𝑓)∩𝐹(𝑔) and, consequently, 𝑀∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔) is singleton.

Corollary 13. Let 𝑀 be a nonempty subset of a metric space (𝑋,𝑑) and (𝑇,𝑓) and (𝑇,𝑔) are Banach operator pairs on 𝑀. Assume that 𝑐𝑙(𝑇(𝑀)) is complete, 𝑇 is generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction, and 𝐹(𝑓)∩𝐹(𝑔) is nonempty and closed. Then 𝑀∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔) is singleton.

Proof. Since (𝑇,𝑓) and (𝑇,𝑔) are Banach operator pairs on 𝑀, therefore 𝑇(𝐹(𝑓)∩𝐹(𝑔))⊆𝐹(𝑓)∩𝐹(𝑔); then by closedness of 𝐹(𝑓)∩𝐹(𝑔) we have cl(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔). By Theorem 12, 𝑇 has a unique fixed point 𝑧 in 𝐹(𝑓)∩𝐹(𝑔) and, consequently, 𝑀∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔) is singleton.

In Theorem 12, if 𝐿=0, then we easily obtain the following results which properly contain Theorem 3.3 of Akbar et al. [22].

Corollary 14. Let 𝑀 be a nonempty subset of a metric space (𝑋,𝑑) and 𝑇,𝑓, and 𝑔 are self-maps of 𝑀. Assume that 𝑐𝑙(𝑇(𝑀)) is complete, 𝑇 is generalized (𝜑,𝜙)_𝑓,𝑔-weak contraction, 𝐹(𝑓)∩𝐹(𝑔) is nonempty, and 𝑐𝑙(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔). Then 𝑀∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔) is singleton.

In Corollary 14, if 𝜑(𝑡)=𝑡 and 𝜙(𝑡)=(1−𝑘)𝑡 for a constant 𝑘 with 0<𝑘<1, then for 𝑔=𝑓 we obtain the following result which improves Lemma 3.1 of Chen and Li [16] and Theorem 2.2 of Al-Thagafi and Shahzad [19].

Corollary 15. Let 𝑀 be a nonempty subset of a metric space (𝑋,𝑑) and 𝑇,𝑓 and 𝑔 are self-maps of 𝑀. Assume that 𝑐𝑙(𝑇(𝑀)) is complete, 𝑇 is generalized 𝑓-contraction, 𝐹(𝑓)∩𝐹(𝑔) is nonempty, and 𝑐𝑙(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔). Then 𝑀∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔) is singleton.

The following theorem properly contains Theorem 3.8 and Corollary 3.9 of Akbar et al. [22].

Theorem 16. Let 𝑀 be a nonempty subset of a normed [resp., Banach] space 𝑋 and 𝑇,𝑓, and 𝑔 are self-maps of 𝑀. Suppose that 𝐹(𝑓)∩𝐹(𝑔) is 𝑞-star shaped, 𝑐𝑙(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔)  [resp., 𝑤𝑐𝑙(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔)], and 𝑇,𝑓, and 𝑔 satisfy 𝜑𝑘𝑚𝑇𝑥−𝑇𝑦≤𝜑₁𝑥,𝑦−𝑘𝜙𝑚₁𝑥,𝑦+𝑘𝐿𝜑𝑛₁𝑥,𝑦, (46) for all 𝑘∈(0,1) and 𝑥,𝑦∈𝑀, where(a)𝜑∶[0,∞)→[0,∞) is a continuous monotone nondecreasing function with 𝜑(𝑡)=0 if and only if 𝑡=0,(b)𝜙∶[0,∞)→[0,∞) is a lower semicontinuous function from right such that 𝜙 is positive on (0,∞) and 𝜙(0)=0,(c)𝑚₁(𝑥,𝑦)=max{‖𝑓𝑥−𝑔𝑦‖,dist(𝑓𝑥,[𝑞,𝑇𝑥]),dist(𝑔𝑦,[𝑞,𝑇𝑦]), (1/2)[dist(𝑔𝑦,[𝑞,𝑇𝑥])+dist(𝑓𝑥,[𝑞,𝑇𝑦])]},(d)𝑛₁(𝑥,𝑦)=min{[dist(𝑓𝑥,[𝑞,𝑇𝑥])+dist(𝑔𝑦,[𝑞,𝑇𝑦])],dist(𝑔𝑦,[𝑞,𝑇𝑥]),dist(𝑓𝑥,[𝑞,𝑇𝑦])}.Then 𝑀∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔)≠𝜙 provided that 𝑐𝑙(𝑇(𝑀)) is compact [resp., 𝑤𝑐𝑙(𝑇(𝑀)) is weakly compact] and 𝑇is continuous [resp., 𝐼−𝑇 is demiclosed at 0, where 𝐼 stands for identity map].

Proof. For each 𝑛∈ℕ, we define 𝑇_𝑛∶𝐹(𝑓)∩𝐹(𝑔)→𝐹(𝑓)∩𝐹(𝑔) by 𝑇_𝑛𝑥=(1−𝑘_𝑛)𝑞+𝑘_𝑛𝑇𝑥 for all 𝑥∈𝐹(𝑓)∩𝐹(𝑔) and a fixed sequence of real numbers 𝑘_𝑛(0<𝑘_𝑛<1) converging to 1. Since 𝐹(𝑓)∩𝐹(𝑔) is 𝑞-star shaped and cl(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔) [resp., wcl(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔)], we have cl(𝑇_𝑛(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔) [resp., wcl(𝑇_𝑛(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔)] for each 𝑛∈ℕ. Let 𝜙_𝑛=𝑘_𝑛𝜙 and 𝐿_𝑛=𝑘_𝑛𝐿. Then by using (46), for all 𝑥,𝑦∈𝐹(𝑓)∩𝐹(𝑔), we have 𝜑𝑇_𝑛𝑥−𝑇_𝑛𝑦=𝜑𝑘_𝑛𝑚𝑇𝑥−𝑇𝑦≤𝜑₁𝑥,𝑦−𝑘_𝑛𝜙𝑚₁𝑥,𝑦+𝑘_𝑛𝐿𝜑𝑛₁𝑚𝑥,𝑦=𝜑₁𝑥,𝑦−𝜙_𝑛𝑚₁𝑥,𝑦+𝐿_𝑛𝜑𝑛₁𝑥,𝑦, (47) where 𝑚₁,1𝑥,𝑦=max𝑓𝑥−𝑔𝑦,dist𝑓𝑥,𝑞,𝑇𝑥,dist𝑔𝑦,𝑞,𝑇𝑦2,dist𝑔𝑦,𝑞,𝑇𝑥+dist𝑓𝑥,𝑞,𝑇𝑦≤max𝑓𝑥−𝑔𝑦𝑓𝑥−𝑇_𝑛𝑥,𝑔𝑦−𝑇_𝑛𝑦,12𝑔𝑦−𝑇_𝑛𝑥+𝑓𝑥−𝑇_𝑛𝑦,𝑛₁,𝑥,𝑦=mindist𝑓𝑥,𝑞,𝑇𝑥+dist𝑔𝑦,𝑞,𝑇𝑦dist𝑔𝑦,𝑞,𝑇𝑥,dist𝑓𝑥,𝑞,𝑇𝑦≤min𝑓𝑥−𝑇_𝑛𝑥+𝑔𝑦−𝑇_𝑛𝑦,𝑔𝑦−𝑇_𝑛𝑥,𝑓𝑥−𝑇_𝑛𝑦. (48) Clearly, 𝜙_𝑛∶ [0,∞)→[0,∞) is a lower semicontinuous function from right such that 𝜙_𝑛 is positive on (0,∞) and 𝜙_𝑛(0)=0 and 𝐿_𝑛≥0. Thus, for each 𝑛∈ℕ,𝑇_𝑛 is generalized (𝜑,𝜙_𝑛,𝐿_𝑛)_𝑓,𝑔-weak contraction. As cl(𝑇(𝑀)) is compact [resp., wcl(𝑇(𝑀)) is weakly compact], therefore, for each 𝑛∈ℕ, cl(𝑇_𝑛(𝑀)) is compact [resp., wcl(𝑇_𝑛(𝑀)) is weakly compact]. Thus, cl(𝑇_𝑛(𝑀)) [resp., wcl(𝑇_𝑛(𝑀))] is complete for each 𝑛∈ℕ. By Theorem 12, for each 𝑛≥1, there exists {𝑥_𝑛} in 𝑀 such that 𝑥_𝑛=𝑓(𝑥_𝑛)=𝑔(𝑥_𝑛)=𝑇_𝑛(𝑥_𝑛).
Again the compactness of cl(𝑇(𝑀)) implies that there exists a subsequence {𝑇𝑥_𝑚} of {𝑇𝑥_𝑛} such that 𝑇𝑥_𝑚→𝑧∈cl(𝑇(𝑀)). Since {𝑇𝑥_𝑚} is a sequence in 𝑇(𝐹(𝑓)∩𝐹(𝑔)) and cl(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔), therefore 𝑧∈𝐹(𝑓)∩𝐹(𝑔). Moreover 𝑥_𝑚=𝑇_𝑚𝑥_𝑚=1−𝑘_𝑚𝑞+𝑘_𝑚𝑇𝑥_𝑚⟶𝑧. (49) As 𝑇 is continuous on 𝑀, we have 𝑇𝑧=𝑧. Thus 𝑀∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔)≠𝜙.
Next, weak compactness of wcl(𝑇(𝑀)) implies that there exists a subsequence {𝑇𝑥_𝑚} of {𝑇𝑥_𝑛} such that 𝑇𝑥_𝑚 converges weakly to𝑧∈wcl(𝑇(𝑀)). Since {𝑇𝑥_𝑚} is a sequence in 𝑇(𝐹(𝑓)∩𝐹(𝑔)) and wcl(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔), therefore 𝑧∈𝐹(𝑓)∩𝐹(𝑔). Also we have (𝐼−𝑇)𝑥_𝑚 → 0 as 𝑚→∞. Further, demiclosedness of 𝐼−𝑇 at 0 implies 𝑧=𝑇𝑧. Thus 𝑀∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔)≠𝜙.

Corollary 17. Let 𝑀 be a nonempty subset of a normed [resp., Banach] space 𝑋 and 𝑇,𝑓, and 𝑔 are self-maps of 𝑀. Suppose that 𝐹(𝑓)∩𝐹(𝑔) is 𝑞-star shaped and closed [resp., weakly closed] and (𝑇,𝑓) and (𝑇,𝑔) are Banach operator pairs and satisfy (46). Then 𝑀∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔)≠𝜙 provided that 𝑐𝑙(𝑇(𝑀)) is compact [resp., 𝑤𝑐𝑙(𝑇(𝑀)) is weakly compact] and 𝑇is continuous [resp., 𝐼−𝑇 is demiclosed at 0, where 𝐼 stands for identity map].

In Theorem 16, if 𝜑(𝑡)=𝑡 and 𝜙(𝑡)=(1/𝑘−1)𝑡 for a constant 𝑘 with 0<𝑘<1, then we easily obtain the following result.

Corollary 18. Let 𝑀 be a nonempty subset of a normed [resp., Banach] space 𝑋 and 𝑇,𝑓, and 𝑔 are self-maps of 𝑀. Suppose that 𝐹(𝑓)∩𝐹(𝑔) is 𝑞-star shaped, 𝑐𝑙(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔) [resp., 𝑤𝑐𝑙(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔)], and 𝑇, 𝑓, and 𝑔 satisfy 𝑇𝑥−𝑇𝑦≤𝑚₁𝑥,𝑦+𝐿𝑛₁𝑥,𝑦∀𝑥,𝑦∈𝑀, (50) where 𝑚₁,1𝑥,𝑦=max𝑓𝑥−𝑔𝑦,dist𝑓𝑥,𝑞,𝑇𝑥,dist𝑔𝑦,𝑞,𝑇𝑦2,𝑛dist𝑔𝑦,𝑞,𝑇𝑥+dist𝑓𝑥,𝑞,𝑇𝑦₁,.𝑥,𝑦=mindist𝑓𝑥,𝑞,𝑇𝑥+dist𝑔𝑦,𝑞,𝑇𝑦dist𝑔𝑦,𝑞,𝑇𝑥,dist𝑓𝑥,𝑞,𝑇𝑦 (51) Then 𝑀∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔)≠𝜙 provided that 𝑐𝑙(𝑇(𝑀)) is compact [resp., wcl(𝑇(𝑀)) is weakly compact] and 𝑇 is continuous [resp., 𝐼−𝑇 is demiclosed at 0, where 𝐼 stands for identity map].

Remark 19. (1) By comparing Theorem 2.3(i) of Abbas and Ilić [12] with the first case of Corollary 18 (when 𝑔=𝑓), their assumptions “𝑀 is 𝑞-star shaped, cl(𝑇(𝑀))⊆𝑓(𝑀), and 𝑓 and 𝑇 are weakly compatible on 𝑀” are replaced with “𝐹(𝑓) is 𝑞-star shaped and cl(𝑇(𝐹(𝑓)))⊆𝐹(𝑓)."
(2) By comparing Theorem 2.3(ii) of Abbas and Ilić [12] with the second case of Corollary 18 (when 𝑔=𝑓), their assumptions “𝑀 is 𝑞-star shaped, cl(𝑇(𝑀))⊆𝑓(𝑀), 𝑓 and 𝑇 are weakly compatible on 𝑀, 𝑓 is weakly continuous, and 𝑓−𝑇 is demiclosed at 0” are replaced with “𝐹(𝑓) is 𝑞-star shaped, wcl(𝑇(𝐹(𝑓)))⊆𝐹(𝑓), and 𝑇 is weakly continuous.”
(3) By comparing Theorem 2.4 of Song [21] with the first case of Corollary 18 (when 𝐿=0), his assumptions “𝑀 is 𝑞-star shaped, cl(𝑇(𝑀))⊆𝑓(𝑀))∩𝑔(𝑀), the pairs (𝑇,𝑓) and (𝑇,𝑔) are 𝐶_𝑞-commuting, and 𝑓 and 𝑔 are 𝑞-affine and continuous on 𝑀” are replaced with 𝐹(𝑓)∩𝐹(𝑔) is 𝑞-star shaped and cl(𝑇(𝐹(𝑓)∩𝐹(𝑔))⊆𝐹(𝑓)∩𝐹(𝑔).”
(4) By comparing Theorem 2.2(i) of Hussain and Jungck [20] with the first case of Corollary 18 ( when 𝐿=0), their assumptions “𝑀 is complete and 𝑞-star shaped, 𝑓 and 𝑔 are continuous and affine on 𝑀,  𝑇(𝑀)⊆𝑓(𝑀)∩𝑔(𝑀),  𝑞∈𝐹(𝑓)∩𝐹(𝑔), and (𝑇,𝑓) and (𝑇,𝑔) are 𝑅-subweakly commuting pair on 𝑀” are replaced with “𝐹(𝑓)∩𝐹(𝑔) is 𝑞-star shaped and cl(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔).”
(5) By comparing Theorem 2.2(ii) of Hussain and Jungck [20] with the second case of Corollary 18 ( when 𝐿=0), their assumptions “𝑀 is weakly compact and 𝑞-star shaped, 𝑓 and 𝑔 are affine and continuous on 𝑀,  𝑇(𝑀)⊆𝑓(𝑀)∩𝑔(𝑀), 𝑞∈𝐹(𝑓)∩𝐹(𝑔), (𝑇,𝑓) and (𝑇,𝑔) are 𝑅-subweakly commuting pair on 𝑀, and 𝑓−𝑇 is demiclosed at 0” are replaced with “wcl(𝑇(𝑀)) is weakly compact, 𝐹(𝑓)∩𝐹(𝑔) is 𝑞-star shaped, wcl(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔), and 𝐼−𝑇 is demiclosed at 0.”

Corollary 20. Let 𝑋 be a normed space [resp., Banach] 𝑋 and 𝑇, 𝑓, and 𝑔 are self-maps of 𝑋. If 𝑝∈𝑋 and 𝐷⊆𝐵_𝑀(𝑝), 𝐷₀∶=𝐷∩𝐹(𝑓)∩𝐹(𝑔) is 𝑞-star shaped, 𝑐𝑙(𝑇(𝐷₀))⊆𝐷₀ [resp., wcl(𝑇(𝐷₀))⊆𝐷₀], 𝑐𝑙(𝑇(𝐷)) is compact [resp., wcl(𝑇(𝐷)) is weakly compact], 𝑇 is continuous on 𝐷 [resp., 𝐼−𝑇 is demiclosed at 0, where 𝐼 stands for identity map], and (46) holds for all 𝑥,𝑦∈𝐷, then 𝐵_𝑀(𝑝)∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔)≠𝜙.

Remark 21. Corollary 20 improves Theorem 2.8 of Hussain and Jungck [20], Theorems 3.1–3.4 of Song [21], and Corollary 2.5 of Al-Thagafi and Shahzad [19]. It is also noted that Corollary 3.13 of Akbar et al. [22] is a special case of Corollary 20.
We denote by £₀ the class of closed convex subsets of 𝑋 containing 0. For 𝑀∈£₀, we define 𝑀_𝑝={𝑥∈𝑀∶‖𝑥‖≤2‖𝑝‖}. Clearly 𝐵_𝑀(𝑝)⊆𝑀_𝑝∈£₀.

Theorem 22. Let 𝑋 be a normed [resp., Banach] space and 𝑇,𝑓,𝑔∶𝑋→𝑋. If 𝑝∈𝑋 and 𝑀∈£₀ such that 𝑇(𝑀_𝑝)⊆𝑀, 𝑐𝑙(𝑇(𝑀_𝑝)) is compact [resp., wcl(𝑇(𝑀_𝑝)) is weakly compact], and ‖𝑇𝑥−𝑝‖≤‖𝑥−𝑝‖ for all 𝑥∈𝑀_𝑝, then 𝐵_𝑀(𝑝) is nonempty, closed, and convex with 𝑇(𝐵_𝑀(𝑝))⊆𝐵_𝑀(𝑝). If, in addition, 𝐷 is a subset of 𝐵_𝑀(𝑝),  𝐷₀∶=𝐷∩𝐹(𝑓)∩𝐹(𝑔) is 𝑞-star shaped, 𝑐𝑙(𝑇(𝐷₀))⊆𝐷₀ [resp., wcl(𝑇(𝐷₀))⊆𝐷₀],  𝑇 is continuous on 𝐷 [resp., 𝐼−𝑇 is demiclosed at 0, where 𝐼 stands for identity map], and (46) holds for all 𝑥,𝑦∈𝐷, then 𝐵_𝑀(𝑝)∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔)≠𝜙.

Proof. We may assume that 𝑝∉𝑀. If 𝑦∈𝑀⧵𝑀_𝑝, then ‖𝑦‖>2‖𝑝‖ and so 𝑦−𝑝≥𝑦−𝑝>𝑝≥dist𝑝,𝑀. (52) Thus dist(𝑝,𝑀_𝑝) = dist(𝑝,𝑀). Assume that cl(𝑇(𝑀_𝑝)) is compact; then by the continuity of norm, there exist 𝑧∈cl(𝑇(𝑀_𝑝)) such that ‖𝑧−𝑝‖=dist(𝑝,cl𝑇(𝑀_𝑝)).
If we assume that wcl(𝑇(𝑀_𝑝)) is weakly compact, then by using Lemma 5.5 of [24, p. 192] we can show the existence of 𝑧∈wcl(𝑇(𝑀_𝑝)) such that ‖𝑧−𝑝‖=dist(𝑝,wcl𝑇(𝑀_𝑝)). Thus, in both cases, we have dist𝑝,𝑀_𝑝≤dist𝑝,cl𝑇𝑀_𝑝≤dist𝑝,𝑇𝑀_𝑝≤𝑇𝑥−𝑝≤𝑥−𝑝, (53) for all 𝑥∈𝑀_𝑝. It follows that ‖𝑧−𝑝‖=dist(𝑝,𝑀). Thus 𝐵_𝑀(𝑝) is nonempty, closed, and convex with 𝑇(𝐵_𝑀(𝑝))⊆𝐵_𝑀(𝑝). The compactness of cl(𝑇(𝑀_𝑝)) [resp., weak compactness of wcl(𝑇(𝑀_𝑝))] implies that cl(𝑇(𝐷)) is compact [resp., wcl(𝑇(𝐷)) is weakly compact]. Then by Corollary 20, 𝐵_𝑀(𝑝)∩𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔)≠𝜙.

Remark 23. (i) Theorem 22 extends Theorems 4.1 and 4.2 of Al-Thagafi [17], Theorem 2.6 of Al-Thagafi and Shahzad [19], Theorem 2.14 of Hussain and Jungck [20], and Theorem 8 of Habiniak [23].
(ii) Theorem 3.15 of Akbar et al. [22] is a particular case of Theorem 22.

4. Examples

Now, in order to support the usability of our results, we present the following nontrivial examples.

Example 24. Let 𝑋=[0,∞) be endowed with the Euclidean metric 𝑑(𝑥,𝑦)=|𝑥−𝑦| and let 𝑀={0,1,2,3} and define 𝑇,𝑓,𝑔∶𝑀→𝑀 by ,.𝑇=01230002,𝑓=01230131𝑔=01230313 (54) Here 𝐹(𝑓)={0,1} and 𝐹(𝑔)={0,3} so that 𝐹(𝑓)∩𝐹(𝑔)={0} is nonempty and cl(𝑇(𝐹(𝑓)∩𝐹(𝑔)))={0}⊆𝐹(𝑓)∩𝐹(𝑔). Also cl(𝑇(𝑀))={0,2}, being a finite set, is complete. Define 𝜑,𝜙∶[0,∞)→[0,∞) by 𝜑(𝑡)=𝑡²(𝑡)=𝑡,𝜙2. (55) Now we will show that 𝑇 is generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction for 𝐿≥7/2. Let 𝑥,𝑦∈𝑀 with 𝑥≠𝑦 and consider the following possible cases.(i)If (𝑥,𝑦)∈{(0,3),(3,1),(2,3)}, then 𝑑(𝑇𝑥,𝑇𝑦)=2, 𝑚(𝑥,𝑦)=3, and 𝑛(𝑥,𝑦)=1. So, 𝜑𝑚𝑚𝑛(3)(3)(1)=𝑥,𝑦−𝜙𝑥,𝑦+𝐿𝜑𝑥,𝑦=𝜑−𝜙+𝐿𝜑152+𝐿>4=𝜑𝑑𝑇𝑥,𝑇𝑦, (56) for all 𝐿≥0.(ii)If 𝑥=3,𝑦=0, then 𝑑(𝑇𝑥,𝑇𝑦)=2, 𝑚(𝑥,𝑦)=3/2, and 𝑛(𝑥,𝑦)=1. So, 𝜑𝑚𝑚𝑛3𝑥,𝑦−𝜙𝑥,𝑦+𝐿𝜑𝑥,𝑦=𝜑23−𝜙2(1)=3+𝐿𝜑2+𝐿≥4=𝜑𝑑𝑇𝑥,𝑇𝑦, (57) for all 𝐿≥5/2.(iii)If 𝑥=1,𝑦=3, then 𝑑(𝑇𝑥,𝑇𝑦)=2, 𝑚(𝑥,𝑦)=2, and 𝑛(𝑥,𝑦)=1. So, 𝜑𝑚𝑚𝑛(2)(2)(1)𝑑,𝑥,𝑦−𝜙𝑥,𝑦+𝐿𝜑𝑥,𝑦=𝜑−𝜙+𝐿𝜑=3+𝐿≥4=𝜑𝑇𝑥,𝑇𝑦 (58) for all 𝐿≥1.(iv)If 𝑥=3,𝑦=2, then 𝑑(𝑇𝑥,𝑇𝑦)=2, 𝑚(𝑥,𝑦)=1, and 𝑛(𝑥,𝑦)=1. So, 𝜑𝑚𝑚𝑛(1)(1)(1)=1𝑥,𝑦−𝜙𝑥,𝑦+𝐿𝜑𝑥,𝑦=𝜑−𝜙+𝐿𝜑2+𝐿≥4=𝜑𝑑𝑇𝑥,𝑇𝑦, (59) for all 𝐿≥7/2.(v)If (𝑥,𝑦)∈{(0,1),(1,0),(0,2),(2,0),(1,2),(2,1)}, then 𝑑(𝑇𝑥,𝑇𝑦)=0 and hence inequality (5) is obviously satisfied.So, in view of the above discussion, we concluded that 𝑇 is generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction for 𝐿≥7/2. Therefore, all the conditions of Theorem 12 are satisfied and hence 𝑇,𝑓, and 𝑔 have a common fixed point. Here it is seen that 0 is the unique common fixed point of 𝑇,𝑓, and 𝑔.

Remark 25. (1) In Example 24, it is to be noted that the maps 𝑇,𝑓, and 𝑔 do not satisfy the conditions (1.1) and (3.1) of [22] at the points 𝑥=3,𝑦=0;𝑥=1,𝑦=3 and 𝑥=3,𝑦=2. Therefore, Theorems 2.2 and 3.3 of [22] cannot apply to Example 24.
(2) In Example 24, the maps 𝑇,𝑓, and 𝑔 do not satisfy the condition (3.1) of [26] at the points 𝑥=3,𝑦=0;𝑥=0,𝑦=3. Therefore, Theorem 3.5 of [26] cannot apply to Example 24.

Example 26. Let 𝑋=𝑀=[0,1/2)∪(1/2,1] be endowed with the Euclidean metric 𝑑(𝑥,𝑦)=|𝑥−𝑦| and define 𝑇: 𝑀→𝑀 by 𝑇(𝑥)=131,𝑥∈0,2∪12,23,252,𝑥∈3,1. (60) We observe that cl(𝑇(𝑀))={1/3,2/5}⊆𝑀 and also being a finite set cl(𝑇(𝑀)) is complete. Define the function 𝜑,  𝜙∶[0,∞)→[0,∞) by 𝜑(𝑡)(𝑡)=3=2𝑡,𝜙2𝑡,∀𝑡≥0. (61) Now, we prove that 𝑇 is generalized (𝜑,𝜙,𝐿)-weak contraction mapping for 𝐿≥0, that is, to verify the following inequality: 𝜑𝑑𝑀𝑀𝑁,𝑇𝑥,𝑇𝑦≤𝜑𝑥,𝑦−𝜙𝑥,𝑦+𝐿𝜑𝑥,𝑦 (62) where 𝑀𝑑(),1𝑥,𝑦=max𝑥,𝑦,𝑑𝑇𝑥,𝑥,𝑑𝑇𝑦,𝑦2𝑑,𝑁𝑑(),𝑑.𝑦,𝑇𝑥+𝑑𝑥,𝑇𝑦𝑥,𝑦=min𝑇𝑥,𝑥+𝑑𝑇𝑦,𝑦𝑦,𝑇𝑥,𝑑𝑥,𝑇𝑦 (63) If 𝑥=𝑦∈𝑀, then 𝑑(𝑇𝑥,𝑇𝑦)=0 and hence inequality (62) trivially holds. Let 𝑥,𝑦∈𝑀 with 𝑥≠𝑦; then we have the following two possibilities.(i)If 𝑥∈[0,1/2)∪(1/2,2/3),𝑦∈[2/3,1], then 𝑑(𝑇𝑥,𝑇𝑦)=1/15 and 𝑑(𝑦,𝑇𝑦)∈[4/15,3/5]. Therefore 𝜑𝑑=2𝑇𝑥,𝑇𝑦≤1152𝑑≤1𝑦,𝑇𝑦2𝑀−3𝑥,𝑦=2𝑀𝑥,𝑦2𝑀𝑀𝑀.𝑥,𝑦=𝜑𝑥,𝑦−𝜙𝑥,𝑦 (64) This yields that 𝜑𝑑𝑀𝑀𝑁𝑇𝑥,𝑇𝑦≤𝜑𝑥,𝑦−𝜙𝑥,𝑦+𝐿𝜑𝑥,𝑦 (65) holds for all 𝐿≥0.(ii)If 𝑥∈[2/3,1],𝑦∈[0,1/2)∪(1/2,2/3), then by interchanging the role of 𝑥 and 𝑦 in case (i), inequality (62) follows.Thus we checked that 𝑇 is generalized (𝜑,𝜙,𝐿)-weak contraction for all 𝐿≥0. Therefore, all the conditions of Corollary 8 are satisfied and hence 𝑇 has a fixed point in 𝑀. Here it is seen that 1/3 is the unique fixed point of 𝑇.

Remark 27. In Example 26, 𝑋 is not complete since the sequence {(1/2)(1−1/𝑛)} is not Cauchy in 𝑋. Therefore, Theorem 2.2 of [5], Theorem 2.1 of [6], Theorem 2.4 of [11], and Corollary 3.1 of [25] cannot apply to Example 26.

Remark 28. Theorem 12 and first case of Theorem 7 are not true if we relax the condition cl(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔) and cl(𝑇(𝑀))⊆𝑓(𝑀)∩𝑔(𝑀), respectively. This can be seen by the following example.

Example 29. Let 𝑋=𝑅 with the usual metric and 𝑀=[0,1]. Define 𝑇,𝑓,𝑔∶𝑀→𝑀 by 𝑇(𝑥)=131,𝑥∈0,2∪12,23,20,𝑥∈3,1,11,𝑥=2,𝑓(𝑥)=122,𝑥∈0,3−13,20,𝑥∈3,1,131,𝑥=3,𝑔(𝑥)=121,𝑥∈0,2∪23,1,1𝑥,𝑥∈2,23. (66) Then 𝐹(𝑓)={1/2,1/3}, 𝐹(𝑔)=[1/2,2/3); therefore 𝐹(𝑓)∩𝐹(𝑔)={1/2} is nonempty. Also, cl(𝑇(𝑀))={0,1/3,1}, being a closed subset of complete metric space, is complete. Further, 𝑓(𝑀)={0,1/3,1/2} and 𝑔(𝑀)=[1/2,2/3) gives 𝑓(𝑀)∩𝑔(𝑀)={1/2}. Here, we observe that cl(𝑇(𝐹(𝑓)∩𝐹(𝑔)))={1}⊆^/𝐹(𝑓)∩𝐹(𝑔) and cl(𝑇(𝑀))⊆^/𝑓(𝑀)∩𝑔(𝑀).
Define the function 𝜑,  𝜙∶[0,∞)→[0,∞) by 𝜑(𝑡)=𝑡,𝜙(𝑡)=𝑡²,∀𝑡≥0. (67) Then by using the routine calculation as is done in Example 24, one can easily check that 𝑇 is generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction for all 𝐿≥6. Hence all the conditions of Theorem 12 and first case of Theorem 7 except cl(𝑇(𝐹(𝑓)∩𝐹(𝑔)))⊆𝐹(𝑓)∩𝐹(𝑔) and cl(𝑇(𝑀))⊆𝑓(𝑀)∩𝑔(𝑀), respectively, are satisfied. Note that 𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔)=𝜙, 𝐶(𝑓,𝑇)={1/3}∪[2/3,1], and 𝐶(𝑔,𝑇)=𝜙.

Remark 30. (1) In Example 24, 𝑇 is generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction for 𝐿≥7/2 but it is not generalized (𝜑,𝜙)_𝑓,𝑔-weak contraction at 𝑥=3,𝑦=0 for any 𝜑 and 𝜙 defined in (5) and hence 𝑇 is not generalized 𝜙_𝑓,𝑔-weak contraction. Note that 𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔)={0}.
(2) In Example 29, 𝑇 is generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction for 𝐿≥6 but it is not generalized (𝜑,𝜙)_𝑓,𝑔-weak contraction at 𝑥=2/3, 𝑦=1/2 for any 𝜑 and 𝜙 defined in (5) and hence 𝑇 is not generalized 𝜙_𝑓,𝑔-weak contraction. Note that 𝐹(𝑇)∩𝐹(𝑓)∩𝐹(𝑔)=𝜙.

5. Conclusions

We introduce a new class of generalized (𝜑,𝜙,𝐿)_𝑓,𝑔-weak contraction as a generalization of several known contractions and present a sufficient condition for the existence and uniqueness of common fixed point. Moreover, we also prove some invariant approximation result as an application and supporting examples are also given.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the anonymous reviewers for their valuable comment which helps to improve the contents and presentation of this paper.