A Common Fixed Point Theorem in Metric Space under General Contractive Condition

Abstract

We prove a common fixed point theorem for two pairs of compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) mappings satisfying a general contractive condition in a metric space. Some illustrative examples are furnished to highlight the realized improvements. Our result improves the main result of Sedghi and Shobe (2007).

1. Introduction

Fixed point theory is one of the most fruitful and effective tools in mathematics which has enormous applications within as well as outside mathematics. Starting from the celebrated Banach contraction principle [1], many authors have obtained its several generalizations in different ways (see, e.g., [2–4]).

A metrical common fixed point theorem generally involves conditions on commutativity, continuity, and contraction of the given mappings, as well as completeness (or closedness) of the underlying space (or subspaces), along with conditions on suitable containment amongst the ranges of involved mappings. Hence, in order to prove a new metrical common fixed point theorem, one is always required to weaken one or more of these conditions. In order to weaken commutativity conditions in common fixed point theorems, Sessa [5] introduced the concept of weakly commuting mappings. Jungck [6] defined the notion of compatible mappings in order to generalize the concept of weak commutativity and showed that weakly commuting mappings are compatible, but the converse is not true. Afterwards Jungck and Rhoades [7] introduced the concept of weak compatibility to the setting of single-valued and multivalued mappings which is more general than compatibility. However, the study of common fixed points of noncompatible mappings is also equally interesting, and it was initiated by Pant [8] in metric spaces. Researchers of this domain introduced several definitions of weak commutativity such as compatible mappings, compatibility of type (𝐴), (𝐵), (𝐶), and (𝑃), and several others, whose systematic comparisons and illustrations are available in Murthy [9] and Singh and Tomar [10].

In 2002, Aamri and El Moutawakil [11] introduced the notion of property (E.A). Further, Al-Thagafi and Shahzad [12] introduced the notion of occasionally weakly compatible mappings. It was used further by many authors (see, e.g., [13]). However, in [14], Đorić et al. observed that the condition of occasionally weak compatibility reduces to weak compatibility in the presence of a unique point of coincidence of the given pair of mappings. Thus, no generalization can be obtained by replacing weak compatibility with occasionally weak compatibility (see also [15–17]).

In 2009, Bouhadjera and Godet-Thobie [18] further enlarged the class of compatible (reciprocally continuous) pairs by introducing the concept of subcompatibility (subsequential continuity) of pairs of mappings, which is substantially weaker than compatibility (reciprocal continuity). Since then, Imdad et al. [19] improved the results of Bouhadjera and Godet-Thobie and showed that these results can easily be recovered by replacing subcompatibility with compatibility or subsequential continuity with reciprocal continuity. Very recently, Chauhan et al. [20] obtained some results of this kind using integral type contractive conditions. Many authors established a number of other fixed point results in metric and related spaces (see, e.g., [21–31]).

In 2007, Sedghi and Shobe [32] proved a common fixed point theorem for weakly compatible mappings satisfying a new general contractive type condition. The aim of this paper is to prove a common fixed point theorem for two pairs of self-mappings by using the notions of compatibility and subsequential continuity (alternately subcompatibility and reciprocal continuity) satisfying general contractive condition in a metric space. Some examples are furnished which demonstrate the validity of our result.

2. Preliminaries

Definition 1. Let 𝐴,𝑆∶𝑋→𝑋 be two self-mappings on a metric space (𝑋,𝑑). The mappings 𝐴 and 𝑆 are said to be (1)weakly commuting if 𝑑(𝐴𝑆𝑥,𝑆𝐴𝑥)≤𝑑(𝐴𝑥,𝑆𝑥) for all 𝑥∈𝑋 [5],(2)compatible if lim_𝑛→∞𝑑(𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛)=0 for each sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛 [6],(3)noncompatible if there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛, but lim_𝑛→∞𝑑(𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛) is either nonzero or nonexistent [8], (4)weakly compatible if they commute at their coincidence points; that is, 𝐴𝑆𝑢=𝑆𝐴𝑢 whenever 𝐴𝑢=𝑆𝑢, for some 𝑢∈𝑋 [7],(5)occasionally weakly compatible if there is a point 𝑥∈𝑋 which is a coincidence point of 𝐴 and 𝑆 at which 𝐴 and 𝑆 commute [12],(6)with the property (E.A) if there exists a sequence {𝑥_𝑛} in 𝑋 and some 𝑧∈𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑧 [11].
It can be noticed that arbitrary noncompatible self-mappings satisfy the property (E.A) but two mappings satisfying the property (E.A), need not be noncompatible (see [33, Example 1]). Also, weak compatibility and property (E.A) are independent of each other (see [31, Examples 2.1 and 2.2]).

Definition 2 (see [34]). A pair (𝐴,𝑆) of self-mappings on a metric space (𝑋,𝑑) is called reciprocally continuous if for a sequence {𝑥_𝑛} in 𝑋, lim_𝑛→∞𝐴𝑆𝑥_𝑛=𝐴𝑧 and lim_𝑛→∞𝑆𝐴𝑥_𝑛=𝑆𝑧, whenever lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑧, for some 𝑧∈𝑋.

It is easy to see that if two self-mappings are continuous, then they are obviously reciprocally continuous, but the converse is not true. Moreover, in the setting of common fixed point theorems for compatible pairs of self-mappings satisfying contractive conditions, continuity of one of the mappings implies their reciprocal continuity but not conversely (see [35]).

Definition 3 (see [18]). A pair (𝐴,𝑆) of self-mappings on a metric space (𝑋,𝑑) is said to be subcompatible if there exists a sequence {𝑥_𝑛} such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑧, for some 𝑧∈𝑋 and lim_𝑛→∞𝑑(𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛)=0.

A pair of subcompatible mappings satisfies the property (E.A). Obviously, compatible mappings which satisfy the property (E.A) are subcompatible, but the converse statement does not hold in general (see [36, Example 2.3]). Two occasionally weakly compatible mappings are subcompatible; however, the converse is not true in general (see [18, Example 1.2]).

Definition 4 (see [18]). A pair (𝐴,𝑆) of self-mappings on a metric space (𝑋,𝑑) is called subsequentially continuous if there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑧, for some 𝑧∈𝑋 such that lim_𝑛→∞𝐴𝑆𝑥_𝑛=𝐴𝑧 and lim_𝑛→∞𝑆𝐴𝑥_𝑛=𝑆𝑧.

One can easily check that if two self-mappings 𝐴 and 𝑆 are both continuous, hence also reciprocally continuous mappings but 𝐴 and 𝑆 are not subsequentially continuous (see [35, Example 1]).

Definition 5 (see [32]). By ⋄∶ℝ⁺×ℝ⁺→ℝ⁺, a binary operation will be denoted, satisfying the following conditions: (1)⋄ is associative and commutative, (2)⋄ is continuous.
Some typical examples of ⋄ are 𝑎⋄𝑏=max{𝑎,𝑏},  𝑎⋄𝑏=𝑎+𝑏,  𝑎⋄𝑏=𝑎𝑏,  𝑎⋄𝑏=𝑎𝑏+𝑎+𝑏, and 𝑎⋄𝑏=𝑎𝑏/max{𝑎,𝑏,1}, for each 𝑎,𝑏∈ℝ⁺.

Definition 6 (see [32]). The binary operation ⋄ is said to satisfy 𝛼-property if there exists a positive real number 𝛼 such that 𝑎⋄𝑏≤𝛼max{𝑎,𝑏}, (1) for all 𝑎,𝑏∈ℝ⁺.

Example 7 (see [32]). (1) If 𝑎⋄𝑏=𝑎+𝑏, for each 𝑎,𝑏∈ℝ⁺, then for 𝛼≥2, we have 𝑎⋄𝑏≤𝛼max{𝑎,𝑏}.
(2) If 𝑎⋄𝑏=𝑎𝑏/max{𝑎,𝑏,1}, for each 𝑎,𝑏∈ℝ⁺, then for 𝛼≥1, we have 𝑎⋄𝑏≤𝛼max{𝑎,𝑏}.

3. Main Results

In 2007, Sedghi and Shobe [32] proved the following result.

Theorem 8 (see [32, Theorem 2.1]). Let (𝑋,𝑑) be a complete metric space such that ⋄ satisfies the 𝛼-property with 𝛼>0. Let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings on 𝑋 satisfying the following conditions: (1)𝐴(𝑋)⊆𝑇(𝑋), 𝐵(𝑋)⊆𝑆(𝑋), and 𝑇(𝑋) or 𝑆(𝑋) is a closed subset of 𝑋, (2) the pairs (𝐴,𝑆) and (𝐵,𝑇) are weakly compatible, (3) for all 𝑥,𝑦∈𝑋, 𝑑𝐴𝑥,𝐵𝑦≤𝑘₁𝑑𝑆𝑥,𝑇𝑦⋄𝑑(𝐴𝑥,𝑆𝑥)+𝑘₂𝑑𝑆𝑥,𝑇𝑦⋄𝑑𝐵𝑦,𝑇𝑦+𝑘₃𝑑⋄𝑑𝑆𝑥,𝑇𝑦𝑆𝑥,𝐵𝑦+𝑑𝐴𝑥,𝑇𝑦2, (2) where 𝑘₁,𝑘₂,𝑘₃>0 and 0<𝛼(𝑘₁+𝑘₂+𝑘₃)<1. Then, 𝐴, 𝐵, 𝑆, and 𝑇 have a unique common fixed point in 𝑋.

Now we prove our main result.

Theorem 9. Let 𝐴, 𝐵, 𝑆, and 𝑇 be four self-mappings on a metric space (𝑋,𝑑), and let the operation ⋄ satisfy the 𝛼-property with 𝛼>0. Suppose that the pairs (𝐴,𝑆) and (𝐵,𝑇) are compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous), satisfying inequality (2) of Theorem 8. Then 𝐴, 𝐵, 𝑆, and 𝑇 have a unique common fixed point in 𝑋.

Proof. If the pair of mappings (𝐴,𝑆) is subsequentially continuous and compatible, there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑧, (3) for some 𝑧∈𝑋, and lim_𝑛→∞𝑑𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛=𝑑(𝐴𝑧,𝑆𝑧)=0; (4) that is, 𝐴𝑧=𝑆𝑧. Similarly, with respect to the pair (𝐵,𝑇), there exists a sequence {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝐵𝑦_𝑛=lim_𝑛→∞𝑇𝑦_𝑛=𝑤, (5) for some 𝑤∈𝑋, and lim_𝑛→∞𝑑𝐵𝑇𝑦_𝑛,𝑇𝐵𝑦_𝑛=𝑑(𝐵𝑤,𝑇𝑤)=0; (6) that is, 𝐵𝑤=𝑇𝑤. Hence 𝑧 is a coincidence point of the pair (𝐴,𝑆) whereas 𝑤 is a coincidence point of the pair (𝐵,𝑇).
Now we assert that 𝑧=𝑤. If 𝑧≠𝑤 then using inequality (2) with 𝑥=𝑥_𝑛 and 𝑦=𝑦_𝑛, we have 𝑑𝐴𝑥_𝑛,𝐵𝑦_𝑛≤𝑘₁𝑑𝑆𝑥_𝑛,𝑇𝑦_𝑛⋄𝑑𝐴𝑥_𝑛,𝑆𝑥_𝑛+𝑘₂𝑑𝑆𝑥_𝑛,𝑇𝑦_𝑛⋄𝑑𝐵𝑦_𝑛,𝑇𝑦_𝑛+𝑘₃𝑑𝑆𝑥_𝑛,𝑇𝑦_𝑛⋄𝑑𝑆𝑥_𝑛,𝐵𝑦_𝑛+𝑑𝐴𝑥_𝑛,𝑇𝑦_𝑛2. (7) Letting 𝑛→∞, we get 𝑑(𝑧,𝑤)≤𝑘₁(𝑑(𝑧,𝑤)⋄𝑑(𝑧,𝑧))+𝑘₂(𝑑(𝑧,𝑤)⋄𝑑(𝑤,𝑤))+𝑘₃𝑑()⋄𝑑()()𝑧,𝑤𝑧,𝑤+𝑑𝑧,𝑤2. (8) Since ⋄ satisfies the 𝛼-property, we obtain 𝑑(𝑧,𝑤)≤𝑘₁𝛼max{𝑑(𝑧,𝑤),𝑑(𝑧,𝑧)}+𝑘₂𝛼max{𝑑(𝑧,𝑤),𝑑(𝑤,𝑤)}+𝑘₃𝑑(),𝑑()()𝛼max𝑧,𝑤𝑧,𝑤+𝑑𝑧,𝑤2=𝑘₁𝛼max{𝑑(𝑧,𝑤),0}+𝑘₂𝛼max{𝑑(𝑧,𝑤),0}+𝑘₃{𝑑()(𝑘𝛼max𝑧,𝑤,𝑑𝑧,𝑤)}=𝛼₁+𝑘₂+𝑘₃𝑑()(),𝑧,𝑤<𝑑𝑧,𝑤 (9) which is a contradiction. Hence 𝑧=𝑤. Now we prove that 𝐴𝑧=𝑧. If we suppose that 𝐴𝑧≠𝑧, then from inequality (2) with 𝑥=𝑧 and 𝑦=𝑦_𝑛, we have 𝑑𝐴𝑧,𝐵𝑦_𝑛≤𝑘₁𝑑𝑆𝑧,𝑇𝑦_𝑛⋄𝑑(𝐴𝑧,𝑆𝑧)+𝑘₂𝑑𝑆𝑧,𝑇𝑦_𝑛⋄𝑑𝐵𝑦_𝑛,𝑇𝑦_𝑛+𝑘₃𝑑𝑆𝑧,𝑇𝑦_𝑛⋄𝑑𝑆𝑧,𝐵𝑦_𝑛+𝑑𝐴𝑧,𝑇𝑦_𝑛2. (10) Taking the limit as 𝑛→∞, we get 𝑑(𝐴𝑧,𝑤)≤𝑘₁(𝑑(𝑆𝑧,𝑤)⋄𝑑(𝐴𝑧,𝐴𝑧))+𝑘₂(𝑑(𝐴𝑧,𝑤)⋄𝑑(𝑤,𝑤))+𝑘₃𝑑()⋄𝑑()()𝐴𝑧,𝑤𝐴𝑧,𝑤+𝑑𝐴𝑧,𝑤2; (11) that is, 𝑑(𝐴𝑧,𝑧)≤𝑘₁(𝑑(𝑆𝑧,𝑧)⋄𝑑(𝐴𝑧,𝐴𝑧))+𝑘₂(𝑑(𝐴𝑧,𝑧)⋄𝑑(𝑧,𝑧))+𝑘₃𝑑()⋄𝑑()()𝐴𝑧,𝑧𝐴𝑧,𝑧+𝑑𝐴𝑧,𝑧2≤𝑘₁𝛼max{𝑑(𝑆𝑧,𝑧),𝑑(𝐴𝑧,𝐴𝑧)}+𝑘₂𝛼max{𝑑(𝐴𝑧,𝑧),𝑑(𝑧,𝑧)}+𝑘₃𝑑(),𝑑()()𝛼max𝐴𝑧,𝑧𝐴𝑧,𝑧+𝑑𝐴𝑧,𝑧2. (12) Then, simplifying, we obtain 𝑑(𝐴𝑧,𝑧)≤𝛼𝑘₁+𝑘₂+𝑘₃𝑑()(),𝐴𝑧,𝑧<𝑑𝐴𝑧,𝑧 (13) a contradiction. Hence, 𝐴𝑧=𝑧. Therefore, 𝐴𝑧=𝑆𝑧=𝑧. Now we show that 𝐵𝑧=𝑧. If 𝐵𝑧≠𝑧 then using (2) with 𝑥=𝑥_𝑛 and 𝑦=𝑧, we have 𝑑𝐴𝑥_𝑛,𝐵𝑧≤𝑘₁𝑑𝑆𝑥_𝑛,𝑇𝑧⋄𝑑𝐴𝑥_𝑛,𝑆𝑥_𝑛+𝑘₂𝑑𝑆𝑥_𝑛,𝑇𝑧⋄𝑑(𝐵𝑧,𝑇𝑧)+𝑘₃𝑑𝑆𝑥_𝑛⋄𝑑,𝑇𝑧𝑆𝑥_𝑛,𝐵𝑧+𝑑𝐴𝑥_𝑛,𝑇𝑧2. (14) As 𝑛→∞, we get 𝑑(𝑧,𝐵𝑧)≤𝑘₁(𝑑(𝑧,𝐵𝑧)⋄𝑑(𝑧,𝑧))+𝑘₂(𝑑(𝑧,𝐵𝑧)⋄𝑑(𝐵𝑧,𝐵𝑧))+𝑘₃𝑑()⋄𝑑()()𝑧,𝐵𝑧𝑧,𝐵𝑧+𝑑𝑧,𝐵𝑧2≤𝑘₁𝛼max{𝑑(𝑧,𝐵𝑧),𝑑(𝑧,𝑧)}+𝑘₂𝛼max{𝑑(𝑧,𝐵𝑧),𝑑(𝐵𝑧,𝐵𝑧)}+𝑘₃𝑑(),𝑑()()𝛼max𝑧,𝐵𝑧𝑧,𝐵𝑧+𝑑𝑧,𝐵𝑧2. (15) Then, simplifying, we obtain 𝑑(𝑧,𝐵𝑧)≤𝛼𝑘₁+𝑘₂+𝑘₃𝑑()(),𝑧,𝐵𝑧<𝑑𝑧,𝐵𝑧 (16) a contradiction. Hence, 𝐵𝑧=𝑧. Therefore, 𝑧=𝐴𝑧=𝑆𝑧=𝐵𝑧=𝑇𝑧; that is, 𝑧 is a common fixed point of 𝐴, 𝐵, 𝑆, and 𝑇. The uniqueness of common fixed point is an easy consequence of inequality (2).
Now suppose that the mappings (𝐴,𝑆) (as well as (𝐵,𝑇)) are subcompatible and reciprocally continuous. Then there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑧, (17) for some 𝑧∈𝑋, and lim_𝑛→∞𝑑𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛=𝑑(𝐴𝑧,𝑆𝑧)=0, (18) whereas in respect of the pair (𝐵,𝑇), there exists a sequence {𝑦_𝑛} in 𝑋 with lim_𝑛→∞𝐵𝑦_𝑛=lim_𝑛→∞𝑇𝑦_𝑛=𝑤, (19) for some 𝑤∈𝑋, and lim_𝑛→∞𝑑𝐵𝑇𝑦_𝑛,𝑇𝐵𝑦_𝑛=𝑑(𝐵𝑤,𝑇𝑤)=0. (20) Therefore, 𝐴𝑧=𝑆𝑧 and 𝐵𝑤=𝑇𝑤; that is, 𝑧 is a coincidence point of the pair (𝐴,𝑆) whereas 𝑤 is a coincidence point of the pair (𝐵,𝑇). The rest of the proof can be completed easily.

Example 10. Let 𝑋=[0,∞), and let 𝑑 be the usual metric on 𝑋. Define self-mappings 𝐴, 𝐵, 𝑆, and 𝑇 by 𝑥𝐴𝑥=𝐵𝑥=7,if𝑥∈[0,1];𝑥+67(),𝑥,if𝑥∈1,∞𝑆𝑥=𝑇𝑥=6,if𝑥∈[0,1];𝑥+56,if𝑥∈(1,∞). (21) Consider the sequence {𝑥_𝑛}={1/𝑛}_𝑛∈ℕ in 𝑋. Then lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞17𝑛=0=lim_𝑛→∞16𝑛=lim_𝑛→∞𝑆𝑥_𝑛. (22) Next, lim_𝑛→∞𝐴𝑆𝑥_𝑛=lim_𝑛→∞𝐴16𝑛=lim_𝑛→∞142𝑛=0=𝐴0,lim_𝑛→∞𝑆𝐴𝑥_𝑛=lim_𝑛→∞𝑆17𝑛=lim_𝑛→∞142𝑛=0=𝑆0,lim_𝑛→∞𝑑𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛=0. (23) Consider another sequence {𝑥_𝑛}={1+1/𝑛}_𝑛∈ℕ in 𝑋. Then lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞11+7𝑛=1=lim_𝑛→∞11+6𝑛=lim_𝑛→∞𝑆𝑥_𝑛. (24) However, lim_𝑛→∞𝐴𝑆𝑥_𝑛=lim_𝑛→∞𝐴11+6𝑛=lim_𝑛→∞11+42𝑛=1≠𝐴1,lim_𝑛→∞𝑆𝐴𝑥_𝑛=lim_𝑛→∞𝑆11+7𝑛=lim_𝑛→∞11+42𝑛=1≠𝑆1, (25) but lim_𝑛→∞𝑑(𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛)=0. Thus, the pair (𝐴,𝑆) is compatible as well as subsequentially continuous but not reciprocally continuous (the same for the pair (𝐵,𝑇)). It is easy to check that condition (2) is satisfied with ⋄=max, 𝛼=1, and 𝑘₁+𝑘₂+𝑘₃=6/7. Therefore, all the conditions of Theorem 9 are satisfied. Here, 0 is a coincidence as well as the unique common fixed point of the mappings 𝐴, 𝐵, 𝑆, and 𝑇.
It can be noted that this example cannot be covered by those fixed point theorems which assume both compatibility and reciprocal continuity or by involving conditions of closedness of respective ranges. Indeed, in this example 𝐴(𝑋)=[0,1/7]∪(1,∞) and 𝑆(𝑋)=[0,1/6]∪(1,∞); hence, neither of 𝐴(𝑋) and 𝑆(𝑋) is closed.

Example 11. Let 𝑋=ℝ (set of real numbers), and let 𝑑 be the usual metric on 𝑋. Define self-mappings 𝐴, 𝐵, 𝑆, and 𝑇 by 𝑥𝐴𝑥=𝐵𝑥=4();[),();[).,if𝑥∈−∞,14𝑥−3,if𝑥∈1,∞𝑆𝑥=𝑇𝑥=𝑥+3,if𝑥∈−∞,15𝑥−4,if𝑥∈1,∞ (26) Consider the sequence {𝑥_𝑛}={1+1/𝑛}_𝑛∈ℕ in 𝑋. Then lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞41+𝑛=1=lim_𝑛→∞51+𝑛=lim_𝑛→∞𝑆𝑥_𝑛. (27) Also, lim_𝑛→∞𝐴𝑆𝑥_𝑛=lim_𝑛→∞𝐴51+𝑛=lim_𝑛→∞1+20𝑛=1=𝐴1,lim_𝑛→∞𝑆𝐴𝑥_𝑛=lim_𝑛→∞𝑆41+𝑛=lim_𝑛→∞1+20𝑛=1=𝑆1,lim_𝑛→∞𝑑𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛=0. (28) Consider another sequence {𝑥_𝑛}={(1/𝑛)−4}_𝑛∈ℕ in 𝑋. Then lim_𝑛→∞𝐴𝑥_𝑛=lim_𝑛→∞14𝑛−1=−1=lim_𝑛→∞1𝑛−1=lim_𝑛→∞𝑆𝑥_𝑛. (29) Next, lim_𝑛→∞𝐴𝑆𝑥_𝑛=lim_𝑛→∞𝐴1𝑛−1=lim_𝑛→∞1−14𝑛41=−4=𝐴(−1),lim_𝑛→∞𝑆𝐴𝑥_𝑛=lim_𝑛→∞𝑆14𝑛−1=lim_𝑛→∞1(),4𝑛−1+3=2=𝑆−1 (30) and lim_𝑛→∞𝑑(𝐴𝑆𝑥_𝑛,𝑆𝐴𝑥_𝑛)≠0. Thus, the pair (𝐴,𝑆) is reciprocally continuous as well as subcompatible but not compatible (the same for the pair (𝐵,𝑇)). It is easy to check that condition (2) is satisfied with ⋄=max, 𝛼=1, and 𝑘₁+𝑘₂+𝑘₃=4/5. Therefore, all the conditions of Theorem 9 are satisfied. Thus, 1 is a coincidence as well as the unique common fixed point of the pair (𝐴,𝑆).
It can be noted that this example cannot be covered by those fixed point theorems which involve both compatibility and reciprocal continuity. Again, 𝐴(𝑋)=(−∞,1/4)∪[1,+∞) is not closed. Note also that the mappings 𝐴 and 𝑆 have two points of coincidence (−1 and 1), which are occasionally weakly compatible but not weakly compatible.

In the next example (taken from [18, Example 1.4]), we demonstrate the situation when conditions of Theorem 9 are not satisfied, and the given pairs have no common fixed points.

Example 12. Let 𝑋=[0,+∞) with the standard metric 𝑑, and let 𝐴,𝐵,𝑆,𝑇∶𝑋→𝑋 be given by 𝐴𝑥=𝐵𝑥=𝑥+1,0≤𝑥≤1,2𝑥−1,𝑥>1,𝑆𝑥=𝑇𝑥=1−𝑥,0≤𝑥<1,3𝑥−2,𝑥≥1. (31) Then, as it was shown in [18, Example 1.4], the pairs (𝐴,𝑆) and (𝐵,𝑇) are subsequentially continuous and subcompatible. However, they are neither reciprocally continuous nor compatible (not even occasionally weakly compatible). We note that (𝐴,𝑆) has no common fixed points, although it has a unique point of coincidence 𝑧=1.

We present an example of different kind, inspired by [20, Example 3].

Example 13. Let 𝑋={0,1,2,…,10}, and define a metric 𝑑 on 𝑋 by 𝑑𝑥,𝑦=0,𝑥=𝑦,max𝑥,𝑦,𝑥≠𝑦. (32) Consider the mappings 𝐴,𝐵,𝑆,𝑇∶𝑋→𝑋 given by 𝐴𝑥=𝐵𝑥=0,𝑥=0,𝑥−1,𝑥≥1;𝑆𝑥=𝑇𝑥=0,𝑥=0,𝑥+1,1≤𝑥≤9,10,𝑥=10. (33) Take ⋄=max (which satisfies 𝛼-condition with 𝛼=1). Then,(1)the pair (𝐴,𝑆) (as well as (𝐵,𝑇)) is compatible and subsequentially continuous, (2)condition (2) is satisfied with 𝑘₁=𝑘₂=𝑘₃=0.3. Indeed, in order to prove (1), take 𝑥_𝑛=0 for all 𝑛 but finitely many (which is the only possibility to obtain the same limit for (𝐴𝑥_𝑛) and (𝑆𝑥_𝑛)). Then 𝑑(𝐴𝑥_𝑛,0)→0 and 𝑑(𝑆𝑥_𝑛,0)→0; also 𝑆𝐴𝑥_𝑛→0=𝑆0 and 𝐴𝑆𝑥_𝑛→0=𝐴0. Hence, the pair (𝐴,𝑆) is compatible and subsequentially continuous.
In order to prove (2), suppose that 𝑥,𝑦∈𝑋, 𝑥≠𝑦 (the case 𝑥=𝑦 is trivial). Since we have 𝐴=𝐵, 𝑆=𝑇, and 𝑘₁=𝑘₂, condition (2) is symmetric in 𝑥,𝑦; hence, without loss of generality, we can suppose that 𝑥≥𝑦. Consider the following possible cases.
Case 1. One has 𝑦=0 and 𝑥=1. Then 𝐴𝑥=𝐵𝑦=0, 𝑑(𝐴𝑥,𝐵𝑦)=0, and (2) is satisfied.
Case 2. One has 𝑦∈{0,1} and 2≤𝑥≤9. Then 𝐴𝑥=𝑥−1, 𝐵𝑦=0, 𝑑(𝐴𝑥,𝐵𝑦)=𝑥−1. The right-hand side of (2) becomes (𝑡∈{0,2}) 𝑅=𝑘₁max{𝑥+1,𝑡}+𝑘₂max{𝑥+1,𝑥+1}+𝑘₃1max𝑥+1,2=𝑘max𝑆𝑥,𝐵𝑦+𝑥+1₁+𝑘₂+𝑘₃(𝑥+1)=0.9(𝑥+1)≥0.9⋅108(𝑥−1)>𝑥−1=𝑑𝐴𝑥,𝐵𝑦. (34)Case 3. One has 𝑦∈{0,1} and 𝑥=10. Then 𝑑𝐴𝑥,𝐵𝑦=9=𝑘₁+𝑘₂+𝑘₃⋅10=𝑅. (35)Case 4. One has 2≤𝑦<𝑥≤9. Then 𝑑(𝐴𝑥,𝐵𝑦)=𝑑(𝑥−1,𝑦−1)=𝑥−1 and 𝑅=𝑘₁max{𝑥+1,𝑥+1}+𝑘₂max𝑥+1,𝑦+1+𝑘₃1max𝑥+1,2=𝑘𝑥+1+max𝐴𝑥,𝑇𝑦₁+𝑘₂+𝑘₃(𝑥+1)=0.9(𝑥+1)≥0.9⋅108(𝑥−1)>𝑥−1=𝑑𝐴𝑥,𝐵𝑦. (36)Case 5. One has 2≤𝑦<𝑥=10. Then (2) again reduces to 9=(𝑘₁+𝑘₂+𝑘₃)⋅10.
All the conditions of Theorem 9 are satisfied, and 𝐴, 𝐵, 𝑆, and 𝑇 have a unique common fixed point (which is 𝑧=0).
By choosing 𝐴, 𝐵, 𝑆, and 𝑇 suitably in Theorem 9, we can deduce corollaries for two or three self-mappings. As a sample, we deduce the following corollary for two self-mappings.

Corollary 14. Let 𝐴 and 𝑆 be two self-mappings on a metric space (𝑋,𝑑) such that ⋄ satisfies the 𝛼-property with 𝛼>0. If the pair (𝐴,𝑆) is compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) satisfying 𝑑𝐴𝑥,𝐴𝑦≤𝑘₁𝑑𝑆𝑥,𝑆𝑦⋄𝑑(𝐴𝑥,𝑆𝑥)+𝑘₂𝑑𝑆𝑥,𝑆𝑦⋄𝑑𝐴𝑦,𝑆𝑦+𝑘₃𝑑⋄𝑑𝑆𝑥,𝑆𝑦𝑆𝑥,𝐴𝑦+𝑑𝐴𝑥,𝑆𝑦2, (37) for all 𝑥,𝑦∈𝑋, where 𝑘₁,𝑘₂,𝑘₃>0 and 0<𝛼(𝑘₁+𝑘₂+𝑘₃)<1, then 𝐴 and 𝑆 have a unique common fixed point in 𝑋.

Remark 15. The conclusion of Theorem 9 remains true if we replace inequality (2) by the following: 𝑑𝐴𝑥,𝐵𝑦≤𝑘₁𝑑𝑆𝑥,𝑇𝑦+𝑑(𝐴𝑥,𝑆𝑥)+𝑘₂𝑑𝑆𝑥,𝑇𝑦+𝑑𝐵𝑦,𝑇𝑦+𝑘₃𝑑+𝑑𝑆𝑥,𝑇𝑦𝑆𝑥,𝐵𝑦+𝑑𝐴𝑥,𝑇𝑦2, (38) for all 𝑥,𝑦∈𝑋, where 𝑘₁,𝑘₂,𝑘₃>0 and 0<𝑘₁+𝑘₂+𝑘₃<1/2.
Similarly, other variants of contractive condition can be obtained by specifying operation ⋄.

Remark 16. Similar results can be obtained if condition (2) is replaced by the following one: 𝑑(𝑢)𝑑(),𝑑,𝑑,𝐴𝑥,𝐵𝑦≤𝜓forsome𝑢∈𝑆𝑥,𝑇𝑦,𝑑𝐴𝑥,𝑆𝑥𝐵𝑦,𝑇𝑦,𝑑𝑆𝑥,𝐵𝑦𝐴𝑥,𝑇𝑦 (39) for a suitable function 𝜓∶[0,+∞)→[0,+∞).

4. Conclusion

Theorem 9 is proved for two pairs of compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) mappings satisfying a general contractive condition. Theorem 9 improves the main result of Sedghi and Shobe [32, Theorem 2.1] as we do not require any condition on the containment of ranges of involved mappings and completeness (or closedness) of the whole space (or any subspace). A natural result is defined in the form of a corollary (see Corollary 14). On the other hand, Remark 15 is developed for a particular case, 𝑎⋄𝑏=𝑎+𝑏, which also improves the result of Sedghi and Shobe [32, Corollary 2.2].

Acknowledgments

The authors would like to thank Professor Mohammad Imdad for the preprint of paper [36] and anonymous referees for their valuable comments that helped them to improve the exposition, in particular for Remark 16. The second author is thankful to the Ministry of Education, Science and Technological Development of Serbia.