Cyclic (𝜙)-Contractions in Uniform Spaces and Related Fixed Point Results

Abstract

First, we define cyclic (𝜙)-contractions of different types in a uniform space. Then, we apply these concepts of cyclic (𝜙)-contractions to establish certain fixed and common point theorems on a Hausdorff uniform space. Some more general results are obtained as corollaries. Moreover, some examples are provided to demonstrate the usability of the proved theorems.

1. Introduction

Let 𝑋 be a nonempty set. A nonempty family, 𝜗, of subsets of 𝑋×𝑋 is called the uniform structure of 𝑋 if it satisfies the following properties:(i)if 𝐺 is in 𝜗, then 𝐺 contains the diagonal {(𝑥,𝑥)∣𝑥∈𝑋};(ii)if 𝐺 is in 𝜗 and 𝐻 is a subset of 𝑋×𝑋 which contains 𝐺, then 𝐻 is in 𝜗;(iii)if 𝐺 and 𝐻 are in 𝜗, then 𝐺∩𝐻 is in 𝜗;(iv)if 𝐺 is in 𝜗, then there exists 𝐻 in 𝜗, such that, whenever (𝑥,𝑦) and (𝑦,𝑧) are in 𝐻, then (𝑥,𝑧) is in 𝐺;(v)if 𝐺 is in 𝜗, then {(𝑦,𝑥)∣(𝑥,𝑦)∈𝐺} is also in 𝜗.The pair (𝑋,𝜗) is called a uniform space and the element of 𝜗 is called entourage or neighbourhood or surrounding. The pair (𝑋,𝜗) is called a quasi-uniform space (see, e.g., [1, 2]) if property (v) is omitted.

Existence and uniqueness of fixed points for various contractive mappings in the setting of uniform spaces have been investigated by several authors; see, for example, [3–12] and the references therein.

Recently, an interesting and remarkable notion of cyclic mapping was introduced and studied by Kirk et al. [13]. Following this paper, a number of authors introduced contractive mapping via the cyclic mappings and reported certain fixed point results in the setting of different type of spaces; see, for example, [13–17].

In this paper, we will give the characterization of cyclic mapping in the context of uniform spaces and, further, prove the existence and uniqueness of fixed and common fixed points of such mappings via 𝐴-distance and 𝐸-distance, introduced by Aamri and El Moutawakil [18].

For the sake of completeness, we recollect some basic definitions and fundamental results. Let Δ={(𝑥,𝑥)∣𝑥∈𝑋} be the diagonal of a nonempty set 𝑋. For 𝑉,𝑊∈𝑋×𝑋, we will use the following setting in the sequel: =∣𝑉∘𝑊𝑥,𝑦thereexists(),𝑉𝑧∈𝑋∶𝑥,𝑧∈𝑊,𝑧,𝑦∈𝑉₋₁=𝑥,𝑦∣𝑦,𝑥∈𝑉. (1) For subset 𝑉∈𝜗, a pair of points 𝑥 and 𝑦 are said to be 𝑉-close if (𝑥,𝑦)∈𝑉 and (𝑦,𝑥)∈𝑉. Moreover, a sequence {𝑥_𝑛} in 𝑋 is called a Cauchy sequence for 𝜗, if for any 𝑉∈𝜗 there exists 𝑁≥1 such that 𝑥_𝑛 and 𝑥_𝑚 are 𝑉-close for 𝑛,𝑚≥𝑁. For (𝑋,𝜗), there is a unique topology 𝜏(𝜗) on 𝑋 generated by 𝑉(𝑥)={𝑦∈𝑋∣(𝑥,𝑦)∈𝑉}, where 𝑉∈𝜗.

A sequence {𝑥_𝑛} in 𝑋 is convergent to 𝑥 for 𝜗, denoted by lim_𝑛→∞𝑥_𝑛=𝑥, if for any 𝑉∈𝜗 there exists 𝑛₀∈ℕ such that 𝑥_𝑛∈𝑉(𝑥) for every 𝑛≥𝑛₀. A uniform space (𝑋,𝜗) is called Hausdorff if the intersection of all the 𝑉∈𝜗 is equal to Δ of 𝑋, that is, if (𝑥,𝑦)∈𝑉 for all 𝑉∈𝜗 implies 𝑥=𝑦. If 𝑉=𝑉⁻¹, then we say that a subset 𝑉∈𝜗 is symmetrical. Throughout the paper, we assume that each 𝑉∈𝜗 is symmetrical. For more details, see, for example, [1, 18–21].

Now, we recall the notions of 𝐴-distance and 𝐸-distance.

Definition 1 (see, e.g., [18, 19]). Let (𝑋,𝜗) be a uniform space. A function 𝑝∶𝑋×𝑋→[0,∞) is said to be an 𝐴-distance if for any 𝑉∈𝜗 there exists 𝛿>0 such that if 𝑝(𝑧,𝑥)≤𝛿 and 𝑝(𝑧,𝑦)≤𝛿 for some 𝑧∈𝑋, then (𝑥,𝑦)∈𝑉.

Definition 2 (see, e.g., [18, 19]). Let (𝑋,𝜗) be a uniform space. A function 𝑝∶𝑋×𝑋→[0,∞) is said to be an 𝐸-distance if (𝑝₁)𝑝 is an 𝐴-distance,(𝑝₂)𝑝(𝑥,𝑦)≤𝑝(𝑥,𝑧)+𝑝(𝑧,𝑦),∀𝑥,𝑦,𝑧∈𝑋.

Example 3 (see, e.g., [18, 19]). Let (𝑋,𝜗) be a uniform space and let 𝑑 be a metric on 𝑋. It is evident that (𝑋,𝜗_𝑑) is a uniform space, where 𝜗_𝑑 is the set of all subsets of 𝑋×𝑋 containing a “band” 𝐵_𝜖={(𝑥,𝑦)∈𝑋²∣𝑑(𝑥,𝑦)<𝜖} for some 𝜖>0. Moreover, if 𝜗⊆𝜗_𝑑, then 𝑑 is an 𝐸-distance on (𝑋,𝜗).

Lemma 4 (see, e.g., [18, 19]). Let (𝑋,𝜗) be a Hausdorff uniform space and let 𝑝 be an 𝐴-distance on X. Let {𝑥_𝑛} and {𝑦_𝑛} be sequences in 𝑋 and {𝛼_𝑛} and let {𝛽_𝑛} be sequences in [0,∞) converging to 0. Then, for 𝑥,𝑦,𝑧∈𝑋, the following hold. (a)If 𝑝(𝑥_𝑛,𝑦)≤𝛼_𝑛 and 𝑝(𝑥_𝑛,𝑧)≤𝛽_𝑛 for all 𝑛∈ℕ, then 𝑦=𝑧. In particular, if 𝑝(𝑥,𝑦)=0 and 𝑝(𝑥,𝑧)=0, then 𝑦=𝑧.(b)If 𝑝(𝑥_𝑛,𝑦_𝑛)≤𝛼_𝑛 and 𝑝(𝑥_𝑛,𝑧)≤𝛽_𝑛 for all 𝑛∈ℕ, then {𝑦_𝑛} converges to 𝑧.(c)If 𝑝(𝑥_𝑛,𝑥_𝑚)≤𝛼_𝑛 for all 𝑛,𝑚∈ℕ with 𝑚>𝑛, then {𝑥_𝑛} is a Cauchy sequence in (𝑋,𝜗).

Let 𝑝 be an 𝐴-distance. A sequence in a uniform space (𝑋,𝜗) with an 𝐴-distance is said to be a 𝑝-Cauchy if for every 𝜖>0 there exists 𝑛₀∈ℕ such that 𝑝(𝑥_𝑛,𝑥_𝑚)<𝜖 for all 𝑛,𝑚≥𝑛₀.

Definition 5 (see, e.g., [18, 19]). Let (𝑋,𝜗) be a uniform space and let 𝑝 be an 𝐴-distance on 𝑋. (1)𝑋 is 𝑆-complete if for every 𝑝-Cauchy sequence {𝑥_𝑛} there exists 𝑥 in 𝑋 with lim_𝑛→∞𝑝(𝑥_𝑛,𝑥)=0.(2)𝑋 is 𝑝-Cauchy complete if for every 𝑝-Cauchy sequence {𝑥_𝑛} there exists 𝑥 in 𝑋 with lim_𝑛→∞𝑥_𝑛=𝑥 with respect to 𝜏(𝜗).

Remark 6. Let (𝑋,𝜗) be a Hausdorff uniform space which is 𝑆-complete. If a sequence {𝑥_𝑛} is a 𝑝-Cauchy sequence, then we have lim_𝑛→∞𝑝(𝑥_𝑛,𝑥)=0. Regarding Lemma 4(b), we derive that lim_𝑛→∞𝑥_𝑛=𝑥 with respect to the topology 𝜏(𝜗) and hence 𝑆-completeness implies 𝑝-Cauchy completeness.

Definition 7. Let (𝑋,𝜗) be a Hausdorff uniform space and let 𝑝 be an 𝐴-distance on 𝑋. Two self-mappings 𝑓 and 𝑔 of 𝑋 are said to be weak compatible if they commute at their coincidence points; that is, 𝑓𝑥=𝑔𝑥 implies that 𝑓𝑔𝑥=𝑔𝑓𝑥.

We denote by ℱ the class of functions 𝜙∶[0,∞)→[0,∞) nondecreasing and continuous satisfying 𝜙(𝑡)>0 for 𝑡∈(0,∞) and 𝜙(0)=0.

Definition 8 (see [17]). A function 𝜙∶[0,∞)→[0,∞) is called a comparison function if it satisfies the following: (i)𝜙 is increasing; that is, 𝑡₁≤𝑡₂ implies 𝜙(𝑡₁)≤𝜙(𝑡₂), for 𝑡₁,𝑡₂∈[0,∞);(ii){𝜙^𝑛(𝑡)}_𝑛∈ℕ converges to 0 as 𝑛→∞, for all 𝑡∈[0,∞).

Definition 9 (see [22]). A function 𝜙∶[0,∞)→[0,∞) is called a (𝑐)-comparison function if (i)𝜙 is increasing,(ii)there exist 𝑘₀∈ℕ,𝑎∈(0,1) and a convergent series of nonnegative terms ∑^∞_𝑘=1𝑣_𝑘 such that 𝜙^𝑘+1(𝑡)≤𝑎𝜙^𝑘(𝑡)+𝑣_𝑘, (2) for 𝑘≥𝑘₀ and any 𝑡∈[0,∞).

Let 𝒞 be the collection of all (𝑐)-comparison functions 𝜙∶[0,∞)→[0,∞) defined in Definition 9.

Lemma 10 (see [22]). If 𝜙∶[0,∞)→[0,∞) is a (𝑐)-comparison function, then the following hold: (i)𝜙 is comparison function,(ii)𝜙(𝑡)<𝑡, for any 𝑡∈[0,∞),(iii)𝜙 is continuous at 0,(iv)the series ∑^∞_𝑘=0𝜙^𝑘(𝑡) converges for any 𝑡∈[0,∞).

In 1922, Banach proved that every contraction in a complete metric space has a unique fixed point. This celebrated result has been generalized and improved by many authors in the context of different abstract spaces for various operators (see [1–28] and the references therein). Recently, fixed point theorems for operators 𝑇 defined on a complete metric space 𝑋 with a cyclic representation of 𝑋 with respect to 𝑇 have appeared in the literature (see, e.g., [13–17]). Now, we present a modification of the main result of [16]. For this, we need the following definitions.

Definition 11 (see [13]). Let 𝑋 be a nonempty set, 𝑚 a positive integer, and 𝑇∶𝑋→𝑋 a mapping. 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is said to be a cyclic representation of 𝑋 with respect to 𝑇 if (i)𝐴_𝑖,𝑖=1,2,…,𝑚, are nonempty sets;(ii)𝑇(𝐴₁)⊂𝐴₂,…,𝑇(𝐴_𝑚−1)⊂𝐴_𝑚,𝑇(𝐴_𝑚)⊂𝐴₁.

Definition 12. Let (𝑋,𝑑) be a metric space, 𝑚 a positive integer, 𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty subsets of 𝑋, and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. An operator 𝑇∶𝑋→𝑋 is a cyclic (𝜙)-contraction if (i)𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇,(ii)𝑑(𝑇𝑥,𝑇𝑦)≤𝜙(𝑑(𝑥,𝑦)), for any 𝑥∈𝐴_𝑖, 𝑦∈𝐴_𝑖+1, 𝑖=1,2,…,𝑚, where 𝐴_𝑚+1=𝐴₁ and 𝜙∈ℱ.

The main result of [14] is the following.

Theorem 13 (Theorem 6 of [14]). Let (𝑋,𝑑) be a complete metric space, m a positive integer, 𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty subsets of 𝑋, and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. Let 𝑇∶𝑋→𝑋 be a cyclic (𝜙−𝜓)-contraction with 𝜙,𝜓∈ℱ. Then, 𝑇 has a unique fixed point 𝑧∈⋂^𝑚_𝑖=1𝐴_𝑖.

The main aim of this paper is to prove results similar to the abovementioned theorems in uniform spaces and to present modifications of Theorem 2.1 [16], Theorems 3.1-3.2 in [18], and other related results.

2. Main Result

First, we present the following definition.

Definition 14. Let (𝑋,𝜗) be a uniform space, 𝑚 a positive integer, 𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty subsets of 𝑋, and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. An operator 𝑇∶𝑋→𝑋 is a cyclic (𝜙)-contraction if (i)𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇,(ii)for any 𝑥∈𝐴_𝑖, 𝑦∈𝐴_𝑖+1, 𝑖=1,2,…,𝑚, 𝑝𝑇𝑥,𝑇𝑦≤𝜙𝑝𝑥,𝑦, (3) where 𝐴_𝑚+1=𝐴₁ and 𝜙∈𝒞.

Our main result is the following.

Theorem 15. Let (𝑋,𝜗) be an 𝑆-complete Hausdorff uniform space such that 𝑝 is an 𝐸-distance on 𝑋, 𝑚 a positive integer, and 𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty closed subsets of 𝑋 with respect to the topological space (𝑋,𝜏(𝜗)), and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. Let 𝑇∶𝑋→𝑋 be a cyclic (𝜙)-contraction. Then, 𝑇 has a unique fixed point 𝑥∈⋂^𝑚_𝑖=1𝐴_𝑖.

Proof. We first show that the fixed point of 𝑇 is unique (if it exists). Suppose, on the contrary, that 𝑦,𝑧∈𝑋 with 𝑦≠𝑧 are fixed points of 𝑇. The cyclic character of 𝑇 and the fact that 𝑦,𝑧∈𝑋 are fixed points of 𝑇 imply that 𝑦,𝑧∈⋂^𝑚_𝑖=1𝐴_𝑖. Using the contractive condition, we obtain 𝑝𝑦,𝑧=𝑝𝑇𝑦,𝑇𝑧≤𝜙𝑝𝑦,𝑧<𝑝𝑦,𝑧 (4) and from the last inequality 𝑝𝑦,𝑧=0. (5) Similarly, we can show that 𝑝(𝑦,𝑦)=0 and, consequently, 𝑦=𝑧.
Now, we prove the existence of a fixed point. Note that 𝑝 is not symmetric. To show that the sequence {𝑥_𝑛} is Cauchy, we will show that both lim_𝑛→∞𝑝(𝑥_𝑛,𝑥_𝑛+𝑞)=0 and lim_𝑛→∞𝑝(𝑥_𝑛+𝑞,𝑥_𝑛)=0, for any 𝑞>1.
For this aim, take 𝑥₀∈𝑋 and consider the sequence given by 𝑥_𝑛+1=𝑇𝑥_𝑛,𝑛=0,1,2,…. (6) If there exists 𝑛₀∈ℕ such that 𝑥_𝑛0+1=𝑥_𝑛0, then the proof is completed. In this case, 𝑥_𝑛0 is the required fixed point of 𝑇. Throughout the proof, we assume that 𝑥_𝑛+1≠𝑥_𝑛forany𝑛=0,1,2,…. (7) Notice that for any 𝑛>0 there exists 𝑖_𝑛∈{1,2,…,𝑚} such that 𝑥_𝑛−1∈𝐴_𝑖𝑛 and 𝑥_𝑛∈𝐴_𝑖𝑛+1, since 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. Due to the fact that 𝑇 is a cyclic (𝜙)-contraction, we have 𝑝𝑥_𝑛,𝑥_𝑛+1=𝑝𝑇𝑥_𝑛−1,𝑇𝑥_𝑛≤𝜙𝑝𝑥_𝑛−1,𝑥_𝑛, (8) by taking 𝑥=𝑥_𝑛 and 𝑦=𝑥_𝑛+1 in (3). From (8) and taking the monotonicity of 𝜙 into account, we derive by induction that 𝑝𝑥_𝑛,𝑥_𝑛+1≤𝜙^𝑛𝑝𝑥₀,𝑥₁forany𝑛=1,2,…. (9) As 𝑝 is an 𝐸-distance, we obtain that 𝑝𝑥_𝑛,𝑥_𝑚≤𝑝𝑥_𝑛,𝑥_𝑛+1+⋯+𝑝𝑥_𝑚−1,𝑥_𝑚, (10) so for 𝑞≥1 we have that 𝑝𝑥_𝑛,𝑥_𝑛+𝑞≤𝜙^𝑛𝑝𝑥₀,𝑥₁+⋯+𝜙^{𝑛+𝑞−1}𝑝𝑥₀,𝑥₁. (11) In the sequel, we will prove that {𝑥_𝑛} is a 𝑝-Cauchy sequence. Denoting 𝑆_𝑛=^𝑛_𝑘=0𝜙^𝑘𝑝𝑥₀,𝑥₁,𝑛≥0, (12) implies that 𝑝𝑥_𝑛,𝑥_𝑛+𝑞≤𝑆_{𝑛+𝑞−1}−𝑆_𝑛−1. (13) As 𝜙 is a (𝑐)-comparison function, supposing 𝑝(𝑥₀,𝑥₁)>0, by Lemma 10, (iv), it follows that ^∞_𝑘=0𝜙^𝑘𝑝𝑥₀,𝑥₁<∞, (14) so there is 𝑆∈[0,∞) such that lim_𝑛→∞𝑆_𝑛=𝑆. (15) Then, by (13) we obtain that lim_𝑛→∞𝑝𝑥_𝑛,𝑥_𝑛+𝑞=0. (16)
By repeating the same arguments in the proof of (16), we conclude that lim_𝑛→∞𝑝𝑥_𝑛+𝑞,𝑥_𝑛=0. (17)
Consequently, we get that the sequence {𝑥_𝑛}_𝑛≥0 is a 𝑝-Cauchy in the 𝑆-complete space 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. Thus, there exists 𝑥∈𝑋 such that lim_𝑛→∞𝑥_𝑛=𝑥. In what follows we prove that 𝑥 is a fixed point of 𝑇. In fact, since lim_𝑛→∞𝑥_𝑛=𝑥, as 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇, the sequence {𝑥_𝑛} has infinite terms in each 𝐴_𝑖 for 𝑖∈{1,2,…,𝑚}.
Since 𝐴_𝑖 is closed for every 𝑖, it follows that 𝑥∈⋂^𝑚_𝑖=1𝐴_𝑖; thus we take a subsequence 𝑥_𝑛𝑘 of {𝑥_𝑛} with 𝑥_𝑛𝑘∈𝐴_𝑖−1. Using the contractive condition, we can obtain 𝑝(𝑥,𝑇𝑥)≤𝑝𝑥,𝑥_𝑛𝑘+1+𝑝𝑥_𝑛𝑘+1,𝑇𝑥=𝑝𝑥,𝑥_𝑛𝑘+1+𝑝𝑇𝑥_𝑛𝑘,𝑇𝑥≤𝑝𝑥,𝑥_𝑛𝑘+1+𝜙𝑝𝑥_𝑛𝑘,𝑥 (18) and since 𝑥_𝑛𝑘→𝑥 and 𝜙 belong to 𝒞, letting 𝑘→∞ in the last inequality, we have 𝑝(𝑥,𝑇𝑥)=0. Analogously, we can derive that 𝑝(𝑥,𝑥)=0 and, therefore, 𝑥 is a fixed point of 𝑇. This finishes the proof.

Corollary 16. Let (𝑋,𝜗) be an 𝑆-complete Hausdorff uniform space such that 𝑝 is an 𝐸-distance on 𝑋, 𝑚 a positive integer, 𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty closed subsets of 𝑋 with respect to the topological space (𝑋,𝜏(𝜗)), and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. Let operator 𝑇∶𝑋→𝑋 satisfy (i)𝑝(𝑇𝑥,𝑇𝑦)≤𝑘𝑝(𝑥,𝑦), for any 𝑥∈𝐴_𝑖, 𝑦∈𝐴_𝑖+1, 𝑖=1,2,…,𝑚, where 𝐴_𝑚+1=𝐴₁ and 0<𝑘<1.Then, 𝑇 has a unique fixed point 𝑧∈⋂^𝑚_𝑖=1𝐴_𝑖.

Proof. By Theorem 15, it is enough to set 𝜙(𝑡)=𝑘𝑡.

Corollary 17 (cf. [16]). Let (𝑋,𝑑) be a complete metric space, 𝑚 a positive integer, 𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty closed subsets of 𝑋, and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. Let 𝑇∶𝑋→𝑋 be a cyclic (𝜙)-contraction. Then, 𝑇 has a unique fixed point 𝑧∈⋂^𝑚_𝑖=1𝐴_𝑖.

Proof. By Theorem 15, it is enough to set 𝜗={𝑈_𝜖∣𝜖>0}.

Corollary 18 (cf. [13]). Let (𝑋,𝑑) be a complete metric space, 𝑚 a positive integer, 𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty closed subsets of 𝑋, and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 a cyclic representation of 𝑋 with respect to 𝑇. Let 𝑇∶𝑋→𝑋 satisfy 𝑝𝑇𝑥,𝑇𝑦≤𝑘𝑝𝑥,𝑦, (19) for any 𝑥∈𝐴_𝑖, 𝑦∈𝐴_𝑖+1, 𝑖=1,2,…,𝑚, where 𝑘∈(0,1) and 𝐴_𝑚+1=𝐴₁. Then, 𝑇 has a unique fixed point 𝑧∈⋂^𝑚_𝑖=1𝐴_𝑖.

Definition 19. Let (𝑋,𝜗) be a uniform space, 𝑚 a positive integer, 𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty subsets of 𝑋, and 𝑇,𝑔∶𝑋→𝑋 self-mappings. An operator 𝑇 is a cyclic (𝜙)-𝑔-contraction if (i)𝑔𝑋=⋃^𝑚_𝑖=1𝑔𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇,(ii)𝑝(𝑇𝑥,𝑇𝑦)≤𝜙(𝑝(𝑔𝑥,𝑔𝑦)), for any 𝑥∈𝐴_𝑖, 𝑦∈𝐴_𝑖+1, 𝑖=1,2,…,𝑚, where 𝐴_𝑚+1=𝐴₁ and 𝜙∈𝒞.

Inspired by [28], we now prove a common fixed point theorem as an application of our Theorem 15.

Theorem 20. Let (𝑋,𝜗) be a uniform space, 𝑇,𝑔∶𝑋→𝑋 self-maps such that 𝑇 is cyclic (𝜙)-𝑔-contraction, and 𝑔𝑋𝑆-complete Hausdorff uniform space together with 𝑝 being an 𝐸-distance on 𝑋. Suppose that 𝑔𝐴₁,𝑔𝐴₂,…,𝑔𝐴_𝑚 are nonempty closed subsets of 𝑔𝑋 with respect to the uniform topology and 𝑇𝑋⊂𝑔𝑋=⋃^𝑚_𝑖=1𝑔𝐴_𝑖. Then, 𝑇 and 𝑔 have a unique coincidence point. Moreover, if 𝑇 and 𝑔 are weakly compatible, then they have a unique common fixed point 𝑧∈⋂^𝑚_𝑖=1𝑔𝐴_𝑖.

Proof. As 𝑔∶𝑋→𝑋, so there exists 𝐸⊂𝑋 such that 𝑔𝐸=𝑔𝑋 and 𝑔∶𝐸→𝑋 is one-to-one. Now, since 𝑇𝑋⊂𝑔𝑋, we define mappings ℎ∶𝑔𝐸→𝑔𝐸 by ℎ(𝑔𝑥)=𝑇𝑥. Since 𝑔 is one-to-one on 𝐸, so ℎ is well defined. As 𝑇 is cyclic (𝜙)-𝑔-contraction, so 𝑝𝑇𝑥,𝑇𝑦≤𝜙𝑝𝑔𝑥,𝑔𝑦, (20) for any 𝑔𝑥∈𝑔𝐴_𝑖, 𝑔𝑦∈𝑔𝐴_𝑖+1, 𝑖=1,2,…,𝑚. Thus, 𝑝ℎ𝑔𝑥,ℎ𝑔𝑦=𝑝𝑇𝑥,𝑇𝑦≤𝜙𝑝𝑔𝑥,𝑔𝑦, (21) for any 𝑔𝑥∈𝑔𝐴_𝑖, 𝑔𝑦∈𝑔𝐴_𝑖+1, 𝑖=1,2,…,𝑚, which implies that ℎ is cyclic (𝜙)-contraction on 𝑔𝑋. Hence, all the conditions of Theorem 15 are satisfied by ℎ, so ℎ has a unique fixed point 𝑧=𝑔𝑥 in 𝑔𝑋. That is, 𝑔𝑥=𝑧=ℎ(𝑧)=ℎ(𝑔𝑥)=𝑇𝑥, so 𝑇 and 𝑔 have a unique coincidence point as required. Moreover, if 𝑇 and 𝑔 are weakly compatible, then they have a unique common fixed point.

Corollary 21 (cf. Theorem 3.2 [18]). Let (𝑋,𝜗) be a uniform space, 𝑇,𝑔∶𝑋→𝑋 self-maps such that 𝑇 is (𝜙)-𝑔-contraction, and 𝑔𝑋𝑆-complete Hausdorff uniform space together with 𝑝 being an 𝐸-distance on 𝑋. Suppose that 𝑇𝑋⊂𝑔𝑋 and 𝑇 and 𝑔 are commuting. Then, 𝑇 and 𝑔 have a unique common fixed point 𝑧∈𝑋.

Proof. Take 𝐴_𝑖=𝑋 for all 𝑖=1,…,𝑚 in Theorem 20.

Example 22. Let (𝑋,𝑑) be a metric space, where 𝑋={1/𝑛}∪{0} and 𝑑=|,|. Set 𝐴₁={1/3𝑛}∪{0,1}, 𝐴₂={1/(3𝑛+1)}∪{0,1}, and 𝐴₃={1/(3𝑛+2)}∪{0,1}. Define 𝜗={𝑈_𝜖∣𝜖>0}. It is easy to see that (𝑋,𝜗) is a uniform space. If we define 𝜙∶[0,∞)→[0,∞) by 𝜙(𝑡)=𝑘𝑡 for 0<𝑘<1 and 𝑇∶𝑋→𝑋 by 𝑇(0)=𝑇(1)=0 and 𝑇(1/𝑛)=1/(4𝑛+1), then for every 𝑥,𝑦≠0,1 we have 𝑑1𝑇𝑥,𝑇𝑦=𝑑,14𝑛+1=|4(4𝑚+1𝑚−𝑛)|||≤||4𝑛+1||4𝑚+1𝑚−𝑛≤14𝑛⋅𝑚41𝑛−1𝑚=14𝑑𝑥,𝑦. (22) Also, for 𝑥,𝑦=0,1 the above inequality obviously holds. This shows that the contractive condition of Corollary 16 is satisfied and 0 is fixed point 𝑇.

Definition 23. Let (𝑋,𝜗) be a uniform space, let 𝑓,𝑔∶𝑋→𝑋 be two mappings, and let 𝐴 and 𝐵 be nonempty closed subsets of 𝑋. The 𝑋=𝐴∪𝐵 is said to be a cyclic representation of 𝑋 with respect to the pair (𝑓,𝑔) if 𝑓(𝐴)⊂𝐵 and 𝑔(𝐵)⊂𝐴.

Definition 24. Let (𝑋,𝜗) be a uniform space, 𝐴,𝐵 nonempty subsets of 𝑋, and 𝑋=𝐴∪𝐵. Two self-maps 𝑓,𝑔∶𝑋→𝑋 are called cyclic (𝜙)-contraction pair if (i)𝑋=𝐴∪𝐵 is a cyclic representation of 𝑋 with respect to the pair (𝑓,𝑔),(ii)max{𝑝(𝑓𝑥,𝑔𝑦),𝑝(𝑔𝑦,𝑓𝑥)}≤𝜙(max{𝑝(𝑥,𝑓𝑥),𝑝(𝑦,𝑔𝑦)}), for any 𝑥∈𝐴, 𝑦∈𝐵, where 𝜙∈𝒞.

Theorem 25. Let (𝑋,𝜗) be an 𝑆-complete Hausdorff uniform space such that 𝑝 is an 𝐸-distance on 𝑋 and let 𝐴,𝐵 be nonempty closed subsets of 𝑋 with respect to the topological space (𝑋,𝜏(𝜗)) and 𝑋=𝐴∪𝐵. Suppose that 𝑓,𝑔∶𝑋→𝑋 are cyclic (𝜙)-contraction pair. Then, 𝑓 and 𝑔 have a unique common fixed point 𝑥∈𝐴∩𝐵.

Proof. Take 𝑥₀∈𝑋 and consider the sequence given by 𝑓𝑥_2𝑛=𝑥_2𝑛+1,𝑔𝑥_2𝑛+1=𝑥_2𝑛+2,𝑛=0,1,2,…. (23) Since 𝑋=𝐴∪𝐵, for any 𝑛>0, 𝑥_2𝑛∈𝐴, and 𝑥_2𝑛+1∈𝐵, and (𝑓,𝑔) are cyclic (𝜙)-contraction pair, we have 𝑝𝑥_2𝑛+1,𝑥_2𝑛+2=𝑝𝑓𝑥_2𝑛,𝑔𝑥_2𝑛+1≤max𝑝𝑓𝑥_2𝑛,𝑔𝑥_2𝑛+1,𝑝𝑔𝑥_2𝑛+1,𝑓𝑥_2𝑛≤𝜙max𝑝𝑥_2𝑛,𝑓𝑥_2𝑛,𝑝𝑥_2𝑛+1,𝑔𝑥_2𝑛+1=𝜙max𝑝𝑥_2𝑛,𝑥_2𝑛+1,𝑝𝑥_2𝑛+1,𝑥_2𝑛+2. (24) Hence, 𝑝𝑥_2𝑛+1,𝑥_2𝑛+2≤𝜙𝑝𝑥_2𝑛,𝑥_2𝑛+1. (25) Similarly, we have 𝑝𝑥_2𝑛,𝑥_2𝑛+1=𝑝𝑔𝑥_2𝑛−1,𝑓𝑥_2𝑛≤max𝑝𝑓𝑥_2𝑛,𝑔𝑥_2𝑛−1,𝑝𝑔𝑥_2𝑛−1,𝑓𝑥_2𝑛≤𝜙max𝑝𝑥_2𝑛,𝑥_2𝑛+1,𝑝𝑥_2𝑛−1,𝑥_2𝑛. (26) Hence, 𝑝𝑥_2𝑛,𝑥_2𝑛+1≤𝜙𝑝𝑥_2𝑛−1,𝑥_2𝑛. (27)
From inequalities (25) and (27) and taking into account the monotonicity of 𝜙, we get by induction that 𝑝𝑥_𝑛,𝑥_𝑛+1≤𝜙^𝑛𝑝𝑥₀,𝑥₁forany𝑛=1,2,…. (28) Since 𝑝 is an 𝐸-distance, we find that 𝑝𝑥_𝑛,𝑥_𝑚≤𝑝𝑥_𝑛,𝑥_𝑛+1+⋯+𝑝𝑥_𝑚−1,𝑥_𝑚, (29) so for 𝑞≥1 we have that 𝑝𝑥_𝑛,𝑥_𝑛+𝑞≤𝜙^𝑛𝑝𝑥₀,𝑥₁+⋯+𝜙^{𝑛+𝑞−1}𝑝𝑥₀,𝑥₁. (30) In the sequel, we will prove that {𝑥_𝑛} is a 𝑝-Cauchy sequence. Denote 𝑆_𝑛=^𝑛_𝑘=0𝜙^𝑘𝑝𝑥₀,𝑥₁,𝑛≥0. (31) By relation (31), we have 𝑝𝑥_𝑛,𝑥_𝑛+𝑞≤𝑆_{𝑛+𝑞−1}−𝑆_𝑛−1. (32) Regarding 𝜙∈𝒞 together with Lemma 10(iv), we get that ^∞_𝑘=0𝜙^𝑘𝑝𝑥₀,𝑥₁<∞, (33) since 𝑝(𝑥₀,𝑥₁)>0. Thus, there is 𝑆∈[0,∞) such that lim_𝑛→∞𝑆_𝑛=𝑆. (34) Then, by (32) we obtain that 𝑝𝑥_𝑛,𝑥_𝑛+𝑞⟶0as𝑛⟶∞. (35) In an analogous way, we derive that 𝑝𝑥_𝑛+𝑞,𝑥_𝑛⟶0as𝑛⟶∞. (36) Hence, we get that {𝑥_𝑛}_𝑛≥0 is a 𝑝-Cauchy sequence in the 𝑆-complete space 𝑋=𝐴∪𝐵. So there exists 𝑥∈𝑋 such that lim_𝑛→∞𝑓𝑥_2𝑛=lim_𝑛→∞𝑔𝑥_2𝑛+1=𝑥. In what follows, we prove that 𝑥 is a fixed point of 𝑓,𝑔. In fact, since lim_𝑛→∞𝑓𝑥_2𝑛=lim_𝑛→∞𝑔𝑥_2𝑛+1=𝑥 and as 𝑋=𝐴∪𝐵 is a cyclic representation of 𝑋 with respect to 𝑓,𝑔, the sequence {𝑥_𝑛} has infinite terms in each 𝐴,𝐵.
Since 𝐴,𝐵 are closed, it follows that 𝑥∈𝐴∩𝐵; thus we take subsequences 𝑥_2𝑛,𝑥_2𝑛+1 of {𝑥_𝑛} with 𝑥_2𝑛∈𝐴 and 𝑥_2𝑛+1∈𝐵. Using the contractive condition, we can obtain 𝑝𝑥,𝑓𝑥≤𝑝𝑥,𝑥_2𝑛+2+𝑝𝑥_2𝑛+2,𝑓𝑥=𝑝𝑥,𝑥_2𝑛+2+𝑝𝑔𝑥_2𝑛+1,𝑓𝑥≤𝑝𝑥,𝑥_2𝑛+2+max𝑝𝑔𝑥_2𝑛+1,𝑝,𝑓𝑥𝑓𝑥,𝑔𝑥_2𝑛+1≤𝑝𝑥,𝑥_2𝑛+2𝑝,𝑝𝑥+𝜙max𝑥,𝑓𝑥_2𝑛+1,𝑥_2𝑛+2, (37) and since 𝑥_𝑛→𝑥 and 𝜙 belong to 𝒞, letting 𝑛→∞ in the last inequality, we have 𝑝(𝑥,𝑓𝑥)≤𝜙(𝑝(𝑥,𝑓𝑥))<𝑝(𝑥,𝑓𝑥); hence𝑝(𝑥,𝑓𝑥)=0. Similarly, we can show that 𝑝(𝑥,𝑥)=0 and, therefore, 𝑥 is a fixed point of 𝑓. Similarly, we can show that 𝑥 is a fixed point of 𝑔. Finally, in order to prove the uniqueness of the fixed point, we have 𝑦,𝑧∈𝑋 with 𝑦 and 𝑧 fixed points of 𝑓,𝑔. The cyclic character of 𝑓,𝑔 and the fact that 𝑦,𝑧∈𝑋 are fixed points of 𝑓,𝑔 imply that 𝑦,𝑧∈𝐴∩𝐵. Using the contractive condition, we obtain 𝑝𝑝𝑝𝑦,𝑧=𝑝𝑓𝑦,𝑔𝑧≤max𝑓𝑦,𝑔𝑧,𝑝𝑔𝑧,𝑓𝑦≤𝜙max𝑦,𝑓𝑦,𝑝𝑧,𝑔𝑧=0, (38) and from the last inequality we get 𝑝𝑦,𝑧=0. (39) Using the same arguments above, we can show that 𝑝(𝑦,𝑦)=0 and, consequently, 𝑦=𝑧. This finishes the proof.

Corollary 26. Let (𝑋,𝑑) be a complete metric space and 𝐴,𝐵 nonempty closed subsets of 𝑋 and 𝑋=𝐴∪𝐵. Let 𝑓,𝑔∶𝑋→𝑋 be cyclic (𝜙)-contraction pair. Then, 𝑓 and 𝑔 have a unique common fixed point 𝑧∈𝐴∩𝐵.

Proof. By Theorem 25, it is enough to set 𝜗={𝑈_𝜖∣𝜖>0}.

Corollary 27. Let (𝑋,𝜗) be an 𝑆-complete Hausdorff uniform space such that 𝑝 is an 𝐸-distance on 𝑋 and let 𝐴,𝐵 be nonempty closed subsets of 𝑋 with respect to the topological space (𝑋,𝜏(𝜗)) and 𝑋=𝐴∪𝐵. Suppose that the maps 𝑓,𝑔∶𝑋→𝑋 satisfy the following inequality: (i)max{𝑝(𝑓𝑥,𝑔𝑦),𝑝(𝑔𝑦,𝑓𝑥)}≤𝑘max{𝑝(𝑥,𝑓𝑥),𝑝(𝑦,𝑔𝑦)}, for any 𝑥∈𝐴, 𝑦∈𝐵, where 0<𝑘<1.Then, 𝑓,𝑔 have a unique common fixed point 𝑧∈𝐴∩𝐵.

Proof. By Theorem 25, it is enough to set 𝜙(𝑡)=𝑘𝑡.

Example 28. Let (𝑋,𝑝) be a partial metric space, where 𝑋={1/𝑛}∪{0,1} and 𝑝(𝑥,𝑦)=max{𝑥,𝑦}. Set 𝐴={1/2𝑛}∪{0,1} and 𝐵={1/(2𝑛+1)}∪{0,1}. Define 𝜗={𝑈_𝜖∣𝜖>0}, where 𝑈_𝜖={(𝑥,𝑦)∈𝑋²∶𝑝(𝑥,𝑦)<𝑝(𝑥,𝑥)+𝜖}. It is easy to see that (𝑋,𝜗) is a uniform space. If we define 𝜙∶[0,∞)→[0,∞) by 𝜙(𝑡)=𝑘𝑡 for 0<𝑘<1 and 𝑓∶𝑋→𝑋 by 𝑓(1/2𝑛)=1/(4𝑛+1), 𝑓(0)=𝑓(1)=0, and 𝑔(1/(2𝑛+1))=1/(4𝑛+2), 𝑔(0)=𝑔(1)=0. Then, for every 𝑥,𝑦≠0,1 we have 𝑝𝑝1max𝑓𝑥,𝑔𝑦,𝑝𝑔𝑦,𝑓𝑥=max,14𝑛+1,𝑝14𝑚+2,14𝑚+214𝑛+1=max,14𝑛+1≤14𝑚+221max,12𝑛=12𝑚+12max𝑝𝑥,𝑓𝑥,𝑝𝑦,𝑔𝑦, (40) for any 𝑥∈𝐴, 𝑦∈𝐵. Also, for 𝑥,𝑦=0,1 the above inequality obviously holds. This shows that the contractive condition of Corollary 27 is satisfied and 0 is a common fixed point of 𝑓 and 𝑔.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU, for the financial support.