Some Fixed and Periodic Points in Abstract Metric Spaces

Abstract

We generalize in this paper some results on common fixed points of two, respectively four, contractive-type mappings in abstract metric spaces by removing condition of normality of the cone in their formulations. Further, some results about periodic points of self-maps are extended to the setting of abstract metric spaces.

1. Introduction and Preliminaries

Ordered normed spaces, cones, and topical functions have applications in applied mathematics, for instance, in using Newton's approximation method [1–4] and in optimization theory [5, 6]. 𝐾-metric and 𝐾-normed spaces were introduced in the mid-20th century ([2], see also [3, 4]) by replacing an ordered Banach space instead of the set of real numbers, as the codomain for a metric. Huang and Zhang [7] reintroduced such spaces under the name of cone metric spaces, but went further, defining convergent and Cauchy sequences in the terms of interior points of the underlying cone. In such a way, nonnormal cones can be used as well (although they used only normal cones), paying attention to the fact that Sandwich Theorem and continuity of the metric may not hold. These and other authors (e.g., [8–14]) proved some fixed point theorems for contractive-type mappings in cone metric spaces, as well as topological vector-space-valued cone metric spaces (e.g., [15, 16]).

The following definitions and results will be needed in the sequel (see, e.g., [2, 3, 5, 17, 18]).

Let 𝐸 be a real topological vector space. A subset 𝐾 of 𝐸 is called a cone if (a) 𝐾 is closed, nonempty and 𝐾≠{𝜃}; (b) 𝑎,𝑏∈ℝ, 𝑎,𝑏≥0, 𝑥,𝑦∈𝐾 imply that 𝑎𝑥+𝑏𝑦∈𝐾; (c) 𝐾∩(−𝐾)={𝜃}.

Given a cone 𝐾, we define the partial ordering ⪯ with respect to 𝐾 by 𝑥⪯𝑦 if and only if 𝑦−𝑥∈𝐾. We will write 𝑥≪𝑦 for 𝑦−𝑥∈int𝐾, where int𝐾 stands for the interior of 𝐾 and use 𝑥≺𝑦 for (𝑥⪯𝑦 and 𝑥≠𝑦). If int𝐾≠∅, then 𝐾 is called a solid cone [3]. Note that the notation 0≪𝑐 for an interior point of a positive cone was first used by Kreĭn and Rutman [19].

The cone 𝐾 in the topological vector space 𝐸 is called normal if 𝐸 has a base of neighborhoods of 𝜃 consisting of order-convex subsets (see [16]). In the case of a normed space, this is equivalent to the condition that there is a number 𝑘>0 such that, for all 𝑥,𝑦∈𝐸, 𝜃⪯𝑥⪯𝑦 implies ‖𝑥‖≤𝑘‖𝑦‖. Equivalently, the cone 𝐾 is normal if (∀𝑛)𝑥_𝑛⪯𝑦_𝑛⪯𝑧_𝑛,lim_𝑛→∞𝑥_𝑛=lim_𝑛→∞𝑧_𝑛=𝑥implylim_𝑛→∞𝑦_𝑛=𝑥. (1.1) For details see [5].

Example 1.1 (see [3]). Let 𝐸=𝐶¹_ℝ[0,1] with ‖𝑥‖=‖𝑥‖_∞+‖𝑥^′‖_∞ and 𝐾={𝑥∈𝐸∶𝑥(𝑡)≥0}. This cone is nonnormal. Consider, for example, 𝑥_𝑛(𝑡)=𝑡^𝑛/𝑛 and 𝑦_𝑛(𝑡)=1/𝑛. Then 𝜃⪯𝑥_𝑛⪯𝑦_𝑛, and lim_𝑛→∞𝑦_𝑛=𝜃, but ‖𝑥_𝑛‖=max_𝑡∈[0,1]|𝑡^𝑛/𝑛|+max_𝑡∈[0,1]|𝑡^𝑛−1|=1/𝑛+1>1; hence 𝑥_𝑛 does not converge to zero. It follows by (1.1) that 𝐾 is a nonnormal cone.

Definition 1.2 (see [4, 15, 16]). Let 𝑋 be a nonempty set and 𝐸 a topological vector space with a cone 𝐾. Suppose that a mapping 𝑑∶𝑋×𝑋→𝐸 satisfies the following:(d₁)𝜃⪯𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 and 𝑑(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦;(d₂)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋;(d₃)𝑑(𝑥,𝑧)⪯𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋. The function 𝑑 is called an abstract metric and (𝑋,𝑑) is called an abstract metric space (or a topological vector-space-valued cone metric space or a 𝐾-metric space); we will use the first mentioned term.

The concept of an abstract metric space is obviously more general than that of a metric space. If 𝐸 is a Banach space then abstract metric space becomes a cone metric space of [7]. For new results in cone metric spaces see [20–26].

Definition 1.3. Let (𝑋,𝑑) be an abstract metric space. We say that a sequence {𝑥_𝑛} in 𝑋 is(i)a Cauchy sequence if, for every 𝑐 in 𝐸 with 𝜃≪𝑐, there is an 𝑛₀∈ℕ such that for all 𝑚,𝑛>𝑛₀, 𝑑(𝑥_𝑚,𝑥_𝑛)≪𝑐;(ii)a convergent sequence if, for every 𝑐 in 𝐸 with 𝜃≪𝑐, there is an 𝑛₀∈ℕ such that for all 𝑛>𝑛₀, 𝑑(𝑥_𝑛,𝑥)≪𝑐 for some fixed 𝑥∈𝑋.
An abstract metric space 𝑋 is said to be complete if every Cauchy sequence in 𝑋 is convergent in 𝑋.

Let (𝑋,𝑑) be an abstract metric space. The following properties are often used, particularly in the case when the underlying cone is nonnormal. The only assumption is that the cone 𝐾 is solid. For details about these properties see, for example, [11].(p₁) If 𝑎⪯ℎ𝑎 where 𝑎∈𝐾 and ℎ∈[0,1), then 𝑎=𝜃.(p₂)If 𝜃⪯𝑢≪𝑐 for each 𝑐, 𝜃≪𝑐, then 𝑢=𝜃.(p₃)If 𝑢⪯𝑣 and 𝑣≪𝑤, then 𝑢≪𝑤.(p₄)If 𝑐∈int𝐾, 𝜃⪯𝑥_𝑛, and 𝑥_𝑛→𝜃, then there exists 𝑘∈ℕ such that, for all 𝑛>𝑘 we have 𝑥_𝑛≪𝑐. (Note that, in general, the converse is not true. Indeed, in Example 1.1, 𝑥_𝑛↛𝜃, but 𝑥_𝑛≪𝑐 for 𝑛 sufficiently large.)

In generalizing some theorems of Huang-Zhang [7], Abbas and Rhoades proved the following result in abstract metric spaces over normal cones.

Theorem 1.4 (see [9]). Let (𝑋,𝑑) be a complete cone metric space over a normal cone. Suppose that 𝑓,𝑔∶𝑋→𝑋 are two self-maps satisfying 𝑑𝑓𝑥,𝑔𝑦⪯𝛼𝑑𝑥,𝑦+𝛽𝑑𝑥,𝑓𝑥+𝑑𝑦,𝑔𝑦+𝛾𝑑𝑥,𝑔𝑦+𝑑𝑦,𝑓𝑥, (1.2) for all 𝑥,𝑦∈𝑋, where 𝛼,𝛽,𝛾≥0, and 𝛼+2𝛽+2𝛾<1. Then 𝑓 and 𝑔 have a unique common fixed point in 𝑋. Moreover, any fixed point of 𝑓 is a fixed point of 𝑔 and conversely.

Sing et al. extended this result of Abbas-Rhoades to four maps. They proved the following theorem.

Theorem 1.5 (see [27]). Let (𝑋,𝑑) be a complete cone metric space over a normal cone. Suppose that the mappings 𝑓, 𝑔, 𝑆, and 𝑇 are four selfmaps on 𝑋 such that 𝑓𝑋⊂𝑇𝑋 and 𝑔𝑋⊂𝑆𝑋 and satisfying 𝑑𝑓𝑥,𝑔𝑦⪯𝛼𝑑𝑆𝑥,𝑇𝑦+𝛽𝑑𝑆𝑥,𝑓𝑥+𝑑𝑇𝑦,𝑔𝑦+𝛾𝑑𝑆𝑥,𝑔𝑦+𝑑𝑇𝑦,𝑓𝑥, (1.3) for all 𝑥,𝑦∈𝑋, where 𝛼,𝛽,𝛾≥0 and 𝛼+2𝛽+2𝛾<1. Suppose that the pairs {𝑓,𝑆} and {𝑔,𝑇} are weakly compatible. Then 𝑓, 𝑔, 𝑆, and 𝑇 have a unique common fixed point.

In 1977, Rhoades proved the following interesting result.

Theorem 1.6 (see [28]). Let (𝑋,𝑑) be a complete metric space. Let 𝑓∶𝑋→𝑋, and suppose that there exist decreasing functions 𝛼_𝑖∶(0,+∞)→[0,1), 𝑖=1,…,5, such that ∑⁵_𝑖=1𝛼_𝑖(𝑡)<1 for each 𝑡∈(0,+∞) and satisfying 𝑑𝑓𝑥,𝑓𝑦≤𝛼₁𝑑𝑥,𝑦𝑑𝑥,𝑦+𝛼₂𝑑𝑥,𝑦𝑑𝑥,𝑓𝑥+𝛼₃𝑑𝑥,𝑦𝑑𝑦,𝑓𝑦+𝛼₄𝑑𝑥,𝑦𝑑𝑓𝑦,𝑥+𝛼₅𝑑𝑥,𝑦𝑑𝑓𝑥,𝑦, (1.4) for all 𝑥,𝑦∈𝑋, 𝑥≠𝑦. Then 𝑓 has a unique fixed point 𝑧 and for each 𝑥₀∈𝑋 the sequence {𝑓^𝑛𝑥₀} converges to 𝑧.

We generalize in this paper Theorems 1.4 and 1.5 by removing normality condition in their formulations. An example will show that these generalizations are proper. Further, some results of Abbas and Rhoades about periodic points of selfmaps from [29] are extended to abstract metric spaces. Theorem 1.6 is also presented in this new setting, with a slightly shorter proof.

Note that it was shown in [15, 30, 31] that some of the fixed point results in abstract metric spaces can be directly reduced to the respective metric results. However, the results of the present paper do not fall into this category, since some of them are new even in the context of metric spaces.

2. Fixed Point Theorems

In this section we will prove Theorems 1.4 and 1.5 by omitting the assumption of normality in the results. We use only the definition of convergence in terms of the relation “≪”. The only assumption is that 𝐾 is a solid cone, so we use neither continuity of the vector metric 𝑑, nor Sandwich Theorem. We begin with the following.

Theorem 2.1. Let (𝑋,𝑑) be an abstract metric space over a solid cone 𝐾. Suppose that 𝑓, 𝑔, 𝑆, and 𝑇 are four self-maps on 𝑋 such that 𝑓𝑋⊂𝑇𝑋 and 𝑔𝑋⊂𝑆𝑋 and suppose that at least one of these four subsets of 𝑋 is complete. Let 𝑑𝑓𝑥,𝑔𝑦⪯𝛼𝑑𝑆𝑥,𝑇𝑦+𝛽𝑑𝑆𝑥,𝑓𝑥+𝑑𝑇𝑦,𝑔𝑦+𝛾𝑑𝑆𝑥,𝑔𝑦+𝑑𝑇𝑦,𝑓𝑥, (2.1) for all 𝑥,𝑦∈𝑋, where 𝛼,𝛽,𝛾≥0 and 𝛼+2𝛽+2𝛾<1. Then the pairs (𝑓,𝑆) and (𝑔,𝑇) have a unique common point of coincidence. If, moreover, pairs (𝑓,𝑆) and (𝑔,𝑇) are weakly compatible, then 𝑓, 𝑔, 𝑆, and 𝑇 have a unique common fixed point.

For definitions of terms like “point of coincidence” and “weakly compatible pair” see, for example, [11].

Remark 2.2. In the papers [9] and [27], the cone 𝐾 is supposed to be normal and solid. In that case the proof is essentially the same as in the setting of usual metric spaces.

We now give the proof of Theorem 2.1.

Proof. Suppose 𝑥₀∈𝑋 is an arbitrary point, and define the sequence {𝑦_𝑛} by 𝑦_2𝑛=𝑓𝑥_2𝑛=𝑇𝑥_2𝑛+1, 𝑦_2𝑛+1=𝑔𝑥_2𝑛+1=𝑆𝑥_2𝑛+2, 𝑛=0,1,2,…. Now, as in [27], by (2.1), we have 𝑑𝑦_2𝑛,𝑦_2𝑛+1=𝑑𝑓𝑥_2𝑛,𝑔𝑥_2𝑛+1⪯𝛼𝑑𝑆𝑥_2𝑛,𝑇𝑥_2𝑛+1+𝛽𝑑𝑆𝑥_2𝑛,𝑓𝑥_2𝑛+𝑑𝑇𝑥_2𝑛+1,𝑔𝑥_2𝑛+1+𝛾𝑑𝑆𝑥_2𝑛,𝑔𝑥_2𝑛+1+𝑑𝑇𝑥_2𝑛+1,𝑓𝑥_2𝑛⪯𝛼+𝛽+𝛾𝑑𝑦_2𝑛−1,𝑦_2𝑛+𝛽+𝛾𝑑𝑦_2𝑛,𝑦_2𝑛+1, (2.2) which implies that 𝑑(𝑦_2𝑛,𝑦_2𝑛+1)⪯𝛿𝑑(𝑦_2𝑛−1,𝑦_2𝑛), where 𝛿=(𝛼+𝛽+𝛾)/(1−(𝛽+𝛾))<1. Similarly it can be shown that 𝑑𝑦_2𝑛+1,𝑦_2𝑛+2⪯𝛿𝑑𝑦_2𝑛,𝑦_2𝑛+1. (2.3) Therefore, for all 𝑛, 𝑑𝑦_𝑛,𝑦_𝑛+1⪯𝛿𝑑𝑦_𝑛−1,𝑦_𝑛⪯⋯⪯𝛿^𝑛𝑑𝑦₀,𝑦₁. (2.4) Now, for any 𝑚>𝑛, 𝑑𝑦_𝑛,𝑦_𝑚⪯𝑑𝑦_𝑛,𝑦_𝑛+1+𝑑𝑦_𝑛+1,𝑦_𝑛+2+⋯+𝑑𝑦_𝑚−1,𝑦_𝑚⪯𝛿^𝑛+𝛿^𝑛+1+⋯+𝛿^𝑚−1𝑑𝑦₀,𝑦₁⪯𝛿^𝑛𝑑𝑦1−𝛿⁰,𝑦₁→𝜃as𝑛→∞. (2.5) Thus, by properties (p₄) and (p₃) and Definition 1.3, {𝑦_𝑛} is a Cauchy sequence.
Suppose, for example, that 𝑆𝑋 is a complete subset of 𝑋. Then 𝑦_𝑛→𝑢=𝑆𝑣, 𝑛→∞, for some 𝑣∈𝑋. Of course, subsequences {𝑦_2𝑛−1} and {𝑦_2𝑛} also converge to 𝑢. Let us prove that 𝑓𝑣=𝑢. Using (2.1) we get that 𝑑𝑓𝑣,𝑢=𝑑𝑓𝑣,𝑔𝑥_2𝑛−1+𝑑𝑔𝑥_2𝑛−1,𝑢⪯𝛼𝑑𝑆𝑣,𝑇𝑥_2𝑛−1+𝛽𝑑𝑆𝑣,𝑓𝑣+𝑑𝑇𝑥_2𝑛−1,𝑔𝑥_2𝑛−1+𝛾𝑑𝑆𝑣,𝑔𝑥_2𝑛−1+𝑑𝑇𝑥_2𝑛−1,𝑓𝑣+𝑑𝑔𝑥_2𝑛−1,𝑢, (2.6) which further implies that 𝑑𝑓𝑣,𝑢⪯𝛼+𝛽+𝛾𝑑1−𝛽+𝛾𝑢,𝑇𝑥_2𝑛−1+1+𝛽+𝛾𝑑1−𝛽+𝛾𝑢,𝑔𝑥_2𝑛−1. (2.7) Let 𝜃≪𝑐 be given. Since 𝑦_𝑛→𝑢 as 𝑛→∞, choose a natural number 𝑛₀ such that for all 𝑛>𝑛₀ (Definition 1.3) we have that 𝑑𝑢,𝑇𝑥_2𝑛−1≪1−𝛽+𝛾2𝛼+𝛽+𝛾𝑐,𝑑𝑢,𝑔𝑥_2𝑛−1≪1−𝛽+𝛾21+𝛽+𝛾𝑐. (2.8) Thus, according to (2.7) we obtain 𝑑(𝑓𝑣,𝑢)≪(𝑐/2)+(𝑐/2)=𝑐. Therefore, 𝑑(𝑓𝑣,𝑢)≪𝑐 for all 𝑐∈int𝐾. Using property (p₂), it follows that 𝑑(𝑓𝑣,𝑢)=𝜃 and so 𝑓𝑣=𝑢=𝑆𝑣. Since 𝑢∈𝑓𝑋⊂𝑇𝑋, we get that there exists 𝑤∈𝑋 such that 𝑇𝑤=𝑢. Let us prove that also 𝑔𝑤=𝑢. By triangle inequality and (2.1), we have 𝑑𝑔𝑤,𝑢⪯𝑑𝑔𝑤,𝑓𝑥_2𝑛+𝑑𝑓𝑥_2𝑛,𝑢⪯𝛼𝑑𝑆𝑥_2𝑛,𝑇𝑤+𝛽𝑑𝑆𝑥_2𝑛,𝑓𝑥_2𝑛𝑑+𝑑𝑇𝑤,𝑔𝑤+𝛾𝑆𝑥_2𝑛,𝑔𝑤+𝑑𝑇𝑤,𝑓𝑥_2𝑛+𝑑𝑔𝑥_2𝑛−1,𝑢, (2.9) which further implies that 𝑑𝑔𝑤,𝑢⪯𝛼+𝛽+𝛾𝑑1−𝛽+𝛾𝑢,𝑆𝑥_2𝑛+1+𝛽+𝛾𝑑1−𝛽+𝛾𝑢,𝑓𝑥_2𝑛. (2.10) Now, for given 𝜃≪𝑐, since 𝑦_𝑛→𝑢 as 𝑛→∞, choose a natural 𝑛₁ such that for all 𝑛>𝑛₁ we have that 𝑑𝑢,𝑆𝑥_2𝑛≪1−𝛽+𝛾2𝛼+𝛽+𝛾𝑐,𝑑𝑢,𝑓𝑥_2𝑛≪1−𝛽+𝛾21+𝛽+𝛾𝑐. (2.11) According to (2.10) we obtain 𝑑(𝑔𝑤,𝑢)≪(𝑐/2)+(𝑐/2)=𝑐. Therefore, 𝑑(𝑔𝑤,𝑢)≪𝑐 for all 𝑐∈int𝐾. Using property (p₂), it follows that 𝑑(𝑔𝑤,𝑢)=𝜃 and so 𝑔𝑤=𝑢=𝑇𝑤. We have proved that 𝑢 is a common point of coincidence for pairs (𝑓,𝑆) and (𝑔,𝑇).
If now these pairs are weakly compatible, then 𝑓𝑢=𝑓𝑆𝑣=𝑆𝑓𝑣=𝑆𝑢=𝑧₁ (say) and 𝑔𝑢=𝑔𝑇𝑤=𝑇𝑔𝑤=𝑇𝑢=𝑧₂ (say). Moreover, 𝑑𝑧₁,𝑧₂()𝑑𝑑𝑧=𝑑𝑓𝑢,𝑔𝑢⪯𝛼𝑑𝑆𝑢,𝑇𝑢+𝛽𝑆𝑢,𝑓𝑢+𝑑𝑇𝑢,𝑔𝑢+𝛾𝑆𝑢,𝑔𝑢+𝑑𝑇𝑢,𝑓𝑢=𝛼𝑑₁,𝑧₂+𝛽𝑑𝑧₁,𝑧₁+𝑑𝑧₂,𝑧₂+𝛾𝑑𝑧₁,𝑧₂+𝑑𝑧₂,𝑧₁=𝛼+2𝛾𝑑𝑧₁,𝑧₂⪯𝛼+𝛽+𝛾𝑑𝑧₁,𝑧₂≺𝑑𝑧₁,𝑧₂ (2.12) implies that 𝑧₁=𝑧₂. So we have that 𝑓𝑢=𝑔𝑢=𝑆𝑢=𝑇𝑢. It remains to prove that, for example, 𝑢=𝑔𝑢. Indeed, 𝑑()𝑑𝑑[]𝑑=𝑑⪯𝑑,𝑢,𝑔𝑢=𝑑𝑓𝑣,𝑔𝑢⪯𝛼𝑑𝑆𝑣,𝑇𝑢+𝛽𝑆𝑣,𝑓𝑣+𝑑𝑇𝑢,𝑔𝑢+𝛾𝑆𝑣,𝑔𝑢+𝑑𝑇𝑢,𝑓𝑣=𝛼𝑑𝑢,𝑔𝑢+𝛽𝜃+𝜃+𝛾𝑢,𝑔𝑢+𝑑𝑔𝑢,𝑢𝛼+2𝛾𝑢,𝑔𝑢𝛼+𝛽+𝛾𝑢,𝑔𝑢≺𝑑𝑢,𝑔𝑢 (2.13) implying that 𝑔𝑢=𝑢. The uniqueness follows from (2.1). The proofs for cases in which 𝑓𝑋, 𝑔𝑋, or 𝑇𝑋 is complete are similar and are therefore omitted. The theorem is proved.

We present now two examples showing that Theorem 2.1 is a proper extension of the known results. In both examples, the conditions of Theorem 2.1 are fulfilled, but in the first one (because of nonnormality of the cone) the main theorems from [9, 27] cannot be applied.

Example 2.3 (the case of a nonnormal cone). Let 𝑋=[0,1], and let 𝐸 be the set of all real-valued functions on 𝑋 which also have continuous derivatives on 𝑋. Note that 𝐸 is a vector space over ℝ under usual function operations. Let 𝜏 be the strongest vector (locally convex) topology on 𝐸. Then (𝐸,𝜏) is a topological vector space which is not normable and is not even metrizable. Let 𝐾={𝜑∈𝐸∶𝜑(𝑡)≥0,𝑡∈ℝ} and 𝑑∶𝑋×𝑋→𝐸 be defined by 𝑑(𝑥,𝑦)(𝑡)=|𝑥−𝑦|⋅𝑒^𝑡. Then (𝑋,𝑑) is an abstract metric space over a nonnormal solid cone (Example 1.1). Consider the four mappings 𝑓,𝑔,𝑇,𝑆∶𝑋→𝑋 defined by 𝑥𝑓𝑥=8𝑥,𝑔𝑥=𝑥12,𝑇𝑥=2𝑥,𝑆𝑥=3. (2.14) Clearly 𝑓(𝑋)⊂𝑇(𝑋) and 𝑔(𝑋)⊂𝑆(𝑋).
For 𝑥,𝑦∈𝑋, 𝑑(𝑡)=𝑥𝑓𝑥,𝑔𝑦8−𝑦𝑒12^𝑡=18𝑥−2𝑦3𝑒^𝑡,𝑑(𝑡)=𝑥𝑆𝑥,𝑇𝑦3−𝑦2𝑒^𝑡,𝑑(𝑡)(𝑡)=𝑥𝑓𝑥,𝑆𝑥+𝑑𝑔𝑦,𝑇𝑦8−𝑥3𝑒^𝑡+𝑦−𝑦122𝑒^𝑡=5𝑥+245𝑦𝑒12^𝑡,𝑑(𝑡)=𝑥𝑓𝑥,𝑇𝑦+𝑑𝑔𝑦,𝑆𝑥8−𝑦2+𝑦−𝑥123𝑒^𝑡. (2.15) Now 𝑑(𝑡)=1𝑓𝑥,𝑔𝑦8𝑥−2𝑦3𝑒^𝑡≤16𝑥3−𝑦2𝑒^𝑡+165𝑥+245𝑦𝑒12^𝑡+16𝑥8−𝑦2+𝑦−𝑥123𝑒^𝑡(𝑡)𝑑(𝑡)(𝑡𝑑(𝑡)(𝑡.=𝛼𝑑𝑆𝑥,𝑇𝑦+𝛽𝑓𝑥,𝑆𝑥+𝑟𝑑𝑔𝑦,𝑇𝑦)+𝛾𝑓𝑥,𝑇𝑦+𝑑𝑔𝑦,𝑆𝑥) (2.16) Thus all the conditions of Theorem 2.1 are satisfied with 𝛼+2𝛽+2𝛾=5/6<1. Note that 0 is the unique common fixed point of 𝑓, 𝑔, 𝑆, and 𝑇.

Example 2.4 (the case of a normal cone). Let 𝐸=𝐶_ℝ[0,1] with 𝐾={𝜑∈𝐸∶𝜑(𝑡)≥0,𝑡∈[0,1]} (this cone is normal; see [5]). Let 𝑋=[0,∞), and let 𝑑∶𝑋×𝑋→𝐸 be defined as 𝑑(𝑥,𝑦)(𝑡)∶=|𝑥−𝑦|𝑒^𝑡. Take the functions 𝑓𝑥=𝑥/3, 𝑔𝑥=0, 𝑆𝑋=𝑇𝑋=𝑥 which map the set 𝑋 into 𝑋. All the conditions of Theorem 2.1 are fulfilled with 𝛾=1/3, 𝛼+2𝛽<1/3. Obviously, 𝑓, 𝑔, 𝑆, and 𝑇 have the unique common fixed point 𝑥=0.

Remark 2.5. Taking 𝑆=𝑇=𝑖_𝑋 and appropriate choices of 𝑓, 𝑔, 𝛼, 𝛽, and 𝛾 in Theorem 2.1, one easily gets [9, Corollaries 2.2–2.8]. In each of the following cases (1)–(7), (𝑋,𝑑) is a complete abstract metric space,   𝐾 a solid cone, and 𝑓 is a selfmap on 𝑋.(1) Let 𝑑𝑓^𝑝𝑥,𝑓^𝑞𝑦⪯𝛼𝑑𝑥,𝑦+𝛽𝑑𝑥,𝑓^𝑝𝑥+𝑑𝑦,𝑓^𝑞𝑦+𝛾𝑑𝑥,𝑓^𝑞𝑦+𝑑𝑦,𝑓^𝑝𝑥 (2.17) for all 𝑥,𝑦∈𝑋, where 𝛼,𝛽,𝛾≥0 and 𝛼+2𝛽+2𝛾<1, and 𝑝 and 𝑞 are fixed positive integers. Then 𝑓 has a unique fixed point in 𝑋.(2) If 𝑑𝑓𝑥,𝑓𝑦⪯𝛼𝑑𝑥,𝑦+𝛽𝑑𝑥,𝑓𝑥+𝑑𝑦,𝑓𝑦+𝛾𝑑𝑥,𝑓𝑦+𝑑𝑦,𝑓𝑥 (2.18) for all 𝑥,𝑦∈𝑋, where 𝛼,𝛽,𝛾≥0 and 𝛼+2𝛽+2𝛾<1, then 𝑓 has a unique fixed point in 𝑋.(3) If 𝑑𝑓𝑥,𝑓𝑦⪯𝑎₁𝑑𝑥,𝑦+𝑎₂𝑑𝑥,𝑓𝑥+𝑎₃𝑑𝑦,𝑓𝑦+𝑎₄𝑑𝑥,𝑓𝑦+𝑎₅𝑑𝑦,𝑓𝑥 (2.19) for all 𝑥,𝑦∈𝑋, where 𝑎_𝑖≥0 for each 𝑖∈{1,2,…,5} and 𝑎₁+𝑎₂+⋯+𝑎₅<1, then 𝑓 has a unique fixed point in 𝑋.(4) If 𝑑𝑓𝑥,𝑓𝑦⪯𝛼𝑑𝑥,𝑦 (2.20) for all 𝑥,𝑦∈𝑋, where 𝛼∈[0,1), then 𝑓 has a unique fixed point in 𝑋.(5) If 𝑑𝑓𝑥,𝑓𝑦⪯𝛽𝑑𝑥,𝑓𝑥+𝑑𝑦,𝑓𝑦 (2.21) for all 𝑥,𝑦∈𝑋, where 𝛽∈[0,1/2), then 𝑓 has a unique fixed point in 𝑋.(6) If 𝑑𝑓𝑥,𝑓𝑦⪯𝛾𝑑𝑥,𝑓𝑦+𝑑𝑦,𝑓𝑥 (2.22) for all 𝑥,𝑦∈𝑋, where 𝛾∈[0,1/2), then 𝑓 has a unique fixed point in 𝑋.(7) If 𝑑𝑓𝑥,𝑓𝑦⪯𝛼𝑑𝑥,𝑦+𝛽𝑑𝑥,𝑓𝑥+𝑑𝑦,𝑓𝑦 (2.23) for all 𝑥,𝑦∈𝑋, where 𝛼,𝛽≥0 and 𝛼+2𝛽<1, then 𝑓 has a unique fixed point in 𝑋.

We add an example of a Banach-type contraction on a nonnormal abstract metric space.

Example 2.6. Let 𝑋=[0,1], 𝐸=𝐶¹_ℛ[0,1], 𝐾={𝜑∈𝐸∶𝜑(𝑡)≥0}. An abstract metric 𝑑 on 𝑋 is defined by 𝑑(𝑥,𝑦)(𝑡)∶=|𝑥−𝑦|⋅𝜙(𝑡) where 𝜙∈𝐾 is an arbitrary function (e.g., 𝜙(𝑡)=2^𝑡). It is easy to see that (𝑋,𝑑) is a complete abstract metric space. Suppose that mapping 𝑓∶𝑋→𝑋 satisfies 𝑑𝑓𝑥,𝑓𝑦⪯𝜆𝑑𝑥,𝑦, (2.24) for all 𝑥,𝑦∈𝑋, where 𝜆∈[0,1). All the conditions from Remark 2.5(4) hold, and 𝑓 has a unique fixed point in 𝑋.
This example verifies that Theorem 2.1 is a proper extension of the results from [7]. Indeed, we know (see Example 1.1) that the cone 𝐾 is nonnormal. So, in this example Theorem 1 from [7] cannot be applied.

Corollary 2.7. Let (𝑋,𝑑) be an abstract metric space and 𝐾 a solid cone. Suppose that the mappings 𝑓, 𝑔, 𝑆 and 𝑇 are four selfmaps of 𝑋 such that 𝑓(𝑋)⊂𝑇(𝑋) and 𝑔(𝑋)⊂𝑆(𝑋), and suppose that at least one of these four subsets of 𝑋 is complete. Let 𝑑𝑓^𝑛𝑥,𝑔^𝑛𝑦⪯𝛼𝑑𝑆𝑥,𝑇𝑦 (2.25) for all 𝑥,𝑦∈𝑋, where 𝛼∈[0,1) and (𝑓,𝑆) and (𝑔,𝑇) commute. Then 𝑓, 𝑔, 𝑆, and 𝑇 have a unique common fixed point.

Proof. By Theorem 2.1, we obtain 𝑢∈𝑋 such that 𝑓^𝑛𝑢=𝑔^𝑛𝑢=𝑆𝑢=𝑇𝑢=𝑢. (2.26) The result then follows from the fact that 𝑑𝑓𝑢,𝑔𝑢=𝑑𝑓^𝑛𝑓𝑢,𝑔^𝑛𝑔𝑢⪯𝛼𝑑𝑆𝑓𝑢,𝑇𝑔𝑢=𝛼𝑑𝑓𝑢,𝑔𝑢 (2.27) since 𝛼<1 so that 𝑓𝑢=𝑔𝑢 by property (p₁). Again 𝑑𝑓𝑢,𝑢=𝑑𝑓^𝑛𝑓𝑢,𝑔^𝑛𝑢⪯𝛼𝑑𝑆𝑓𝑢,𝑇𝑢=𝛼𝑑𝑓𝑢,𝑢, (2.28) implies that 𝑓𝑢=𝑢. And hence 𝑢 is the unique common fixed point of 𝑓, 𝑔, 𝑆, and 𝑇.

3. Periodic Point Theorems

It is clear that if 𝑓 is a map which has a fixed point 𝑝, then 𝑝 is also a fixed point of 𝑓^𝑛 for every 𝑛∈ℕ. However the converse is not true. For example, consider 𝑋=ℝ and 𝑓 defined by 𝑓𝑥=𝑎−𝑥, 𝑎≠0. Then 𝑓 has a unique fixed point at 𝑎/2, but every even iterate of 𝑓 is the identity map, which has each point of ℝ as its fixed point. On the other hand, if 𝑋=[0,+∞), 𝑓𝑥=𝑥², then every iterate of 𝑓 has the same fixed point as 𝑓. If a map 𝑓 satisfies 𝐹(𝑓)=𝐹(𝑓^𝑛) for each 𝑛∈ℕ, where 𝐹(𝑓) stands for the set of all fixed points of 𝑓, then it is said to have property 𝑃 [29]. We will say that 𝑓,𝑔∶𝑋→𝑋 have property 𝑄 if 𝐹(𝑓)∩𝐹(𝑔)=𝐹(𝑓^𝑛)∩𝐹(𝑔^𝑛) for each 𝑛∈ℕ.

The next result is a generalization of the corresponding result in metric spaces (see [29, Theorem 1.1]). It will be deduced also without using normality of the cone.

Theorem 3.1. Let (𝑋,𝑑) be an abstract metric space over a solid cone 𝐾, and let 𝑓∶𝑋→𝑋 be such that 𝐹(𝑓)≠∅ and that 𝑑𝑓𝑥,𝑓²𝑥⪯𝜆𝑑𝑥,𝑓𝑥 (3.1) holds for some 𝜆∈(0,1) and either (i) for all 𝑥∈𝑋 or (ii) for all 𝑥∈𝑋, 𝑥≠𝑓𝑥. Then 𝑓 has property 𝑃.

Proof. We will always assume that 𝑛>1, since the statement for 𝑛=1 is trivial. Let 𝑢∈𝐹(𝑓^𝑛). Suppose that 𝑓 satisfies (i). Then 𝑑𝑢,𝑓𝑢=𝑑𝑓𝑓^𝑛−1𝑢,𝑓²𝑓^𝑛−1𝑢⪯𝜆𝑑𝑓^𝑛−1𝑢,𝑓^𝑛𝑢=𝜆𝑑𝑓𝑓^𝑛−2𝑢,𝑓²𝑓^𝑛−2𝑢⪯𝜆²𝑑𝑓^𝑛−2𝑢,𝑓^𝑛−1𝑢⪯⋯⪯𝜆^𝑛𝑑𝑢,𝑓𝑢⪯𝜆𝑑𝑢,𝑓𝑢. (3.2) According to property (p₁) it follows that 𝑑(𝑢,𝑓𝑢)=𝜃, that is, 𝑓𝑢=𝑢. Suppose that 𝑓 satisfies (ii). If 𝑓𝑢=𝑢, then there is nothing to prove. Suppose, if possible, that 𝑓𝑢≠𝑢. Then, similarly as in case (i) we get that 𝑑(𝑢,𝑓𝑢)=𝑑(𝑓𝑓^𝑛−1𝑢,𝑓²𝑓^𝑛−1𝑢). In order to use (3.1) we need that 𝑓^𝑛−1𝑢≠𝑓𝑓^𝑛−1𝑢=𝑓^𝑛𝑢. But, if this is not the case, then 𝑓^𝑛−1𝑢=𝑢 and so 𝑢=𝑓^𝑛𝑢=𝑓𝑢, a contradiction. Hence, applying (3.1) we obtain that 𝑑𝑢,𝑓𝑢=𝑑𝑓𝑓^𝑛−1𝑢,𝑓²𝑓^𝑛−1𝑢⪯𝜆𝑑𝑓^𝑛−1𝑢,𝑓^𝑛𝑢=𝜆𝑑𝑓𝑓^𝑛−2𝑢,𝑓²𝑓^𝑛−2𝑢. (3.3) Repeating the same argument several times we finally obtain, similarly as in case (i), that 𝑑(𝑢,𝑓𝑢)⪯𝜆^𝑛𝑑(𝑢,𝑓𝑢), which again implies 𝑢=𝑓𝑢 since 𝜆∈(0,1), a contradiction.

Corollary 3.2. Let (𝑋,𝑑) be an abstract metric space over a solid cone 𝐾. Suppose that a mapping 𝑓∶𝑋→𝑋 satisfies 𝑑𝑓𝑥,𝑓𝑦⪯𝛼𝑑𝑥,𝑦+𝛽𝑑𝑥,𝑓𝑥+𝑑𝑦,𝑓𝑦+𝛾𝑑𝑥,𝑓𝑦+𝑑𝑦,𝑓𝑥 (3.4) for all 𝑥,𝑦∈𝑋, where 𝛼,𝛽,𝛾≥0 and 𝛼+2𝛽+2𝛾<1. Then 𝑓 has property 𝑃.

Proof. From Remark 2.5(2), 𝐹(𝑓)≠∅. We will prove that 𝑓 satisfies the condition (i) of Theorem 3.1. Indeed, 𝑑𝑓𝑥,𝑓²𝑥=𝑑𝑓𝑥,𝑓𝑓𝑥⪯𝛼𝑑𝑥,𝑓𝑥+𝛽𝑑𝑥,𝑓𝑥+𝑑𝑓𝑥,𝑓²𝑥+𝛾𝑑𝑥,𝑓²𝑥𝑑+𝑑𝑓𝑥,𝑓𝑥⪯𝛼𝑑𝑥,𝑓𝑥+𝛽𝑥,𝑓𝑥+𝑑𝑓𝑥,𝑓²𝑥+𝛾𝑑𝑥,𝑓𝑥+𝑑𝑓𝑥,𝑓²𝑥, (3.5) which implies that 𝑑(𝑓𝑥,𝑓²𝑥)⪯𝜆𝑑(𝑥,𝑓𝑥), where 𝜆=(𝛼+𝛽+𝛾)/(1−(𝛽+𝛾))<1. Hence, 𝑓 has property 𝑃.

The method of proof of the following result differs to the one from [29] (see also [9, Theorem 3.2]).

Theorem 3.3. Let (𝑋,𝑑) be a complete abstract metric space over a solid cone 𝐾. Suppose that mappings 𝑓,𝑔∶𝑋→𝑋 satisfy (2.1) (with 𝑆=𝑇=𝑖_𝑋). Then 𝑓 and 𝑔 have property 𝑄.

Proof. By Theorem 2.1, we have that 𝐹(𝑓)∩𝐹(𝑔)={𝑢}, where 𝑢 is the unique common fixed point of 𝑓 and 𝑔. So 𝐹(𝑓^𝑛)∩𝐹(𝑔^𝑛)≠∅ for each 𝑛∈ℕ. Let 𝑣∈𝐹(𝑓^𝑛)∩𝐹(𝑔^𝑛), where 𝑛>1 is arbitrary. Then, we obtain 𝑑(𝑢,𝑣)=𝑑𝑓^𝑛𝑢,𝑔^𝑛𝑣=𝑑𝑓𝑓^𝑛−1𝑢,𝑔𝑔^𝑛−1𝑣⪯𝛼𝑑𝑓^𝑛−1𝑢,𝑔^𝑛−1𝑣+𝛽𝑑𝑓^𝑛−1𝑢,𝑓^𝑛𝑢+𝑑𝑔^𝑛−1𝑣,𝑔^𝑛𝑣+𝛾𝑑𝑓^𝑛−1𝑢,𝑔^𝑛𝑣+𝑑𝑔^𝑛−1𝑣,𝑓^𝑛𝑢=𝛼𝑑𝑢,𝑔^𝑛−1𝑣+𝛽𝑑𝑔^𝑛−1𝑣,𝑔^𝑛𝑣+𝛾𝑑𝑢,𝑔^𝑛𝑣+𝑑𝑢,𝑔^𝑛−1𝑣⪯𝛼𝑑𝑢,𝑔^𝑛−1𝑣+𝛽𝑑𝑔^𝑛−1𝑣,𝑢+𝛽𝑑𝑢,𝑔^𝑛𝑣+𝛾𝑑𝑢,𝑔^𝑛𝑣+𝑑𝑢,𝑔^𝑛−1𝑣, (3.6) wherefrom it follows that 𝑑(𝑢,𝑔^𝑛𝑣)⪯𝛿𝑑(𝑢,𝑔^𝑛−1𝑣), where 𝛿=(𝛼+𝛽+𝛾)/(1−(𝛽+𝛾)).
Further, we have that 𝑑𝑢,𝑔^𝑛−1𝑣⪯𝛿𝑑𝑢,𝑔^𝑛−2𝑣. (3.7) Indeed, 𝑑𝑢,𝑔^𝑛−1𝑣=𝑑𝑓^𝑛−1𝑢,𝑔^𝑛−1𝑣=𝑑𝑓𝑓^𝑛−2𝑢,𝑔𝑔^𝑛−2𝑣⪯𝛼𝑑𝑓^𝑛−2𝑢,𝑔^𝑛−2𝑣+𝛽𝑑𝑓^𝑛−2𝑢,𝑓^𝑛−1𝑢+𝑑𝑔^𝑛−2𝑣,𝑔^𝑛−1𝑣+𝛾𝑑𝑓^𝑛−2𝑢,𝑔^𝑛−1𝑣+𝑑𝑔^𝑛−2𝑣,𝑓^𝑛−1𝑢=𝛼𝑑𝑢,𝑔^𝑛−2𝑣+𝛽𝑑𝑔^𝑛−2𝑣,𝑔^𝑛−1𝑣+𝛾𝑑𝑢,𝑔^𝑛−1𝑣+𝑑𝑢,𝑔^𝑛−2𝑣⪯𝛼𝑑𝑢,𝑔^𝑛−2𝑣+𝛽𝑑𝑔^𝑛−2𝑣,𝑢+𝛽𝑑𝑢,𝑔^𝑛−1𝑣+𝛾𝑑𝑢,𝑔^𝑛−1𝑣+𝑑𝑢,𝑔^𝑛−2𝑣, (3.8) which implies (3.7). Hence, 𝑑(𝑢,𝑣)=𝑑𝑢,𝑔^𝑛𝑣⪯𝛿𝑑𝑢,𝑔^𝑛−1𝑣⪯𝛿²𝑑𝑢,𝑔^𝑛−2𝑣⪯⋯⪯𝛿^𝑛𝑑(𝑢,𝑣). (3.9) Since 𝛿^𝑛∈[0,1), according to property (p₁) it follows 𝑑(𝑢,𝑣)=𝜃. Hence 𝑣=𝑢, which implies that 𝑓 and 𝑔 have property 𝑄.

Corollary 3.4. Let (𝑋,𝑑) be a complete abstract metric space over a solid cone 𝐾. Suppose that the mapping 𝑓∶𝑋→𝑋 satisfies one of the conditions (3)–(6) of Remark 2.5. Then 𝑓 has property 𝑃.

Remark 3.5. In the paper [9], the space (𝑋,𝑑) is supposed to be a complete cone metric space over a normal and solid cone 𝐾. Hence, our Theorems 3.1, 3.3, and Corollary 3.2 are proper extensions of Theorems  3.1, 3.2.  and  3.3 from [9].

In the following result the cone 𝐾 is regular, hence also normal (for the definition see, e.g., [5]).

Theorem 3.6. Let (𝑋,𝑑) be an abstract metric space over a regular cone 𝐾. Let 𝑓,𝑔∶𝑋→𝑋 be two mappings such that 𝑓𝑋⊂𝑔𝑋 and one of these subset of 𝑋 is complete. Suppose that there exist decreasing functions 𝛼_𝑖∶𝐾→[0,1), 𝑖=1,…,5, such that ∑⁵_𝑖=1𝛼_𝑖(𝑡)<1 for each 𝑡∈𝐾 and satisfying 𝑑𝑓𝑥,𝑓𝑦⪯𝛼₁𝑑𝑔𝑥,𝑔𝑦𝑑𝑔𝑥,𝑔𝑦+𝛼₂𝑑𝑔𝑥,𝑔𝑦𝑑𝑔𝑥,𝑓𝑥+𝛼₃𝑑𝑔𝑥,𝑔𝑦𝑑𝑔𝑦,𝑓𝑦+𝛼₄𝑑𝑔𝑥,𝑔𝑦𝑑𝑓𝑦,𝑔𝑥+𝛼₅𝑑𝑔𝑥,𝑔𝑦𝑑𝑓𝑥,𝑔𝑦 (3.10) for all 𝑥,𝑦∈𝑋, 𝑥≠𝑦. Then 𝑓 and 𝑔 have a unique point of coincidence. If, moreover, the pair (𝑓,𝑔) is weakly compatible, then 𝑓 and 𝑔 have a unique common fixed point.

Proof. Suppose, for example, that 𝑔𝑋 is complete. Take an arbitrary 𝑥₀∈𝑋 and, using that 𝑓𝑋⊂𝑔𝑋, construct a Jungck sequence {𝑦_𝑛} defined by 𝑦_𝑛=𝑓𝑥_𝑛=𝑔𝑥_𝑛+1, 𝑛=0,1,2,…. Let us prove that this is a Cauchy sequence. If 𝑦_𝑛=𝑦_𝑛−1 for some 𝑛, then it is easy to prove that the sequence {𝑦_𝑛} becomes eventually constant and so convergent.
Suppose that 𝑦_𝑛≠𝑦_𝑛−1 for each 𝑛∈ℕ. Using (3.10), we obtain that 𝑑𝑦_𝑛,𝑦_𝑛+1=𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1⪯𝛼₁𝑑𝑦_𝑛−1,𝑦_𝑛𝑑𝑦_𝑛−1,𝑦_𝑛+𝛼₂𝑑𝑦_𝑛−1,𝑦_𝑛𝑑𝑦_𝑛−1,𝑦_𝑛+𝛼₃𝑑𝑦_𝑛−1,𝑦_𝑛𝑑𝑦_𝑛,𝑦_𝑛+1+𝛼₄𝑑𝑦_𝑛−1,𝑦_𝑛𝑑𝑦_𝑛+1,𝑦_𝑛−1+𝛼₅𝑑𝑦_𝑛−1,𝑦_𝑛𝑑𝑦_𝑛,𝑦_𝑛, (3.11) for each 𝑛∈ℕ. Also, 𝑑𝑦_𝑛+1,𝑦_𝑛=𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1⪯𝛼₁𝑑𝑦_𝑛−1,𝑦_𝑛𝑑𝑦_𝑛,𝑦_𝑛−1+𝛼₂𝑑𝑦_𝑛−1,𝑦_𝑛𝑑𝑦_𝑛+1,𝑦_𝑛+𝛼₃𝑑𝑦_𝑛−1,𝑦_𝑛𝑑𝑦_𝑛−1,𝑦_𝑛+𝛼₄𝑑𝑦_𝑛−1,𝑦_𝑛𝑑𝑦_𝑛,𝑦_𝑛+𝛼₅𝑑𝑦_𝑛−1,𝑦_𝑛𝑑𝑦_𝑛+1,𝑦_𝑛−1. (3.12) Adding the last two relations (and putting temporarily 𝛼_𝑖=𝛼_𝑖(𝑑(𝑦_𝑛−1,𝑦_𝑛)), 𝑖=1,…,5) we obtain 𝑑𝑦_𝑛,𝑦_𝑛+1⪯𝛽𝑑𝑦_𝑛−1,𝑦_𝑛𝑑𝑦_𝑛−1,𝑦_𝑛, (3.13) where 𝛽(𝑡)=2𝛼₁(𝑡)+𝛼₂(𝑡)+𝛼₃(𝑡)+𝛼₄(𝑡)+𝛼₅(𝑡)2−𝛼₂(𝑡)+𝛼₃(𝑡)+𝛼₄(𝑡)+𝛼₅(𝑡). (3.14) It is easy to see that monotonicity of all 𝛼_𝑖's implies that 𝛽 is also a decreasing function and that 0<𝛽(𝑡)<1 for each 𝑡∈𝐾. In particular, 𝑑(𝑦_𝑛,𝑦_𝑛+1)≺𝑑(𝑦_𝑛−1,𝑦_𝑛) and so the sequence {𝑑(𝑦_𝑛,𝑦_𝑛+1)} is strictly decreasing (and bounded from below).
Since the cone 𝐾 is regular, there exists lim_𝑛→∞𝑑(𝑦_𝑛,𝑦_𝑛+1)=𝑝 and 𝜃⪯𝑝⪯𝑑(𝑦_𝑛,𝑦_𝑛+1) for each 𝑛. Then 1>𝛽(𝑝)>𝛽(𝑑(𝑦_𝑛,𝑦_𝑛+1)) for each 𝑛, and hence 𝑑𝑦_𝑛,𝑦_𝑛+1⪯𝛽𝑝𝑑𝑦_𝑛−1,𝑦_𝑛⪯𝛽𝑝²𝑑𝑦_𝑛−2,𝑦_𝑛−1⪯⋯⪯𝛽𝑝^𝑛𝑑𝑦₀,𝑦₁, (3.15) where 𝛽(𝑝) is a fixed scalar belonging to [0,1).
Now we prove that {𝑦_𝑛} is a Cauchy sequence in the usual way: for 𝑚>𝑛 it is 𝑑𝑦_𝑛,𝑦_𝑚⪯𝑑𝑦_𝑛,𝑦_𝑛+1+⋅⋅⋅+𝑑𝑦_𝑚−1,𝑦_𝑚⪯𝛽𝑝^𝑛+⋯𝛽𝑝^𝑚−1𝑑𝑦₀,𝑦₁⪯𝛽𝑝^𝑛𝑝𝑑𝑦1−𝛽⁰,𝑦₁→𝜃as𝑛→∞. (3.16) Thus, by properties (p₄) and (p₃) and Definition 1.3, {𝑦_𝑛} is a Cauchy sequence in 𝑔𝑋 and so there is 𝑧∈𝑋 such that 𝑓𝑥_𝑛=𝑔𝑥_𝑛+1→𝑔𝑧 when 𝑛→∞. We will prove that 𝑓𝑧=𝑔𝑧.
Put 𝑥=𝑥_𝑛, 𝑦=𝑧 in the contractive condition. We obtain (writing temporarily 𝛼_𝑖=𝛼_𝑖(𝑑(𝑥_𝑛,𝑧))) that 𝑑𝑓𝑥_𝑛,𝑓𝑧⪯𝛼₁𝑑𝑔𝑥_𝑛,𝑔𝑧+𝛼₂𝑑𝑔𝑥_𝑛,𝑓𝑥_𝑛+𝛼₃𝑑𝑔𝑧,𝑓𝑧+𝛼₄𝑑𝑓𝑧,𝑔𝑥_𝑛+𝛼₅𝑑𝑓𝑥_𝑛,𝑔𝑧⪯𝛼₁𝑑𝑔𝑥_𝑛,𝑔𝑧+𝛼₂+𝛼₄𝑑𝑔𝑥_𝑛,𝑓𝑥_𝑛+𝛼₃𝑑𝑔𝑧,𝑓𝑧+𝛼₄𝑑𝑓𝑧,𝑓𝑥_𝑛+𝛼₅𝑑𝑓𝑥_𝑛,𝑔𝑧. (3.17) Taking into account that all 𝛼_𝑖's are bounded in [0,1) and that the abstract metric 𝑑 is continuous (because the cone 𝐾 is normal), passing to the limit in the last vector inequality, we obtain that 𝑑𝑔𝑧,𝑓𝑧⪯𝛼₁⋅𝜃+𝛼₂+𝛼₄⋅𝜃+𝛼₃𝑑𝑔𝑧,𝑓𝑧+𝛼₄𝑑𝑓𝑧,𝑔𝑧+𝛼₅⋅𝜃, (3.18) that is, 𝑑(𝑔𝑧,𝑓𝑧)⪯(𝛼₃+𝛼₄)𝑑(𝑔𝑧,𝑓𝑧). Since 𝛼₃+𝛼₄<1, it follows that 𝑔𝑧=𝑓𝑧=𝑤 and 𝑓 and 𝑔 have a point of coincidence 𝑤.
Suppose that 𝑤₁=𝑓𝑧₁=𝑔𝑧₁ is another point of coincidence for 𝑓 and 𝑔. Then (3.10) implies that 𝑑𝑤₁,𝑤₂=𝑑𝑓𝑧,𝑓𝑧₁⪯𝛼₁𝑑𝑔𝑧,𝑔𝑧₁+𝛼₂𝑑𝑔𝑧,𝑓𝑧+𝛼₃𝑑𝑔𝑧₁,𝑓𝑧₁+𝛼₄𝑑𝑓𝑧₁,𝑔𝑧+𝛼₅𝑑𝑓𝑧,𝑔𝑧₁=𝛼₁𝑑𝑤,𝑤₁+𝛼₂⋅𝜃+𝛼₃⋅𝜃+𝛼₄𝑑𝑤₁,𝑤+𝛼₅𝑑𝑤,𝑤₁=𝛼₁+𝛼₄+𝛼₅𝑑𝑤,𝑤₁. (3.19) Since 𝛼₁+𝛼₄+𝛼₅<1, the last relation is possible only if 𝑤=𝑤₁. So, the point of coincidence is unique.
If (𝑓,𝑔) is weakly compatible, then [8, Proposition 1.4] implies that 𝑓 and 𝑔 have a unique common fixed point. The proof for the case in which 𝑓𝑋 is complete is similar and is therefore omitted.

Remark 3.7. Taking 𝐸=ℝ, 𝐾=[0,+∞), 𝑔(𝑥)=𝑥, we obtain a shorter proof of Theorem 1.6 (i.e., [28, Theorem 4]).

Remark 3.8. Taking appropriate choices of 𝑓, 𝑔 and 𝛼_𝑖, 𝑖=1,…,5 in Theorem 3.6, one can easily get the results of Reich (see relations (7), (8) in [28]), Hardy-Rogers (see relation (18) in [28]) and Ćirić (see relation (21) in [28]) in the setting of abstract metric spaces.

Acknowledgments

The authors (the first and the second) would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research Grant OUP-UKM-FST-2012. The fourth and fifth authors are thankful to the Ministry of Science and Technological Development of Serbia.