Common Fixed Points of Weak Contractions in Cone Metric Spaces

Abstract

Results on common fixed points of mappings in cone metric spaces under weak contractive conditions (B. S. Choudhury and N. Metiya (2010)) are unified and generalized. Also, cone metric versions of some other related results on weak contractions are proved. Examples show that our results are different than the existing ones.

1. Introduction

The idea to use an ordered Banach space instead of the set of real numbers, as the codomain for a metric, goes back to the mid-20th century (see, e.g., Kurepa [1], Kreĭn and Rutman [2], Kantorovič [3]). Fixed point theory in 𝐾-metric and 𝐾-normed spaces was developed by Perov [4], Vandergraft [5], and others. For more details we refer the reader to survey papers of Zabrejko [6] and Proinov [7]. In 2007, Huang and Zhang [8] reintroduced such spaces under the name of cone metric spaces and gave definitions of convergent and Cauchy sequences in the terms of interior points of the underlying cone, proving some fixed point theorems in such spaces. After that, fixed-points in cone metric spaces have been a subject of intensive research (see [9] for a survey of these results, and also [10–12]).

Fixed point results under so-called weak contractive conditions were first obtained in [13, 14]. They were generalized by various authors (see, e.g., [15]), in particular, using a pair of control functions 𝜑 and 𝜓 [16–21]. Note, however, that it was shown in [22] that in a certain sense the usage of function 𝜓 is superfluous. Weak contractions in cone metric spaces were treated in [23–25], results from [24] being the most general ones.

In this paper we generalize and unify the results on weak contractions in cone metric spaces from [24]. Examples show that these generalizations are proper. Further, we extend theorems from [16] and some related results to the case of cone metric spaces and give examples of applications of the obtained results.

2. Preliminaries

Let 𝐸 be a real Banach space with 𝜃 as the zero element, and let 𝑃 be a subset of 𝐸 with the interior int𝑃. The subset 𝑃 is called a cone if (a) 𝑃 is closed, nonempty, and 𝑃≠{∅}; (b) 𝑎,𝑏∈ℝ, 𝑎,𝑏≥0, and 𝑡,𝑢∈𝑃 imply 𝑎𝑡+𝑏𝑢∈𝑃; (c) 𝑃∩(−𝑃)={𝜃}. For the given cone 𝑃, a partial ordering ⪯ with respect to 𝑃 is introduced in the following way: 𝑡⪯𝑢 if and only if 𝑢−𝑡∈𝑃. If 𝑢−𝑡∈int𝑃, we write 𝑡≪𝑢.

If int𝑃≠∅, the cone 𝑃 is called solid (we will always assume that the given cone has this property). It is called normal if there is a number 𝐾>0, such that, for all 𝑡,𝑢∈𝐸, 𝜃⪯𝑡⪯𝑢 implies ‖𝑡‖≤𝐾‖𝑢‖ or, equivalently, if forall(𝑛)𝑡_𝑛⪯𝑢_𝑛⪯𝑣_𝑛, and lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞𝑣_𝑛=𝑡 imply lim_𝑛→∞𝑢_𝑛=𝑡. The cone 𝑃 is called regular if every decreasing sequence in 𝐸 which is order-bounded from below is convergent, that is, if whenever {𝑡_𝑛} is a sequence in 𝐸 such that 𝑡₁≽𝑡₂≽⋯≽𝑡_𝑛≽⋯≽𝑢 for some 𝑢∈𝐸, then there exists 𝑡∈𝐸 such that 𝑡_𝑛→𝑡, 𝑛→∞. Finally, 𝑃 is called strongly minihedral if every subset 𝐴 of 𝐸 has an infimum in 𝐸, provided it is order-bounded from below. It is well known that every strongly minihedral cone is regular and every regular cone is normal. The converses are not true [26].

Let 𝑋 be a nonempty set. Suppose that a mapping 𝑑∶𝑋×𝑋→𝐸 satisfies (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,𝑦)⪯𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦) for all 𝑥,𝑦,𝑧∈𝑋. Then 𝑑 is called a cone metric on 𝑋, and (𝑋,𝑑) is called a cone metric space [8]. The concept of a cone metric space is obviously more general than that of a metric space.

Sometimes the following additional property of the cone metric will be needed:(𝑑₄)𝑑(𝑥,𝑦)≫𝜃, for all 𝑥,𝑦∈𝑋 with 𝑥≠𝑦.

For definitions of notions such as convergent and Cauchy sequences, completeness, and so forth, we refer to [8] and for a survey of fixed point results in such spaces to [9].

Definition 2.1 (see [24]). Let (𝑋,𝑑) be a cone metric space over a solid cone 𝑃. Denote by Φ the class of functions 𝜑∶int𝑃∪{𝜃}→int𝑃∪{𝜃} satisfying the following conditions:(1)𝜑(𝑡)=𝜃 if and only if 𝑡=𝜃;(2)𝜑(𝑡)≪𝑡 for 𝑡∈int𝑃;(3)for all 𝑡∈int𝑃 and 𝑥,𝑦∈𝑋, either 𝜑(𝑡)⪯𝑑(𝑥,𝑦) or 𝑑(𝑥,𝑦)≪𝜑(𝑡) holds.Denote by Ψ the class of functions 𝜓∶𝑃→𝑃 satisfying the following conditions:(4)𝜓 is strictly increasing; that is, 𝑡₁≺𝑡₂ if and only if 𝜓(𝑡₁)≺𝜓(𝑡₂);(5)𝜓(𝑡)=𝜃 if and only if 𝑡=𝜃.Let two mappings 𝑓,𝑔∶𝑋→𝑋 and two arbitrary points 𝑥,𝑦∈𝑋 be given. The following four sets of vectors will be used: 𝑀⁴_𝑓,𝑔=𝑑,1𝑥,𝑦𝑔𝑥,𝑔𝑦,𝑑𝑔𝑥,𝑓𝑥,𝑑𝑔𝑦,𝑓𝑦2𝑑,𝑀𝑔𝑥,𝑓𝑦+𝑑𝑔𝑦,𝑓𝑥_3𝑓,𝑔=𝑑,1𝑥,𝑦𝑔𝑥,𝑔𝑦2𝑑,1𝑔𝑥,𝑓𝑥+𝑑𝑔𝑦,𝑓𝑦2𝑑,𝑁𝑔𝑥,𝑓𝑦+𝑑𝑔𝑦,𝑓𝑥_4𝑓,𝑔=𝑑,1𝑥,𝑦𝑥,𝑦,𝑑𝑥,𝑓𝑥,𝑑𝑦,𝑔𝑦2𝑑,𝑁𝑥,𝑔𝑦+𝑑𝑦,𝑓𝑥_3𝑓,𝑔=𝑑,1𝑥,𝑦𝑥,𝑦2𝑑,1𝑥,𝑓𝑥+𝑑𝑦,𝑔𝑦2𝑑𝑥,𝑔𝑦+𝑑𝑦,𝑓𝑥. (2.1) If 𝑖∶𝑋→𝑋 is the identity mapping, we will write 𝑀^𝑘_𝑓(𝑥,𝑦)∶=𝑀^𝑘_𝑓,𝑖(𝑥,𝑦) for 𝑘∈{3,4}.
Let 𝑓,𝑔∶𝑋→𝑋 be two self-maps on a nonempty set 𝑋. Recall that a point 𝑥∈𝑋 is called a coincidence point of the pair (𝑓,𝑔) and 𝑦 is its point of coincidence if 𝑓𝑥=𝑔𝑥=𝑦. The pair (𝑓,𝑔) is said to be weakly compatible if for each 𝑥∈𝑋, 𝑓𝑥=𝑔𝑥 implies 𝑓𝑔𝑥=𝑔𝑓𝑥. A classical result of Jungck states that if, two weakly compatible maps have a unique point of coincidence 𝑦, then 𝑦 is their unique common fixed point.
Roughly speaking, there are two types of common fixed point results with weak contractive conditions. Those of the first type use conditions with 𝑑(𝑓𝑥,𝑓𝑦) on the left-hand side and some element of the 𝑀-set on the right-hand one. The other use conditions with 𝑑(𝑓𝑥,𝑔𝑦) on the left-hand side and some element of the 𝑁-set on the right-hand side. An example of the first type is the following results in cone metric spaces that were proved by Choudhury and Metiya.

Theorem 2.2 (see [24], Theorems  3.1, 3.2, and 3.3). Let (𝑋,𝑑) be a cone metric space over a regular cone 𝑃 such that (𝑑₄) holds. Let 𝑓,𝑔∶𝑋→𝑋 be such that one of the following inequalities holds for all 𝑥,𝑦∈𝑋: 𝜓𝑑𝑑𝑑,𝜓𝑑1𝑓𝑥,𝑓𝑦⪯𝜓𝑔𝑥,𝑔𝑦−𝜑𝑔𝑥,𝑔𝑦𝑓𝑥,𝑓𝑦⪯𝜓2𝑑𝑑𝜓𝑑1𝑓𝑥,𝑔𝑥+𝑑𝑓𝑦,𝑔𝑦−𝜑𝑔𝑥,𝑔𝑦,𝑓𝑥,𝑓𝑦⪯𝜓2𝑑𝑓𝑥,𝑔𝑦+𝑑𝑓𝑦,𝑔𝑥−𝜑𝑑𝑔𝑥,𝑔𝑦, (2.2)(2.3)(2.4) where 𝜑∈Φ and 𝜓∈Ψ are continuous. If 𝑓𝑋⊂𝑔𝑋 and 𝑔𝑋 is complete, then 𝑓 and 𝑔 have a unique point of coincidence in 𝑋 (and so they have a unique common fixed point in 𝑋 if the pair (𝑓,𝑔) is weakly compatible).

On the other hand, in the case of metric spaces, the following result was proved by Đorić.

Theorem 2.3 (see [16], Theorem  2.1). Let (𝑋,𝑑) be a complete metric space, 𝜓∈Ψ be continuous, and 𝜑∈Φ be lower semicontinuous (here 𝑃=[0,+∞)). Let 𝑓,𝑔∶𝑋→𝑋 be two self-maps satisfying the inequality 𝜓𝑑𝑓𝑥,𝑔𝑦≤𝜓𝑚𝑥,𝑦−𝜑𝑚𝑥,𝑦, (2.5) for all 𝑥,𝑦∈𝑋, where 𝑚(𝑥,𝑦)=max𝑁⁴_𝑓,𝑔(𝑥,𝑦). Then 𝑓 and 𝑔 have a unique common fixed point in 𝑋.

In this paper we generalize and unify results of Theorem 2.2 (we will call the conditions that we use “weak contractive conditions of the first type”). Examples show that these generalizations are proper. Further, we extend Theorem 2.3 and some related results to the case of cone metric spaces (the respective conditions will be called “weak contractive conditions of the second type”) and give examples of applications of the obtained results.

3. Auxiliary Results

We will make use of the following result of Choudhury and Metiya.

Lemma 3.1 (see [24]). Let (𝑋,𝑑) be a cone metric space over a regular cone 𝑃 such that(𝑑₄) holds and suppose that there exists𝜑∈Φ (see Definition 2.1). If{𝑦_𝑛} is a sequence in 𝑋such that{𝑑(𝑦_𝑛,𝑦_𝑛+1)} is decreasing, then {𝑑(𝑦_𝑛,𝑦_𝑛+1)} converges either to 𝜃 or to 𝑟∈int𝑃.

Note that 𝜑∈Φ is not supposed to be continuous. It is easy to show that without the existence of function 𝜑 the conclusion of Lemma 3.1 may fail to hold.

The following result is a cone metric version of [21, lemma  2.1].

Lemma 3.2. Let (𝑋,𝑑) be a cone metric space over a regular cone 𝑃 such that (𝑑₄) holds and suppose that there exists 𝜑∈Φ (see Definition 2.1). Let {𝑦_𝑛} be a sequence in 𝑋 such that {𝑑(𝑦_𝑛,𝑦_𝑛+1)} is decreasing w.r.t. ⪯ and that lim_𝑛→∞𝑑𝑦_𝑛,𝑦_𝑛+1=𝜃. (3.1) If {𝑦_2𝑛} is not a Cauchy sequence, then there exists 𝑐∈int𝑃 and two sequences {𝑚_𝑘} and {𝑛_𝑘} of positive integers such that the following five sequences tend to 𝜑(𝑐) when 𝑘→∞: 𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘,𝑑𝑦_2𝑚𝑘+1,𝑦_2𝑛𝑘+1,𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘+1,𝑑𝑦_{2𝑚𝑘−1},𝑦_2𝑛𝑘,𝑑𝑦_{2𝑚𝑘−1},𝑦_2𝑛𝑘+1. (3.2)

Proof. Suppose that {𝑦_2𝑛} is not a Cauchy sequence. Then there exists 𝑐∈int𝑃 such that for each 𝑛₀∈ℕ there exist 𝑛,𝑚∈ℕ with 𝑛>𝑚≥𝑛₀ and 𝜑(𝑐)−𝑑(𝑦_2𝑚,𝑦_2𝑛)∉int𝑃. Hence, by property (2.4) of function 𝜑, 𝜑(𝑐)⪯𝑑(𝑦_2𝑚,𝑦_2𝑛) holds for 𝑛>𝑚≥𝑛₀. Therefore, there exist sequences {𝑚_𝑘} and {𝑛_𝑘} of positive integers such that 𝑛_𝑘>𝑚_𝑘>𝑘,𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘⪰𝜑(𝑐),𝑑𝑦_2𝑚𝑘,𝑦_{2𝑛𝑘−2}≪𝜑(𝑐) (3.3) (the last inequality is obtained by taking the smallest possible 𝑛_𝑘). Now we have 𝜑(𝑐)⪯𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘⪯𝑑𝑦_2𝑚𝑘,𝑦_{2𝑛𝑘−2}+𝑑𝑦_{2𝑛𝑘−2},𝑦_{2𝑛𝑘−1}+𝑑𝑦_{2𝑛𝑘−1},𝑦_2𝑛𝑘≪𝜑(𝑐)+𝑑𝑦_{2𝑛𝑘−2},𝑦_{2𝑛𝑘−1}+𝑑𝑦_{2𝑛𝑘−1},𝑦_2𝑛𝑘. (3.4) Letting 𝑘→∞ and using assumption (3.1) and the normality of the cone, we obtain that lim_𝑘→∞𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘=𝜑(𝑐). (3.5) Further, 𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘⪯𝑑𝑦_2𝑚𝑘,𝑦_2𝑚𝑘+1+𝑑𝑦_2𝑚𝑘+1,𝑦_2𝑛𝑘+1+𝑑𝑦_2𝑛𝑘+1,𝑦_2𝑛𝑘,𝑑𝑦_2𝑚𝑘+1,𝑦_2𝑛𝑘+1⪯𝑑𝑦_2𝑚𝑘+1,𝑦_2𝑚𝑘+𝑑𝑦_2𝑚𝑘,𝑦_2𝑛𝑘+𝑑𝑦_2𝑛𝑘,𝑦_2𝑛𝑘+1, (3.6) where 𝑑(𝑦_2𝑚𝑘,𝑦_2𝑛𝑘)→𝜑(𝑐), imply that lim_𝑘→∞𝑑𝑦_2𝑚𝑘+1,𝑦_2𝑛𝑘+1=𝜑(𝑐). (3.7) The other three limits can be obtained similarly.

4. Weak Contractions of the First Type in Cone Metric Spaces

Theorem 4.1. Let (𝑋,𝑑) be a cone metric space over a regular cone 𝑃 such that (𝑑₄) holds and suppose that there exists a continuous function 𝜑∈Φ. Let 𝑓,𝑔∶𝑋→𝑋 be two selfmaps such that 𝑓𝑋⊂𝑔𝑋 and let one of these subsets of 𝑋 be complete. Suppose that for all 𝑥,𝑦∈𝑋 there exists 𝑢=𝑢𝑥,𝑦∈𝑀⁴_𝑓,𝑔𝑥,𝑦 (4.1) such that 𝑑𝑓𝑥,𝑓𝑦⪯𝑢𝑥,𝑦−𝜑𝑢𝑥,𝑦 (4.2) holds true. 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.

Remark 4.2. Theorem 4.1 remains true if condition (4.2) is replaced by 𝜓𝑑𝑓𝑥,𝑓𝑦⪯𝜓𝑢𝑥,𝑦−𝜑𝑢𝑥,𝑦 (4.3) for some continuous 𝜓∈Ψ (see Definition 2.1). The proof is essentially the same, and so, for the sake of simplicity, we stay within the given version. The same remark applies to all other results in the rest of the paper. See also paper [22] where it is shown that practically each weak contractive condition with function 𝜓 can be replaced by an equivalent condition without 𝜓.

Proof. Starting from arbitrary 𝑥₁∈𝑋 and using the assumption 𝑓𝑋⊂𝑔𝑋, construct a Jungck sequence {𝑦_𝑛} satisfying 𝑦_𝑛=𝑓𝑥_𝑛=𝑔𝑥_𝑛+1 for 𝑛∈ℕ. If 𝑦_𝑛0=𝑦_𝑛0+1 for some 𝑛₀∈ℕ, then 𝑔𝑥_𝑛0+1=𝑦_𝑛0=𝑦_𝑛0+1=𝑓𝑥_𝑛0+1 and 𝑓 and 𝑔 have a point of coincidence. Suppose, further, that 𝑦_𝑛≠𝑦_𝑛+1 for 𝑛∈ℕ. Putting 𝑥=𝑥_𝑛+1, 𝑦=𝑥_𝑛 in (4.2) we obtain that 𝑑𝑦_𝑛+1,𝑦_𝑛=𝑑𝑓𝑥_𝑛+1,𝑓𝑥_𝑛⪯𝑢−𝜑(𝑢), (4.4) where 𝑢=𝑢𝑥_𝑛+1,𝑥_𝑛∈𝑑𝑔𝑥_𝑛+1,𝑔𝑥_𝑛,𝑑𝑔𝑥_𝑛+1,𝑓𝑥_𝑛+1,𝑑𝑔𝑥_𝑛,𝑓𝑥_𝑛,12𝑑𝑔𝑥_𝑛+1,𝑓𝑥_𝑛+𝑑𝑔𝑥_𝑛,𝑓𝑥_𝑛+1=𝑑𝑦_𝑛,𝑦_𝑛−1,𝑑𝑦_𝑛,𝑦_𝑛+1,𝑑𝑦_𝑛−1,𝑦_𝑛,12𝑑𝑦_𝑛−1,𝑦_𝑛+1. (4.5) The case 𝑢=𝑑(𝑦_𝑛,𝑦_𝑛+1) is impossible, since it would imply 𝑢⪯𝑢−𝜑(𝑢) and 𝑢=𝜃 (using properties of the function 𝜑), which is already excluded. In all other cases we get that 𝑑(𝑦_𝑛+1,𝑦_𝑛)⪯𝑑(𝑦_𝑛,𝑦_𝑛−1), and, more precisely, 𝑑𝑦_𝑛+1,𝑦_𝑛⪯𝑢𝑥_𝑛+1,𝑥_𝑛⪯𝑑𝑦_𝑛,𝑦_𝑛−1. (4.6) Indeed, the right-hand inequality is trivial in the case when 𝑢=𝑑(𝑦_𝑛,𝑦_𝑛−1), and in the case 𝑢=(1/2)𝑑(𝑦_𝑛−1,𝑦_𝑛+1), then 𝑑(𝑦_𝑛+1,𝑦_𝑛)⪯𝑑(𝑦_𝑛,𝑦_𝑛−1) and 𝑢⪯(1/2)𝑑(𝑦_𝑛−1,𝑦_𝑛)+(1/2)𝑑(𝑦_𝑛,𝑦_𝑛+1)⪯(1/2)𝑑(𝑦_𝑛−1,𝑦_𝑛)+(1/2)𝑑(𝑦_𝑛−1,𝑦_𝑛)=𝑑(𝑦_𝑛−1,𝑦_𝑛).
We have proved that the sequence {𝑑(𝑦_𝑛,𝑦_𝑛+1)} is decreasing w.r.t. ⪯ and so Lemma 3.1 implies that it converges to some 𝑟, where either 𝑟=𝜃 or 𝑟∈int𝑃. But, if 𝑟∈int𝑃, then (4.6) implies that also 𝑢(𝑥_𝑛+1,𝑥_𝑛)→𝑟 as 𝑛→∞. Hence, passing to the limit in (4.4) we get that 𝑟⪯𝑟−𝜑(𝑟) and 𝑟=𝜃, a contradiction. Thus, 𝑟=lim_𝑛→∞𝑑(𝑦_𝑛,𝑦_𝑛+1)=𝜃.
Let us prove that {𝑦_𝑛} is a Cauchy sequence in 𝑋. Suppose that it is not. It follows from monotonicity of the sequence {𝑑(𝑦_𝑛,𝑦_𝑛+1)} and lim_𝑛→∞𝑑(𝑦_𝑛,𝑦_𝑛+1)=𝜃 that neither {𝑦_2𝑛} is a Cauchy sequence. Lemma 3.2 implies that there exist sequences {𝑚_𝑘} and {𝑛_𝑘} of positive integers such that the sequences (3.2) all tend to 𝜑(𝑐) for some 𝑐≫0. Using (4.6) and putting 𝑥=𝑥_2𝑛𝑘, 𝑦=𝑥_{2𝑚𝑘−1} in (4.2) we get that 𝑑𝑦_2𝑛𝑘+1,𝑦_2𝑚𝑘=𝑑𝑓𝑥_2𝑛𝑘,𝑓𝑥_{2𝑚𝑘−1}⪯𝑢𝑥_2𝑛𝑘,𝑥_{2𝑚𝑘−1}−𝜑𝑢𝑥_2𝑛𝑘,𝑥_{2𝑚𝑘−1}. (4.7) Letting 𝑘→∞ we get that 𝜑(𝑐)⪯𝜑(𝑐)−𝜑(𝜑(𝑐)). Properties of function 𝜑 imply that 𝑐=𝜃, a contradiction. Hence, {𝑦_𝑛} is a Cauchy sequence.
By the assumption, there exists lim_𝑛→∞𝑦_𝑛=𝑔𝑝 for some 𝑝∈𝑋. Let us prove that 𝑓𝑝=𝑔𝑝. Putting 𝑥=𝑥_𝑛, 𝑦=𝑝 in (4.2) we get that 𝑑𝑓𝑥_𝑛,𝑓𝑝⪯𝑢−𝜑(𝑢), (4.8) where 𝑢=𝑢𝑥_𝑛,𝑝∈𝑑𝑔𝑥_𝑛,𝑔𝑝,𝑑𝑔𝑥_𝑛,𝑓𝑥_𝑛,1,𝑑𝑔𝑝,𝑓𝑝2𝑑𝑔𝑥_𝑛,𝑓𝑝+𝑑𝑔𝑝,𝑓𝑥_𝑛. (4.9) In other words, at least one of four possible inequalities holds for infinitely many 𝑛∈ℕ. Hence, passing to the limit, we obtain that 𝑑(𝑔𝑝,𝑓𝑝)⪯𝜃−𝜑(𝜃)=𝜃 or 𝑑(𝑔𝑝,𝑓𝑝)⪯𝑑(𝑔𝑝,𝑓𝑝)−𝜑(𝑑(𝑔𝑝,𝑓𝑝)) or 𝑑(𝑔𝑝,𝑓𝑝)⪯1/2𝑑(𝑔𝑝,𝑓𝑝)−𝜑(1/2𝑑(𝑔𝑝,𝑓𝑝))⪯1/2𝑑(𝑔𝑝,𝑓𝑝). By the properties of 𝜑 it follows that 𝑑(𝑔𝑝,𝑓𝑝)=𝜃 and 𝑔𝑝=𝑓𝑝=𝑞; hence 𝑞 is a point of coincidence for the pair (𝑓,𝑔).
To prove that this point of coincidence is unique, assume that there is another 𝑞₁∈𝑋 such that 𝑞₁=𝑓𝑝₁=𝑔𝑝₁ for some 𝑝₁∈𝑋. Then 𝑑𝑞,𝑞₁=𝑑𝑓𝑝,𝑓𝑝₁⪯𝑢−𝜑(𝑢), (4.10) where 𝑢=𝑢𝑝,𝑝₁∈𝑑𝑔𝑝,𝑔𝑝₁,𝑑𝑔𝑝,𝑓𝑝,𝑑𝑔𝑝₁,𝑓𝑝₁,12𝑑𝑔𝑝,𝑓𝑝₁+𝑑𝑓𝑝,𝑔𝑝₁=𝑑𝑞,𝑞₁,𝜃. (4.11) In both cases we get that 𝑑(𝑞,𝑞₁)=𝜃, that is, the point of coincidence is unique.

Obviously, the theorem in [24, Theorem  3.1] (Theorem 2.2 with condition (2.2)) is a special case of Theorem 4.1.

Remark 4.3. The previous theorem can be modified so that continuity of 𝜑 is substituted by its lower semicontinuity; however, in this case it has to be assumed that the cone 𝑃 is strongly minihedral. For details see [10]. The same applies to other assertions to the end of the paper.

The following example shows that there are cases when the existence of a common fixed point can be deduced using Theorem 4.1, but cannot be obtained using the theorem in [24, Theorems  3.1, 3.2, and  3.3] (Theorem 2.2 with either of the conditions (2.2), (2.3), or (2.4)).

Example 4.4. Let 𝑋=[0,1]∪[3/2,2], 𝐸=ℝ² (𝜃=(0,0)), 𝑃={(𝑥,𝑦)∈𝐸∶𝑥≥0,𝑦≥0} and 𝑑(𝑥,𝑦)=(|𝑥−𝑦|,𝛼|𝑥−𝑦|), where 𝛼>0 is fixed. 𝑑 is obviously a cone metric satisfying property (𝑑₄) and the cone 𝑃 is regular (even minihedral). Function 𝜑∶int𝑃∪{𝜃}→int𝑃∪{𝜃} defined by 𝜑(𝜃)=𝜃 and 𝜑(𝑡₁,𝑡₂)=((1/4)𝑡₁,(1/4)𝑡₂) for 𝑡₁>0, 𝑡₂>0 belongs to the respective class Φ. Consider the mappings 𝑓,𝑔∶𝑋→𝑋 defined by: 𝑔𝑥=𝑥 for 𝑥∈𝑋, 𝑓𝑥=3/4 for 𝑥∈[0,1] and 𝑓𝑥=1/4 for 𝑥∈[3/2,2]. We will show that, taking, for example, 𝜓(𝑡)=𝑡, neither of conditions (2.2), (2.3), (2.4) is satisfied; hence neither of Theorems 3.1, 3.2, and 3.3 from [24] can be used to conclude that there exists a common fixed point of 𝑓 and 𝑔 (which is obviously 𝑝=3/4).
Indeed, for 𝑥=1, 𝑦=3/2, we get that 𝑑(𝑓𝑥,𝑓𝑦)=(1/2,(1/2)𝛼). On the other hand 𝑑(𝑔𝑥,𝑔𝑦)−𝜑(𝑑(𝑔𝑥,𝑔𝑦))=𝑑(1,3/2)−𝜑(𝑑(1,3/2))=(1/2,(1/2)𝛼)−1/4(1/2,(1/2)𝛼)=(3/8,(3/8)𝛼)≺(1/2,(1/2)𝛼). Hence, condition (2.2) is not satisfied. Further, for 𝑥=3/4, 𝑦=3/2, we obtain that 𝑑(𝑓𝑥,𝑓𝑦)=(1/2,(1/2)𝛼) and (1/2)(𝑑(𝑔𝑥,𝑓𝑥)+𝑑(𝑔𝑦,𝑓𝑦))−𝜑(𝑑(𝑔𝑥,𝑔𝑦))=(1/2)(𝑑(3/4,3/4)+𝑑(3/2,1/4))−𝜑(𝑑(3/4,3/2))=(1/2)(5/4,(5/4)𝛼)−(1/4)(3/4,(3/4)𝛼)=(7/16,(7/16)𝛼)≺(1/2,(1/2)𝛼). Hence condition (2.3) is not satisfied. Finally, taking 𝑥=1/4, 𝑦=3/2, we have that 𝑑(𝑓𝑥,𝑓𝑦)=(1/2,(1/2)𝛼), but (1/2)(𝑑(𝑓𝑥,𝑔𝑦)+𝑑(𝑓𝑦,𝑔𝑥))−𝜑(𝑑(𝑔𝑥,𝑔𝑦))=(1/2)(𝑑(3/4,3/2)+𝑑(1/4,1/4))−𝜑(𝑑(1/4,3/2))=(1/2)(3/4,(3/4)𝛼)−(1/4)(5/4,(5/4)𝛼)=(1/16,(1/16)𝛼)≺(1/2,(1/2)𝛼). Hence, neither (2.4) is satisfied.
On the other hand, condition (4.2) of Theorem 4.1 is satisfied. Indeed, if 𝑥,𝑦∈[0,1] or 𝑥,𝑦∈[1,3/2], then 𝑑(𝑓𝑥,𝑓𝑦)=𝜃 and the condition is trivially satisfied. If 𝑥∈[0,1] and 𝑦∈[3/2,2] (or vice versa), take 𝑢=𝑢(𝑥,𝑦)=𝑑(𝑔𝑦,𝑓𝑦)=(𝑦−1/4,(𝑦−1/4)𝛼) and we obtain that 𝑑(𝑓𝑥,𝑓𝑦)=(1/2,(1/2)𝛼) and 𝑢−𝜑(𝑢)=(𝑦−1/4,(𝑦−1/4)𝛼)−(1/4)(𝑦−1/4,(𝑦−1/4)𝛼)=3/4(𝑦−1/4,(𝑦−1/4)𝛼)≽(3/4)(5/4,(5/4)𝛼)=(15/16,(15/16)𝛼)≻(1/2,(1/2)𝛼). Thus, conclusion about the existence of a common fixed point can be deduced from Theorem 4.1.

In the next theorem the set 𝑀³_𝑓,𝑔(𝑥,𝑦) is used instead of 𝑀⁴_𝑓,𝑔(𝑥,𝑦). The proof is essentially the same as for Theorem 4.1 and so is omitted.

Theorem 4.5. Let (𝑋,𝑑) be a cone metric space over a regular cone 𝑃 such that (𝑑₄) holds and suppose that there exists a continuous function 𝜑∈Φ. Let 𝑓,𝑔∶𝑋→𝑋 be two selfmaps such that 𝑓𝑋⊂𝑔𝑋 and let one of these subsets of 𝑋 be complete. Suppose that for all 𝑥,𝑦∈𝑋 there exists 𝑢=𝑢𝑥,𝑦∈𝑀³_𝑓,𝑔𝑥,𝑦, (4.12) such that 𝑑𝑓𝑥,𝑓𝑦⪯𝑢𝑥,𝑦−𝜑𝑢𝑥,𝑦 (4.13) holds true. 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.

In the following theorem we unify Theorems 3.1, 3.2, and 3.3 of [24] (Theorem 2.2 with conditions (2.2), (2.3), or (2.4)).

Theorem 4.6. Let (𝑋,𝑑) be a cone metric space over a regular cone 𝑃 such that (𝑑₄) holds and suppose that there exists a continuous function 𝜑∈Φ. Let 𝑓,𝑔∶𝑋→𝑋 be two selfmaps such that 𝑓𝑋⊂𝑔𝑋 and let one of these subsets of 𝑋 be complete. Suppose that for all 𝑥,𝑦∈𝑋 there exists 𝑢=𝑢𝑥,𝑦∈𝑀³_𝑓,𝑔𝑥,𝑦, (4.14) such that 𝑑𝑓𝑥,𝑓𝑦⪯𝑢𝑥,𝑦−𝜑𝑑𝑔𝑥,𝑔𝑦 (4.15) holds true. 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. As usual, form a Jungck sequence by 𝑦_𝑛=𝑓𝑥_𝑛=𝑔𝑥_𝑛+1. If 𝑦_𝑛0=𝑦_𝑛0+1, then it can be proved as in Theorem 4.1 that (𝑓,𝑔) has a point of coincidence. Assume that 𝑦_𝑛≠𝑦_𝑛+1 for all 𝑛∈ℕ. Then 𝑑𝑦_𝑛+1,𝑦_𝑛=𝑑𝑓𝑥_𝑛+1,𝑓𝑥_𝑛⪯𝑢𝑥_𝑛+1,𝑥_𝑛−𝜑𝑑𝑦_𝑛+1,𝑦_𝑛, (4.16) where 𝑢𝑥_𝑛+1,𝑥_𝑛∈𝑑𝑦_𝑛,𝑦_𝑛1,12𝑑𝑦_𝑛,𝑦_𝑛+1+𝑑𝑦_𝑛−1,𝑦_𝑛,12𝜃+𝑑𝑦_𝑛−1,𝑦_𝑛+1. (4.17) In each of the three possible cases it is easy to obtain that {𝑑(𝑦_𝑛+1,𝑦_𝑛)} is a decreasing sequence and that 𝑑𝑦_𝑛+1,𝑦_𝑛⪯𝑢𝑥_𝑛+1,𝑥_𝑛⪯𝑑𝑦_𝑛,𝑦_𝑛−1. (4.18) Hence, all these three terms tend to some 𝑟∈int𝑃∪{𝜃}. Passing to the limit in relation (4.16) we get that 𝑟⪯𝑟−𝜑(𝑟), wherefrom it follows that 𝑟=𝜃.
That {𝑦_𝑛} is a Cauchy sequence can be proved using Lemma 3.2 similarly as in the proof of Theorem 4.1. Hence, there exists 𝑝∈𝑋 such that 𝑦_𝑛=𝑓𝑥_𝑛=𝑔𝑥_𝑛+1→𝑔𝑝 when 𝑛→∞. Let us prove that 𝑔𝑝 is a point of coincidence of the pair (𝑓,𝑔).
Putting 𝑥=𝑥_𝑛, 𝑦=𝑝 in (4.15) we get that 𝑑𝑓𝑥_𝑛,𝑓𝑝⪯𝑢𝑥_𝑛,𝑝−𝜑𝑑𝑔𝑥_𝑛,𝑔𝑝, (4.19) where 𝑢𝑥_𝑛,𝑝∈𝑑𝑔𝑥_𝑛,1,𝑔𝑝2𝑑𝑔𝑥_𝑛,𝑓𝑥_𝑛,1+𝑑𝑔𝑝,𝑓𝑝2𝑑𝑔𝑥_𝑛,𝑓𝑝+𝑑𝑓𝑥_𝑛,𝑔𝑝. (4.20) Passing to the limit (more precisely, considering one of the inequalities that holds for infinitely many 𝑛 as in the proof of Theorem 4.1) we get either 𝑑(𝑔𝑝,𝑓𝑝)⪯𝜃 or 𝑑(𝑔𝑝,𝑓𝑝)⪯(1/2)𝑑(𝑔𝑝,𝑓𝑝) and both are possible only if 𝑓𝑝=𝑔𝑝. Hence 𝑔𝑝 is a point of coincidence of (𝑓,𝑔).
The proof that the point of coincidence is unique is essentially the same as in Theorem 4.1.

The next example illustrates how Theorem 4.6 can be used to prove the existence of a common fixed point, while either Theorem  3.2 or 3.3 of [24] cannot.

Example 4.7. Let 𝑋=[0,1], and let 𝐸=ℝ², 𝑃={(𝑥,𝑦)∈𝐸∶𝑥≥0,𝑦≥0} and 𝑑(𝑥,𝑦)=(|𝑥−𝑦|,𝛼|𝑥−𝑦|) be as in the previous example. Take function 𝜑∈Φ defined by 𝜑(𝜃)=𝜃 and 𝜑(𝑡₁,𝑡₂)=((1/2)𝑡²₁,(1/2)𝑡²₂). Mappings 𝑓,𝑔∶𝑋→𝑋 are defined as 𝑓𝑥=𝑥−(1/2)𝑥² and 𝑔𝑥=𝑥.

Condition (2.3) is not satisfied. Indeed, take 𝑥=1 and 𝑦=0 to obtain that 𝑑(𝑓𝑥,𝑓𝑦)=𝑑(1/2,0)=(1/2,(1/2)𝛼) and (1/2)(𝑑(𝑔𝑥,𝑓𝑥)+𝑑(𝑔𝑦,𝑓𝑦))−𝜑(𝑑(𝑔𝑥,𝑔𝑦))=(1/2)𝑑(1,1/2)−𝜑(𝑑(1,0))=(1/4,(1/4)𝛼)−(1/2,(1/2)𝛼)=(−1/4,−(1/4)𝛼)≺(1/2,(1/2)𝛼). Similarly, condition (2.4) is not satisfied, for taking again 𝑥=1 and 𝑦=0 one obtains that 𝑑(𝑓𝑥,𝑓𝑦)=𝑑(1/2,0)=(1/2,(1/2)𝛼), but (1/2)(𝑑(𝑔𝑥,𝑓𝑦)+𝑑(𝑔𝑦,𝑓𝑥))−𝜑(𝑑(𝑔𝑥,𝑔𝑦))=(1/2)(𝑑(1,0)+𝑑(0,1/2))−𝜑(𝑑(1,0))=(3/4,(3/4)𝛼)−(1/2,(1/2)𝛼)=(1/4,(1/4)𝛼)≺(1/2,(1/2)𝛼).

We show that, however, condition (4.15) is satisfied and so Theorem 4.6 can be used to conclude that there exists a common fixed point of 𝑓 and 𝑔 (which is obviously 𝑝=0) (note that this can also be done using condition (2.2)). Indeed, take 𝑢=𝑢(𝑥,𝑦)=𝑑(𝑔𝑥,𝑔𝑦)=𝑑(𝑥,𝑦)∈𝑀³_𝑓,𝑔(𝑥,𝑦). In order to prove inequality (4.15) it is enough to consider the first coordinates of respective vectors, that is, we have to prove that 1𝑥−2𝑥²1−𝑦+2𝑦²≤−1𝑥−𝑦2|𝑥−𝑦|² (4.21) holds for all 𝑥,𝑦∈[0,1]. But, it is an easy consequence of |𝑥−𝑦|≤𝑥+𝑦.

Finally, we state (proof can be deduced similarly as for the previous theorems) the following cone metric version of [18, Theorems  3.1 and 4.1] (see also [21, Theorem  3.6]).

Theorem 4.8. Let (𝑋,𝑑) be a cone metric space over a regular cone 𝑃 such that (𝑑₄) holds and suppose that there exists a continuous function 𝜑∈Φ. Let 𝑓∶𝑋→𝑋 be a selfmap such that for all 𝑥,𝑦∈𝑋 there exist 𝑢𝑥,𝑦∈𝑀⁴_𝑓𝑥,𝑦,𝑣𝑥,𝑦∈𝑑𝑥,𝑦,𝑑𝑦,𝑓𝑦, (4.22) such that 𝑑(𝑓𝑥,𝑓𝑦)⪯𝑢(𝑥,𝑦)−𝜑(𝑣(𝑥,𝑦)). Then 𝑓 has a fixed point.

Note that in this case fixed point of 𝑓 need not be unique. It is enough to consider the identity mapping 𝑓=𝑖_𝑋 and take 𝑣=𝜃.

5. Weak Contractions of the Second Type in Cone Metric Spaces

In this section we consider weak contractions which we have called “of the second type” (see the end of Section 2).

Theorem 5.1. Let (𝑋,𝑑) be a complete cone metric space over a regular cone 𝑃 such that (𝑑₄) holds and suppose that there exists a continuous function 𝜑∈Φ. Let 𝑓,𝑔∶𝑋→𝑋 be two mappings such that for all 𝑥,𝑦∈𝑋 there exists 𝑢𝑥,𝑦∈𝑁⁴_𝑓,𝑔𝑥,𝑦 (5.1) such that 𝑑𝑓𝑥,𝑔𝑦⪯𝑢𝑥,𝑦−𝜑𝑢𝑥,𝑦. (5.2) Then 𝑓 and 𝑔 have a unique common fixed point.

Proof. Let us prove first that the common fixed point of 𝑓 and 𝑔 is unique (if it exists). Suppose that 𝑝≠𝑞 are two distinct common fixed points of 𝑓 and 𝑔. Then (5.2) implies that 𝑑𝑝,𝑞=𝑑𝑓𝑝,𝑔𝑞⪯𝑢𝑝,𝑞−𝜑𝑢𝑝,𝑞, (5.3) where 𝑢(𝑝,𝑞)∈𝑁⁴_𝑓,𝑔(𝑝,𝑞)={𝑑(𝑝,𝑞),𝜃,𝜃,𝑑(𝑝,𝑞)}={𝜃,𝑑(𝑝,𝑞)}. Checking both possible cases and using the properties of function 𝜑, we readily obtain that 𝑑(𝑝,𝑞)=𝜃, that is, 𝑝=𝑞.
In order to prove the existence of a common fixed point, proceed this time constructing a Jungck sequence by 𝑥_2𝑛+1=𝑓𝑥_2𝑛, 𝑥_2𝑛+2=𝑔𝑥_2𝑛+1, for arbitrary 𝑥₀∈𝑋. Consider the two possible cases.
Suppose that 𝑥_𝑛=𝑥_𝑛+1 for some 𝑛∈ℕ. Then 𝑥_𝑛+1=𝑥_𝑛+2 and it follows that the sequence is eventually constant, and so convergent. Indeed, let, for example, 𝑛=2𝑘 (in the case 𝑛=2𝑘+1 the proof is similar). Then, putting 𝑥=𝑥_2𝑘, 𝑦=𝑥_2𝑘+1 in (5.2), we get that there exists 𝑢∈𝑑𝑥_2𝑘,𝑥_2𝑘+1,𝑑𝑥_2𝑘,𝑓𝑥_2𝑘,𝑑𝑥_2𝑘+1,𝑔𝑥_2𝑘+1,12𝑑𝑥_2𝑘,𝑔𝑥_2𝑘+1+𝑑𝑥_2𝑘+1,𝑓𝑥_2𝑘=𝜃,𝑑𝑥_2𝑘+1,𝑥_2𝑘+2,12𝑑𝑥_2𝑘,𝑥_2𝑘+2, (5.4) such that 𝑑(𝑥_2𝑘+1,𝑥_2𝑘+2)⪯𝑢−𝜑(𝑢). Consider the three possible cases:(1°)𝑢=𝜃; it trivially follows that 𝑥_2𝑘=𝑥_2𝑘+1.(2°)𝑢=𝑑(𝑥_2𝑘+1,𝑥_2𝑘+2); it follows that 𝑑𝑥_2𝑘+1,𝑥_2𝑘+2⪯𝑑𝑥_2𝑘+1,𝑥_2𝑘+2−𝜑𝑑𝑥_2𝑘+1,𝑥_2𝑘+2, (5.5) and by the properties of function 𝜑 that 𝑥_2𝑘=𝑥_2𝑘+1.(3°)𝑢=(1/2)𝑑(𝑥_2𝑘,𝑥_2𝑘+2); since 𝑢⪯(1/2)(𝑑(𝑥_2𝑘,𝑥_2𝑘+1)+𝑑(𝑥_2𝑘+1,𝑥_2𝑘+2))=(1/2)𝑑(𝑥_2𝑘+1,𝑥_2𝑘+2), it follows that 𝑑𝑥_2𝑘+1,𝑥_2𝑘+2⪯12𝑑𝑥_2𝑘+1,𝑥_2𝑘+21−𝜑2𝑑𝑥_2𝑘,𝑥_2𝑘+2⪯12𝑑𝑥_2𝑘+1,𝑥_2𝑘+2, (5.6) wherefrom 𝑑(𝑥_2𝑘+1,𝑥_2𝑘+2)⪯(1/2)𝑑(𝑥_2𝑘+1,𝑥_2𝑘+2) which is only possible if 𝑥_2𝑘=𝑥_2𝑘+1.Suppose now that 𝑥_𝑛≠𝑥_𝑛+1 for all 𝑛∈ℕ. Putting 𝑥=𝑥_2𝑛, 𝑦=𝑥_2𝑛−1 in (5.2), we get that there exists 𝑢∈𝑑𝑥_2𝑛,𝑥_2𝑛−1,𝑑𝑥_2𝑛,𝑓𝑥_2𝑛,𝑑𝑥_2𝑛−1,𝑔𝑥_2𝑛−1,12𝑑𝑥_2𝑛,𝑔𝑥_2𝑛−1+𝑑𝑥_2𝑛−1,𝑓𝑥_2𝑛=𝑑𝑥_2𝑛,𝑥_2𝑛−1,𝑑𝑥_2𝑛,𝑥_2𝑛+1,12𝑑𝑥_2𝑛−1,𝑥_2𝑛+1, (5.7) such that 𝑑(𝑥_2𝑛+1,𝑥_2𝑛)⪯𝑢−𝜑(𝑢). Consider the three possible cases:(1°)𝑢=𝑑(𝑥_2𝑛,𝑥_2𝑛−1); it follows that 𝑑𝑥_2𝑛+1,𝑥_2𝑛⪯𝑑𝑥_2𝑛,𝑥_2𝑛−1−𝜑𝑑𝑥_2𝑛,𝑥_2𝑛−1≺𝑑𝑥_2𝑛,𝑥_2𝑛−1 (5.8) and 𝑑(𝑥_2𝑛+1,𝑥_2𝑛)≺𝑑(𝑥_2𝑛,𝑥_2𝑛−1).(2°)𝑢=𝑑(𝑥_2𝑛,𝑥_2𝑛+1); it follows that 𝑑𝑥_2𝑛+1,𝑥_2𝑛⪯𝑑𝑥_2𝑛,𝑥_2𝑛+1−𝜑𝑑𝑥_2𝑛,𝑥_2𝑛+1≺𝑑𝑥_2𝑛,𝑥_2𝑛+1, (5.9) which is impossible.(3°)𝑢=(1/2)𝑑(𝑥_2𝑛−1,𝑥_2𝑛+1); it follows that 𝑑𝑥_2𝑛+1,𝑥_2𝑛⪯12𝑑𝑥_2𝑛−1,𝑥_2𝑛+11−𝜑2𝑑𝑥_2𝑛−1,𝑥_2𝑛+1. (5.10)By the properties of function 𝜑 we obtain that 𝑑(𝑥_2𝑛+1,𝑥_2𝑛)⪯(1/2)(𝑑(𝑥_2𝑛−1,𝑥_2𝑛)+𝑑(𝑥_2𝑛,𝑥_2𝑛+1)) and 𝑑(𝑥_2𝑛+1,𝑥_2𝑛)⪯𝑑(𝑥_2𝑛,𝑥_2𝑛−1).
Hence, in any possible case, 𝑑(𝑥_2𝑛+1,𝑥_2𝑛)⪯𝑑(𝑥_2𝑛,𝑥_2𝑛−1) and, similarly, 𝑑(𝑥_2𝑛+2,𝑥_2𝑛+1)⪯𝑑(𝑥_2𝑛+1,𝑥_2𝑛). Thus, the sequence {𝑑(𝑥_𝑛,𝑥_𝑛+1)} is decreasing; moreover, 𝑑𝑥_2𝑛+2,𝑥_2𝑛+1⪯𝑢𝑥_2𝑛+1,𝑥_2𝑛⪯𝑑𝑥_2𝑛+1,𝑥_2𝑛,𝑑𝑥_2𝑛+1,𝑥_2𝑛⪯𝑢𝑥_2𝑛,𝑥_2𝑛−1⪯𝑑𝑥_2𝑛,𝑥_2𝑛−1. (5.11)
We prove now that 𝑑𝑥_𝑛,𝑥_𝑛+1⟶𝜃,𝑛⟶∞. (5.12) Indeed, passing to the limit in (5.11) when 𝑛→∞ (and using regularity of the cone), we obtain that 𝑑(𝑥_𝑛+1,𝑥_𝑛)→𝑟 and 𝑢(𝑥_𝑛+1,𝑥_𝑛)→𝑟 (𝑛→∞) for some 𝑟∈int𝑃∪{𝜃}. If 𝑟∈int𝑃, then passing to the limit in 𝑑𝑥_2𝑛+1,𝑥_2𝑛+2⪯𝑢𝑥_2𝑛,𝑥_2𝑛+1−𝜑𝑢𝑥_2𝑛,𝑥_2𝑛+1, (5.13) we obtain that 𝑟⪯𝑟−𝜑(𝑟) and 𝑟=𝜃 by the properties of function 𝜑∈Φ. Hence, (5.12) holds.
We next prove that {𝑥_𝑛} is a Cauchy sequence. According to monotonicity of {𝑑(𝑥_𝑛,𝑥_𝑛+1)} and (5.12), it is sufficient to show that the subsequence {𝑥_2𝑛} is a Cauchy sequence. Suppose that this is not the case. Applying Lemma 3.2 we obtain that there exist 𝑐≫𝜃 and two sequences of positive integers {𝑚_𝑘} and {𝑛_𝑘} such that the sequences 𝑑𝑥_2𝑚𝑘,𝑥_2𝑛𝑘,𝑑𝑥_2𝑚𝑘,𝑥_2𝑛𝑘+1,𝑑𝑥_{2𝑚𝑘−1},𝑥_2𝑛𝑘,𝑑𝑥_{2𝑚𝑘−1},𝑥_2𝑛𝑘+1 (5.14) all tend to 𝜑(𝑐) when 𝑘→∞.
Now, from (5.11) and the obtained limits, we have that lim_𝑘→∞𝑢𝑥_{2𝑚𝑘−1},𝑥_2𝑛𝑘=𝜑(𝑐), (5.15) for any 𝑢(𝑥_{2𝑚𝑘−1},𝑥_2𝑛𝑘)∈𝑁⁴_𝑓,𝑔(𝑥_{2𝑚𝑘−1},𝑥_2𝑛𝑘). Letting 𝑘→∞, utilizing (5.15) and the obtained limits, we get 𝜑(𝑐)⪯𝜑(𝑐)−𝜑𝜑(𝑐), (5.16) which is a contradiction if 𝑐≫𝜃. This shows that {𝑥_2𝑛} is a Cauchy sequence and hence {𝑥_𝑛} is a Cauchy sequence.
Since the space (𝑋,𝑑) is complete, there exists 𝑝∈𝑋 such that lim_𝑛→∞𝑥_𝑛=𝑝. Then also 𝑥_2𝑛+1=𝑓𝑥_2𝑛→𝑝 and 𝑥_2𝑛=𝑔𝑥_2𝑛−1→𝑝 (𝑛→∞). Putting 𝑥=𝑥_2𝑛 and 𝑦=𝑝 in (5.2), we get 𝑑(𝑓𝑥_2𝑛,𝑔𝑝)⪯𝑢−𝜑(𝑢), where 𝑢∈𝑑𝑥_2𝑛,𝑝,𝑑𝑥_2𝑛,𝑓𝑥_2𝑛,1,𝑑𝑝,𝑔𝑝2𝑑𝑥_2𝑛,𝑔𝑝+𝑑𝑝,𝑓𝑥_2𝑛. (5.17)
So, in this case we have four possibilities:(1°)𝑑(𝑓𝑥_2𝑛,𝑔𝑝)⪯𝑑(𝑥_2𝑛,𝑝)−𝜑(𝑑(𝑥_2𝑛,𝑝));(2°)𝑑(𝑓𝑥_2𝑛,𝑔𝑝)⪯𝑑(𝑥_2𝑛,𝑓𝑥_2𝑛)−𝜑(𝑑(𝑥_2𝑛,𝑓𝑥_2𝑛));(3°)𝑑(𝑓𝑥_2𝑛,𝑔𝑝)⪯𝑑(𝑝,𝑔𝑝)−𝜑(𝑑(𝑝,𝑔𝑝));(4°)𝑑(𝑓𝑥_2𝑛,𝑔𝑝)⪯(1/2)(𝑑(𝑥_2𝑛,𝑔𝑝)+𝑑(𝑝,𝑓𝑥_2𝑛))−𝜑((1/2)(𝑑(𝑥_2𝑛,𝑔𝑝)+𝑑(𝑝,𝑓𝑥_2𝑛))).Passing to the limit when 𝑛→∞ in these four relations, we obtain one of the next three inequalities: 𝑑(𝜃),𝑑𝑑,𝑑⪯1𝑝,𝑔𝑝⪯𝜃−𝜑𝑝,𝑔𝑝⪯𝑑𝑝,𝑔𝑝−𝜑𝑝,𝑔𝑝𝑝,𝑔𝑝2𝑑1𝑝,𝑔𝑝−𝜑2𝑑⪯1𝑝,𝑔𝑝2𝑑𝑝,𝑔𝑝. (5.18) In each of the cases it easily follows that 𝑔𝑝=𝑝.
Now, putting 𝑥=𝑦=𝑝 in (5.2), one gets 𝑑(𝑓𝑝,𝑔𝑝)⪯𝑢−𝜑(𝑢), where 𝑢∈{𝜃,𝑑(𝑝,𝑓𝑝),(1/2)𝑑(𝑝,𝑓𝑝)} and in each of the possible three cases it easily follows that 𝑓𝑝=𝑝. Hence, 𝑝 is a common fixed point of 𝑓 and 𝑔.

Putting 𝑔=𝑓 in Theorem 5.1, one obtains

Corollary 5.2. Let (𝑋,𝑑) be a complete cone metric space over a regular cone 𝑃 such that (𝑑₄) holds and suppose that there exists a continuous function 𝜑∈Φ. Let 𝑓∶𝑋→𝑋 be such that for all 𝑥,𝑦∈𝑋 there exists 𝑢𝑥,𝑦∈𝑀⁴_𝑓𝑥,𝑦 (5.19) such that 𝑑𝑓𝑥,𝑓𝑦⪯𝑢𝑥,𝑦−𝜑𝑢𝑥,𝑦. (5.20) Then 𝑓 has a unique common fixed point.

Note that putting 𝐸=ℝ and 𝑃=[0,+∞) in Theorem 5.1 and Corollary 5.2 we obtain as corollaries [16, Theorems  2.1 and 2.2], [15, Theorem  2.1], and [21, Theorem  4.1 and Corollary  4.2].

Adapting an example from [16] we give an example when Theorem 5.1 (modified to use a function 𝜓∈Ψ according to Remark 4.2) can be used to deduce the existence of a common fixed point.

Example 5.3. Let 𝑋=[0,1], and let 𝐸=ℝ², 𝑃={(𝑥,𝑦)∈𝐸∶𝑥≥0,𝑦≥0} and 𝑑(𝑥,𝑦)=(|𝑥−𝑦|,𝛼|𝑥−𝑦|) be as in Examples 4.4 and 4.7. Take 𝜑∈Φ defined by 𝜑(𝜃)=𝜃 and 𝜑(𝑡₁,𝑡₂)=(𝑡₁,𝑡₂) for 𝑡₁,𝑡₂>0; take 𝜓∈Ψ defined by 𝜓(𝑡₁,𝑡₂)=(3𝑡₁,3𝑡₂) for 𝑡₁,𝑡₂≥0 (they satisfy the conditions of Definition 2.1). Consider the mappings 𝑓,𝑔∶𝑋→𝑋 given as 𝑓𝑥=(1/3)𝑥 and 𝑔𝑥=0. Condition 𝜓𝑑𝑓𝑥,𝑔𝑦⪯𝜓𝑢𝑥,𝑦−𝜑𝑢𝑥,𝑦, (5.21) reduced to the first coordinates of respective vectors, has the form which was checked to be true in [16]. Hence, the existence of a common fixed point (𝑝=0) of mappings 𝑓 and 𝑔 follows from Theorem 5.1.

The next is a kind of Hardy-Rogers-type result with weak condition. It can be considered as a cone metric version of results from [19, 21]. For the sake of simplicity we take only one mapping 𝑓∶𝑋→𝑋 and for 𝑥,𝑦∈𝑋 denote Θ⁵_𝑓𝑥,𝑦=𝐴𝑑𝑥,𝑦+𝐵𝑑𝑥,𝑓𝑥+𝐶𝑑𝑦,𝑓𝑦+𝐷𝑑𝑥,𝑓𝑦+𝐸𝑑𝑦,𝑓𝑥, (5.22) where 𝐴>0, 𝐵,𝐶,𝐷,𝐸≥0, 𝐴+𝐵+𝐶+𝐷+𝐸≤1.

Theorem 5.4. Let (𝑋,𝑑) be a complete cone metric space over a regular cone 𝑃 such that (𝑑₄) holds and suppose that there exists a continuous function 𝜑∈Φ. Let 𝑓∶𝑋→𝑋 and suppose that for all 𝑥,𝑦∈𝑋, 𝑑𝑓𝑥,𝑓𝑦⪯Θ⁵_𝑓𝑥,𝑦−𝜑Θ⁵_𝑓𝑥,𝑦, (5.23) holds. Then 𝑓 has a unique fixed point.

Proof. The given condition (5.23) and properties of function 𝜑 imply that 𝑑𝑓𝑥,𝑓𝑦⪯Θ⁵_𝑓𝑥,𝑦 (5.24) for each 𝑥,𝑦∈𝑋. Starting with arbitrary 𝑥₀∈𝑋 construct the Picard sequence by 𝑥_𝑛+1=𝑓𝑥_𝑛. Condition (5.24) implies that 𝑑𝑥_𝑛+1,𝑥_𝑛+2=𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1⪯𝐴𝑑𝑥_𝑛,𝑥_𝑛+1+𝐵𝑑𝑥_𝑛,𝑥_𝑛+1+𝐶𝑑𝑥_𝑛+1,𝑥_𝑛+2+𝐷𝑑𝑥_𝑛,𝑥_𝑛+2+𝐸𝑑𝑥_𝑛+1,𝑥_𝑛+1⪯(𝐴+𝐵+𝐷)𝑑𝑥_𝑛,𝑥_𝑛+1+(𝐶+𝐷)𝑑𝑥_𝑛+1,𝑥_𝑛+2, (5.25) wherefrom (1−𝐶−𝐷)𝑑𝑥_𝑛+1,𝑥_𝑛+2⪯(𝐴+𝐵+𝐷)𝑑𝑥_𝑛,𝑥_𝑛+1, (5.26) and, similarly, (1−𝐵−𝐸)𝑑𝑥_𝑛+2,𝑥_𝑛+1⪯(𝐴+𝐶+𝐸)𝑑𝑥_𝑛+1,𝑥_𝑛. (5.27) Adding up, one obtains that 𝑑𝑥_𝑛+1,𝑥_𝑛+2⪯𝜆𝑑𝑥_𝑛,𝑥_𝑛+1, (5.28) where 𝜆=(2𝐴+𝐵+𝐶+𝐷+𝐸)/(2−𝐵−𝐶−𝐷−𝐸)≤1. It follows that {𝑑(𝑥_𝑛+1,𝑥_𝑛)} is a decreasing sequence which (by the regularity of cone 𝑃) tends to some 𝑟∈int𝑃∪{𝜃}. In order to prove that 𝑟=𝜃, put 𝑥=𝑥_𝑛+1 and 𝑦=𝑥_𝑛 in (5.23) to obtain 𝑑𝑥_𝑛+2,𝑥_𝑛+1⪯Θ⁵_𝑓𝑥_𝑛+1,𝑥_𝑛−𝜑Θ⁵_𝑓𝑥_𝑛+1,𝑥_𝑛, (5.29) where Θ⁵_𝑓𝑥_𝑛+1,𝑥_𝑛=𝐴𝑑𝑥_𝑛+1,𝑥_𝑛+𝐵𝑑𝑥_𝑛+1,𝑥_𝑛+2+𝐶𝑑𝑥_𝑛,𝑥_𝑛+1+𝐷𝑑𝑥_𝑛+1,𝑥_𝑛+1+𝐸𝑑𝑥_𝑛,𝑥_𝑛+2⪯(𝐴+𝐶+𝐸)𝑑𝑥_𝑛,𝑥_𝑛+1+(𝐵+𝐸)𝑑𝑥_𝑛+1,𝑥_𝑛+2. (5.30) Similarly, Θ⁵_𝑓𝑥_𝑛,𝑥_𝑛+1⪯(𝐴+𝐶+𝐷)𝑑𝑥_𝑛,𝑥_𝑛+1+(𝐵+𝐷)𝑑𝑥_𝑛+1,𝑥_𝑛+2. (5.31) On the other hand, (5.24) implies that Θ⁵_𝑓𝑥_𝑛+1,𝑥_𝑛⪰𝑑𝑥_𝑛+1,𝑥_𝑛+2. (5.32) In the case when 𝐷=𝐸, passing to the limit when 𝑛→∞, we obtain that lim_𝑛→∞Θ⁵_𝑓(𝑥_𝑛+1,𝑥_𝑛)=𝑟; the same conclusion is obtained if 𝐷<𝐸 (or 𝐷>𝐸). Hence, passing to the limit in (5.29), we get that 𝑟⪯𝑟−𝜑(𝑟), wherefrom 𝑟=𝜃.
As in some previous proofs, in order to obtain that {𝑥_𝑛} is a Cauchy sequence, suppose that it is not the case and using Lemma 3.2 deduce that there exist 𝑐≫𝜃 and two sequences {𝑚_𝑘} and {𝑛_𝑘} of positive integers such that 𝑛_𝑘>𝑚_𝑘>𝑘 and the sequences 𝑑𝑥_2𝑚𝑘,𝑥_2𝑛𝑘,𝑑𝑥_2𝑚𝑘,𝑥_2𝑛𝑘+1,𝑑𝑥_{2𝑚𝑘−1},𝑥_2𝑛𝑘,𝑑𝑥_{2𝑚𝑘−1},𝑥_2𝑛𝑘+1 (5.33) all tend to 𝜑(𝑐). Putting 𝑥=𝑥_2𝑛𝑘 and 𝑦=𝑥_{2𝑚𝑘−1} in (5.23) gives 𝑑𝑥_2𝑛𝑘+1,𝑥_2𝑚𝑘=𝑑𝑓𝑥_2𝑛𝑘,𝑓𝑥_{2𝑚𝑘−1}⪯Θ⁵_𝑓𝑥_2𝑛𝑘,𝑥_{2𝑚𝑘−1}−𝜑Θ⁵_𝑓𝑥_2𝑛𝑘,𝑥_{2𝑚𝑘−1}. (5.34) Here Θ⁵_𝑓𝑥_2𝑛𝑘,𝑥_{2𝑚𝑘−1}=𝐴𝑑𝑥_2𝑛𝑘,𝑥_{2𝑚𝑘−1}+𝐵𝑑𝑥_2𝑛𝑘,𝑥_2𝑛𝑘+1+𝐶𝑑𝑥_{2𝑚𝑘−1},𝑥_2𝑛𝑘+𝐷𝑑𝑥_2𝑛𝑘,𝑥_2𝑚𝑘+𝐸𝑑𝑥_{2𝑚𝑘−1},𝑥_2𝑛𝑘+1⟶𝐴𝜑(𝑐)+𝐵⋅𝜃+𝐶⋅𝜃+𝐷𝜑(𝑐)+𝐸𝜑(𝑐)=(𝐴+𝐷+𝐸)𝜑(𝑐), (5.35) when 𝑘→∞. Since also 𝑑(𝑥_2𝑛𝑘+1,𝑥_2𝑚𝑘)→𝜑(𝑐) when 𝑘→∞, we obtain that 𝜑(𝑐)⪯(𝐴+𝐷+𝐸)𝜑(𝑐)−𝜑(𝐴+𝐷+𝐸)𝜑(𝑐)⪯𝜑(𝑐)−𝜑(𝐴+𝐷+𝐸)𝜑(𝑐), (5.36) implying that 𝜑(𝑐)=𝜃 (because 𝐴>0).
Thus, the sequence {𝑥_𝑛} converges to some 𝑧 in the complete metric space 𝑋. In order to prove that 𝑓𝑧=𝑧, suppose the contrary and put 𝑥=𝑥_𝑛 and 𝑦=𝑧 in (5.24). It follows that 𝑑𝑓𝑥_𝑛,𝑓𝑧⪯𝐴𝑑𝑥_𝑛,𝑧+𝐵𝑑𝑥_𝑛,𝑥_𝑛+1+𝐶𝑑𝑧,𝑓𝑧+𝐷𝑑𝑥_𝑛,𝑓𝑧+𝐸𝑑𝑧,𝑥_𝑛+1. (5.37) Passing to the limit when 𝑛→∞ gives that 𝑑𝑧,𝑓𝑧⪯(𝐶+𝐷)𝑑𝑧,𝑓𝑧≺(𝐴+𝐵+𝐶+𝐷+𝐸)𝑑𝑧,𝑓𝑧⪯𝑑𝑧,𝑓𝑧, (5.38) a contradiction, since 𝐴>0.
The proof that the fixed point of 𝑓 is unique is standard.

In a similar way one can obtain a version of the previous theorem containing two selfmaps 𝑓 and 𝑔 (see [21, Theorem  5.2]).

At the end, we again state a cone metric version of a result from [18, Theorems  3.2 and 4.2] (see also [21, Theorem  3.7]).

Theorem 5.5. Let (𝑋,𝑑) be a cone metric space over a regular cone 𝑃 such that (𝑑₄) holds and suppose that there exists a continuous function 𝜑∈Φ. Let 𝑓,𝑔∶𝑋→𝑋 be two selfmaps such that for all 𝑥,𝑦∈𝑋 there exist 𝑢𝑥,𝑦∈𝑁⁴_𝑓𝑥,𝑦,𝑣𝑥,𝑦∈𝑑𝑥,𝑦,𝑑𝑥,𝑓𝑥,𝑑𝑦,𝑔𝑦, (5.39) such that 𝑑(𝑓𝑥,𝑔𝑦)⪯𝑢(𝑥,𝑦)−𝜑(𝑣(𝑥,𝑦)). Then 𝑓 and 𝑔 have a common fixed point.

Here also common fixed point of 𝑓 and 𝑔 need not be unique.

Acknowledgments

The first author acknowledges support from the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province of China (20114BAB211002), the Foundation of Jiangxi Provincial Education Department (GJJ12173), and the Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University. The second and fourth authors are thankful to the Ministry of Science and Technological Development of Serbia.