Coincidence Points for Expansive Mappings under c-Distance in Cone Metric Spaces

Abstract

We establish some fixed (common fixed) and coincidence point results for mappings verifying some expansive type contractions in cone metric spaces with the help of the concept of a c-distance. Our results generalize, extend, and unify several well-known comparable results in the literature. Some examples are also presented.

1. Introduction and Preliminaries

Huang and Zhang [1] reintroduced the notion of cone metric spaces and established fixed point theorems for mappings on this space. After that, many fixed point theorems have been proved in normal or nonnormal cone metric spaces by some authors (see e.g., [1–26] and references contained therein).

We need to recall some basic notations, definitions, and necessary results from literature. Let ℝ₊ be the set of nonnegative real numbers. Let 𝐸 be a real Banach space and 0_𝐸 is the zero vector of 𝐸.

Definition 1.1 (see [1]). A nonempty subset 𝑃 of 𝐸 is called a cone if the following conditions hold:(i)𝑃 is closed and 𝑃≠{0_𝐸},(ii)𝑎,𝑏∈ℝ,𝑎,𝑏≥0,𝑥,𝑦∈𝑃⇒𝑎𝑥+𝑏𝑦∈𝑃,(iii)𝑥∈𝑃,−𝑥∈𝑃⇒𝑥=0_𝐸.
Given a cone 𝑃⊂𝐸, a partial ordering ≤_𝐸 with respect to 𝑃 is naturally defined by 𝑥≤_𝐸𝑦 if and only if 𝑦−𝑥∈𝑃, for 𝑥,𝑦∈𝐸. We will write 𝑥<_𝐸𝑦 to indicate that 𝑥≤_𝐸𝑦 but 𝑥≠𝑦, while 𝑥≪𝑦 will stand for 𝑦−𝑥∈int𝑃, where int𝑃 denotes the interior of 𝑃. A cone 𝑃 is said solid if int𝑃 is non-empty.

Definition 1.2 (see [1]). Let 𝑋 be a non-empty set and 𝑑∶𝑋×𝑋→𝑃 satisfies(i)𝑑(𝑥,𝑦)=0_𝐸 if and only if 𝑥=𝑦,(ii)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋,(iii)𝑑(𝑥,𝑦)≤_𝐸𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦) for all 𝑥,𝑦,𝑧∈𝐸.Then, the mapping 𝑑 is called a cone metric on 𝑋 and the pair (𝑋,𝑑) is called a cone metric space.

Definition 1.3 (see [1]). Let (𝑋,𝑑) be a cone metric space, {𝑥_𝑛} is a sequence in 𝑋 and 𝑥∈𝑋.(i)If for every 𝑐∈𝐸 with 0_𝐸≪_𝐸𝑐 there is 𝑁∈ℕ such that 𝑑(𝑥_𝑛,𝑥)≪_𝐸𝑐 for all 𝑛≥𝑁, then {𝑥_𝑛} is said to be convergent to 𝑥. This limit is denoted by lim_𝑛→+∞𝑥_𝑛=𝑥 or 𝑥_𝑛→𝑥 as 𝑛→+∞.(ii)If for every 𝑐∈𝐸 with 0_𝐸≪_𝐸𝑐, there is 𝑁∈ℕ such that 𝑑(𝑥_𝑛,𝑥_𝑚)≪_𝐸𝑐 for all 𝑛,𝑚>𝑁, then {𝑥_𝑛} is called a Cauchy sequence in 𝑋.(iii)If every Cauchy sequence in 𝑋 is convergent in 𝑋, then (𝑋,𝑑) is called a complete cone metric space.

Definition 1.4. Let (𝑋,𝑑) be a cone metric space and let 𝑇∶𝑋→𝑋 be a given mapping. We say that 𝑇 is continuous on 𝑥₀∈𝑋 if for every sequence {𝑥_𝑛} is 𝑋, we have 𝑥_𝑛⟶𝑥₀as𝑛⟶∞⟹𝑇𝑥_𝑛⟶𝑇𝑥₀as𝑛⟶∞. (1.1) If 𝑇 is continuous on each point 𝑥₀∈𝑋, then we say that 𝑇 is continuous on 𝑋.

In 2011, Cho et al. [11] and Wang and Guo [27] introduced a new concept of 𝑐-distance in cone metric spaces, which is a cone version of 𝑤-distance of Kada et al. [16] and proved some fixed point theorems for some contractive-type mappings in partially ordered cone metric spaces using the 𝑐-distance. For other results, see [13, 24].

Definition 1.5 (see [11, 27]). Let (𝑋,𝑑) be a cone metric space. Then, a function 𝑞∶𝑋→𝐸 is called a 𝑐-distance on 𝑋 if the following are satisfied:(q1)𝑞(𝑥,𝑦)≥_𝐸0_𝐸 for all 𝑥,𝑦∈𝑋,(q2)𝑞(𝑥,𝑧)≤_𝐸𝑞(𝑥,𝑦)+𝑞(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋,(q3) for each 𝑥∈𝑋 and 𝑛≥1, if 𝑞(𝑥,𝑦_𝑛)≤_𝐸𝑢 for some 𝑢=𝑢_𝑥∈𝑃, then 𝑞(𝑥,𝑦)≤_𝐸𝑢 whenever {𝑦_𝑛} is a sequence in 𝑋 converging to a point 𝑦∈𝑋,(q4) for all 𝑐∈𝐸 with 0_𝐸≪𝑐, there exists 𝑒∈𝐸 with 0_𝐸≪𝑒 such that 𝑞(𝑧,𝑥)≪𝑒 and 𝑞(𝑧,𝑦)≪𝑒 imply 𝑑(𝑥,𝑦)≪𝑐.

Remark 1.6 (see [11, 27]). The 𝑐-distance 𝑞 is a 𝑤-distance on 𝑋 if we take (𝑋,𝑑) is a metric space, 𝐸=ℝ₊, 𝑃=[0,∞), and (𝑞3) is replaced by the following condition. For any 𝑥∈𝑋, 𝑞(𝑥,⋅)∶𝑋→ℝ₊ is lower semicontinuous. Moreover, (𝑞3) holds whenever 𝑞(𝑥,⋅) is lower semicontinuous. Thus, if (𝑋,𝑑) is a metric space, 𝐸=ℝ₊ and 𝑃=[0,∞), then every 𝑤-distance is a 𝑐-distance. But the converse is not true in general case. Therefore, the 𝑐-distance is a generalization of the 𝑤-distance.

Example 1.7 (see [11, 27]). Let (𝑋,𝑑) be a cone metric space and let 𝑃 be a normal cone. Define a mapping 𝑞∶𝑋×𝑋→𝐸 by 𝑞(𝑥,𝑦)=𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋. Then, 𝑞 is a 𝑐-distance.

Example 1.8 (see [11, 27]). Let (𝑋,𝑑) be a cone metric space and let 𝑃 be a normal cone. Define a mapping 𝑞∶𝑋×𝑋→𝐸 by 𝑞(𝑥,𝑦)=𝑑(𝑢,𝑦) for all 𝑥,𝑦∈𝑋, where 𝑢 is a fixed point in X. Then, 𝑞 is a 𝑐-distance.

Example 1.9. Let (𝑋,𝑑) be a cone metric space and let 𝑃 be a normal cone. Define a mapping 𝑞∶𝑋×𝑋→𝐸 by 𝑞(𝑥,𝑦)=(𝑑(𝑥,𝑢)+𝑑(𝑢,𝑦))/2 for all 𝑥,𝑦∈𝑋, where 𝑢 is a fixed point in X. Then 𝑞 is a 𝑐-distance.

Remark 1.10 (see [11, 27]). (1) 𝑞(𝑥,𝑦)=𝑞(𝑦,𝑥) does not necessarily hold for all 𝑥,𝑦∈𝑋.
(2) 𝑞(𝑥,𝑦)=0_𝐸 is not necessarily equivalent to 𝑥=𝑦 for all 𝑥,𝑦∈𝑋.

Lemma 1.11 (see [11, 27]). Let (𝑋,𝑑) be a cone metric space and let 𝑞 be a 𝑐-distance on 𝑋. Let {𝑥_𝑛} and {𝑦_𝑛} be a sequences in 𝑋 and 𝑥,𝑦,𝑧∈𝑋. Suppose that {𝑢_𝑛} is a sequence in 𝑃 converging to 0_𝐸. The following hold. (1)If 𝑞(𝑥_𝑛,𝑦)≤_𝐸𝑢_𝑛 and 𝑞(𝑥_𝑛,𝑧)≤_𝐸𝑢_𝑛, then 𝑦=𝑧. (2)If 𝑞(𝑥_𝑛,𝑦_𝑛)≤_𝐸𝑢_𝑛 and 𝑞(𝑥_𝑛,𝑧)≤_𝐸𝑢_𝑛, then {𝑦_𝑛} converges to 𝑧. (3)If 𝑞(𝑥_𝑛,𝑥_𝑚)≤_𝐸𝑢_𝑛 for all 𝑚>𝑛, then {𝑥_𝑛} is a Cauchy sequence in 𝑋. (4)If 𝑞(𝑦,𝑥_𝑛)≤_𝐸𝑢_𝑛, then {𝑥_𝑛} is a Cauchy sequence in 𝑋.

Let 𝑓,𝑔∶𝑋→𝑋 be two selfmaps on a nonempty set 𝑋. Recall that a point 𝑥∈𝑋 is called a coincidence point of the pair (𝑓,𝑔) if 𝑓𝑥=𝑔𝑥. The point 𝑦=𝑓𝑥=𝑔𝑥 is called a point of coincidence. If 𝑥=𝑓𝑥=𝑔𝑥, then 𝑥 is said a common fixed point of 𝑓 and 𝑔.

The purpose of this paper is to give some common fixed and coincidence point theorems for mappings verifying some expansive type contractions on cone metric spaces via a 𝑐-distance. Also, some examples are presented.

2. Main results

First, we present the following useful lemma, which is a variant of (2.2) in Lemma 1.11.

Lemma 2.1. Let (𝑋,𝑑) be a cone metric space and let 𝑞 be a 𝑐-distance on 𝑋. Let {𝑥_𝑛} be a sequence in 𝑋. Suppose that {𝛼_𝑛} and {𝛽_𝑛} are tow sequences in 𝑃 converging to 0_𝐸. If 𝑞(𝑥_𝑛,𝑦)≤_𝐸𝛼_𝑛 and 𝑞(𝑥_𝑛,𝑧)≤_𝐸𝛽_𝑛, then 𝑦=𝑧.

Proof. Let 𝑐≫0_𝐸 be arbitrary. Since 𝛼_𝑛→0_𝐸, so there exists 𝑁₁∈ℕ such that 𝛼_𝑛≪𝑐/2 for all 𝑛≥𝑁₁. Similarly, there exists 𝑁₂∈ℕ such that 𝛽_𝑛≪𝑐/2 for all 𝑛≥𝑁₁. Thus, for all 𝑁≥max{𝑁₁,𝑁₂}, we have 𝑞𝑥_𝑛≪𝑐,𝑦2,𝑞𝑥_𝑛≪𝑐,𝑧2. (2.1) Take 𝑒=𝑐/2, so by (𝑞4), we get that 𝑑(𝑦,𝑧)≪𝑐 for each 𝑐≫0_𝐸, hence 𝑦=𝑧.

Now, we present coincidence point results in the frame work of cone metric spaces in terms of a 𝑐-distance. Note that (2.2) is called an expansive type contraction.

Theorem 2.2. Let (𝑋,𝑑) be a cone metric space and let 𝑞 be a 𝑐-distance on 𝑋. Let 𝑓,𝑇∶𝑋→𝑋 be two functions such that there are three nonnegative real numbers 𝑎, 𝑏, and 𝑐 with 𝑎+𝑏+𝑐>1 such that 𝑞𝑇𝑥,𝑇𝑦≥_𝐸𝑎𝑞𝑓𝑥,𝑓𝑦+𝑏𝑞𝑇𝑥,𝑓𝑥+𝑐𝑞𝑇𝑦,𝑓𝑦. (2.2) Assume the following hypotheses: (1)𝑏<1 and 𝑎≠0, (2)𝑓(𝑋)⊆𝑇(𝑋), (3)(𝑇(𝑋),𝑑) is a complete subset of (𝑋,𝑑). Then 𝑇 and 𝑓 have a coincidence point, say 𝑢∈𝑋. One has 𝑞(𝑇𝑢,𝑇𝑢)=0_𝐸.
Also, if 𝑎>1, then the point of coincidence 𝑥=𝑇𝑢=𝑓𝑢 is unique.

Proof. Let 𝑥₀∈𝑋. Since 𝑓(𝑋)⊆𝑇(𝑋), we can choose 𝑥₁∈𝑋 such that 𝑇𝑥₁=𝑓𝑥₀. Again since 𝑓(𝑋)⊆𝑇(𝑋), we can choose 𝑥₂∈𝑋 such that 𝑇𝑥₂=𝑓𝑥₁. Continuing this process, we can construct a sequence {𝑥_𝑛} in 𝑋 such that 𝑇𝑥_𝑛=𝑓𝑥_𝑛−1 for all 𝑛≥1.
By (2.2), we have 𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛=𝑞𝑇𝑥_𝑛,𝑇𝑥_𝑛+1≥_𝐸𝑎𝑞𝑓𝑥_𝑛,𝑓𝑥_𝑛+1+𝑏𝑞𝑇𝑥_𝑛,𝑓𝑥_𝑛+𝑐𝑞𝑇𝑥_𝑛+1,𝑓𝑥_𝑛+1=𝑎𝑞𝑓𝑥_𝑛,𝑓𝑥_𝑛+1+𝑏𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛+𝑐𝑞𝑓𝑥_𝑛,𝑓𝑥_𝑛+1. (2.3) Therefore, (1−𝑏)𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛≤_𝐸(𝑎+𝑐)𝑞𝑓𝑥_𝑛,𝑓𝑥_𝑛+1. (2.4) Set 𝜆=(1−𝑏)/(𝑎+𝑐). By hypotheses, we have 𝜆∈(0,1). Also, 𝑞𝑓𝑥_𝑛,𝑓𝑥_𝑛+1≤_𝐸𝜆𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛∀𝑛≥1. (2.5) By induction, we get that 𝑞𝑓𝑥_𝑛,𝑓𝑥_𝑛+1≤_𝐸𝜆^𝑛𝑞𝑓𝑥₀,𝑓𝑥₁∀𝑛≥0. (2.6) Let 𝑚>𝑛. By (𝑞2), we have 𝑞𝑓𝑥_𝑛,𝑓𝑥_𝑚≤_𝐸^𝑚−1_𝑖=𝑛𝑞𝑓𝑥_𝑖,𝑓𝑥_𝑖+1≤_𝐸𝜆^𝑛𝑞1−𝜆𝑓𝑥₀,𝑓𝑥₁. (2.7) Since 𝜆<1, so by Lemma 1.11(3), the sequence {𝑇𝑥_𝑛=𝑓𝑥_𝑛−1} is Cauchy in (𝑇(𝑋),𝑑), which is complete, hence there exists 𝑢∈𝑋 such that 𝑇𝑥_𝑛→𝑇𝑢. Thus, 𝑓𝑥_𝑛→𝑇𝑢 as 𝑛→∞, that is, lim_𝑛→+∞𝑑𝑇𝑥_𝑛,𝑇𝑢=lim_𝑛→+∞𝑑𝑓𝑥_𝑛,𝑇𝑢=0. (2.8) We claim that 𝑇𝑢=𝑓𝑢. Recall that 𝑞(𝑓𝑥_𝑛,𝑓𝑥_𝑚)≤_𝐸(𝜆^𝑛/(1−𝜆))𝑞(𝑓𝑥₀,𝑓𝑥₁) and 𝑓𝑥_𝑛→𝑇𝑢, so by (𝑞3) and as 𝑚→∞, we get that 𝑞𝑓𝑥_𝑛,𝑇𝑢≤_𝐸𝜆^𝑛𝑞1−𝜆𝑓𝑥₀,𝑓𝑥₁. (2.9) From (2.2), we have 𝑞𝑓𝑥_𝑛−1,𝑇𝑢=𝑞𝑇𝑥_𝑛≥,𝑇𝑢_𝐸𝑎𝑞𝑓𝑥_𝑛,𝑓𝑢+𝑏𝑞𝑇𝑥_𝑛,𝑓𝑥_𝑛≥+𝑐𝑞𝑇𝑢,𝑓𝑢_𝐸𝑎𝑞𝑓𝑥_𝑛,𝑓𝑢. (2.10) Since 𝑎≠0, so it follows that 𝑞𝑓𝑥_𝑛,𝑓𝑢≤_𝐸1𝑎𝑞𝑓𝑥_𝑛−1,𝑇𝑢. (2.11) By (2.9), we obtain 𝑞𝑓𝑥_𝑛,𝑓𝑢≤_𝐸1𝑎𝑞𝑓𝑥_𝑛−1,𝑇𝑢≤_𝐸1𝑎𝜆^𝑛−1𝑞1−𝜆𝑓𝑥₀,𝑓𝑥₁. (2.12) Set 𝛼_𝑛=∶(𝜆^𝑛/(1−𝜆))𝑞(𝑓𝑥₀,𝑓𝑥₁) and 𝛽_𝑛=∶(1/𝑎)(𝜆^𝑛−1/(1−𝜆))𝑞(𝑓𝑥₀,𝑓𝑥₁). Since 𝜆<1, so 𝛼_𝑛,𝛽_𝑛→0_𝐸 as 𝑛→∞. Thus, by (2.9), (2.12), and Lemma 2.1, get that 𝑇𝑢=𝑓𝑢.
Using (2.2), we get 𝑞(𝑇𝑢,𝑇𝑢)≥_𝐸(𝑎+𝑏+𝑐)𝑞(𝑇𝑢,𝑇𝑢). (2.13) Since 𝑎+𝑏+𝑐>1, so 𝑞(𝑇𝑢,𝑇𝑢)=0_𝐸. Now, we prove that if 𝑎>1, then the point of coincidence 𝑥=𝑇𝑢=𝑓𝑢 is unique.
Let 𝑥 and 𝑦 be two points of coincidence of 𝑇 and 𝑓, that is, there exist 𝑢,𝑣∈𝑋 such that 𝑥=𝑇𝑢=𝑓𝑢 and 𝑦=𝑇𝑣=𝑓𝑣. By the above, we have 𝑞(𝑇𝑢,𝑇𝑢)=𝑞(𝑇𝑣,𝑇𝑣)=0_𝐸. By (2.2), the following holds: 𝑞(𝑇𝑢,𝑇𝑣)≥_𝐸()()()().𝑎𝑞𝑓𝑢,𝑓𝑣+𝑏𝑞𝑇𝑢,𝑓𝑢+𝑐𝑞𝑇𝑣,𝑓𝑣=𝑎𝑞𝑇𝑢,𝑇𝑣+𝑏𝑞𝑇𝑢,𝑇𝑢+𝑐𝑞𝑇𝑣,𝑇𝑣=𝑎𝑞𝑇𝑢,𝑇𝑣 (2.14) If 𝑎>1, we conclude that 𝑞(𝑇𝑢,𝑇𝑣)=0.
Let 𝑐≫0_𝐸 be arbitrary. Take 𝑒=𝑐≫0_𝐸. Since 𝑞(𝑇𝑢,𝑇𝑢)=𝑞(𝑇𝑢,𝑇𝑣)=0_𝐸≪𝑒, then by the condition (𝑞4), we get that 𝑑(𝑇𝑢,𝑇𝑣)≪𝑐. Thus, 𝑥=𝑇𝑢=𝑇𝑣=𝑦. This completes the proof.

Now, we state the following corollaries.

Corollary 2.3. Let (𝑋,𝑑) be a cone metric space and let 𝑞 be a 𝑐-distance on 𝑋. Let 𝑓,𝑇∶𝑋→𝑋 be two functions. Assume there exists 𝑎>1 such that 𝑞𝑇𝑥,𝑇𝑦≥_𝐸𝑎𝑞𝑓𝑥,𝑓𝑦. (2.15) Assume the following hypotheses: (1)𝑓(𝑋)⊆𝑇(𝑋), (2)(𝑇(𝑋),𝑑) is a complete subset of 𝑋. Then, 𝑇 and 𝑓 have a coincidence point, say 𝑢∈𝑋. One has 𝑞(𝑇𝑢,𝑓𝑢)=0_𝐸 and 𝑥=𝑇𝑢=𝑓𝑢 is the unique point of coincidence of 𝑇 and 𝑓.

Proof. It follows by taking 𝑏=𝑐=0 in Theorem 2.2.

Corollary 2.4. Let (𝑋,𝑑) be a cone metric space and let 𝑞 be a 𝑐-distance on 𝑋. Let 𝑓,𝑇∶𝑋→𝑋 be two functions. Assume there exist two nonnegative real numbers 𝑎 and 𝑏, with 𝑎+𝑏>1 such that 𝑞𝑇𝑥,𝑇𝑦≥_𝐸𝑎𝑞𝑓𝑥,𝑓𝑦+𝑏𝑞𝑇𝑥,𝑓𝑥. (2.16) Assume the following hypotheses: (1)𝑏<1, (2)𝑓(𝑋)⊆𝑇(𝑋), (3)(𝑇(𝑋),𝑑) is a complete subset of 𝑋. Then, 𝑇 and 𝑓 have a coincidence point, say 𝑢∈𝑋. One has 𝑞(𝑇𝑢,𝑇𝑢)=0_𝐸 and 𝑥=𝑇𝑢=𝑓𝑢 is the unique point of coincidence of 𝑇 and 𝑓.

Proof. It follows by taking 𝑐=0 in Theorem 2.2.

Corollary 2.5. Let (𝑋,𝑑) be a complete cone metric space and let 𝑞 be a 𝑐-distance on 𝑋. Let 𝑇∶𝑋→𝑋 be a surjective function. Assume there are three nonnegative real numbers 𝑎, 𝑏 and 𝑐 with 𝑎+𝑏+𝑐>1 such that 𝑞𝑇𝑥,𝑇𝑦≥_𝐸𝑎𝑞𝑥,𝑦+𝑏𝑞(𝑇𝑥,𝑥)+𝑐𝑞𝑇𝑦,𝑦. (2.17) If 𝑏<1 and 𝑎≠0, then 𝑇 has a fixed point, say 𝑢∈𝑋. One has 𝑞(𝑇𝑢,𝑇𝑢)=0_𝐸. Also if 𝑎>1, then the fixed point 𝑢 is unique.

Proof. It follows by taking 𝑓=Id_𝑋, the identity on 𝑋, in Theorem 2.2. Note that when 𝑇 is surjective, so 𝑇(𝑋)=𝑋, that is, the hypothesis (2) in Theorem 2.2 holds.

The next result is similar to Theorem 2.2, except that the contractive condition (2.2) is replaced by 𝑞𝑇𝑦,𝑇𝑥≥_𝐸𝑎𝑞𝑓𝑦,𝑓𝑥+𝑏𝑞𝑇𝑥,𝑓𝑥+𝑐𝑞𝑇𝑦,𝑓𝑦. (2.18) Note that this contractive condition is studied since it is different to (2.2) because of Remark 1.10.(2). Its proof is essentially the same as for Theorem 2.2 and so is omitted.

Theorem 2.6. Let (𝑋,𝑑) be a cone metric space and let 𝑞 be a 𝑐-distance on 𝑋. Let 𝑓,𝑔∶𝑋→𝑋 be two functions such that there are three nonnegative real numbers 𝑎, 𝑏, and 𝑐 with 𝑎+𝑏+𝑐>1 such that 𝑞𝑇𝑦,𝑇𝑥≥_𝐸𝑎𝑞𝑓𝑦,𝑓𝑥+𝑏𝑞𝑇𝑥,𝑓𝑥+𝑐𝑞𝑇𝑦,𝑓𝑦. (2.19) Assume the following hypotheses: (1)𝑐<1 and 𝑎≠0, (2)𝑓(𝑋)⊆𝑇(𝑋), (3)(𝑇(𝑋),𝑑) is a complete subset of 𝑋. Then, 𝑇 and 𝑓 have a coincidence point say 𝑢∈𝑋. One has 𝑞(𝑇𝑢,𝑇𝑢)=0_𝐸. If 𝑎>1, 𝑥=𝑇𝑢=𝑓𝑢 is the unique point of coincidence of 𝑇 and 𝑓.

Remark 2.7. Let (𝑋,𝑑) be a cone metric space and let 𝑃 be a normal cone. Take in Theorem 2.2 or Theorem 2.6 the 𝑐-distance 𝑞∶𝑋×𝑋→𝐸 defined by 𝑞(𝑥,𝑦)=𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋. Then, the inequalities (2.2) and (2.19) correspond to the contractive condition given in Theorem 2.1 of Shatanawi and Awawdah [28]. Thus, our results (Theorems 2.2 and 2.6) extend and generalize the results in [28].

Remark 2.8. Some similar results as above corollaries could be derived from Theorem 2.6.

Now, we present the following example.

Example 2.9. Let 𝐸=𝐶¹_ℝ[0,1] with ‖𝑥‖_𝐸=‖𝑥‖_∞+‖𝑥^′‖_∞ and 𝑃={𝑥∈𝐸,𝑥(𝑡)≥0,𝑡∈[0,1]}. Let 𝑋=[0,∞) and let 𝑑∶𝑋×𝑋→𝐸 be defined by 𝑑(𝑥,𝑦)(𝑡)=|𝑥−𝑦|2^𝑡. Then, (𝑋,𝑑) is a cone metric space. Let, further, 𝑞∶𝑋×𝑋→𝐸 be defined by 𝑞(𝑥,𝑦)(𝑡)=2^𝑡𝑦. It is easy to check that 𝑞 is a 𝑐-distance. Consider the mappings 𝑓,𝑇∶𝑋→𝑋 defined by 3𝑇𝑥=49𝑥,𝑓𝑥=20𝑥. (2.20) Take 𝑎=4/3 and 𝑏=𝑐=0 (we have 𝑎+𝑏+𝑐>1, 𝑏<1 and 𝑎>1). For all 𝑥,𝑦∈𝑋, we have 𝑞𝑇𝑥,𝑇𝑦=2^𝑡34𝑦≥2^𝑡35𝑦=𝑎𝑞𝑓𝑥,𝑓𝑦. (2.21) All hypotheses of Corollary 2.3 are satisfied, and 𝑢=0 is a coincidence point of 𝑓 and 𝑇. Also, 𝑞(𝑇𝑢,𝑇𝑢)=𝑞(0,0)=0 and 𝑥=𝑇𝑢=𝑇0=0 is the unique point of coincidence of 𝑇 and 𝑓.

Next, we present a common fixed point theorem for two maps involving some expansive type contractions given by the conditions (2.22).

Theorem 2.10. Let (𝑋,𝑑) be a complete cone metric space and let 𝑞 be a 𝑐-distance on 𝑋. Let 𝑇,𝑆∶𝑋→𝑋 be two mappings. Suppose that 𝑇 and 𝑆 satisfy the following inequalities: 𝑞(𝑇(𝑆𝑥),𝑆𝑥)+𝑘𝑞(𝑇(𝑆𝑥),𝑥)≥_𝐸(),𝑞(𝑆())(𝑆())≥𝑎𝑞𝑆𝑥,𝑥𝑇𝑥,𝑇𝑥+𝑘𝑞𝑇𝑥,𝑥_𝐸𝑏𝑞(𝑇𝑥,𝑥) (2.22) for all 𝑥∈𝑋 and some nonnegative real numbers 𝑎, 𝑏, and 𝑘 with 𝑎>1+2𝑘 and 𝑏>1+2𝑘. If 𝑇 and 𝑆 are continuous and surjective, then 𝑇 and 𝑆 have a common fixed point.

Proof. Let 𝑥₀ be an arbitrary point in 𝑋. Since 𝑇 is surjective, there exists 𝑥₁∈𝑋 such that 𝑥₀=𝑇𝑥₁. Also, since 𝑆 is surjective, there exists 𝑥₂∈𝑋 such that 𝑥₂=𝑆𝑥₁. Continuing this process, we construct a sequence (𝑥_𝑛) in 𝑋 such that 𝑥_2𝑛=𝑇𝑥_2𝑛+1 and 𝑥_2𝑛+1=𝑆𝑥_2𝑛+2 for all 𝑛∈ℕ. Now, for 𝑛∈ℕ, we have 𝑞𝑇𝑆𝑥_2𝑛+2,𝑆𝑥_2𝑛+2+𝑘𝑞𝑇𝑆𝑥_2𝑛+2,𝑥_2𝑛+2≥_𝐸𝑎𝑞𝑆𝑥_2𝑛+2,𝑥_2𝑛+2. (2.23) Thus, we have 𝑞𝑥_2𝑛,𝑥_2𝑛+1+𝑘𝑞𝑥_2𝑛,𝑥_2𝑛+2≥_𝐸𝑎𝑞𝑥_2𝑛+1,𝑥_2𝑛+2. (2.24) By (𝑞2), we have 𝑞(𝑥_2𝑛,𝑥_2𝑛+1)+𝑞(𝑥_2𝑛+1,𝑥_2𝑛+2)≥_𝐸𝑝(𝑥_2𝑛,𝑥_2𝑛+2). Hence, we get that 𝑞𝑥_2𝑛,𝑥_2𝑛+1+𝑘𝑞𝑥_2𝑛,𝑥_2𝑛+1+𝑘𝑞𝑥_2𝑛+1,𝑥_2𝑛+2≥_𝐸𝑎𝑞𝑥_2𝑛+1,𝑥_2𝑛+2. (2.25) Therefore, 𝑞𝑥_2𝑛+1,𝑥_2𝑛+2≤_𝐸1+𝑘𝑞𝑥𝑎−𝑘^2𝑛,𝑥_2𝑛+1. (2.26) On other hand, we have 𝑞𝑆𝑇𝑥_2𝑛+1,𝑇𝑥_2𝑛+1+𝑘𝑞𝑆𝑇𝑥_2𝑛+1,𝑥_2𝑛+1≥_𝐸𝑏𝑞𝑇𝑥_2𝑛+1,𝑥_2𝑛+1. (2.27) Thus, 𝑞𝑥_2𝑛−1,𝑥_2𝑛+𝑘𝑞𝑥_2𝑛−1,𝑥_2𝑛+1≥_𝐸𝑏𝑞𝑥_2𝑛,𝑥_2𝑛+1. (2.28) Again, using (𝑞2), we have 𝑞𝑥_2𝑛−1,𝑥_2𝑛+𝑘𝑞𝑥_2𝑛−1,𝑥_2𝑛+𝑘𝑞𝑥_2𝑛,𝑥_2𝑛+1≥_𝐸𝑏𝑞𝑥_2𝑛,𝑥_2𝑛+1. (2.29) Hence, 𝑞𝑥_2𝑛,𝑥_2𝑛+1≤_𝐸1+𝑘𝑞𝑥𝑏−𝑘^2𝑛−1,𝑥_2𝑛. (2.30) Let 𝜆=max1+𝑘,𝑎−𝑘1+𝑘𝑏−𝑘. (2.31) Then, by combining (2.26) and (2.30), we have 𝑞𝑥_𝑛,𝑥_𝑛+1≤_𝐸𝜆𝑞𝑥_𝑛−1,𝑥_𝑛∀𝑛≥1. (2.32) Repeating (2.32) 𝑛-times, we get 𝑞𝑥_𝑛,𝑥_𝑛+1≤_𝐸𝜆^𝑛𝑞𝑥₀,𝑥₁. (2.33) Thus, for 𝑚>𝑛, we have 𝑞𝑥_𝑛,𝑥_𝑚≤_𝐸𝑞𝑥_𝑛,𝑥_𝑛+1⋯+𝑞𝑥_𝑚−1,𝑥_𝑚≤_𝐸𝜆^𝑛+⋯+𝜆^𝑚−1𝑞𝑥₀,𝑥₁≤_𝐸𝜆^𝑛𝑞𝑥1−𝜆⁰,𝑥₁. (2.34) By assumption, we get that 0≤𝜆<1. By Lemma 1.11(3), {𝑥_𝑛} is a Cauchy sequence in the complete cone metric space (𝑋,𝑑). Then, there exists 𝑣∈𝑋 such that 𝑥_𝑛→𝑣 as 𝑛→+∞. Since 𝑥_2𝑛+1→𝑣 and 𝑥_2𝑛→𝑣 as 𝑛→+∞, so clearly, the fact that 𝑇 and 𝑆 are continuous and uniqueness of limit yields that 𝑣=𝑇𝑣=𝑆𝑣, that is, 𝑣 is a common fixed point of 𝑇 and 𝑆.

Corollary 2.11. Let (𝑋,𝑑) be a complete cone metric space and let 𝑞 be a 𝑐-distance on 𝑋. Let 𝑇∶𝑋→𝑋 be a continuous surjective mapping satisfying 𝑞(𝑇(𝑇𝑥),𝑇𝑥)+𝑘𝑞(𝑇(𝑇𝑥),𝑥)≥_𝐸𝑎𝑞(𝑇𝑥,𝑥) (2.35) for all 𝑥∈𝑋 and some nonnegative real numbers 𝑎 and 𝑘 with 𝑎>1+2𝑘. Then, 𝑇 has a fixed point.

Proof. It follows from Theorem 2.10 by taking 𝑆=𝑇 and 𝑏=𝑎.

Remark 2.12. Corollary 4.1 of [29], Theorem 4 of [30], and Corollary 2.8 of [28] are particular cases of Corollary 2.11.

We give the following examples illustrating our result obtained by Theorem 2.10.

Example 2.13. Let 𝐸=ℝ with ‖𝑥‖_𝐸=|𝑥|, 𝑃=[0,∞), and 𝑋=[0,∞). Let 𝑑,𝑞∶𝑋×𝑋→𝐸 be defined by 𝑞(𝑥,𝑦)=𝑑(𝑥,𝑦)=|𝑥−𝑦|. Then, obviously, (𝑋,𝑑) is a cone metric space and 𝑞 is a 𝑐-distance. Consider two mappings 𝑆,𝑇∶𝑋→𝑋 defined by 𝑇𝑥=2𝑥 and 𝑆𝑥=3𝑥. For every 𝑘>0, take 𝑎=(2+5𝑘)/2 and 𝑏=4+5𝑘. We have 𝑎,𝑏>1+2𝑘 and the conditions (2.22) hold. So 𝑇 and 𝑆 have a (unique) common fixed point, which is 𝑣=0.