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Some fixed and coincidence point theorems for expansive maps in cone metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 19 (2012)
Abstract
In this article, we establish some common fixed and common coincidence point theorems for expansive type mappings in the setting of cone metric spaces. Our results extend some known results in metric spaces to cone metric spaces. Also, we introduce some examples the support the validity of our results.
Mathematics Subject Classification: 54H25; 47H10; 54E50.
1. Introduction
Huang and Zhang [1] introduced the notion of cone metric spaces as a generalization of metric spaces. They replacing the set of real numbers by an ordered Banach space. Huang and Zhang [1] presented the notion of convergence of sequences in cone metric spaces and proved some fixed point theorems. Then after, many authors established many fixed point theorems in cone metric spaces. For some fixed point theorems in cone metric spaces we refer the reader to [2–30].
In the present article, E stands for a real Banach space.
Definition 1.1. Let P be a subset of E with\text{Int}\left(P\right)\ne \mathrm{0\u0338}. Then P is called a cone if the following conditions are satisfied:

(1)
P is closed and P ≠ {θ}.

(2)
a, b ∈ R ^{+} , x, y ∈ P implies ax + by ∈ P.

(3)
x ∈ P ∩ P implies x = θ.
For a cone P, define a partial ordering ≼ with respect to P by x ≼ y if and only if y  x ∈ P. We shall write x ≺ y to indicate that x ≼ y but x ≠ y, while x ≪ y will stand for y  x ∈ Int P. It can be easily shown that λInt(P) ⊆ Int(P) for all positive scalar λ.
Definition 1.2. [1]Let X be a nonempty set. Suppose the mapping d : X × X → E satisfies

(1)
θ ≺ d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y.

(2)
d(x, y) = d(y, x) for all x, y ∈ X.

(3)
d(x, y) ≼ d(x, z) + d(y, z) for all x, y ∈ X.
Then d is called a cone metric on X, and (X, d) is called a cone metric space.
Definition 1.3. [1]Let (X, d) be a cone metric space. Let (x_{ n }) be a sequence in X and x ∈ X. If for every c ∈ E with θ ≪ c, there is an N ∈ N such that d(x_{ n }, x) ≪ c for all n ≥ N, then (x_{ n }) is said to be convergent and (x_{ n }) converges to x and x is the limit of (x_{ n }). We denote this by lim_{n→+∞}x_{ n }= x or x_{ n }→ x as n → +∞. If for every c ∈ E with θ ≪ c there is an N ∈ N such that d(x_{ n },x_{ m }) ≪ c for all n,m ≥ N, then (x_{ n }) is called a Cauchy sequence in X. The space (X,d) is called a complete cone metric space if every Cauchy sequence is convergent.
The cone P in a real Banach space E is called normal if there is a number λ > 0 such that for all x,y ∈ E,
Iiker Ṣahin and Mustafa Telci [30] studied a theorem on common fixed points of expansive type mappings in cone metric spaces.
Definition 1.4. [30]Let E and F be real Banach spaces and P and Q be cones in E and F, respectively. Let (X, d) and (Y, ρ) be cone metric spaces, where d : X × X → E and ρ : Y × Y → F. A function f : X → Y is said to be continuous at x_{0} ∈ X, if for every c ∈ F with 0 ≪ c, there exists b ∈ E with 0 ≪ b such that ρ(fx, fx_{0}) ≪ c whenever x ∈ X and d(x, x_{0}) ≪ b.
Lemma 1.1. [30]Let (X, d) and (Y, ρ) be cone metric spaces. A function f : X → Y is continuous at a point x_{0} ∈ X if and only if whenever a sequence (x_{ n }) in X converges to x_{0}, the sequence (fx_{ n }) converges to fx_{0}.
Theorem 1.1. [30]Let (X, d) be a complete cone metric space and P be a cone. Let f and g be surjective selfmappings of X satisfying the following inequalities
for all x ∈ X, where a,b > 1. If either f or g is continuous, then f and g have a common fixed point.
The aim of this article is to study new theorems of common fixed and coincidence point for expansive mappings in cone metric spaces under a set of conditions. Our results generalize several well known comparable results in the literature. Also, we introduced some examples to support the validity of our results.
2. Main results
We start with the following theorem
Theorem 2.1. Let (X, d) be a cone metric space with a solid cone P. Let T, f : X → X be mappings satisfying:
for all x, y ∈ X where a,b,c ≥ 0 with a + b +c > 1. Suppose the following hypotheses:

(1)
b < 1 or c < 1.

(2)
fX ⊆ TX.

(3)
TX is a complete subspace of X.
Then T and f have a coincidence point.
Proof. Let x_{0} ∈ X. Since fX ⊆ TX, we choose x_{1} ∈ X such that Tx_{1} = fx_{0}. Again, we can choose x_{2} ∈ X such that Tx_{2} = fx_{1}. Continuing in the same way, we construct a sequence (x_{ n }) in X such that Tx_{n+ 1}= fx_{ n }for all n ∈ N ∪ {0}.
If fx_{m1}= fx_{ m }for some m ∈ N, then Tx_{ m }= fx_{ m }. Thus x_{ m }is a coincidence point of T and f.
Now, assume that x_{n1}≠ x_{ n }for all n ∈ N.
Case (1): Suppose b < 1.
By (1), we have
Thus, we have
Hence
Case (2): Suppose c < 1.
By (1), we have
Thus, we have
Hence
In Case (1), we let
and in Case (2), we let
Thus in both cases, we have λ < 1 and
By (4), we have
So for m > n, we have
Let θ ≪ c be given, choose δ > 0 such that c + N_{δ}(0) ⊆ P, where
Also, choose a natural number N_{1} such that
for m ≥ N_{1}. Then
for all m ≥ N_{1}. Thus,
for all m > n. Therefore (Tx_{ n }) is a cauchy sequence in (TX, d). Since (TX, d) is a complete cone metric space, there is u ∈ X such that (Tx_{ n }) converges to Tu as n → +∞. Hence fx_{ n }converges to Tu as n → +∞. Since a + b + c > 1, we have a, b and c are not all 0. So we have the following cases.
Case 1: If a ≠ 0, then
Hence
Let θ ≪ c be given, choose δ > 0 such that c + N_{δ}(0) ⊆ P, where
Since \frac{1}{a}d\left(T{x}_{n},Tu\right)\to \theta. We choose a natural number n_{0} ∈ N such that
for n ≥ n_{0}. Then
for all n ≥ n_{0}. Thus,
for all n ≥ n_{0}. Thus fx_{ n }→ fu as n → +∞. By uniqueness of limit, we have Tu = fu. Therefore T and f have a coincidence point.
Case 2: If b ≠ 0, then
Hence
As similar proof of Case (1), we can show that fu = Tu. Thus f and T have a coincidence point.
Case 3: If c ≠ 0, then
Hence
As similar proof of Case (1), we can show that fu = Tu. Thus f and T have a coincidence point.
Corollary 2.1. Let (X,d) be a cone metric space with a solid cone P. Let T, f : X → X be mappings satisfying:
for all x, y ∈ X where a, b ≥ 0 with a + b > 1 and b < 1. Suppose the following hypotheses:

(1)
fX ⊆ TX.

(2)
TX is a complete subspace of X.
Then T and f have a coincidence point.
Corollary 2.2. Let (X, d) be a complete cone metric space with a solid cone P. Let T, f :X → X be mappings satisfying:
for all x, y ∈ X where a > 1. Suppose the following hypotheses:

(1)
fX ⊆ TX.

(2)
TX is a complete subspace of X.
Then T and f have a coincidence point.
Corollary 2.3. Let (X, d) be a complete cone metric space with a solid cone P. Let T :X → X be a surjective mapping satisfying:
for all x, y ∈ X where a,b,c ≥ 0 with a + b + c > 1. Suppose b < 1 or c < 1. Then T has a fixed point.
Proof. Follows from Theorem 2.1 by taking f = I, the identity map.
Corollary 2.4. Let (X, d) be a complete cone metric space with a solid cone P. Let T :X → X be a surjective mapping satisfying:
for all x, y ∈ X where with a > 1. Then T has a fixed point.
Putting E = R, P = {x ∈ R : x ≥ 0} and d : X × X → R in Corollaries 2.1 and 2.2, we have the following results:
Corollary 2.5. Let (X, d) be a complete metric space. Let T : X → X be a surjective mapping satisfying:
for all x, y ∈ X where a, b ≥ 0 with a + b > 1 and b < 1. Then T has a fixed point.
Corollary 2.6. Let (X, d) be a complete metric space. Let T : X → X be a surjective mapping satisfying:
for all x, y ∈ X where a, b ≥ 0 with a + b > 1 and b < 1. Then T has a fixed point.
Now, we present a fixed point theorem for two maps.
Theorem 2.2. Let T, S : X → X be two surjective mappings of a complete cone metric space (X, d) with a solid cone P. Suppose that T and S satisfying the following inequalities
and
for all x ∈ X and some nonnegative real numbers a, b and k with a > 1 + 2k and b >1 + 2k. If T or S is continuous, then T and S have a common fixed point
Proof. Let x_{0} be an arbitrary point in X. Since T is surjective, there exists x_{1} ∈ X such that x_{0} = Tx_{1}. Also, since S is surjective, there exists x_{2} ∈ X such that x_{2}= Sx_{1}. 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 ∪ {0}. Now, for n ∈ N ∪ {0}, we have
Thus, we have
which implies that
Hence
On other hand, we have
Thus, we have
Since d(x_{2n 1},x_{2n}) + d(x_{2n}, x_{2n+1}) ≽ d(x_{2n 1},x_{2n+ 1}), we have
Hence
Let
Then by combining (7) and (8), we have
Repeating (9) ntimes, we get
Thus, for m > n, we have
As similar arguments to proof of Theorem 2.1, we can show that (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 → +∞. Therefore x_{2n+ 1}→ v and x_{2n+ 2}→ v as n → +∞. Without loss of generality, we may assume that T is continuous, then Tx_{2n+ 1}→ Tv as n → +∞. But Tx_{2n+ 1}= x_{ 2n }→ v as n → +∞. Thus, we have Tv = v. Since S is surjective, there exists w ∈ X such that Sw = v. Now,
implies that kd(v,w) ≽ ad(v,w). Thus
Since a > k, we conclude that d(v, w) = θ. So v = w. Hence Tv = Sv = v. Therefore v is a common fixed point of T and S.
By taking b = a in Theorem 2.2, we have the following result.
Corollary 2.7. Let T, S : X → X be two surjective mappings of a complete cone metric space (X, d) with a solid cone P. Suppose that T and S satisfying the following inequalities
and
for all x ∈ X and some nonnegative real numbers a and k with a > 1 + 2k. If T or S is continuous, then T and f have a common fixed point
By taking S = T in Corollary 2.7, we have the following corollary.
Corollary 2.8. Let T : X → X be a surjective mapping of a complete cone metric space (X, d) with a solid cone P. Suppose that T satisfying
for all x ∈ X and some nonnegative real number a and k with a > 1 + 2k. If T is continuous, then T has a fixed point.
Now, we present some examples to illustrate the useability of our results.
Example 2.1. (The case of normal cone) Let X = [0,+∞), E = R^{2}. Let P = {(a, b) :a ≥ 0,b ≥ 0} be the cone with d(x, y) = (x  y, x  y). Then (X, d) is a complete cone metric space. Define T : X → X by Tx = 2x. Then T has a fixed point.
Proof. Note that
for all x ∈ X. Thus T satisfies all the hypotheses of Corollary 2.8 and hence T has a fixed point. Here 0 is the fixed point of T.
Example 2.2. (The case of nonnormal cone) Let X = [0, 1], E={C}_{\mathbf{R}}^{1}\left(\left[0,1\right]\right). Let P = {ϕ ∈ E: ϕ(t) ≥ 0, t ∈ [0, 1]}. Define the mapping d : X × X → E by
where ϕ ∈ P is a fixed function, for example ϕ(t) = e^{t}. Define T, f : X → X byTx=\frac{1}{4}xandfx=\frac{1}{16}x. Then T and f have a coincidence point.
Proof. Note that
for all x, y ∈ X and t ∈ [0, 1]. Thus T and f satisfy all the hypotheses of Corollary 2.2 and hence T and f have a coincidence point. Here 0 is the coincidence point of T and f.
Remarks:
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Shatanawi, W., Awawdeh, F. Some fixed and coincidence point theorems for expansive maps in cone metric spaces. Fixed Point Theory Appl 2012, 19 (2012). https://doi.org/10.1186/16871812201219
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DOI: https://doi.org/10.1186/16871812201219
Keywords
 common fixed point
 cone metric space
 coincidence fixed point