Some Fixed Point Theorems of Integral Type Contraction in Cone Metric Spaces

In 2007, Huang and Zhang in 1 introduced cone metric space by substituting an ordered Banach space for the real numbers and proved some fixed point theorems in this space. Many authors study this subject and many fixed point theorems are proved; see 2–5 . In this paper, the concept of integral in this space is introduced and a fixed point theorem is proved. In order to do this, we recall some definitions, examples, and lemmas from 1, 4 as follows. Let E be a real Banach space. A subset P of E is called a cone if and only if the following hold:


Introduction
In 2007, Huang and Zhang in 1 introduced cone metric space by substituting an ordered Banach space for the real numbers and proved some fixed point theorems in this space. Many authors study this subject and many fixed point theorems are proved; see 2-5 . In this paper, the concept of integral in this space is introduced and a fixed point theorem is proved. In order to do this, we recall some definitions, examples, and lemmas from 1, 4 as follows.
Let E be a real Banach space. A subset P of E is called a cone if and only if the following hold: i P is closed, nonempty, and P / {0}, ii a, b ∈ R, a, b ≥ 0, and x, y ∈ P imply that ax by ∈ P, iii x ∈ P and −x ∈ P imply that x 0.
Given a cone P ⊂ E, we define a partial ordering ≤ with respect to P by x ≤ y if and only if y − x ∈ P. We will write x < y to indicate that x ≤ y but x / y, while x y will stand for 2 Fixed Point Theory and Applications y − x ∈ int P, where int P denotes the interior of P. The cone P is called normal if there is a number K > 0 such that 0 ≤ x ≤ y implies x ≤ K y , for all x, y ∈ E. The least positive number satisfying above is called the normal constant 1 . The cone P is called regular if every increasing sequence which is bounded from above is convergent. That is, if {x n } n≥1 is a sequence such that x 1 ≤ x 2 ≤ · · · ≤ y for some y ∈ E, then there is x ∈ E such that lim n → ∞ x n − x 0. Equivalently, the cone P is regular if and only if every decreasing sequence which is bounded from below is convergent 1 . Also every regular cone is normal 4 . In addition, there are some nonnormal cones.
. Then 0 ≤ g ≤ f, f 2 and g 2K 1. Since K f < g , K is not normal constant of P. Therefore, P is non-normal cone.
From now on, we suppose that E is a real Banach space, P is a cone in E with int P / ∅, and ≤ is partial ordering with respect to P . Let X be a nonempty set. As it has been defined in 1 , a function d : X × X → E is called a cone metric on X if it satisfies the following conditions: i d x, y ≥ 0 for all x, y ∈ X and d x, y 0 if and only if x y, ii d x, y d y, x , for all x, y ∈ X, iii d x, y ≤ d x, z d y, z , for all x, y, z ∈ X.
Then X, d is called a cone metric space.
Example 1.2. Suppose E l 1 , P {{x n } n∈N ∈ E : x n ≥ 0, for all n, X, ρ is a metric space and d : X × X → E is defined by d x, y {ρ x, y /2 n } n∈N . Then X, d is a cone metric space and the normal constant of P is equal to 1. Definition 1.3. Let X, d be a cone metric space. Let {x n } n∈N be a sequence in X and x ∈ X. If for any c ∈ E with 0 c, there is n 0 ∈ N such that for all n > n 0 , d x n , x c, then {x n } n∈N is said to be convergent to x, and x is the limit of {x n } n∈N . We denote this by Definition 1.4. Let X, d be a cone metric space and {x n } n∈N be a sequence in X. If for any c ∈ E with 0 c, there is n 0 ∈ N such that for all m, n > n 0 , d x n , x m c, then {x n } n∈N is called a Cauchy sequence in X. Definition 1.5. Let X, d be a cone metric space, if every Cauchy sequence is convergent in X, then X is called a complete cone metric space.
then T is called continuous on X.
Fixed Point Theory and Applications 3 The following lemmas are useful for us to prove the main result.
The following example is a cone metric space. for all x, y ∈ X, where β ∈ 0, 1 is a constant. Then f has a unique fixed point x 0 ∈ X. Also, for all x ∈ X, the sequence {f n x } ∞ n 1 converges to x 0 .

Certain Integral Type Contraction Mapping in Cone Metric Space
In 2002, Branciari in 6 introduced a general contractive condition of integral type as follows.
Theorem 2.1. Let X, d be a complete metric space, α ∈ 0, 1 , and f : X → X is a mapping such that for all x, y ∈ X,

Fixed Point Theory and Applications
where φ : 0, ∞ → 0, ∞ is nonnegative and Lebesgue-integrable mapping which is summable (i.e., with finite integral) on each compact subset of 0, ∞ such that for each > 0, 0 φ t dt > 0, then f has a unique fixed point a ∈ X, such that for each x ∈ X, lim n → ∞ f n x a.
In this section we define a new concept of integral with respect to a cone and introduce the Branciari's result in cone metric spaces.
respectively. where S Con must be unique. We show the common value S Con by b a φ x d P x or to simplicity b a φd p .

2.5
We denote the set of all cone integrable function φ : a, b → P by L 1 a, b , P .

2.14
This shows that φ is an example of subadditive cone integrable function.
Theorem 2.9. Let X, d be a complete cone metric space and P a normal cone. Suppose that φ : P → P is a nonvanishing map and a subadditive cone integrable on each a, b ⊂ P such that for each for some α ∈ 0, 1 , then f has a unique fixed point in X.
Proof. Let x 1 ∈ P. Choose x n 1 f x n . We have

2.16
Since α ∈ 0, 1 thus This means that {x n } n∈N is a Cauchy sequence and since X is a complete cone metric space, thus {x n } n∈N is convergent to x 0 ∈ X. Finally, since which is a contradiction. Thus f has a unique fixed point x 0 ∈ X.

2.26
Thus a,b