New Fixed Point Results of Single-Valued Mapping for c-Distance in Cone Metric Spaces

and Applied Analysis 3 for all x, y ∈ E. The least positive number K satisfying the above condition is called the normal constant of P . It is clear that K ≥ 1. Definition 2.1 see 9 . Let X be a nonempty set and E a real Banach space equipped with the partial ordering with respect to the cone P . Suppose that the mapping d : X × X → E satisfies the following conditions: 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, y d y, z for all x, y, z ∈ X. Then d is called a cone metric on X and X, d is called a cone metric space. Definition 2.2 see 9 . Let X, d be a cone metric space, {xn} a sequence in X, and x ∈ X. 1 For all c ∈ E with θ c, if there exists a positive integer N such that d xn, x c for all n > N, then xn is said to be convergent and x is the limit of {xn}. We denote this by xn → x. 2 For all c ∈ E with θ c, if there exists a positive integer N such that d xn, xm c for all n,m > N, then {xn} is called a Cauchy sequence in X. 3 A cone metric space X, d is called complete if every Cauchy sequence in X is convergent. Lemma 2.3 see 25 . (1) If E be a real Banach space with a cone P and a λa where a ∈ P and 0 ≤ λ < 1, then a θ. (2) If c ∈ intP , θ an and an → θ, then there exists a positive integer N such that an c for all n ≥ N. Nextwe give the notation of c-distance on a conemetric spacewhich is a generalization of ω-distance of Kada et al. 26 with some properties. Definition 2.4 see 16 . Let X, d be a cone metric space. A function q : X ×X → E is called a c-distance on X if the following conditions hold: q1 θ q x, y for all x, y ∈ X, q2 q x, y q x, y q y, z for all x, y, z ∈ X, q3 for each x ∈ X and n ≥ 1, if q x, yn u for some u ux ∈ P , then q x, y u whenever {yn} is a sequence in X converging to a point y ∈ X, q4 for all c ∈ E with θ c, there exists e ∈ E with θ e such that q z, x e and q z, y e imply d x, y c. Example 2.5 see 16 . Let E R and P {x ∈ E : x ≥ 0}. Let X 0,∞ and define a mapping d : X × X → E by d x, y |x − y| for all x, y ∈ X. Then X, d is a cone metric space. Define a mapping q : X ×X → E by q x, y y for all x, y ∈ X. Then q is a c-distance on X. 4 Abstract and Applied Analysis Lemma 2.6 see 16 . Let X, d be a cone metric space and q is a c-distance on X. Let {xn} and {yn} be sequences in X and x, y, z ∈ X. Suppose that un is a sequences in P converging to θ. Then the following hold. 1 If q xn, y un and q xn, z un, then y z. 2 If q xn, yn un and q xn, z un, then {yn} converges to z. 3 If q xn, xm un form > n, then {xn}is a Cauchy sequence in X. 4 If q y, xn un, then {xn} is a Cauchy sequence in X. Remark 2.7 see 16 . 1 q x, y q y, x does not necessarily for all x, y ∈ X. 2 q x, y θ is not necessarily equivalent to x y for all x, y ∈ X. 3. Main Results In this section, we generalize some fixed point results from 21 by replacing the constants in contractive conditions with functions. Theorem 3.1. Let X, d be a complete cone metric space and q is a c-distance on X. Let f : X → X be a mapping and suppose that there exists mapping k : X → 0, 1 such that the following hold: a k fx ≤ k x for all x ∈ X, b q fx, fy k x q x, y for all x, y ∈ X. Then f has a fixed point x∗ ∈ X and for any x ∈ X, iterative sequence {fnx} converges to the fixed point. If v fv, then q v, v θ. The fixed point is unique. Proof. Choose x0 ∈ X. Set x1 fx0, x2 fx1 fx0, . . . , xn 1 fxn f x0. Then we have q xn, xn 1 q ( fxn−1, fxn )