Some fixed and coincidence point theorems for expansive maps in cone metric spaces
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Some fixed and coincidence point theorems for expansive maps in cone metric spaces Wasfi Shatanawi and Fadi Awawdeh* * Correspondence: awawdeh@hu. edu.jo Department of Mathematics, Hashemite University, Zarqa 13115, Jordan
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. Keywords: common fixed point, cone metric space, coincidence fixed point
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 Int(P) = 0. Then P is called a cone if the following conditions are satisfied: (1) P is closed and P ≠ {θ}. (2) a, b Î R+, x, y Î P implies ax + by Î P. (3) x Î 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 Î P. We shall write x ≺ y to indicate that x ≼ y but x ≠ y, while x ≪ y will stand for y - x Î Int P. It can be easily shown that lInt(P) ⊆ Int(P) for all positive scalar l. 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 Î X and d(x, y) = θ if and only if x = y. (2) d(x, y) = d(y, x) for all x, y Î X. (3) d(x, y) ≼ d(x, z) + d(y, z) for all x, y Î X. Then d is called a cone metric on X, and (X, d) is called a cone metric space. © 2012 Shatanawi and Awawdeh; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Shatanawi and Awawdeh Fixed Point Theory and Applications 2012, 2012:19 http://www.fixedpointtheoryandapplications.com/content/2012/1/19
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Definition 1.3. [1]Let (X, d) be a cone metric space. Let (xn) be a sequence in X and x Î X. If for every c Î E with θ ≪ c, there is an N Î N such that d(xn, x) ≪ c for all n ≥ N, then (xn) is said to be convergent and (xn) converges to x and x is the limit of (xn). We denote this by limn®+∞ xn = x or xn ® x as n ® +∞. If for every c Î E with θ ≪ c there is an N Î N such that d(xn,xm) ≪ c for all n,m ≥ N, then (xn) is called a Cauchy sequence in X. The space (X,d) is called a complete cone metric space if every Cauchy sequence is converg
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