Some Generalizations of Fixed Point Theorems in Cone Metric Spaces
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Review Article Some Generalizations of Fixed Point Theorems in Cone Metric Spaces J. O. Olaleru Mathematics Department, University of Lagos, Yaba, Lagos, Nigeria Correspondence should be addressed to J. O. Olaleru, [email protected] Received 17 March 2009; Revised 15 July 2009; Accepted 29 August 2009 Recommended by Mohamed A. Khamsi We generalize, extend, and improve some recent fixed point results in cone metric spaces including the results of H. Guang and Z. Xian 2007; P. Vetro 2007; M. Abbas and G. Jungck 2008; Sh. Rezapour and R. Hamlbarani 2008. In all our results, the normality assumption, which is a characteristic of most of the previous results, is dispensed. Consequently, the results generalize several fixed results in metric spaces including the results of G. E. Hardy and T. D. Rogers 1973, R. Kannan 1969, G. Jungck, S. Radenovic, S. Radojevic, and V. Rakocevic 2009, and all the references therein. Copyright q 2009 J. O. Olaleru. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction The recently discovered applications of ordered topological vector spaces, normal cones and topical functions in optimization theory have generated a lot of interest and research in ordered topological vector spaces e.g., see 1, 2. Recently, Huang and Zhang 3 introduced cone metric spaces, which is a generalization of metric spaces, by replacing the real numbers with ordered Banach spaces. They later proved some fixed point theorems for different contractive mappings. Their results have been generalized by different authors e.g. see 4– 7. This paper generalizes, extends and improves the results of all those authors. The following definitions are given in 3. Let E be a real Banach space and P a subset of E. P is called a cone if and only if i P is closed, nonempty, and P / {0}; ii a, b ∈ R, a, b ≥ 0, x, y ∈ P ⇒ ax by ∈ P ; iii P −P {0}. For a given cone P ⊆ E, we can define a partial ordering ≤ with respect to P by x ≤ y y, while x y will stand for if and only if y − x ∈ P . x < y will stand for x ≤ y and x / y − x ∈ int P , where int P denotes the interior of P .
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Fixed Point Theory and Applications
The cone P is called normal if there is M > 0 such that for all x, y ∈ E, 0 ≤ x ≤ y implies x ≤ My. The least positive number M satisfying the above is called the normal constant of P . The cone P is called regular if every increasing sequence which is bounded from above is convergent. That is, if {xn }n≥1 is a sequence such that x1 ≤ x2 ≤ · · · ≤ y for some y ∈ E, then there is x ∈ E such that limn → ∞ xn − x 0. Equivalently, the cone P is regular if and only if every decreasing sequence which is bounded from below is convergent. In 5 it was shown that every regular cone is normal. In the sequel we will suppose that E is a metrizable linear topological space whose topology is defined by a real
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