Nonlinear Contractive Conditions for Coupled Cone Fixed Point Theorems
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Research Article Nonlinear Contractive Conditions for Coupled Cone Fixed Point Theorems Wei-Shih Du Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan Correspondence should be addressed to Wei-Shih Du, [email protected] Received 19 April 2010; Revised 8 June 2010; Accepted 5 July 2010 Academic Editor: Juan J. Nieto Copyright q 2010 Wei-Shih Du. 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. We establish some new coupled fixed point theorems for various types of nonlinear contractive maps in the setting of quasiordered cone metric spaces which not only obtain several coupled fixed point theorems announced by many authors but also generalize them under weaker assumptions.
1. Introduction The existence of fixed point in partially ordered sets has been studied and investigated recently in 1–13 and references therein. Since the various contractive conditions are important in metric fixed point theory, there is a trend to weaken the requirement on contractions. Nieto and Rodr´ıguez-L´opez in 8, 10 used Tarski’s theorem to show the existence of solutions for fuzzy equations and fuzzy differential equations, respectively. The existence of solutions for matrix equations or ordinary differential equations by applying fixed point theorems are presented in 2, 6, 9, 11, 12. In 3, 13, the authors proved some fixed point theorems for a mixed monotone mapping in a metric space endowed with partial order and applied their results to problems of existence and uniqueness of solutions for some boundary value problems. In 2006, Bhaskar and Lakshmikantham 2 first proved the following interesting coupled fixed point theorem in partially ordered metric spaces. Theorem BL Bhaskar and Lakshmikantham. Let X, be a partially ordered set and d a metric on X such that X, d is a complete metric space. Let F : X ×X → X be a continuous mapping having the mixed monotone property on X. Assume that there exists a k ∈ 0, 1 with k dx, u d y, v , d F x, y , Fu, v ≤ 2
∀u x, y v.
1.1
2
Fixed Point Theory and Applications
y ∈ X, If there exist x0 , y0 ∈ X such that x0 Fx0 , y0 and Fy0 , x0 y0 , then, there exist x, such that x Fx, y and y Fy, x. Let E be a topological vector space (t.v.s. for short) with its zero vector θE . A nonempty subset K of E is called a convex cone if K K ⊆ K and λK ⊆ K for λ ≥ 0. A convex cone K is said to be pointed if K ∩ −K {θE }. For a given proper, pointed, and convex cone K in E, we can define a partial ordering K with respect to K by x K y ⇐⇒ y − x ∈ K.
1.2
y while x K y will stand for y−x ∈ int K, where int K denotes x ≺K y will stand for x K y and x / the interior of K. In the following, unless otherwise specified, we always assume that Y is a locally convex Hausdorff t.v.s. with its zero vector θ, K a proper, closed, conv
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