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Abstract: In this paper, we extend the coupled fixed point theorems for mixed monotone operators obtained by Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379–1393] and Luong and Thuan [N.V. Luong and N.X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal. 74 (2011) 983–992], by weakening the involved contractive condition. An example as well as an application to nonlinear Fredholm integral equations is also given in order to illustrate the effectiveness of our generalizations. [Copyright &y& Elsevier]

Abstract: In this paper, we extend the coupled fixed point theorems for mixed monotone operators obtained in [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7) (2006) 1379–1393] by significantly weakening the contractive condition involved. Our technique of proof is essentially different and more natural. An example as well as an application to periodic BVP is also given in order to illustrate the effectiveness of our generalizations. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce the concept of tripled fixed point for nonlinear mappings in partially ordered complete metric spaces and obtain existence, and existence and uniqueness theorems for contractive type mappings. Our results generalize and extend recent coupled fixed point theorems established by Gnana Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7) (2006) 1379–1393]. Examples to support our new results are given. [Copyright &y& Elsevier]

Abstract: We establish a geometric lemma giving a list of equivalent conditions for some subsets of the plane. As its application, we get that various contractive conditions using the so-called altering distance functions coincide with classical ones. We consider several classes of mappings both on metric spaces and ordered metric spaces. In particular, we show that unexpectedly, some very recent fixed point theorems for generalized contractions on ordered metric spaces obtained by Harjani and Sadarangani [J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72 (2010) 1188–1197], and Amini-Harandi and Emami [A. Amini-Harandi, H. Emami A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. 72 (2010) 2238–2242] do follow from an earlier result of O’Regan and Petruşel [D. O’Regan and A. Petruşel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341 (2008) 1241–1252]. [Copyright &y& Elsevier]

Abstract: We provide a positive answer to a question raised by Eldred and Veeramani [A.A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006) 1001–1006] about the existence of a best proximity point for a cyclic contraction map in a reflexive Banach space. Moreover, we introduce a new class of maps, called cyclic -contractions, which contains the cyclic contraction maps as a subclass. Convergence and existence results of best proximity points for cyclic -contraction maps are also obtained. [Copyright &y& Elsevier]