Báo cáo sinh học: Fixed point-type results for a class of extended cyclic self-mappings under three general weak contractive conditions of rational type

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Báo cáo sinh học: "Fixed point-type results for a class of extended cyclic self-mappings under three general weak contractive conditions of rational type"Fixed Point Theory andApplications This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon.Fixed point-type results for a class of extended cyclic self-mappings under three general weak contractive conditions of rational type Fixed Point Theory and Applications 2011, 2011:102 doi:10.1186/1687-1812-2011-102 Manuel De la Sen (wepdepam@lg.ehu.es) Ravi P Agarwal (Agarwal@tamuk.edu) ISSN 1687-1812 Article type Research Submission date 12 September 2011 Acceptance date 21 December 2011 Publication date 21 December 2011 Article URL http://www.fixedpointtheoryandapplications.com/content/2011/1/102 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Fixed Point Theory and Applications go to http://www.fixedpointtheoryandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 De la Sen and Agarwal ; 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.Fixed point-type results for a class of extended cyclic self-mappings under threegeneral weak contractive conditions of rational typeManuel De la Sen*1 and Ravi P Agarwal21 Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campusof Leioa (Bizkaia) – Aptdo. 644-Bilbao, 48080-Bilbao, Spain2 Department of Mathematics, Texas A&M University - Kingsville, 700 University Blvd.,Kingsville, TX 78363-8202, USA*Corresponding author: manuel.delasen@ehu.esEmail address:RPA: Agarwal@tamuk.eduAbstractThis article discusses three weak φ-contractive conditions of rational type for a class of 2-cyclic self-mappings defined on the union of two non-empty subsets of a metric space toitself. If the space is uniformly convex and the subsets are non-empty, closed, andconvex, then the iterates of points obtained through the self-mapping converge to uniquebest proximity points in each of the subsets.1. IntroductionA general contractive condition has been proposed in [1, 2] for mappings on a partiallyordered metric space. Some results about the existence of a fixed point and then itsuniqueness under supplementary conditions are proved in those articles. The rational 1contractive condition proposed in [3] includes as particular cases several of thepreviously proposed ones [1, 4–12], including Banach principle [5] and Kannan fixedpoint theorems [4, 8, 9, 11]. The rational contractive conditions of [1, 2] are applicableonly on distinct points of the considered metric spaces. In particular, the fixed pointtheory for Kannan mappings is extended in [4] by the use of a non-increasing functionaffecting the contractive condition and the best constant to ensure a fixed point is alsoobtained. Three fixed point theorems which extended the fixed point theory for Kannanmappings were stated and proved in [11]. More attention has been paid to theinvestigation of standard contractive and Meir-Keeler-type contractive 2-cyclic self-mappings T : A ∪ B → A ∪ B defined on subsets A , B ⊆ X and, in general, p-cyclic self-mappings defined on any number of subsets , T : U i∈ p A i → U i∈ p A i Ai ⊂ Xi ∈ p := {1 , 2 , ... , p } , where ( X , d ) is a metric space (see, for instance [13–22]). More recentinvestigation about cyclic self-mappings is being devoted to its characterization inpartially ordered spaces and also to the formal extension of the contractive conditionthrough the use of more general strictly increasing functions of the ...