Fixed and periodic points of generalized contractions in metric spaces
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RESEARCH
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Fixed and periodic points of generalized contractions in metric spaces Mujahid Abbas1 , Basit Ali2 and Salvador Romaguera3* * Correspondence: [email protected] 3 Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, 46022, Spain Full list of author information is available at the end of the article
Abstract Wardowski (Fixed Point Theory Appl. 2012:94, 2012, doi:10.1186/1687-1812-2012-94) introduced a new type of contraction called F-contraction and proved a fixed point result in complete metric spaces, which in turn generalizes the Banach contraction principle. The aim of this paper is to introduce F-contractions with respect to a self-mapping on a metric space and to obtain common fixed point results. Examples are provided to support results and concepts presented herein. As an application of our results, periodic point results for the F-contractions in metric spaces are proved. MSC: 47H10; 47H07; 54H25 Keywords: F-contraction; property P; property Q; common fixed point
1 Introduction and preliminaries The Banach contraction principle [] is a popular tool in solving existence problems in many branches of mathematics (see, e.g., [–]). Extensions of this principle were obtained either by generalizing the domain of the mapping or by extending the contractive condition on the mappings [–]. Initially, existence of fixed points in ordered metric spaces was investigated and applied by Ran and Reurings []. Since then, a number of results have been proved in the framework of ordered metric spaces (see [–]). Contractive conditions involving a pair of mappings are further additions to the metric fixed point theory and its applications (for details, see [–]). Recently, Wardowski [] introduced a new contraction called F-contraction and proved a fixed point result as a generalization of the Banach contraction principle []. In this paper, we introduce an F-contraction with respect to a self-mapping on a metric space and obtain common fixed point results in an ordered metric space. In the last section, we give some results on periodic point properties of a mapping and a pair of mappings in a metric space. We begin with some basic known definitions and results which will be used in the sequel. Throughout this article, N, R+ , R denote the set of natural numbers, the set of positive real numbers and the set of real numbers, respectively. Definition Let f and g be self-mappings on a set X. If fx = gx = w for some x in X, then x is called a coincidence point of f and g and w is called a coincidence point of f and g. Furthermore, if fgx = gfx whenever x is a coincidence point of f and g, then f and g are called weakly compatible mappings []. ©2013 Abbas et al.; 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.
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