Common fixed points of set-valued F-contraction mappings on domain of sets endowed with directed graph
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Common fixed points of set-valued F-contraction mappings on domain of sets endowed with directed graph Mujahid Abbas1,2 · Talat Nazir3,4 · Tatjana Aleksi´c Lampert5 · Stojan Radenovi´c6
Received: 12 May 2015 / Revised: 23 January 2016 / Accepted: 27 January 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract The aim of this paper is to present common fixed point results of set-valued graphic F-contraction mappings on a family of sets endowed with a graph. Some examples are presented to support the results proved herein. Our results unify, generalize and extend various results in the existing literature. Keywords Set-valued mapping · Domain of sets · Common fixed point · Graph F-contraction pair · Directed graph Mathematics Subject Classification
47H10 · 54E50 · 54H25
Communicated by Hector Ramirez.
B
Stojan Radenovi´c [email protected]; [email protected] Mujahid Abbas [email protected] Talat Nazir [email protected] Tatjana Aleksi´c Lampert [email protected]
1
Nonlinear Analysis Research Group Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3
Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, 72123 Västerås, Sweden
4
Department of Mathematics, COMSATS Institute of Information Technology, 22060 Abbottabad, Pakistan
5
Faculty of Sciences, Department of Mathematics, Radoja Domanoví´ca 12, 34 000 Kragujevac, Serbia
6
Department of Mathematical Sciences, State University of Novi Pazar, Novi Pazar, Serbia
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T. Nazir et al.
1 Introduction and preliminaries The order oriented fixed point theory is studied in an environment created by a class of partially ordered sets with appropriate mappings satisfying certain order condition such as monotonicity, expansivity or order continuity. Existence of fixed points in partially ordered metric spaces has been studied by Ran and Reurings (2004). Recently, many researchers have obtained fixed point results for single and multi-valued mappings de defined on partially ordered metrics spaces (see, e.g., Amini-Harandi and Emami 2010; Beg and Butt 2013a; Harjani and Sadarangani 2009; Nieto and López 2005). Jachymski and Jozwik (2007) introduced a new approach in metric fixed point theory by replacing the order structure with a graph structure on a metric space. In this way, the results proved in ordered metric spaces are generalized (see also Jachymski 2008 and the reference therein); in fact, in 2010, Gwozdz-Lukawska and Jachymski (2009), developed the Hutchinson-Barnsley theory for finite families of mappings on a metric space endowed with a directed graph. Abbas and Nazir (2013) obtained some fixed point results for a power graph contraction pair endowed with a graph. Bojor (2012) proved fixed point theorems for Reich type contractions on metric spaces with a graph. For more results in this direction, we refer to Abbas et al. (2014); Aleomraninejad et al
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