Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (2016), 5245–5251 Research Article On the fixed point theory in bicomplete quasi-metric spaces Carmen Alegre a ,HacerDa˘g b , Salvador Romaguera a,b , Pedro Tirado a,* a Instituto Universitario de Matem´ atica Pura y Aplicada, Universitat Polit` ecnica de Val` encia, 46022 Valencia, Spain. b Departamento de Matem´ atica Aplicada, Universitat Polit` ecnica de Val` encia, 46022 Valencia, Spain. Communicated by B. Samet Abstract We show that some important fixed point theorems on complete metric spaces as Browder’s fixed point theorem and Matkowski’s fixed point theorem can be easily generalized to the framework of bicomplete quasi-metric spaces. From these generalizations we deduce quasi-metric versions of well-known fixed point theorems due to Krasnoselski˘ ı and Stetsenko; Khan, Swalesh and Sessa; and Dutta and Choudhury, re- spectively. In fact, our approach shows that many fixed point theorems for ϕ-contractions on bicomplete quasi-metric spaces, and hence on complete G-metric spaces, are actually consequences of the corresponding fixed point theorems for complete metric spaces. c 2016 All rights reserved. Keywords: Quasi-metric space, bicomplete, ϕ-contraction, fixed point. 2010 MSC: 47H10, 54H25, 54E50. 1. Introduction and preliminaries The study of the fixed point theory in quasi-metric spaces has received an increasing attention in the last years (see e.g. [1–4, 8, 10, 15, 20, 21, 27]) due, in great part, to the usefulness of these spaces and other related structures, as the so-called partial metric spaces, to the theory of computation, the complexity analysis of algorithm (see e.g. [5, 25, 26, 28]), as well as to the fixed point theory for G-metric spaces [1, 14]. The purpose of this paper is to show that some important fixed point theorems on complete metric spaces as Browder’s fixed point theorem and Matkowski’s fixed point theorem can be easily generalized * Corresponding author Email addresses: calegre@mat.upv.es (Carmen Alegre), hada@doctor.upv.es (HacerDa˘g), sromague@mat.upv.es (Salvador Romaguera), pedtipe@mat.upv.es (Pedro Tirado) Received 2016-07-12