Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

Article

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Commun. Korean Math. Soc. 2012; 27(1): 117-128

Printed March 1, 2012

https://doi.org/10.4134/CKMS.2012.27.1.117

Copyright © The Korean Mathematical Society.

Nonlinear contractions in partially ordered quasi $b$-metric spaces

Masood Hussain Shah and Nawab Hussain

Department of Mathematical Sciences, King Abdulaziz University

Abstract

Using the concept of a $g$-monotone mapping we prove some common fixed point theorems for $g$-non-decreasing mappings which satisfy some generalized nonlinear contractions in partially ordered complete quasi $b$-metric spaces. The new theorems are generalizations of very recent fixed point theorems due to L. Ciric, N. Cakic, M. Rojovic, and J. S. Ume, [$Monotone$ $generalized$ $nonlinear$ $contractions$ $in$ $partailly$ $ordered$ $metric$ $spaces$, Fixed Point Theory Appl. (2008), article, ID-131294] and R. P. Agarwal, M. A. El-Gebeily, and D. O'Regan [$Generalized$ $contractions$ $in$ $partially$ $ordered$ $metric$ $spaces$, Appl. Anal. $\bf 87$ (2008), 1--8].

Keywords: $g$-monotone mapping, $g$-non-decreasing mapping, coincidence, fixed point, common fixed point, complete quasi $b$-metric space

MSC numbers: Primary 54H25; Secondary 47H10