%0 Journal Article
%T Remarks on Some Recent Coupled Coincidence Point Results in Symmetric -Metric Spaces
%A Stojan Radenovi£¿
%J Journal of Operators
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/290525
%X We use a method of reducing coupled coincidence point results in (ordered) symmetric -metric spaces to the respective results for mappings with one variable, even obtaining (in some cases) more general theorems. Our results generalize, extend, unify, and complement recent coupled coincidence point theorems in this frame, established by Cho et al. (2012), Aydi et al. (2011), and Choudhury and Maity (2011). Also, by using our method several recent tripled coincidence point results in ordered symmetric -metric spaces can be reduced to the coincidence point results with one variable. 1. Introduction and Preliminaries In 2004, Mustafa and Sims introduced a new notion of generalized metric space called G-metric space, where to every triplet of elements a nonnegative real number is assigned [1]. Fixed point theory, as well as coupled and tripled cases, in such spaces were studied in [2¨C6]. In particular, Banach contraction mapping principle was established in these works. Fixed point theory has also developed rapidly in metric and cone metric spaces endowed with a partial ordering (see [7, 8] and references therein). Fixed point problems have also been considered in partially ordered G-metric spaces [9¨C11]. For more details on the following definitions and results concerning G-metric spaces, we refer the reader to [1, 9, 12¨C20]. Definition 1. Let be a nonempty set, and let be a function satisfying the following properties:(a) if ;(b) for all , , and with ;(c) for all , , and , with ;(d) (symmetry in all three variables); and(e) for all , , , and . Then the function is called a G-metric on and the pair is called a G-metric space. Definition 2. Let be a G-metric space and let be a sequence of points in .(i)A point is said to be the limit of a sequence if , and one says that the sequence is G-convergent to .(ii)The sequence is said to be a G-Cauchy sequence if, for every , there is a positive integer such that , for all ; that is, , as .(iii) is said to be G-complete (or a complete G-metric space) if every G-Cauchy sequence in is G-convergent in . Proposition 3 (see [1]). Let be a G-metric space, and let be a sequence of points in . Then the following are equivalent.(1)The sequence £¿£¿is G-convergent to .(2) £¿£¿as£¿£¿ .(3) £¿£¿as£¿£¿ .(4) £¿£¿as£¿£¿ . Definition 4 (see [1, 10]). A G-metric on is said to be symmetric if for all . Every G-metric on defines a metric on by For a symmetric G-metric space, one obtains However, for an arbitrary G-metric on , just the following inequality holds: , for all . Definition 5. In this work, one will consider the following three classes of
%U http://www.hindawi.com/journals/joper/2013/290525/