Some Remarks on Completeness and Compactness in G-Metric Spaces

Abstract

Complete metric spaces have great importance in functional analysis and its applications. The purpose of this paper is to introduce and study on some types of completeness in generalized metric spaces by the aid of Bourbaki Cauchy and cofinally Bourbaki-Cauchy sequences which are belonging to the class bigger than the class of Cauchy sequences. Moreover, by transporting some topological concepts to generalized metric spaces, the relations between these concepts and these new types of completeness properties are given.

Keywords

• generalized metric spaces,compactness,completeness

References

1. [1] M. Abbas, T. Nazir and D. Doric, Common fixed point of mappings satisfying (E.A) propertyin generalized metric spaces, Appl. Math. Comput. 218 (2012), 7665-7670.[2] M. Abbas, T. Nazir and P. Vetro, Common fixed point results for three maps in G-metricspaces, Filomat 25(4) (2011), 1-17.[3] M. Abbas and B. E. Rhoades, Common fixed point results for noncommuting mappingswithout continuity in generalized metric spaces, Appl. Math. Comput. 215 (2009), 262-269.[4] T. V. An, N. V. Dung and V. T. L. Hang, A new approach to fixed point theorems on G-metricspaces, Topology Appl. 160(12) (2013), 1486-1493.[5] G. Beer, Between compactness and completeness, Topology Appl. 155(6) (2008), 503-514.[6] B. S. Choudhury, and P. Maity, Coupled fixed point results in generalized metric spaces,Math. Comput. Model. 54 (2011), 73-79.[7] M. I. Garrido, and A. S. Mero˜no, New types of completeness in metric spaces, Ann. Acad.Sci. Fenn. Math. 39 (2014), 733–758.[8] M. ˙Ilkhan and E.E. Kara Uniform continuity and Cauchy continuity in G-metric spaces, J.Inequal. Spec. Funct. 8(3) (2017), 59-68.[9] A. Kaewcharoen, Common fixed point theorems for contractive mappings satisfying Φ-mapsin G-metric spaces, Banach J. Math. Anal. 6(1) (2012), 101-111.[10] S. A. Mohiuddine and A. Alotaibi, Some results on a triplet fixed point for nonlinear contractionsin partially ordered G-metric spaces, Fixed Point Theory Appl. 2012(179) (2012),12 pages.[11] F. Moradlou, P. Salimi, and P. Vetro, Some new extensions of Edelstein-Suzuki-type fixed point theorem to G-metric and G-cone metric spaces, Acta Math. Sci. 33(4) (2013), 1049-1058.[12] Z. Mustafa, F. Awawdeh, and W. Shatanawi, Fixed Point Theorem for Expansive Mappingsin G-Metric Spaces, Int. J. Contemp. Math. Sciences 5(50) (2010), 2463-2472.[13] Z. Mustafa, H. Aydi and E. Karapınar, On common fixed points in G-metric spaces using(E.A) property, Comput. Math. Appl. 64 (2012), 1944-1956.[14] Z. Mustafa, M. Khandagji, and W. Shatanawi, Fixed point results on complete G-metricspaces, Studia Sci. Math. Hungar. 48(3) (2011), 304-319.[15] Z. Mustafa, and B. Sims, A new approach to generalized metric spaces, J. Nonlinear ConvexAnal. 7(2) (2006), 289-297.[16] M. Ozt¨urk and M. Ba¸sarır, ¨ On some common fixed point theorems with φ-maps on G-conemetric spaces, Bull. Math. Anal. Appl. 3(1) (2011), 121-133.