Some remarks on completeness and compactness in G-metric spaces

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.

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M. Abbas, T. Nazir and D. Doric, “Common fixed point of mappings satisfying (E.A) property in generalized metric spaces,” Applied Mathematics and Computation, vol.218, pp. 7665- 7670, 2012.

M. Abbas, T. Nazir and P. Vetro, “Common fixed point results for three maps in G-metric spaces”, Filomat, vol.25, no.4, pp.1-17, 2011.

M. Abbas and B. E. Rhoades, “Common fixed point results for noncommuting mappings without continuity in generalized metric spaces,” Applied Mathematics and Computation, vol.215, pp.262- 269, 2009.

T. V. An, N. V. Dung and V. T. L. Hang, “A new approach to fixed point theorems on G-metric spaces,” Topology and its Applications, vol.160, no.12, pp.1486-1493, 2013.

G. Beer, “Between compactness and completeness,” Topology and its Applications, vol.155, no.6, pp.503-514, 2008.

B. S. Choudhury, and P. Maity, “Coupled fixed point results in generalized metric spaces,” Mathematical and Computer Modelling, vol.54, pp.73-79, 2011.

M. I. Garrido, and A. S. Merono, “New types of completeness in metric spaces,” Annales Academiæ Scientiarum Fennicæ Mathematica, vol.39, pp.733–758, 2014.

M. İlkhan and E.E. Kara, “Uniform continuity and Cauchy continuity in G-metric spaces,” Journal of Inequalities and Special Functions, vol.8, no.3, pp.59-68, 2017.

A. Kaewcharoen, “Common fixed point theorems for contractive mappings satisfying ?- maps in G-metric spaces,” Banach Journal of Mathematical Analysis, vol.6, no.1, pp.101-111, 2012.

S. A. Mohiuddine and A. Alotaibi, “Some results on a triplet fixed point for nonlinear contractions in partially ordered G-metric spaces,” Fixed Point Theory and Applications, vol.2012, no.179, 12 pages, 2012.

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 Mathematica Scientia, vol.33, no.4, pp.1049- 1058, 2013.

Z. Mustafa, F. Awawdeh, and W. Shatanawi, “Fixed Point Theorem for Expansive Mappings in G-Metric Spaces,” International Journal of Contemporary Mathematical Sciences, vol.5, no.50, pp.2463-2472, 2010.

Z. Mustafa, H. Aydi and E. Karapınar, “On common fixed points in G-metric spaces using (E.A) property,” Computers & Mathematics with Applications, vol.64, pp.1944-1956, 2012.

Z. Mustafa, M. Khandagji, and W. Shatanawi, “Fixed point results on complete G-metric spaces,” Studia Scientiarum Mathematicarum Hungarica, vol.48, no.3, pp.304-319, 2011.

Z. Mustafa, and B. Sims, “A new approach to generalized metric spaces,” Journal of Nonlinear and Convex Analysis, vol.7, no.2, pp.289-297, 2006.

M. Öztürk and M. Başarır, “On some common fixed point theorems with φ-maps on G-cone metric spaces,” Bulletin of Mathematical Analysis and Applications, vol.3, no.1, pp.121-133, 2011.