Remarks on G-Metric Spaces

Abstract

In 2005, Mustafa and Sims (2006) introduced and studied a new class of generalized metric spaces, which are called G-metric spaces, as a generalization of metric spaces. We establish some useful propositions to show that many �xed point theorems on (nonsymmetric) G-metric spaces given recently by many authors follow directly from well-known theorems on metric spaces. Our technique can be easily extended to other results as shown in application.

Common fixed point for mappings satisfying new contractive condition and applications to integral equations

“…Recently, Jleli-Samet [8] and Samet et al [13] observed that some fixed point theorems in the context of a G-metric space can be proved (by simple transformation) using related existing results in the setting of a (quasi-) metric space. Namely, if the contraction condition of the fixed point theorem on G-metric space can be reduced to two variables, then one can construct an equivalent fixed point theorem in setting of usual metric space.…”

“…This idea is not completely new, but it was not successfully used before (see [11]). Very recently, Karapinar and Agarwal [9] suggested new contraction conditions in G-metric space in a way that the techniques in [8,13] are not applicable. In this approach [9], contraction conditions cannot be expressed in two variables.…”

“…Abbas and Rhoades [2] initiated the study of common fixed point in generalized metric space. Recently, many fixed point theorems for certain contractive conditions have been established in G-metric spaces, and for more details one can refer to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Recently, Gu and Zhang [7] proved a common fixed point theorems for six self-mappings with twice power type contractive condition in metric space.…”

Fixed and common fixed point results for cyclic mappings of Ω -distance

“…After that this notion was fructified by several scientists which proved valuable fixed point theorems in G-metric spaces; please, see Aydi et al [3,4]; Chandok et al [6]; Chough et al [8], Karapinar and Agarwal [13], Popa and Patriciu [18], Shatanawi and Postolache [24]. Jleli and Samet [11] and Samet et al [20] proved that some of fixed point theorems in G-metric spaces can be obtained from usual metric spaces or from quasi metric spaces. Karapinar and Agarwal [13] proved that the approach of Jleli and Samet [11] and Samet et al [20] cannot be applied if the contraction condition in the statement of the theorem is not reducible to two variables and they introduced and proved diverse interesting results in G-metric spaces.…”

“…Jleli and Samet [11] and Samet et al [20] proved that some of fixed point theorems in G-metric spaces can be obtained from usual metric spaces or from quasi metric spaces. Karapinar and Agarwal [13] proved that the approach of Jleli and Samet [11] and Samet et al [20] cannot be applied if the contraction condition in the statement of the theorem is not reducible to two variables and they introduced and proved diverse interesting results in G-metric spaces.…”