Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”

Abstract

Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476]. We shall prove that there are no normal cones with normal constant M < 1 and for each k > 1 there are cones with normal constant M > k. Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476], we obtain generalizations of the results.

Common fixed points for multivalued mappings in G-metric spaces with applications

“…On the other hand, the study of metric spaces expressed the most important role to many fields both in pure and applied science such as biology, medicine, physics and computer science (see [2,[20][21][22][23][24][25][26][27][28][29][30][31][32][33] and references therein). Some generalizations of the notion of a metric space have been proposed by some authors, such as, rectangular metric spaces, semi metric spaces, pseudo metric spaces, probabilistic metric spaces, fuzzy metric spaces, quasi metric spaces, quasi semi metric spaces, D-metric spaces, cone metric spaces, and partial metric spaces (see [3,9,13,27,28,30,32]).…”

Common fixed point under contractive condition of Ćirić’s type on cone metric type spaces

“…Cones and ordered normed spaces have some applications in optimization theory (see [5,6]). The initial study of Huang and Zhang [4] inspired many authors to prove fixed point theorems, as well as common fixed point theorems for two or more mappings on cone metric space, e.g., [7][8][9][10][11][12][13][14][15][16][17][18].…”

Tripled coincidence points for mixed comparable mappings in partially ordered cone metric spaces over Banach algebras

“…The results given by V. Berinde and M. Borcut [1] generalized and extended the works of Bhaskar and Lakshmikantham and Sabetghadam. In 2007, Huang and Zhang [7] introduced the concept of cone metric spaces as a generalization of general metric spaces, in which the distance d(x, y) of x and y is defined to be a vector in an ordered Banach space E and proved that the Banach contraction principle remains true in the setting of cone metric spaces. Since then, many fixed point results of the mappings with certain contractive property on cone metric spaces have been proved on the basis of the work of Huang and Zhang [7] (see [2,3,4,5,8,9,10,11,12,13,14,15,16,17,18,20] and the references therein). Among those works, the results of [15] attract much attention since the authors of [15] introduced the concept of cone metric spaces over Banach algebras by replacing Banach spaces with Banach algebras in order to generalize the Banach contraction principle to a more general form.…”