Strict Contractive Conditions and Common Fixed Point Theorems in Cone Metric Spaces

Abstract

A lot of authors have proved various common fixed-point results for pairs of self-mappings under strict contractive conditions in metric spaces. In the case of cone metric spaces, fixed point results are usually proved under assumption that the cone is normal. In the present paper we prove common fixed point results under strict contractive conditions in cone metric spaces using only the assumption that the cone interior is nonempty. We modify the definition of property E.A , introduced recently in the work by Aamri and Moutawakil 2002 , and use it instead of usual assumptions about commutativity or compatibility of the given pair. Examples show that the obtained results are proper extensions of the existing ones.

Topological Vector Space-Valued Cone Metric Spaces and Fixed Point Theorems

“…Huang and Zhang 7 reintroduced such spaces under the name of cone metric spaces but went further, defining convergent and Cauchy sequences in the terms of interior points of the underlying cone. These and other authors see, e.g., [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] proved some fixed point and common fixed point theorems for contractive-type mappings in cone metric spaces and cone uniform spaces.…”

Fixed Point in Topological Vector Space-Valued Cone Metric Spaces

“…In a recent paper 17 the authors obtained common fixed points of a pair of mapping satisfying generalized contractive type conditions without the assumption of normality in a class of topological vector space-valued cone metric spaces which is bigger than that of studied in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] . In this paper we continue to study fixed point results in topological vector space valued cone metric spaces.…”

Common fixed point theorems for mappings satisfying property (E.A) on cone metric spaces

“…In [8], the authors generalized the results of [9] to a pair of self-mappings defined on cone metric spaces. The authors [17] proved common fixed point results under strict contractive conditions in cone metric spaces. Also, the authors [18] obtained fixed point theorems for set-valued mappings in cone metric spaces.…”