Common Fixed Point Theorems for Weakly Compatible Pairs on Cone Metric Spaces

Abstract

We prove several fixed point theorems on cone metric spaces in which the cone does not need to be normal.

Common fixed point theorems in C∗ -algebra-valued metric spaces

“…The Banach contraction principle which shows that every contractive mapping has a unique fixed point in a complete metric space has been extended in many directions ( [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][18][19][20][21][22]). One of the branches of this theory is devoted to the study of common fixed points.…”

“…One such notion which is weaker than commuting is the concept of compatibility introduced by Jungck [13]. Subsequently, several authors have obtained coincidence and common fixed point results for mappings, utilizing this concept and its generalizations, see [1,4,7,11,14,19] and references therein.…”

“…In 2007, Huang and Zhang [10] introduced cone metric spaces which generalized metric spaces, and obtained various fixed point theorems for contractive mappings. The existence of a common fixed point on cone metric spaces was investigated recently in [1,6,11,14]. Very recently, Ma et al in [16] introduced a concept of C * -algebra-valued metric spaces and presented some fixed point results for mappings under contractive or expansive conditions in these spaces.…”

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].…”

Fixed Point Theorems of Weakly Monotone Prešić Type Mappings in Ordered Cone Metric Spaces

“…Remark 2.1 ( [7]). Let P be a cone in a real Banach space E with zero vector 0 E and a, b, c ∈ P. Then 1.…”