Existence of fixed points in complete cone metric spaces. (English) Zbl 1203.54034

Azam, Akbar; Arshad, Muhammad; Beg, Ismat

Int. J. Mod. Math. 5, No. 1, 91-99 (2010).

The article deals with mappings \(S,T:X \to X\) satisfying the condition
\[ d(Sx,Ty) \leq Ad(x,y) + Bd(x,Sx) + Cd(y,Ty) + Dd(x,Ty) + Ed(y,Sx) \quad (x,y \in X), \]
where \((X,d)\) is a complete cone metric space, \(A,B,C,D\) are nonnegative real numbers with \(A + B + C + D + E < 1\), \(B = C\) or \(D = E\). Recall that a cone metric space is a pair \((X,d)\), where \(X\) is an arbitrary set, and \(d:X \times X \to B\) is a function with values in a Banach space \(B\) ordered by a cone \(P\) and satisfying the standard properties of a usual metric; the authors assume that \(P\) is normal (in the sense of M. G. Kreĭn) with the normal constant \(\kappa\). Under the assumptions above, the authors prove the existence of a unique common fixed point for \(S\) and \(T\).
Reviewer’s remark. Strangely, the authors ascribe the notion of cone metric space and fixed point theorems for operators in these spaces to L. D. Huang and X. Zhang [J. Math. Anal. Appl. 332, No. 2, 1468–1476 (2007; Zbl 1118.54022)]. Yet this notion (as a generalized metric, pseudo-metric or \(K\)-metric space) and numerous fixed point theorems in these spaces have been known since the first half of the twentieth century and were researched by G. Kurepa, L. V. Kantorovich, J. Schröder, and others.

Reviewer: Peter Zabreiko (Minsk)

MSC:

54H25

Fixed-point and coincidence theorems (topological aspects)

Keywords:

fixed points; common fixed points; contractive type mappings; cone metric space; quasi-metric space; generalized metric spaces

Citations:

Zbl 1118.54022