Assad-Kirk-type fixed point theorems for a pair of nonself mappings on cone metric spaces. (English) Zbl 1186.54035

Janković, S.; Kadelburg, Z.; Radenović, S.; Rhoades, B. E.

Fixed Point Theory Appl. 2009, Article ID 761086, 16 p. (2009).

Let \(E\) be a real Banach space. A subset \(P\) of \(E\) is a cone [L. G. Huang and X. Zhang, J. Math. Anal. Appl. 332, No. 2, 1468–1476 (2007; Zbl 1118.54022)] if 7mm

(i)

\(P\) is closed, nonempty and \(P\neq\{0\}\);

(ii)

\(a,b\in \mathbb R\), \(a,b\geq 0\) and \(x,y\in P\) imply \(ax+ by\in P\);

(iii)

\(P\cap (-P)= \{0\}\).

In this paper new Assad-Kirk type fixed point theorems [N. A. Assad and W. A. Kirk, Pac. J. Math. 43, 553–562 (1972; Zbl 0239.54032)] for a pair of non-self mappings defined on a closed subset of a metrically convex cone metric space are obtained.

Reviewer: Valeriu Popa (Bacău)

MSC:

54H25

Fixed-point and coincidence theorems (topological aspects)

Keywords:

cone metric space; metrically convex metric space; Assad-Kirk-type fixed point theorem

Citations:

Zbl 1118.54022; Zbl 0239.54032

References:

[1]	Huang, L-G; Zhang, X, Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications, 332, 1468-1476, (2007) · Zbl 1118.54022
[2]	Abbas, M; Jungck, G, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, Journal of Mathematical Analysis and Applications, 341, 416-420, (2008) · Zbl 1147.54022
[3]	Ilic, D; Rakocevic, V, Common fixed points for maps on cone metric space, Journal of Mathematical Analysis and Applications, 341, 876-882, (2008) · Zbl 1156.54023
[4]	Rezapour, Sh; Hamlbarani, R, Some notes on the paper: “Cone metric spaces and fixed point theorems of contractive mappings”, Journal of Mathematical Analysis and Applications, 345, 719-724, (2008) · Zbl 1145.54045
[5]	Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450. · Zbl 1257.47059
[6]	Rhoades, BE, A fixed point theorem for some non-self-mappings, Mathematica Japonica, 23, 457-459, (1978) · Zbl 0396.47038
[7]	Assad, NA, On a fixed point theorem of Kannan in Banach spaces, Tamkang Journal of Mathematics, 7, 91-94, (1976) · Zbl 0356.47027
[8]	Assad, NA; Kirk, WA, Fixed point theorems for set-valued mappings of contractive type, Pacific Journal of Mathematics, 43, 553-562, (1972) · Zbl 0239.54032
[9]	Imdad, M; Kumar, S, Rhoades-type fixed-point theorems for a pair of nonself mappings, Computers & Mathematics with Applications, 46, 919-927, (2003) · Zbl 1065.47059
[10]	Ciric, Lj, Non-self mappings satisfying non-linear contractive condition with applications, Nonlinear Analysis, Theory, Methods and Applications, 71, 2927-2935, (2009) · Zbl 1166.47052
[11]	Gajic, Lj; Rakocevic, V, Pair of non-self-mappings and common fixed points, Applied Mathematics and Computation, 187, 999-1006, (2007) · Zbl 1118.54304
[12]	Radenovic, S; Rhoades, BE, Fixed point theorem for two non-self mappings in cone metric spaces, Computers and Mathematics with Applications, 57, 1701-1707, (2009) · Zbl 1186.65073
[13]	Jungck, G; Radenovic, S; Radojevic, S; Rakocevic, V, Common fixed point theorems for weakly compatible pairs on cone metric spaces, Fixed Point Theory and Applications, 2009, 13, (2009) · Zbl 1190.54032
[14]	Rhoades, BE, A comparison of various definitions of contractive mappings, Transactions of the American Mathematical Society, 226, 257-290, (1977) · Zbl 0365.54023
[15]	Kadelburg Z, Radenovic S, Rakocvic V: Remarks on “Quasi-contractions on cone metric spaces”.Applied Mathematics Letters, (2009). In press · Zbl 0396.47038
[16]	Rezapour Sh: A review on topological properties of cone metric spaces.Proceedings of the International Conference Analysis, Topology and Applications (ATA ’08), May-June 2008, Vrnjacka Banja, Serbia