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Zbl 1147.54022
Abbas, M.; Jungck, G.
Common fixed point results for noncommuting mappings without continuity in cone metric spaces. (English)
[J] J. Math. Anal. Appl. 341, No. 1, 416-420 (2008). ISSN 0022-247X

Let $(X, d)$ be a cone metric space and $P$ a normal cone with a constant. Let maps $f, g: X \rightarrow X$ be such that $f(X) \subseteq g(X), g(X)$ is a complete subspace of $X$ and $f, g$ are commuting at their coincidence points. Further let for any $x, y$ in $X$, $$d(fx, fy) \leq ad(gx, gy) + b[d(fx, gx) + d(fy, gy)] + c[d(fx, gy) + d(fy, gx)],$$ where $a\geq 0, b\geq 0, c\geq 0$ and $a + 2b + 2c < 1.$ Then the authors, extending a result of {\it G.\,Jungck} [Am.\ Math.\ Mon.\ 83, 261--263 (1976; Zbl 0321.54025)] (respectively, {\it R.\,Kannan} [Bull.\ Calcutta Math.\ Soc.\ 60, 71--76 (1968; Zbl 0209.27104)]), show in Theorem~2.1 with $a = k, b = c = 0$ (respectively, in Theorem~2.3, with $a = c = 0, b = k$) that $f$ and $g$ have a unique common fixed point. They obtain the same conclusion in Theorem~2.4 with $a = b = 0, c = k$. (In Theorem~2.4, ``$d(fx, fy)\leq$" is misprinted as "$d(fx, fy)<$".)
[S. L. Singh (Rishikesh)]

MSC 2000:: *54H25 Fixed-point theorems in topological spaces
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: cone metric space; common fixed point; coincidence point; weakly compatible maps

Citations: Zbl 0321.54025; Zbl 0209.27104

Cited in: Zbl pre06129370 Zbl 1217.54036 Zbl 1210.54054 Zbl 1203.54043 Zbl 1190.54032 Zbl 1189.65119

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