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Common fixed point results for noncommuting mappings without continuity in cone metric spaces. (English) Zbl 1147.54022
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 ”$d(fx, fy)<$”.)

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
[1] Beg, I.; Abbas, M.: Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed point theory appl. 2006, 1-7 (2006) · Zbl 1133.54024 · doi:10.1155/FPTA/2006/74503
[2] Deimling, K.: Nonlinear functional analysis, (1985) · Zbl 0559.47040
[3] Huang, L. -G.; Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings, J. math. Anal. appl. 332, No. 2, 1468-1476 (2007) · Zbl 1118.54022 · doi:10.1016/j.jmaa.2005.03.087
[4] Jungck, G.: Commuting maps and fixed points, Amer. math. Monthly 83, 261-263 (1976) · Zbl 0321.54025 · doi:10.2307/2318216
[5] Jungck, G.: Compatible mappings and common fixed points, Internat. J. Math. math. Sci. 9, No. 4, 771-779 (1986) · Zbl 0613.54029 · doi:10.1155/S0161171286000935
[6] Jungck, G.: Common fixed points for commuting and compatible maps on compacta, Proc. amer. Math. soc. 103, 977-983 (1988) · Zbl 0661.54043 · doi:10.2307/2046888
[7] Jungck, G.: Common fixed points for noncontinuous nonself maps on non-metric spaces, Far east J. Math. sci. (FJMS) 4, 199-215 (1996) · Zbl 0928.54043
[8] Jungck, G.; Rhoades, B. E.: Fixed point for set valued functions without continuity, Indian J. Pure appl. Math. 29, No. 3, 227-238 (1998) · Zbl 0904.54034
[9] Pant, R. P.: Common fixed points of noncommuting mappings, J. math. Anal. appl. 188, 436-440 (1994) · Zbl 0830.54031 · doi:10.1006/jmaa.1994.1437
[10] Rhoades, B. E.: A comparison of various definitions of contractive mappings, Trans. amer. Math. soc. 26, 257-290 (1977) · Zbl 0365.54023 · doi:10.2307/1997954
[11] Sessa, S.: On a weak commutativity condition of mappings in fixed point consideration, Publ. inst. Math. soc. 32, 149-153 (1982) · Zbl 0523.54030
[12] Kannan, R.: Some results on fixed points, Bull. Calcutta math. Soc. 60, 71-76 (1968) · Zbl 0209.27104