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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 G. Jungck [Am. Math. Mon. 83, 261–263 (1976; Zbl 0321.54025)] (respectively, 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 ”$$d(fx, fy)<$$”.)

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems
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##### References:
  Beg, I.; Abbas, M., Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed point theory appl., 2006, 1-7, (2006), Article ID 74503 · Zbl 1133.54024  Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag · Zbl 0559.47040  Huang, L.-G.; Zhang, X., Cone metric spaces and fixed point theorems of contractive mappings, J. math. anal. appl., 332, 2, 1468-1476, (2007) · Zbl 1118.54022  Jungck, G., Commuting maps and fixed points, Amer. math. monthly, 83, 261-263, (1976) · Zbl 0321.54025  Jungck, G., Compatible mappings and common fixed points, Internat. J. math. math. sci., 9, 4, 771-779, (1986) · Zbl 0613.54029  Jungck, G., Common fixed points for commuting and compatible maps on compacta, Proc. amer. math. soc., 103, 977-983, (1988) · Zbl 0661.54043  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  Jungck, G.; Rhoades, B.E., Fixed point for set valued functions without continuity, Indian J. pure appl. math., 29, 3, 227-238, (1998) · Zbl 0904.54034  Pant, R.P., Common fixed points of noncommuting mappings, J. math. anal. appl., 188, 436-440, (1994) · Zbl 0830.54031  Rhoades, B.E., A comparison of various definitions of contractive mappings, Trans. amer. math. soc., 26, 257-290, (1977) · Zbl 0365.54023  Sessa, S., On a weak commutativity condition of mappings in fixed point consideration, Publ. inst. math. soc., 32, 149-153, (1982) · Zbl 0523.54030  Kannan, R., Some results on fixed points, Bull. Calcutta math. soc., 60, 71-76, (1968) · Zbl 0209.27104
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