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 ”\(d(fx, fy)<\)”.)
Reviewer: S. L. Singh (Rishikesh)
References:
| [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), Article ID 74503 · Zbl 1133.54024 |
| [2] | Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag · Zbl 0559.47040 |
| [3] | 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 |
| [4] | Jungck, G., Commuting maps and fixed points, Amer. Math. Monthly, 83, 261-263 (1976) · Zbl 0321.54025 |
| [5] | Jungck, G., Compatible mappings and common fixed points, Internat. J. Math. Math. Sci., 9, 4, 771-779 (1986) · Zbl 0613.54029 |
| [6] | Jungck, G., Common fixed points for commuting and compatible maps on compacta, Proc. Amer. Math. Soc., 103, 977-983 (1988) · Zbl 0661.54043 |
| [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, 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 |
| [10] | Rhoades, B. E., A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 26, 257-290 (1977) · Zbl 0365.54023 |
| [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 |
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