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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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