## 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

### Citations:

Zbl 0321.54025; Zbl 0209.27104
Full Text:

### References:

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