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Coincidences and fixed points of nonself hybrid contractions. (English) Zbl 0985.47046
Let $$(X, d)$$ be a complete metrically convex metric space, $$K$$ a non-empty closed subset of $$X$$, $$F, G: K \to CL(X), S, T$$ selfmaps of $$K$$. The hybrid contractions in this paper satisfy the inequality $$H(Fx, Gy) \leq \alpha d(Tx, Sy) + \beta[d(Tx, Fx) + d(Sy, Gy)] + \gamma[d(Tx, Gy) + d(Sy, Fx)]$$ for each $$x,y \in X$$, where $$\alpha, \beta, \gamma$$ are nonnegative and satisfy the conditions $$\alpha + 2\beta + 2\gamma < 1$$ and $$(\alpha + \beta + \gamma)((1 + \beta + \gamma)/(1 - \beta - \gamma))^2 < 1$$. The authors obtain sufficient conditions for the pairs $$F$$ and $$T$$ and $$G$$ and $$S$$ to have a coincidence. By adding some additional conditions, it is shown that the maps have a common fixed point. The results of this paper extend and correct similar results stated in A. Ahmad and M. Imdad [J. Math. Anal. Appl. 218, No. 2, 546-560 (1998; Zbl 0888.54046)] and A. Ahmed and A. R. Khan [ibid. 213, No. 1, 275-286 (1997; Zbl 0902.47047)].

MSC:
 47H10 Fixed-point theorems 47J05 Equations involving nonlinear operators (general) 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H04 Set-valued operators
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