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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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