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Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert spaces. (English) Zbl 1272.47068
Let \(C\) be a subset of a real Hilbert space \(H\) and \(T:C\to H\) be a mapping. The authors study the following two concepts: (i) \(z\in C\) is a fixed point of \(T\) if \(z=Tz\); (ii) \(z\in H\) is an attractive point of \(T\) if \(\|z-Tx\|\leq\|z-x\|\) for all \(x\in C\). They prove the following results: Suppose that \(T:C\to C\) is a generic generalized hybrid mapping, that is, there are real numbers \(\alpha,\beta,\gamma,\delta\) such that (a) \(\alpha+\beta+\gamma+\delta\geq0\); (b) \(\alpha+\gamma>0\) or \(\alpha+\beta>0\); and (c) \(\alpha\|Tx-Ty\|^2+\beta\|x-Ty\|^2+\gamma\|Tx-y\|^2+\delta\|x-y\|^2\leq0\) for all \(x,y\in C\). Then \(T\) has an attractive point if and only if \(\{T^nz:n=0,1,2\dots\}\) is bounded for some \(z\in C\). It is noted that the assertion is proved without the convexity on \(C\). Moreover, the authors prove a mean convergence theorem of Baillon’s type and a weak convergence theorem of Mann’s type for generic generalized hybrid mappings.

47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J25 Iterative procedures involving nonlinear operators
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