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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.

MSC:
 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47J25 Iterative procedures involving nonlinear operators
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