## Convergence of implicit and explicit schemes for common fixed points for finite families of asymptotically nonexpansive mappings. (Convergence of implicit and explicit schemes for common fixed-points for finite families of asymptotically nonexpansive mappings.)(English)Zbl 1239.49008

Nonlinear Anal., Hybrid Syst. 5, No. 3, 492-501 (2011); corrigendum ibid. 5, No. 3, 603-604 (2011).
Summary: Let $$H$$ be a real Hilbert space and $$T_1,T_2,\dots,T_N$$ be a family of asymptotically nonexpansive self-mappings of $$H$$ with sequences $$\{1+k^{i(n)}_{p(n)}\}$$, such that $$k^{i(n)}_{p(n)}\to 0$$ as $$n\to\infty$$ where $$p(n)=j+1$$ if $$jN<n\leq(j+1)N$$, $$j=0,1,2,\dots$$ and $$n=jN+i(n)$$, $$i(n)\in\{1,2,\dots,N\}$$. Let $$F:=\cap^N_{i=1}\text{Fix}(T_i)\neq\emptyset$$ and let $$f:H\to H$$ be a contraction mapping with coefficient $$\alpha\in(0,1)$$, also let $$A$$ be a strongly positive bounded linear operator with coefficient $$\overline\gamma>0$$, and $$0<\gamma<\frac{\overline\gamma}{\alpha}$$. Let $$\{\alpha_n\}$$, $$\{\beta_n\}$$ be sequences in $$(0,1)$$ satisfying some conditions. Strong convergence of the implicit and explicit schemes are proved for a common fixed-point of the family $$T_1,T_2,\dots,T_n$$, which solves the variational inequality $$\langle(A-\gamma f)\overline x,\overline x-x\rangle\leq 0$$ $$\forall x\in F$$. Our result generalizes and improves several recent results.

### MSC:

 49J40 Variational inequalities 47J25 Iterative procedures involving nonlinear operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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### References:

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