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

Ali, Bashir; Ugwunnadi, G. C.

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.

Keywords:

asymptotically nonexpansive mapping; fixed-point; positive bounded linear mappings; variational inequality

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