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Implicit Mann fixed point iterations for pseudo-contractive mappings. (English) Zbl 1219.47108
Summary: Let $K$ be a compact convex subset of a real Hilbert space $H$ and $T:K\rightarrow K$ be a continuous hemi-contractive map. Let $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ be real sequences in $[0,1]$ such that $a_n+b_n+c_n=1$, and $\{u_n\}$ and $\{v_n\}$ be sequences in $K$. In this paper, we prove that, if $\{b_n\}$, $\{c_n\}$ and $\{v_n\}$ satisfy some appropriate conditions, then, for arbitrary $x_{0}\in K$, the sequence $\{x_n\}$ defined iteratively by $x_n=a_nx_{n - 1}+b_nTv_n+c_nu_n$, $n\geq 1$, converges strongly to a fixed point of $T$.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
Full Text: DOI
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