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A general iterative scheme for $$k$$-strictly pseudo-contractive mappings and optimization problems. (English) Zbl 1213.65080
The article deals with fixed points of $$k$$-strictly pseudo-contractive mappings $$T:\;C (\subset H) \to H$$ in a real Hilbert space $$H$$, and where $$C$$ is a nonempty closed convex subset of $$H$$ with the following properties $$C \pm C \subset C$$. (A mapping $$T$$ is said to be $$k$$-strictly pseudocontractive ($$0 \leq k < 1$$) if $\|Tx - Ty\|^2 \leq \|x - y\|^2 + k\|(I - T)x - (I - T)y\|^2 \quad (x, y \in C).)$
It is assumed that $$F(T) \neq \emptyset$$ and considered the iteration scheme
$x_{n+1} = \alpha_n(u + \gamma f(x_n)) + \beta_nx_n + (1 - \beta_n)I - \alpha_n(I + \mu A))P_CSx_n, \quad n = 0,1,2,\dots,$
where $$S:\;C \to H$$ is a mapping defined by $$Sx = kx + (1 - k)Tx$$, $$P_C$$ is the metric projection of $$H$$ onto $$C$$, $$A$$ is a strongly positive bounded linear operator on $$C$$ satisfying the inequality $$(Ax,x) \geq \overline{\gamma}\|x\|^2$$ with some $$\overline{\gamma} \in (0,1)$$, $$f:\;C \to C$$ is a contraction with constant $$\alpha \in (0,1)$$ and such that $$0 < \gamma < \frac{(1 + \mu)\overline{\gamma}}{\alpha}$$ ($$\mu > 0$$), and, at last, the sequences $$\{\alpha_n\}$$ and $$\{\beta_n\}$$ satisfying the conditions $$\lim\limits_{n \to\infty} \alpha_n = 0$$, $$\sum\limits_{n=0}^\infty \alpha_n = \infty$$, $$0 < \liminf\limits_{n \to \infty} \beta_n \leq \limsup\limits_{n \to \infty} \beta_n < 1$$. Under these assumptions it is stated that $$\{x_n\}$$ converges strongly to a fixed point of $$T$$, which is a solution of the optimization problem
$\min_{x \in F(T)} \;\frac\mu2 (Ax,x) + \frac12 \|x - u\|^2 - h(x),$
where $$h$$ is a potential function for $$\gamma f$$. The analogous statement is proved for a family $$T_i:\;C \to H$$, $$i = 1,\dots,N$$, of $$k_i$$-strictly pseudo-contractive mappings; in this case $$S$$ is defined with the equation $$Sx = kx + (1 - k)\sum\limits_{i=1}^N \eta_iT_ix$$.
Reviewer’s remark: It should be mentioned that in the article there are several places and misprints; in particular, $$C$$ is a closed convex subset of $$H$$ satisfying the property $$C \pm C \subset C$$; such subsets are simply (closed) subspaces in $$H$$ with corresponding consequences.

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 47J25 Iterative procedures involving nonlinear operators
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