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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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