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Iterative algorithms for nonlinear operators. (English) Zbl 1013.47032
This article deals with the following approximations $$x_{n+1} := \alpha_n x_0 + (1 - \alpha_n)(I + c_n T)^{-1}(x_n) +e_n,\quad n = 0, 1, 2,\dots,$$ to a solution $x^*$ of the inclusion $0\in Tx$ with a maximal monotone operator $T$ in a Hilbert space $H$. Here $(\alpha_n)$ and $(c_n)$ are sequences of reals, $(e_n)$ a sequence of errors. The main result is the following: if the conditions (i) $\alpha_n\to 0$; (ii) $\sum_{n=1}^\infty \alpha_n= \infty$; (iii) $c_n\to\infty$; (iv) $\sum_{n=1}^\infty\|e_n\|<\infty$ hold, then the approximations $x_n$ strongly converge to $Px_0$ ($P$ is the projection from $H$ onto the nonempty closed convex set $T^{(-1)}(0)$). A similar result is formulated for weak convergence of approximations $x_n$. The special case of the equation $x = Sx$ with a nonexpansive operator $S$ (and the problem of finding a common fixed point for operators from a contraction semigroup) is also studied. As application, the problem $$\min_{x\in K}\left\{\tfrac\mu 2\langle Ax,x\rangle+\tfrac 12 \|x- u\|^2-\langle x,b\rangle\right\}$$ is considered.

47J25Iterative procedures (nonlinear operator equations)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
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