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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.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H10 Fixed-point theorems
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