Iterative algorithms for nonlinear operators. (English) Zbl 1013.47032

Xu, Hong-Kun

J. Lond. Math. Soc., II. Ser. 66, No. 1, 240-256 (2002).

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.

Reviewer: Peter Zabreiko (Minsk)

MSC:

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

Keywords:

strong convergence; approximations; inclusion; maximal monotone operator; Hilbert space; weak convergence