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Approximating solutions of maximal monotone operators in Hilbert spaces. (English) Zbl 0992.47022

This article deals with iterates

x n+1 =α n x+(1-α n )J r n x n (n=1,2,)(1)


x n-1 =α n x n +(1-α n )J r n x n (n=1,2,),(2)

where J r =(I+rT) -1 , {α n } is a sequence from [0,1], {r n } a sequence from (0,), T:H2 H a maximal monotone operator in a real Hilbert space. The basic results are

(a) a theorem about strong convergence of iterates (1) to Px, where P is the metric projection onto T -1 0;

(b) a theorem about weak convergence of iterates (2) to vT -1 0=lim n Px n , where P is the metric projection onto T -1 0.

In the end of the article the special case when T=f is considered, where f is a proper lower-semicontinuous convex function. The corresponding results is interpreted as theorems of finding a minimizer of f.

47H05Monotone operators (with respect to duality) and generalizations
47J25Iterative procedures (nonlinear operator equations)