Abstract
AbstractThe purpose of this paper is to introduce and investigate two kinds of iterative algorithms for the problem of finding zeros of maximal monotone operators. Weak and strong convergence theorems are established in a real Hilbert space. As applications, we consider a problem of finding a minimizer of a convex function.
Highlights
Let C be a nonempty, closed, and convex subset of a real Hilbert space H
We always assume that T : C → 2H is a maximal monotone operator
Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and T : Ω → 2H a maximal monotone operator with T−1 0 / ∅
Summary
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Qin et al 6 extended 1.7 and 1.10 to the iterative scheme x0 ∈ H, xn 1 αnu βnPC xn − en γnPCfn, ∀n ≥ 0, 1.11 and the iterative one x0 ∈ C, xn 1 αnxn βnPC xn − en γnPCfn, ∀n ≥ 0, 1.12 respectively, where αn βn γn 1, supn≥0 fn < ∞, and en ≤ ηn xn−xn with supn≥0ηn η < 1 Under appropriate conditions, they derived one strong convergence theorem for 1.11 and another weak convergence theorem for 1.12. For other recent research works on approximate proximal point methods and their variants for finding zeros of monotone maximal operators, see, for example, 7–10 and the references therein. We consider the problem of finding zeros of maximal monotone operators by hybrid proximal point method. We consider a problem of finding a minimizer of a convex function
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