科研之友
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摘要
Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C -> H be a -Lipschitzian and.-strongly monotone operator with constants kappa,eta > 0, V, T : C -> C be nonexpansive mappings with Fix (T) not equal 0 where Fix (T) denotes the fixed-point set of T, and f : C -> H be a rho-contraction with coefficient rho is an element of [0, 1). Let 0 < mu < 2 eta/kappa(2) and 0 < gamma <= tau, where tau = 1 - root 1 - mu(2 eta - mu kappa(2)). For each s, t is an element of (0, 1), let x(s,t) t be a unique solution of the fixed-point equation x(st) = P-C[s gamma f(x(s,t)) + (I -s mu F) (tV + (1 - t)T)x(s,t)]. We derive the following conclusions on the behavior of the net {x(s,t)} along the curve t = t(s) : (i) if t(s) = O(s), as s -> 0, then x(s,t(s)) -> z(infinity) strongly, which is the unique solution of the variational inequality of finding z(infinity) is an element of Fix(T) such that <[(mu F - gamma f) + l(I - V)]z(infinity)> for all x is an element of Fix(T) and (ii) if t(s)/s -> infinity, as s -> 0, then x(s,t(s)) -> x(infinity) strongly, which is the unique solution of some hierarchical variational inequality problem.
