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Asymptotic behavior of relatively nonexpansive operators in Banach spaces

Butnariu D., Reich S., Zaslavski A. J.

Journal of Applied Analysis

2001

Vol. 7, nr 2

151-174

Let K be a closed convex subset of a Banach space X and let F be a nonempty closed convex subset of K. We consider complete metric spaces of self-mappings of K which fix all the points of F and are relatively nonexpansive with respect to a given convex function f on X. We prove (under certain assumptions on f) that the iterates of a generic mapping in these spaces converge strongly to a retraction onto F.

Set-valued implicit Wiener-Hopf equations and generalized strongly nonlinear quasivariational inequalities

Ding X. P.

Journal of Applied Analysis

1998

Vol. 4, nr 1

75-87

In this paper, we study a class of generalized strongly nonlinear quasivariational inequality problem (GSNQVIP(T,A,g,D,K(x))) which include the most of quasivariational inequalities and quasicomplementarity problems as special cases. We prove that the generalized strongly nonlinear quasivariational inequality problem is equivalent to solving the set-valued implicit Wiener-Hopf equation. By using the equivalence, a new iterative algorithm for finding the approximate solutions of the generalized strongly nonlinear quasivariational inequality problems are suggested and analyzed. The convergence criteria for the algorithm is also discussed. These new results include many known results for generalized quasivariational inequalities and generalized quasicomplementarity problems as special cases.