Non-strictly convex minimization over the fixed point set of an asymptotically shrinking nonexpansive mapping

Suppose that T is a nonexpansive mapping on a real Hilbert space H satisfying sup(\x\greater than or equal toR)(\\T(x)\\/\\x\\) < 1 for some R > 0. Suppose also that a mapping F: H --> H is kappa-Lipschitzian over T(H) and paramonotone over Fix(T). Then it is shown that a variation of the hybrid steepest descent method (Yamada, Ogura, Yamashita and Sakaniwa (1998), Deutschland Yamada (1998) and Yamada (1999-2001)): u(n+1) := T(u(n)) - lambda(n+1)x F(T(u(n))) (n = 0, 1, 2,...) generates a sequence (u(n)) satisfying lim(n-->infinity) d(u(n), Gamma) = 0, when R is finite dimensional, where Gamma := {u is an element of Fix(T) \ greater than or equal to 0 for all v is an element of Fix(T)} not equal theta is the solution set of the variational inequality problem VIP(F, Fix(T)), This result relaxes the condition on F and (lambda(n)) of the hybrid steepest descent method (Yamada (2001)), and makes the method applicable to the significantly wider class of convexly constrained inverse problems as well as the non-strictly convex minimization over the fixed point set of asymptotically shrinking nonexpansive mapping.