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 satisfying for some R > 0. Suppose also that a mapping is κ-Lipschitzian over and paramonotone over . Then it is shown that a variation of the hybrid steepest descent method (Yamada, Ogura, Yamashita and Sakaniwa (1998), Deutsch and Yamada (1998) and Yamada (1999–2001)): generates a sequence (un ) satisfying , when is finite dimensional, where for all is the solution set of the variational inequality problem . This result relaxes the condition on and (λ 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.