CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES
Jong Soo Jung · Korean Journal of Mathematics · 2008
Let E be a uniformly convex Banach space with a uni- formly G^ateaux difierentiable norm, C a nonempty closed convex subset of E, and T : C ! K(E) a multivalued nonself-mapping such that PT is nonexpansive, where PT(x) = fux 2 Tx : kx i uxk = d(x;Tx)g. For f : C ! C a contraction and t 2 (0;1), let xt be a flxed point of a contraction St : C ! K(E), deflned by Stx := tPT(x)+(1it)f(x); x 2 C. It is proved that if C is a nonexpansive retract of E and fxtg is bounded, then the strong limt!1 xt exists and belongs to the flxed point set of T. Moreover, we study the strong convergence of fxtg with the weak inwardness condition on T in a re∞exive Banach space with a uniformly G^ateaux difierentiable norm. Our results provide a partial answer to Jung's question.