Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: QUADRATIC OPTIMIZATION OF FIXED POINTS FOR A FAMILY OF NONEXPANSIVE MAPPINGS IN HILBERT SPACE | QUADRATIC OPTIMIZATION OF FIXED POINTS FOR A FAMILY OF NONEXPANSIVE MAPPINGS IN HILBERT SPACE B. E. RHOADES Received 10 September 2003 Given a finite family of nonexpansive self-mappings of a Hilbert space a particular quadratic functional and a strongly positive selfadjoint bounded linear operator Yamada et al. defined an iteration scheme which converges to the unique minimizer of the quadratic functional over the common fixed point set of the mappings. In order to obtain their result they needed to assume that the maps satisfy a commutative type condition. In this paper we establish their conclusion without the assumption of any type of commutativity. Finding an optimal point in the intersection F of the fixed point sets of a family of nonexpansive maps is one that occurs frequently in various areas of mathematical sciences and engineering. For example the well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpan-sive maps. See . 3 4 . The problem of finding an optimal point that minimizes a given cost function 0 X R over F is of wide interdisciplinary interest and practical importance. See . 2 4 5 7 14 . A simple algorithmic solution to the problem of minimizing a quadratic function over F is of extreme value in many applications including the set-theoretic signal estimation. See . 5 6 10 14 . The best approximation problem of finding the projection PF a in the norm induced by the inner product of X from any given point a in GX is the simplest case of our problem. Some papers dealing with this best approximation problem are 2 9 11 . Let GX be a Hilbert space C a closed convex subset of GX and Tị where i 1 2 . N a finite family of nonexpansive self-maps of C with F n 1 Fix Ti 0. Define a quadratic function 0 X R by 0 u - Au u b u Vu e X 1 where b e X and A is a selfadjoint strongly positive operator. We will also assume that B I - A satisfies B 1 although this is not restrictive .
