ON THE STABILITY OF THE EXISTENCE OF FIXED POINTS FOR PROJECTION-ITERATIVE METHODS WITH RELAXATION

We consider an a-relaxed projection P-A(alpha) : H -> H given by P-A(alpha)(x) = alpha P-A(x) + (1 - alpha)x where alpha is an element of [0, 1] and P-A is the projection onto a non-empty, convex and closed subset A of a real Hilbert space H. We characterise all the sets F C [0, 1] such that for some non-empty, convex and closed subsets A(1), A(2), ... , A(k) subset of H the composition P-Ak(alpha) P-Ak-1(alpha), ... P-A1(alpha) has a fixed point if alpha is an element of F. We prove, that if dim H >= 3 and k >= 3 then the class of the derscribed above sets F of coefficients a is exactly the class of F-sigma. subsets of [0, 1] containing 0.