A necessary and sufficient condition for the convergence of a sequence of iterates for quasi-nonexpansive mappings
W. V. Petryshyn, Todd E Williamson · Bulletin of the American Mathematical Society · 1972
Introduction.The purpose of this note is to present two theorems which provide necessary and sufficient conditions for the convergence of the successive approximation method (Theorem 1) and of the convex combination iteration method (Theorem 2) for quasi-nonexpansive mappings defined on suitable subsets of Banach spaces and with nonempty sets of fixed points.We also indicate briefly how these theorems are used to deduce a number of known, as well as some new, convergence results for various special classes of mappings of nonexpansive, P-compact, and 1-set-contractive type which recently have been extensively studied by a number of authors.Complete proofs and detailed discussion of the results presented in this note will be given in [15].THEOREM 1.Let X be a real Banach space, D a closed subset of X, and T a quasi-nonexpansive mapping of D into X.Suppose there exists a AM S 1970 subject classifications.Primary 47H15; Secondary 47H05.