The Strong Convergence Of Iterations For Nonexpansive Nonself Mappings

In this thesis we deal with the strong convergence of iterations for nonexpansive nonself mappings.In the first chapter, we introduce some definitions, lemmas and recent results of strong convergence of iterations for nonexpansive nonself map-pings.In the second chapter, we study a new iteration with error which is constructed by the asymptotically quasi-nonexpansive nonself mappings. Based on it, we introduce the asymtotically quasi-nonexpansive nonself mappings in the intermediate sense and prove the convergence theorem in Banach space of the more general modified Ishikawa iterative sequence with error under the two mappings In this iteration, P is nonexpansive retraction mapping.In the third chapter, we study a finite family of nonexpansive nonself mappings and prove the strong convergence of sequence{xn} which is defined by iteration under a finite family of nonexpansive nonself mappings. In this iteration, Q is sunny nonexpansive retraction mapping.In the fourth chapter, based on the study about the congvergence of sin-gle iteration,we go on studying the equivalence of convergence for the two classic iterations. In normal circumstances the convergence between Mann iteration and Ishikawa iteration is unequl. However, under the certain con-dition of iteration parameters:an,αn,βn∈[0,1], limn→∞βn=limn→∞αn= 0,∑n=0+∞αn=+∞, initial point x0=u0, Rhoades and Soltuz has proved that the equivalence of convergence of iterative program for the operator which is Lipschitz continuous andΦ-uniformly pseudo-contractive in Banach space. On the basis of the above research, we study the equiva-lence of convergence of two classic iterations under the lower condition of iteration parameters.