Investigation On Fixed Point Iteration Algorithms For Some Nonlinear Operators

The fixed point theory on nonlinear operator is the main content of nonlinear functional analysis and it has become an important research direction in recent decades.This is mainly due to the following reasons:on the one hand,the fixed point on nonlinear operator is widely used in mathematics?economics and other subjects;on the other hand,there are many problems in the practical applications that are closely related to the fixed point theory on nonlinear operators.By far the fixed point theory of nonlinear operator has formed a systematic theory system.The convergence for fixed point algorithm which approximates various nonlinear operators has been a hot topic that many scholars are studying.In 2016,Marino et.al^[21]gave a new iterative algorithm for strict pseudo-contractions in Hilbert space and proved the strong convergence of the algorithm.They posed the following three open questions at the end of the article,?1?Does the result could be extended to Banach spaces??2?Does the result could be applied to families of strict pseudo-contractive??3?Does the result could be applied to Lipschitzian pseudo-contractive?Inspired by the above research,in this paper we will study the open questions and give the following conclusions:?1?The iterative algorithm designed by Marino^[21]is extended from Hilbert space to q-uniformly smooth Banach space,and the strong convergence of the algorithm is proved.Besides,a new implicit iterative scheme is established and the strong convergence for the implicit iterative is proved.Finally,an example is given to illustrate the designed algorithm.?2?The iterative algorithm designed by Marino^[21]is extended from a single strict pseudo-contraction is extended to a family of strict pseudo-contractions in Hilbert space.Finally,an example is given to illustrate the algorithm.?3?An application on inverse-strongly monotone operator is considered.