Research On Iterative Algorithm Of Nonexpansive Operator And Its Application In Variational Inequality Problem And Multi-valued Variational Inequality Problem

This thesis is mainly of three folds: introduce a new hybrid algorithm for finding a fixed point of a nonexpansive mapping and a cyclic hybrid algorithm for approximating a common fixed point of a finite family of nonexpansive mappings, an inertial extrogradient projection algorithm for variational inequality problems and a projection and contraction algorithm for multi-values variational inequality problems, and a self-adaptive projection algorithm and a relaxed self-adaptive projection algorithm for split equality problems. It mainly divided into the following several parts:In the first chapter, the author presents introduction and preliminary. The author introduces the historical background and research status of the fixed point problems of nonexpansive mappings, variational inequalities problems and split equality problems.Research purpose and meaning are also discussed.In the second chapter, the author provides some related definitions and lemmas needed in the proof of main results.In the third chapter, first of all, the author introduces a new hybrid algorithm to find a fixed point of nonexpansive mapping, which is not based on the modification to any weak convergence algorithms. Secondly, the author proposes a cyclic hybrid method to approximate a common fixed point of a finite family of nonexpansive mappings. Numerical examples show that two algorithms both greatly improve the convergence rate.In the fourth chapter, firstly the author introduces an inertial extragradient algorithm to solve the variational inequality problems by incorporating inertial terms in the extragradient algorithm. Secondly, a projection and contraction algorithm for solving multi-valued variational inequality problem is proposed. The convergence of two algorithms is analyzed.Preliminary computational experiments illustrate the advantage of the proposed algorithms.In the fifth chapter, the author introduces a selfadaptive projection-type algorithm, which computes the stepsizes by adopting a line search and doesn't need to compute(or, at least,estimate) operator norms of A and B. The convergence of the propose algorithm is analyzed.The calculation steps are greatly simplified.In the sixth chapter, the author summarizes the main content of this paper and putforward some problems to be solved.