Fast Algorithms Of Nonexpansive Mapping And The Application In Multiple-sets Split Feasibility Problem

The theory of fixed points is an important topic of modern mathematics and it is widely used in many subjects such as optimization and control theory, nonlinear operator etc. This paper studies the fast algorithms of the nonexpansive mappings and the applications in multi ple-sets split feasibility problem. We combined the geometry of Banach spaces, Banach space non-linear approximation theory, fixed point theory and variational principle and applied the mathematical tools of the metric projection, conjugate gradient method, extrapolation of inertia, projection and contraction method and parallel direction methods. The results in this paper can be viewed as the improvement and supplementation of the corresponding results in some references. The full text is divided into six chapters.1. The historic background and the research status of the fixed point theory are recalled briefly. Then the summary of the work in this paper is given.2. We recall some basic concepts and theories.3. We firstly construct the accelerated Mann iterative and the accelerated CQ iterative fast algorithms by applying the conjugate gradient method in the Hilbert space. Secondly, we prove convergence theorems of the new proposed iterative algorithms and demonstrate its effectiveness by the numerical simulation. The results show that accelerated Mann iterative algorithm is effective. Eespecially for large data iteration, acceleration effect is more obvious. Unfortunately, the accelerated CQ iterative algorithm doesn’t achieve the desired acceleration effect.4. By using the extrapolation of inertia, we modify the acceleration of the algorithms introduced in third chapter to propose an inertia accelerated Mann iterative algorithm. We also present an inertia CQ iterative algorithm. Then, we prove the convergence theorems of the algorithms and verify the acceleration effect of two fast algorithms by numerical simulation experiments.5. Based on the relation of the multiple-sets split feasibility problem and the structured variational inequality, we construct a parallel direction format to solve the multiple-sets split feasibility problem by using the method of projection and contraction algorithm. We presentthe proof of its convergence.6. This chapter summarizes the mainly results of this thesis and raises some problems which need to be solved.