Some Iterative Algorithms And Their Applications For Split Feasibility Problem And Other Relative Problems

This thesis mainly investigate the split feasibility problem,the split fixed point problem and the null problem of maximal monotone operators in Hilbert spaces.We propose several new methods for solving these problems,and establish their weak or strong convergence theorems under certain conditions.In Chapter 1,we first recall the research background of the split feasibility prob-lem,the split fixed point problem and the null problem of maximal monotone oper-ators.Then we introduce the main contribution of this thesis.In Chapter 2,we recall some preliminary concepts and useful lemmas.We first recall the concepts and properties of non-expansive operators and monotone opera-tors.Then we recall several useful lemmas including the Fejer monotonicity,sub-differential inequality and Young inequality.In Chapter 3,we mainly study the split feasibility problem constrained by level sets.Level sets is a class of complex subsets when concerning projection calculation.To overcome this difficulty,we introduce a new relaxed method.Unlike the usual idea of constructing half-spaces,we construct a class of closed balls to approximate the original level sets.Since the projections onto closed balls have closed form,the proposed algorithm is much easier to implement in practice.We establish the conver-gence theorems of the proposed algorithm under several kinds of stepsize.Compared with the existing relaxed algorithm,the proposed algorithm has better approximation accuracy and faster rate of convergence.In Chapter 4,we mainly study the split fixed point problem and the CS' algorith-m.For demi-contractive operators,we prove the convergence of the CS' algorithm under the condition:?n?n = ?,?n?n2<?,which is posed on the stepsize ?n.For the strictly psudo-contractive operators,we conclude that the convergence of the CS'algorithm is guaranteed under a weaker condition:limn?n=0,?n?n=?.The work formulated above generalizes and improves the existing convergence result of the CS'algorithm.Furthermore,by considering product space,we further extend the above results to the split equality problems and the generalized multiple split fixed point problems.In Chapter 5,we mainly study the null problem of maximal monotone operators and the contraction proximal point algorithm.The existing work mainly concerns the convergence of the contraction proximal point algorithm with under-relaxed factors.By making use of properties of firmly nonexpansive operators,we state the strong convergence of the over-relaxed contraction proximal point algorithm under two d-iff erent error criterions.More specifically,we extend the value range of the factor?n from(0,1)to(0,2),which clearly includes the case of over-relaxed factor.Nu-merical experiments conducted indicate that over-relaxed contraction proximal point algorithm converges faster than that in under-relaxed case.