Strong Convergence Theories For Different Split Inverse Problems

Split inverse problem is a general term for a class of problems,which includes many specific problems,such as split variational inequality problem,split feasibility problem,as well as split equality problem,multiple-sets split feasibility problem,multiple-sets split equality problem,proximal split feasibility problem that derived from split feasibility problem and so on.The purpose of this paper is to study the strong convergence of the split feasibility problem,the split equality problem and the multiple-sets split equality problem,which have been widely applied in science and technology,such as intensity-modulated radiation therapy,signal processing,image reconstruction and so on.So,it is of great practical significance to study the algorithms of these problems.And this paper is divided into five parts:In chapter one,we will respectively introduce the development process of split feasibility problem,split equality problem and multiple-sets splitting equality problem,and know about their research background and research status.In chapter two,we will review the basic knowledge and known conclusions that will be used in the next proof.In chapter three,based on the existing algorithms,combining a new kind of mapping(2-mapping with viscous algorithm,we will give a new viscous algorithm to solve the split feasibility problem,and prove it’s strong convergence which will illustrate the effectiveness of the algorithm that we constructed to solve the split feasibility problem.In chapter four,firstly,we will transform the split feasibility problem into solving a fixed point equations.When solving the split equation problem,Mr.Wang Fenghui have constructed strong and weak convergence algorithms respectively by turning the problem into a problem of solving a fixed point equations.Inspired by his idea,by changing the selection method of parameter,we will construct a new fixed point equations,and prove that looking for the solution of the new equations is equivalent to solve the split equation problem;Nextly,based on the new equations given by us,a relaxed alternating algorithm will be given and its weak convergence will be proved;Finally,in order to get a stronger conclusion,we will improve the weak convergence algorithm and prove the strong convergence of the improved algorithm.In chapter five,in order to solve the multiple-sets split equality problem,through special selection method,professor jiang lining et al.have constructed a strong convergent iterative algorithm.On the basis of them,by incorporating inertia technique into their algorithm,we will construct a new inertial relaxation CQ algorithm which generalizes the conclusion given by professor jiang lining et al.,and prove it’s strong convergence.In chapter six,we will give a summary of our work and a prospect for future work.