The Convergence For Split Variational Inclusion And Fixed Point Of Bregman Nonexpansive Mapping

In this paper,we made a more deep analysis for the current two hot issues in the field of nonlinear analysis.one of them is split the variational inclusion,the other is the fixed point problem of Bregman nonexpansive mapping.The split variational inclusion is an extension of split problem in the field of nonlinear analysis.Because it provides us with a unified framework of split the variational inclusion for studing these problems which include the split feasibility problem,split equilibrium problem,split variational inequality problem,split common fixed point problem and others problem through appropriate techniques.The fixed point theory of nonexpansive operators is extended by Bregman nonexpansive mapping.In the second chapter,thethesis study the convergence of iterative sequence for split variational inclusion problem and fixed point problem of nonexpansive mapping.In the third and the fourth chapters,this thesis study Bregman nonexpansive style mappings which are more generally mappings.For the above two issues,we mainly obtained the following research results:Chapter One:a brief introduction about the split variational inclusions and the Bregman nonexpansive type operators,which include the background,significance,research status,contents and the thesis structure.Chapter Two:this thesis construct a Mann-Halpern iterative algorithm for finding a common solution of split variational inclusion problem and fixed point problem of nonexpansive mapping in Hilbert space.we prove a strong convergence theorem under suitable control conditions.Finally,the application to split optimition problem is given by the result.Chapter Three:motivated and inspired by literature[23],in reflexive Banach spaces,we combine the fixed point of Bregman totally quasi-asymptotically nonexpansive mapping with equilibrium problem to study common solution by a new hybrid iterative scheme.At the same time,we obtain that the iterative sequence generated by hybrid iterative algorithm converges strongly to a certain point in F(T)?EP(g)under appropriate conditions.Finally,the application to zero point problem of maximal monotone operators is given by the result.Chapter Four:this thesis discuss a single operator whether can be extended to countable operators for equilibrivum problem in reflexive Banach spaces.Moreover,we obtain the strong convergence theorems for equilibrium problem and fixed point of a countable family Bregman weak relatively nonexpansive mapping under some suitable assumptions.