Research On The Problem Of Splitting Common Fixed Points In HILBERT Space

The split common fixed point problem(namely SCFPP)as a union of split feasibility problem and fixed point problem was first proposed in 2009 by Censor and Sega.l when they did the work of split feasibility problem.Let H1 and H2 be two real Hilbert spaces equipped with its inner<·,·)and norm ?· ?|,respectively.Let U:H1? H1 and T:H2 ?4 H2 be two nonlinear operators.We use Fix(U)and Fix(T)to denote the fixed point sets of U and T,respectively.Let A:H1?H2 be a bounded linear operator with its adjoint A*.The split common fixed point problem can be formulated as:finding an element x ?H1 satisfyingx ? Fix(U),Ax e Fix(T).For this problem,many authors have given various of methods.Consider-ing the wide and important applications of SCFPP,we have a need to do more reasearch about it.In this paper,we investigate some related issues about the split common fixed point problem and two-sets split common fixed point problem in real Hilbert spaces.This paper totally includes five parts:In the first part,we recal-1 the background,history and present situation of the split common fixed point problem,more especially,we introduced several classical but important iterative algorithms.In the second part,based on the Mann-type algorithm we extend the SCFPP to directed operators.By the equivalence relation between SCFPP and the fixed point problem,we construct effective iterative schemes to approximate the set of fixed points of the director mapping,then we obtain the corresponding convergence theorems.In the third part,we deeply study demi-contractive map-pings in Hilbert space,construct different effective iterative schemes,obtain the corresponding strong convergence theorems and finally apply our conclusions to the split feasibility problem.In the fourth part,we study two uniformly Lipschitzian asymptotically pseudocontractive operators in Hilbert space.A unified framework for the study of this class problem and class of operators is provided.An iterative algorithm is constructed and strong convergence analysis is given.The last part draws a conclusion and future work.