The Splitting Equation Problem And Its Related Problems In The Infinite Dimensional HILBERT Space

The split feasibility problem is a class of significant inverse problem arising from signal processing,radiation therapy treatment and medical image reconstruction.Let H1,H2 be two real Hilbert spaces,and C(?)H1,Q(?)H2 be two nonempty closed and convex sets,A:H1?H2 is a bounded and linear operator.Then the split feasibility problem can be formulated as:finding x?C such that Ax?Q.For this problem,many authors have provided kinds of methods.In 2012,Moudafi proposed the split equality problem which is a generalized form of the split feasibility problem.Let H1,H2,H3 be three real Hilbert spaces,and C(?)H1,Q(?)H2 be two nonempty closed and convex sets,A:H1?H3,B:H2?H3 are two bounded and linear operators.Then the split equality problem can be formulated as:finding x?C,y?Q such that Ax = By.Obviously,when H2=H3,B=I(unit operator),the split equality problem reduces to the split feasibility problem.For this problem,many authors have provided relevant algorithms.In view of the significant applications of the split equality problem and related issues,it's worth further study.In the paper,we investigate several related issues of the split equality problem.First,the multiple-sets split equality problem:finding x ??(i=1)t Ci,y??(j=1)r Qj such that Ax=By.Where H1,H2,H3 are three real Hilbert spaces,and {Ci}(i=1)t(?)H1,{Qj}(j=1)r(?)H2 are two groups nonempty closed and convex sets.A:H1?H3,B:H2?H3 are two bounded and linear operators.We provide two algorithms in this paper to solve this problem-the algorithm with split self-adaptive step-size and the algorithm with constructing direction,the main ideas of the algorithms are reducing calculated quantity and improving convergence rate.Second,the split equality fixed point problem:finding x?Fix(T1),y?Fix(T2)such that Ax = By.Where H1,H2,H3 are three real Hilbert spaces.T1:H1?H1,T2:H2?H2 are two nonlinear operators,and Fix(T1),Fix(T2)are the fixed points sets of operators T1,T2,respectively.A:H1?H3,B:H2?H3 are two bounded and linear operators.And the iterative algorithm we present has strong convergence without needing the semi-compactness of the related operators.Third,the split equality variational inclusion problem:finding x E Hi,y E H2 such that 0?U(x),0?K(y),Ax=By.Where H1,H2,H3 are three real Hilbert spaces,U:H1?2H1,K:H2?2H2 are two set-valued maximal monotone operators,A:H1?H3,B:H2?H3 are two bounded and linear operators.And the iterative algorithm we present converges strongly to the minimal norm solution of the split equality variational inclusion problem.