Two Iterative Algorithms For Splitting Common Fixed Point And Splitting Equality Problems

In this paper,we study the split common fixed point problem and the split equa-tion problem in Hilbert spaces.By using of the weighted average iteration method,a new iterative algorithm about two countable families of strict pseudo-contractive mappings is established to approximate the solution of the fixed points.Under some weaker conditions for parameters,the strong convergence of the iterative algorithm is proved,which makes some known results more widely adaptable,and at the same time which solves the split equation problem.In addition,the hybrid projection method is used to establish an iterative algorithm for a quasi-pseudo-contractive mapping to solve the split common fixed point problem.The strong convergence of the iterative algorithm is proved under appropriate conditions.The main purpose of this paper is to study the split common fixed point problem in Hilbert space.Some known results of other scholars are extended and improveced.So the results in this paper have a certain theoretical siginificance and application value.The main results here are as follows:Result one.:Let H₁,H₂ be real Hilbert spaces,and let A:H₁? H₂ be a linear-ly bounded mapping,Ui:H₁?H₁,Ti:H₂?H₂ be two countable families strictly pseudo-contractive mappings with {k_i},{l1} respectively(0<k_i<1,0<li<1).As-suming that k=sup{k_i:i?N}<1,l=sup{li:i?E N}<1,F1:=?i=1?F(Ui)?(?),F2:=ni=1?F(Ti)?(?),S:={z ?F1:Az ?F2} ?(?).Let {xn} is a sequence defined by the following:#12 we suppose that the following conditions are satisfied:#12 Then the sequence {xn} converges weakly to the split common fixed point p ?F1,Ap ?Result two.:Let H₁,H₂ be Real Hilbert spaces,let S:H₁?H₁ and T:H₂?H₂ be two L-Lipschitz continuous and quasi-pseudo-contractive mappings(L>1),U:H₁?H₁ and V:H₂?H₂ are defined as U:(1-?)I+?S(1-?)I+?S)V:(1-?)I+?T(1-?)I+?T).Let A:H₁?H₂ is A linear bounded mapping,A*is its adjoint operator.Let {xn} is a sequence defined by the following:#12 We suppose that the following conditions are satisfied:#12 operators S,T is demiclosed at 0,and the solution set of split common fixed points is not empty.Then the sequence {xn} converges strongly to the split common fixed points p?F(S),Ap?F(T).The two results improve and extend mainly some relevant results of other authors.