The aim of this paper is to construct a novel implicit iterative algorithm for the split common fixed point problem for the demicontractive operatorsU,T, and xn=αnfxn+1-αnUλxn-ρnA*I-TAxn,n≥0, whereUλ=(1-λ)I+λU, and we obtain the sequence which strongly converges to a solutionx^of this problem, and the solutionx^satisfies the variational inequality.〈x^-f(x^),x^-z〉≤0,∀z∈S, whereSdenotes the set of all solutions of the split common fixed point problem.

Some variable Krasnonsel'skiĭ-Mann iteration algorithms generate some sequences "Equation missing"<!-- image only, no MathML or LaTex -->, "Equation missing"<!-- image only, no MathML or LaTex -->, and"Equation missing"<!-- image only, no MathML or LaTex -->, respectively, via the formula "Equation missing"<!-- image only, no MathML or LaTex -->, "Equation missing"<!-- image only, no MathML or LaTex -->, "Equation missing"<!-- image only, no MathML or LaTex -->, where "Equation missing"<!-- image only, no MathML or LaTex --> and the mod function takes values in "Equation missing"<!-- image only, no MathML or LaTex -->, "Equation missing"<!-- image only, no MathML or LaTex -->, "Equation missing"<!-- image only, no MathML or LaTex -->, and"Equation missing"<!-- image only, no MathML or LaTex --> are sequences in "Equation missing"<!-- image only, no MathML or LaTex --> and "Equation missing"<!-- image only, no MathML or LaTex --> are sequences of nonexpansive mappings. We will show, in a fairly general Banach space, that the sequence "Equation missing"<!-- image only, no MathML or LaTex -->, "Equation missing"<!-- image only, no MathML or LaTex -->, "Equation missing"<!-- image only, no MathML or LaTex --> generated by the above formulas converge weakly to the common fixed point of "Equation missing"<!-- image only, no MathML or LaTex -->, respectively. These results are used to solve the multiple-set split feasibility problem recently introduced by Censor et al. (2005). The purpose of this paper is to introduce convergence theorems of some variable Krasnonsel'skiĭ-Mann iteration algorithms in Banach space and their applications which solve the multiple-set split feasibility problem.

The purpose of this paper is to establish two new iteration schemes as follows: xn=αnxn−1+(1−αn)yn, yn∈Txn, n≥1, xn′=βnu+αnxn−1′+(1−αn−βn)yn′, yn′∈Txn′, n≥1, for a multi-valued nonexpansive mapping T in a uniformly convex Banach space and prove that {xn} and {xn′} converge strongly to a fixed point of T under some suitable conditions, respectively. Moreover, a gap in Sahu (Nonlinear Anal. 37:401-407, 1999) is found and revised.

In this paper, we aim to construct a new strong convergence algorithm for a split common fixed point problem involving the demicontractive operators. It is proved that the vector sequence generated via the Halpern-like algorithm converges to a solution of the split common fixed point problem in norm. The main convergence results presented in this paper extend and improve some corresponding results announced recently. The highlights of this paper shed on the novel algorithm and the new analysis techniques.