Rudong Chen

Iteration Scheme with Perturbed Mapping for Common Fixed Points of a Finite Family of Nonexpansive Mappings

We propose an iteration scheme with perturbed mapping for approximation of common fixed points of a finite family of nonexpansive mappings . We show that the proposed iteration scheme converges to the common fixed point which solves some variational inequality.

Iterative algorithm for approximating solutions of maximal monotone operators in Hilbert spaces.

We first introduce and analyze an algorithm of approximating solutions of maximal monotone operators in Hilbert spaces. Using this result, we consider the convex minimization problem of finding a minimizer of a proper lower-semicontinuous convex function and the variational problem of finding a solution of a variational inequality.

Split equality problem and multiple-sets split equality problem for quasi-nonexpansive multi-valued mappings

The multiple-sets split equality problem (MSSEP) requires finding a point x∈⋂i=1NCi, y∈⋂j=1MQj, such that Ax=By, where N and M are positive integers, {C1,C2,…,CN} and {Q1,Q2,…,QM} are closed convex subsets of Hilbert spaces H1, H2, respectively, and A:H1→H3, B:H2→H3 are two bounded linear operators. When N=M=1, the MSSEP is called the split equality problem (SEP). If let B=I, then the MSSEP and SEP reduce to the well-known multiple-sets split feasibility problem (MSSFP) and split feasibility problem (SFP), respectively. Recently, some authors proposed many algorithms to solve the SEP and MSSEP. However, to implement these algorithms, one has to find the projection on the closed convex sets, which is not possible except in simple cases. One of the purposes of this paper is to study the SEP and MSSEP for a family of quasi-nonexpansive multi-valued mappings in the framework of infinite-dimensional Hilbert spaces, and propose an algorithm to solve the SEP and MSSEP without the need to compute the projection on the closed convex sets.

Hybrid methods for accretive variational inequalities involving pseudocontractions in Banach spaces

We use strongly pseudocontractions to regularize a class of accretive variational inequalities in Banach spaces, where the accretive operators are complements of pseudocontractions and the solutions are sought in the set of fixed points of another pseudocontraction. In this paper, we consider an implicit scheme that can be used to find a solution of a class of accretive variational inequalities.Our results improve and generalize some recent results of Yao et al. (Fixed Point Theory Appl, doi:10.1155/2011/180534, 2011) and Lu et al. (Nonlinear Anal, 71(3-4), 1032-1041, 2009).2000 Mathematics subject classification 47H05; 47H09; 65J15

Convergence results of multi-valued nonexpansive mappings in Banach spaces

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.MSC:47H04, 47H10, 47J25.

Convergence analysis of an iterative algorithm for the extended regularized nonconvex variational inequalities

In this paper, we suggest and analyze a new system of extended regularized nonconvex variational inequalities and prove the equivalence between the aforesaid system and a fixed point problem. We introduce a new perturbed projection iterative algorithm with mixed errors to find the solution of the system of extended regularized nonconvex variational inequalities. Furthermore, under moderate assumptions, we research the convergence analysis of the suggested iterative algorithm.

Strong convergence theorems for a common zero of a finite family of H -accretive operators in Banach space

The aim of this paper is to study a finite family of H-accretive operators and prove common zero point theorems of them in Banach space. The results presented in this paper extend and improve the corresponding results of Zegeye and Shahzad (Nonlinear Anal 66:1161–1169, 2007), Liu and He (J Math Anal Appl 385:466–476, 2012) and the related results.

Iterative algorithms for finding the zeroes of sums of operators

Let H1, H2 be real Hilbert spaces, C⊆H1 be a nonempty closed convex set, and 0∉C. Let A:H1→H2, B:H1→H2 be two bounded linear operators. We consider the problem to find x∈C such that Ax=−Bx (0=Ax+Bx). Recently, Eckstein and Svaiter presented some splitting methods for finding a zero of the sum of monotone operator A and B. However, the algorithms are largely dependent on the maximal monotonicity of A and B. In this paper, we describe some algorithms for finding a zero of the sum of A and B which ignore the conditions of the maximal monotonicity of A and B.

An improved method for solving multiple-sets split feasibility problem

The multiple-sets split feasibility problem (MSSFP) has a variety of applications in the real world such as medical care, image reconstruction and signal processing. Censor et al. proposed solving the MSSFP by a proximity function, and then developed a class of simultaneous methods for solving split feasibility. In our paper, we improve a simultaneous method for solving the MSSFP and prove its convergence.

Mann Type Implicit Iteration Approximation for Multivalued Mappings in Banach Spaces

Let be a nonempty compact convex subset of a uniformly convex Banach space and let be a multivalued nonexpansive mapping. For the implicit iterates , , , . We proved that converges strongly to a fixed point of under some suitable conditions. Our results extended corresponding ones and revised a gap in the work of Panyanak (2007).