Songnian He

Solving the split equality problem without prior knowledge of operator norms

The split equality problem has extraordinary utility and broad applicability in many areas of applied mathematics. Recently, Moudafi proposed an alternating CQ algorithm and its relaxed variant to solve it. However, to employ Moudafi’s algorithms, one needs to know a priori norm (or at least an estimate of the norm) of the bounded linear operators (matrices in the finite-dimensional framework). To estimate the norm of an operator is very difficult, but not an impossible task. It is the purpose of this paper to introduce a projection algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any priori information about the operator norms. We also practise this way of selecting stepsizes for variants of the projection algorithm, including a relaxed projection algorithm where the two closed convex sets are both level sets of convex functions, and a viscosity algorithm. Both weak and strong convergence are investigated.

Weak and Strong Convergence Theorems for Nonexpansive Mappings in Banach Spaces

The purpose of this paper is to introduce two implicit iteration schemes for approximating fixed points of nonexpansive mapping and a finite family of nonexpansive mappings , respectively, in Banach spaces and to prove weak and strong convergence theorems. The results presented in this paper improve and extend the corresponding ones of H.-K. Xu and R. Ori, 2001, Z. Opial, 1967, and others.

Generalized viscosity approximation methods for nonexpansive mappings

AbstractWe combine a sequence of contractive mappings {fn} and propose a generalized viscosity approximation method. One side, we consider a nonexpansive mapping S with the nonempty fixed point set defined on a nonempty closed convex subset C of a real Hilbert space H and design a new iterative method to approximate some fixed point of S, which is also a unique solution of the variational inequality. On the other hand, using similar ideas, we consider N nonexpansive mappings {Si}i=1N with the nonempty common fixed point set defined on a nonempty closed convex subset C. Under reasonable conditions, strong convergence theorems are proven. The results presented in this paper improve and extend the corresponding results reported by some authors recently. MSC:47H09, 47H10, 47J20, 47J25.

Boundary point algorithms for minimum norm fixed points of nonexpansive mappings

AbstractLet H be a real Hilbert space and C be a closed convex subset of H. Let T:C→C be a nonexpansive mapping with a nonempty set of fixed points Fix(T). If 0∉C, then Halpern’s iteration process xn+1=(1−tn)Txn cannot be used for finding a minimum norm fixed point of T since xn may not belong to C. To overcome this weakness, Wang and Xu introduced the iteration process xn+1=PC(1−tn)Txn for finding the minimum norm fixed point of T, where the sequence {tn}⊂(0,1), x0∈C arbitrarily and PC is the metric projection from H onto C. However, it is difficult to implement this iteration process in actual computing programs because the specific expression of PC cannot be obtained, in general. In this paper, three new algorithms (called boundary point algorithms due to using certain boundary points of C at each iterative step) for finding the minimum norm fixed point of T are proposed and strong convergence theorems are proved under some assumptions. Since the algorithms in this paper do not involve PC, they are easy to implement in actual computing programs. MSC:47H09, 47H10, 65K10.

A modified subgradient extragradient method for solving monotone variational inequalities

In the setting of Hilbert space, a modified subgradient extragradient method is proposed for solving Lipschitz-continuous and monotone variational inequalities defined on a level set of a convex function. Our iterative process is relaxed and self-adaptive, that is, in each iteration, calculating two metric projections onto some half-spaces containing the domain is involved only and the step size can be selected in some adaptive ways. A weak convergence theorem for our algorithm is proved. We also prove that our method has O(1n)

Strong Convergence Theorems by Shrinking Projection Methods for Class Mappings

O(\frac{1}{n})

Convergence theorems of shrinking projection methods for equilibrium problem, variational inequality problem and a finite family of relatively quasi-nonexpansive mappings

convergence rate.

A general iterative algorithm for an infinite family of nonexpansive operators in Hilbert spaces

We prove a strong convergence theorem by a shrinking projection method for the class of mappings. Using this theorem, we get a new result. We also describe a shrinking projection method for a nonexpansive mapping on Hilbert spaces, which is the same as that of Takahashi et al. (2008).

A hybrid iteration scheme for equilibrium problems and common fixed point problems of generalized quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces

We introduce a W-mapping for a finite family of relatively quasi-nonexpansive mappings and construct an iterative scheme for finding a common element of the solution set of equilibrium problem, the solution set of the variational inequality problem for an inverse-strongly-monotone operator and set of common fixed points of a finite family of relatively quasi-nonexpansive mappings. Strong convergence theorems are presented in a 2-uniformly convex and uniformly smooth Banach space. Our results generalize and extend relative results.

Weak and Strong Convergence Theorems for Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense

In this paper, we introduce a new general iterative algorithm for an infinite family of nonexpansive operators in Hilbert spaces. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to a common point of the sets of fixed points, which solves a variational inequality. Our results improve and extend the corresponding results announced by many others. As applications, at the end of the paper, we apply our results to the split common fixed point problem.