Haiyun Zhou

Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings

Abstract In the present paper, several weak and strong convergence theorems are established for the three-step iterative schemes with errors for asymptotically nonexpansive mappings. Our results extend and improve the recent ones announced by Tan and Xu, Xu and Noor, and many others.

Convergence of a modified Halpern-type iteration algorithm for quasi-ϕ-nonexpansive mappings

The purpose of this work is to modify the Halpern-type iteration algorithm to have strong convergence under a limit condition only in the framework of Banach spaces. The results presented in this work improve on the corresponding ones announced by many others.

Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive Banach spaces

In this paper, by using some new analysis techniques, we study the approximation problems of common fixed points of Halperns iterative sequence for a class of finite nonexpansive mappings in strictly convex and reflexive Banach spaces by using Banachs limit. The main results presented in this paper generalize, extend and improve the corresponding results of Bauschke [The approximation of fixed points of compositions of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 202 (1996) 150-159], Halpern [Fixed points of nonexpansive maps, Bull. Am. Math. Soc. 73 (1967) 957-961], Shioji and Takahashi [Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Am. Math. Soc. 125 (1997) 3641-3645], Takahashi et al. [Approximation of common fixed points of a family of finite nonexpansive mappings in Banach spaces, Sci. Math. Jpn. 56 (2002) 475-480], Wittmann [Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486-491], Xu [Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc. 65 (2002) 109-113, Remarks on an iterative method for nonexpansive mappings, Commun. Appl. Nonlinear Anal. 10 (2003) 67-75] and others.

Nonexpansive Mappings and Iterative Methods in Uniformly Convex Banach Spaces

Abstract In this paper, most of classical and modern convergence theorems of iterative schemes for nonexpansive mappings are presented and the main results in the paper generalize and improve the corresponding results given by many authors.

Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces

Let be a real Hilbert space, a nonempty closed convex subset of , and a maximal monotone operator with . Let be the metric projection of onto . Suppose that, for any given , , and , there exists satisfying the following set-valued mapping equation: for all , where with as and is regarded as an error sequence such that . Let be a real sequence such that as and . For any fixed , define a sequence iteratively as for all . Then converges strongly to a point as , where .

Characteristic conditions for convergence of generalized steepest descent approximation to multivalued accretive operator equations

Abstract In the present paper, sufficient and necessary conditions for convergence of generalized steepest descent approximation to multivalued accretive operator equations are established for the sufficiency part, and a specific error estimation is also given.

Iterative processes with mixed errors for nonlinear equations with perturbed m-accretive operators in Banach spaces

Let X be a real uniformly smooth Banach space of which the dual X^* has a Frechet differentiable norm. Let A:D(A)@?X->2^X be an m-accretive operator with closed domain D(A) and bounded range R(A) and S:X->X a continuous and @a-strongly accretive operator with bounded range R(I-S). It is proved that the Ishikawa and Mann iterative processes with mixed errors converge strongly to the unique solution of the equation z@?Sx+@lAx for given z@?X and @l>0.

Weak stability of the Ishikawa iteration procedures for ∅-hemicontractions and accretive operators

Abstract Let X be an arbitrary Banach space, K be a nonempty closed convex subset of X, and T : K → K be a Lipschitzian and hemicontractive mapping with the property lim inft→∞(∅(t)/t) > 0. It is shown that the Ishikawa iteration procedures are weakly T-stable. As consequences, several related results deal with the weak stability of these procedures for the iteration proximation of solutions of nonlinear equations involving accretive operators. Our results improve and extend those corresponding results announced by Osilike.

ITERATIVE APPROXIMATIONS OF ZEROES FOR ACCRETIVE OPERATORS IN BANACH SPACES

In this paper, we introduce and study a new iterative algorithm for approximating zeroes of accretive operators in Banach spaces.

Approximations for fixed points of φ-hemicontractive mappings by the Ishikawa iterative process with mixed errors

Let X be a real uniformly smooth Banach space, K be a nonempty closed convex subset of X and T : K -> K be a generalized Lipschitzian and hemicontractive mapping. It is shown that the Ishikawa iterative process with mixed errors converges strongly to the unique fixed point of the mapping T. As consequences, several new strong convergence results are deduced and some known results are improved.