Convergence Theorems For Nonexpansive Semigroups Research Articles

Approximation of fixed points for nonexpansive semigroups in Hilbert spaces

Abstract In this paper, we propose two new algorithms for finding a common fixed point of a nonexpansive semigroup in Hilbert spaces and prove some strong convergence theorems for nonexpansive semigroups. Our results improve and generalize the corresponding results given by Shimizu and Takahashi (J. Math. Anal. Appl. 211:71-83, 1997), Shioji and Takahashi (Nonlinear Anal. TMA 34:87-99, 1998), Lau et al. (Nonlinear Anal. TMA 67:1211-1225, 2007) and many others. MSC:47H05, 47H10, 47H17.

Strong Convergence Theorems for Nonexpansive Semigroups and Variational Inequalities in Banach Spaces

Let X be a uniformly convex Banach space and 𝒮 = {T(s) : 0 ≤ s < ∞} be a nonexpansive semigroup such that F(𝒮) = ⋂s>0F(T(s)) ≠ ∅. Consider the iterative method that generates the sequence {xn} by the algorithm , where {αn}, {βn}, and {sn} are three sequences satisfying certain conditions, f : C → C is a contraction mapping. Strong convergence of the algorithm {xn} is proved assuming X either has a weakly continuous duality map or has a uniformly Gâteaux differentiable norm.

Fixed points of non-expansive mappings associated with invariant means in a Banach space

In this paper, we study the fixed point set of the non-expansive mapping T μ for a Banach space with uniformly Gâteaux differentiable norm when μ is a multiplicative left invariant mean on l ∞ ( S ) . As an application, we establish nonlinear ergodic properties for an extremely amenable semigroup of non-expansive mappings in a Banach space with uniformly Gâteaux differentiable norm. Furthermore, we improve a recent result of Atsushiba and Takahashi [S. Atsushiba, W. Takahashi, Weak and strong convergence theorems for non-expansive semigroups in a Banach spaces satisfying Opial’s condition, Sci. Math. Jpn. (in press)] on the fixed point set of non-expansive mappings associated with a left invariant mean on a left amenable semigroup.

Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals

In this paper, we prove Krasnoselskii and Mann's type convergence theorems for nonexpansive semigroups without using Bochner integral and without assuming the strict convexity of Banach spaces. One of our main results is the following: let C be a compact convex subset of a Banach space E and let { T ( t ) : t ⩾ 0 } be a one-parameter strongly continuous semigroup of nonexpansive mappings on C. Let { t n } be a sequence in [ 0 , ∞ ) satisfying lim inf n → ∞ t n < lim sup n → ∞ t n and lim n → ∞ ( t n + 1 − t n ) = 0 . Let λ ∈ ( 0 , 1 ) . Define a sequence { x n } in C by x 1 ∈ C and x n + 1 = λ T ( t n ) x n + ( 1 − λ ) x n for n ∈ N . Then { x n } converges strongly to a common fixed point of { T ( t ) : t ⩾ 0 } .

Weak and strong convergence theorems for nonexpansive semigroups in Banach spaces

We introduce an implicit iterative process for a nonexpansive semigroup and then we prove a weak convergence theorem for the nonexpansive semigroup in a uniformly convex Banach space which satisfies Opial's condition. Further, we discuss the strong convergence of the implicit iterative process.