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Strong convergence of implicit iteration processes for nonexpansive semigroups in Banach spaces

Kozlowski W. M.

Commentationes Mathematicae

2014

Vol 54, No. 2

203--208

Let C be a convex compact subset of a uniformly convex Banach space. Let {Tt}t≥0 be a strongly-continuous nonexpansive semigroup on C. Consider the iterative process defined by the sequence of equations xk+1=ckTtk+1(xk+1)+(1−ck)xk. We prove that, under certain conditions on {ck} and {tk}, the sequence {xk}∞n=1 converges strongly to a common fixed point of the semigroup {Tt}t≥0. There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property, and also weak convergence results in Banach spaces that are simultaneously uniformly convex and uniformly smooth. In this paper, we do not assume the Opial property or uniform smoothness of the norm.

Modified Mann iterative algorithms by hybrid projection methods for nonexpansive semigroups and mixed equilibrium problems

Katchang P., Kumam P.

Journal of Applied Analysis

2012

Vol. 18, nr 2

259-273

In this paper, we propose a modified Mann iterative algorithm by two hybrid projection methods for finding a common element of the set of fixed points of nonexpansive semigroups and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then, we obtain interesting and new strong convergence theorems for the sequences generated by these processes by using the hybrid projection methods in the mathematical programming. The results presented in this paper extend and improve the corresponding one by Nakajo and Takahashi [J. Math. Anal. Appl. 279 (2003), 372-379].

An algorithm for finding a common solution for a system of mixed equilibrium problem, quasi-variational inclusion problem and fixed point problem of nonexpansive semigroup

Min L., Chang S., Zuo P.

Opuscula Mathematica

2010

Vol. 30, no. 4

465-484

In this paper, we introduce a hybrid iterative scheme for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed points for a nonexpansive semigroup and the set of solutions of the quasi-variational inclusion problem with multi-valued maximal monotone mappings and inverse-strongly monotone mappings in a Hilbert space. Under suitable conditions, some strong convergence theorems are proved. Our results extend some recent results in the literature.