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

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Kozlowski W. M.

Commentationes Mathematicae

2014

tom 54

nr 2

Let $C$ be a convex compact subset of a uniformly convex Banach space. Let $\{T_t\}_{t \geq0}$ be a strongly-continuous nonexpansive semigroup on $C$. Consider the iterative process defined by the sequence of equations $$x_{k+1} =c_k T_{t_{k+1}}(x_{k+1})+(1-c_k)x_k.$$ We prove that, under certain conditions on $\{c_k\}$ and $\{t_k\}$, the sequence $\{x_k\}_{n=1}^\infty$ converges strongly to a common fixed point of the semigroup $\{T_t\}_{t \geq0}$. 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.

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1 Commentationes Mathematicae

1 Kozlowski W. M.

1 2014

Wyniki wyszukiwania

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