The general iterative methods for nonexpansive semigroups in Banach spaces

Rattanaporn Wangkeeree () and Rabian Wangkeeree ()

Journal of Global Optimization, 2013, vol. 55, issue 2, 417-436

Abstract: Let E be a real reflexive strictly convex Banach space which has uniformly Gâteaux differentiable norm. Let $${\mathcal{S}=\{T(s): 0 \leq s > \infty\}}$$ be a nonexpansive semigroup on E such that $${Fix(\mathcal{S}) := \cap_{t\geq 0}Fix( T(t) ) \not= \emptyset}$$ , and f is a contraction on E with coefficient 0 > α > 1. Let F be δ-strongly accretive and λ-strictly pseudo-contractive with δ + λ > 1 and $${0 > \gamma > \min\left\{\frac{\delta}{\alpha}, \frac{1-\sqrt{ \frac{1-\delta}{\lambda} }}{\alpha} \right\} }$$ . When the sequences of real numbers {α n } and {t n } satisfy some appropriate conditions, the three iterative processes given as follows : $${\left.\begin{array}{ll}{x_{n+1}=\alpha_n \gamma f(x_n) + (I - \alpha_n F)T(t_n)x_n,\quad n\geq 0,}\\ {y_{n+1}=\alpha_n \gamma f(T(t_n)y_n) + (I - \alpha_n F)T(t_n)y_n,\quad n\geq 0,}\end{array}\right.}$$ and $$ z_{n+1}=T(t_n)( \alpha_n \gamma f(z_n) + (I - \alpha_n F)z_n),\quad n\geq 0 $$ converge strongly to $${\tilde{x}}$$ , where $${\tilde{x}}$$ is the unique solution in $${Fix(\mathcal{S})}$$ of the variational inequality $${ \langle (F - \gamma f)\tilde {x}, j(x - \tilde{x}) \rangle \geq 0,\quad x\in Fix(\mathcal{S}).}$$ Our results extend and improve corresponding ones of Li et al. (Nonlinear Anal 70:3065–3071, 2009 ) and Chen and He (Appl Math Lett 20:751–757, 2007 ) and many others. Copyright Springer Science+Business Media, LLC. 2013

Keywords: General iterative method; Nonexpansive semigroup; Reflexive Banach space; Uniformly Gâteaux differentiable norm; Fixed point; 47H05; 47H09; 47J25; 65J15 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10898-011-9835-6

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