The general iterative methods for nonexpansive semigroups in Banach spaces

Let E be a real reflexive strictly convex Banach space which has uniformly Gâteaux differentiable norm. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${\mathcal{S}=\{T(s): 0 \leq s > \infty\}}$$</EquationSource> </InlineEquation> be a nonexpansive semigroup on E such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${Fix(\mathcal{S}) := \cap_{t\geq 0}Fix( T(t) ) \not= \emptyset}$$</EquationSource> </InlineEquation> , 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${0 > \gamma > \min\left\{\frac{\delta}{\alpha}, \frac{1-\sqrt{ \frac{1-\delta}{\lambda} }}{\alpha} \right\} }$$</EquationSource> </InlineEquation> . When the sequences of real numbers {α<Subscript> n </Subscript>} and {t <Subscript> n </Subscript>} satisfy some appropriate conditions, the three iterative processes given as follows :<Equation ID="Equa"> <EquationSource Format="TEX">$${\left.\begin{array}{ll}{x_{n+1}=\alpha_n \gamma f(x_n) + (I - \alpha_n F)T(t_n)x_n,\quad n\geq 0,}\\ {y_{n+1}=\alpha_n \gamma f(T(t_n)y_n) + (I - \alpha_n F)T(t_n)y_n,\quad n\geq 0,}\end{array}\right.}$$</EquationSource> </Equation>and<Equation ID="Equb"> <EquationSource Format="TEX">$$ z_{n+1}=T(t_n)( \alpha_n \gamma f(z_n) + (I - \alpha_n F)z_n),\quad n\geq 0 $$</EquationSource> </Equation>converge strongly to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${\tilde{x}}$$</EquationSource> </InlineEquation> , where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$${\tilde{x}}$$</EquationSource> </InlineEquation> is the unique solution in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${Fix(\mathcal{S})}$$</EquationSource> </InlineEquation> of the variational inequality <Equation ID="Equc"> <EquationSource Format="TEX">$${ \langle (F - \gamma f)\tilde {x}, j(x - \tilde{x}) \rangle \geq 0,\quad x\in Fix(\mathcal{S}).}$$</EquationSource> </Equation>Our results extend and improve corresponding ones of Li et al. (Nonlinear Anal 70:3065–3071, <CitationRef CitationID="CR17">2009</CitationRef>) and Chen and He (Appl Math Lett 20:751–757, <CitationRef CitationID="CR7">2007</CitationRef>) and many others. Copyright Springer Science+Business Media, LLC. 2013