Viscosity Approximation for Nonexpansive Nonself-Mappings InReflexive Banach Spaces
自反Banach空间中非扩张非自映射的粘滞迭代逼近方法

Let $E$ be a reflexive and strictly convex Banach space with a uniformly G\^ateaux differentiable norm, and $K$ be a nonempty closed convex subset of $E$ which is also a sunny nonexpansive retract of $E$. Assume that $T:K\to E$ is a nonexpansive mapping with $F(T)\neq\emptyset$, and $f:K\to K$ is a fixed contractive mapping. The implicit iterative sequence $\{x_t\}$ is defined by $x_t=P(tf(x_t)+(1-t)Tx_t)$ for $t\in (0,1).$ The explicit iterative sequence$\{x_n\}$ is given by $x_{n+1}=P(\alpha_nf(x_n)+(1-\alpha_n)Tx_n)$, where $\alpha_n\in(0,1)$ satisfies appropriate conditions and $P$ is nonexpansive retraction of $E$ onto $K$. It is shown that $\{x_t\}$ and $\{x_n\}$ strongly converges to a fixed point of $T$.