自反Banach空间中非扩张非自映射的粘滞迭代逼近方法

主要在自反和严格凸的且具有一致G(a)teaux可微范数的Banach空间中研究了非扩张非自映射的粘滞迭代逼近过程,证明了此映射的隐格式与显格式粘滞迭代序列均强收敛到它的某个不动点.

Viscosity Approximation for Nonexpansive Nonself-Mappings InReflexive Banach Spaces

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 $T$.