关于广义{\Phi}-增生算子的最速下降法的收敛定理

设$E$为一致光滑Banach空间,$A:E\to E$为有界次连续广义${\it \Phi} $-增生算子满足:对任意$x_0\in E$,选取$m\ge 1$,使得$\| x_0 - x^* \| \le m$且$\mathop {\underline {\lim } }\limits_{r \to \infty } {\it \Phi} (r) > m\left\| {Ax_0 } \right\|$.设$\{C_n\}$为$0,1]$中数列满足控制条件: i)$C_n\to 0\,(n\to\infty)$; ii)$\sum\limits_{n = 0}^\infty {C_n } = \infty $.设$\{x_n\}_{n\ge0}$由下式产生x_{n + 1} = x_n - C_n Ax_n ,\q n \ge 0, \eqno{(@)}$$则存在常数$a>0$,当$C_n < a$时,$\{x_n\}$强收敛于$A$的唯一零点$x^{*}$.

Convergence Theorems of the Steepest Descent Method for Generalized {\Phi}-Accretive Operators

Let $E$ be a real uniformly smooth Banach space and $A:E\to E$ be a bounded demicontinuous generalized ${\it \Phi} $-accretive operator such that $\left\| {x_0 -x^* } \right\|\le m$, $\mathop {\underline {\lim } }\limits_{r \to \infty } {\it \Phi} (r) > m\left\| {Ax_0 } \right\| $ for arbitrary initial value $x_0 \in E$ and for $m>0$ large enough. Let $\{C_n\}$ be a real sequence in $0,1]$ satisfying control conditions: i)$C_n\to 0,(n\to\infty)$; ii)$\sum\limits_{n = 0}^\infty {C_n } = \infty $. Let $\{x_n\}_{n\ge0}$ be a sequence generated by x_{n + 1} = x_n - C_n Ax_n ,\q n \ge 0. \eqno{(@)}Then there exists a constant $a>0$ such that when $C_n<a$, $\{x_n\}$ converges strongly to the unique zero$x^{*}$ of $A$.