论文标题
关于CAT（0）空间中平均操作员的概念
On a notion of averaged operators in CAT(0) spaces
论文作者
论文摘要
平均操作员在希尔伯特空间的固定点理论中发挥了重要作用。它们出现是为了获得解决方案问题的解决方案，而基础操作员不承包，因此使Banach固定点定理无法访问。我们在$ \ text {cat}（0）$ spaces的更广泛类中介绍了平均运算符的概念。我们称这些运营商$α$ -FORMENTIND非专业，并为准$α$ -FORMEND的非专业运算符制定基本的微积分规则。尤其是准专业运算符的准$α$ - $α$ - 固定的非专业和凸组合的有限型$α$α$α$的组合，又不固定运算符，又是quasi $α$α$ notimny notimny normally normally normally normally normally normally。对于非专业运算符$ t：x \ to x $作用于$ \ text {cat}（0）$ space $ x $，我们表明，迭代$ x_n：= tx_ {n-1} $在固定点设置$ \ text {fix {fix} t $ t $ t $ nontan nontan nontan nontan nontan nontan nontan nontan nontan nontan nontan nontimd y quaSi $ -firm中弱地收敛到某个元素。此外，在特定规律性条件下，投影$ p _ {\ text {fix} t} x_n $强烈收敛到此弱极限。我们的理论用两个经典的循环和平均预测示例进行了说明。
Averaged operators have played an important role in fixed point theory in Hilbert spaces. They emerged as a necessity to obtain solutions to fixed point problems where the underlying operator is not contractive and thus renders Banach fixed point theorem inaccessible. We introduce a notion of averaged operator in the broader class of $\text{CAT}(0)$ spaces. We call these operators $α$-firmly nonexpansive and develop basic calculus rules for the quasi $α$-firmly nonexpansive operators. In particular compositions of quasi $α$-firmly nonexpansive operators is quasi $α$-firmly nonexpansive and convex combination of a finite family of quasi $α$-firmly nonexpansive operators is again quasi $α$-firmly nonexpansive. For a nonexpansive operator $T:X\to X$ acting on a $\text{CAT}(0)$ space $X$ we show that the iterates $x_n:=Tx_{n-1}$ converge weakly to some element in the fixed point set $\text{Fix} T$ whenever $T$ is quasi $α$-firmly nonexpansive. Moreover under a certain regularity condition the projections $P_{\text{Fix} T}x_n$ converge strongly to this weak limit. Our theory is illustrated with two classical examples of cyclic and averaged projections.
