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Search: MSC category 47H20 ( Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07] )

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1. CMB Online first

Bačák, Miroslav; Kovalev, Leonid V.
Lipschitz retractions in Hadamard spaces via gradient flow semigroups
Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions $r\colon X(n)\to X(n-1)$ for $n\ge2.$ It is known that such retractions do not exist if $X$ is the one-dimensional sphere. On the other hand L. Kovalev has recently established their existence in case $X$ is a Hilbert space and he also posed a question as to whether or not such Lipschitz retractions exist for $X$ being a Hadamard space. In the present paper we answer this question in the positive.

Keywords:finite subset space, gradient flow, Hadamard space, Lie-Trotter-Kato formula, Lipschitz retraction
Categories:53C23, 47H20, 54E40, 58D07

2. CMB 2011 (vol 55 pp. 15)

Akiyama, Shigeki; Suzuki, Tomonari
Browder's Convergence for One-Parameter Nonexpansive Semigroups
We give the sufficient and necessary conditions of Browder's convergence theorem for one-parameter nonexpansive semigroups which was proved by Suzuki. We also discuss the perfect kernels of topological spaces.

Keywords:nonexpansive semigroup, common fixed point, Browder's convergence, perfect kernel
Category:47H20

3. CMB 2011 (vol 55 pp. 882)

Xueli, Song; Jigen, Peng
Equivalence of $L_p$ Stability and Exponential Stability of Nonlinear Lipschitzian Semigroups
$L_p$ stability and exponential stability are two important concepts for nonlinear dynamic systems. In this paper, we prove that a nonlinear exponentially bounded Lipschitzian semigroup is exponentially stable if and only if the semigroup is $L_p$ stable for some $p>0$. Based on the equivalence, we derive two sufficient conditions for exponential stability of the nonlinear semigroup. The results obtained extend and improve some existing ones.

Keywords:exponentially stable, $L_p$ stable, nonlinear Lipschitzian semigroups
Categories:34D05, 47H20

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