## Fractional powers of monotone operators in Hilbert spaces

### Daniel Hauer, Yuan He, Dehui Liu

#### Abstract

In this article, we show that for maximal monotone operators $$A$$ on $$H$$ with $$0$$ in the range $$\textrm{Rg}(A)$$ of $$A$$ and every $$0 < \sigma\le 1/2$$, the Dirichlet problem associated with the Bessel-type equation $A_{1-2\sigma u}:=-\frac{1-2\sigma} tu_t-u_{tt}+Au\ni 0$ is well-posed and the associated Dirichlet-to-Neumann map $$\Lambda_{\sigma}$$ is a maximal monotone operator on $$H$$. This allows to defined fractional powers $$A^{\alpha}$$ of nonlinear operators.

Keywords: Monotone operators, Hilbert space, evolution equations, fractional operators.

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 Monday, February 19, 2018 10