Boundary value problems for fractional elliptic equations with parameter in Banach spaces are studied. Uniform \(L_{p}\)-separability properties and sharp resolvent estimates are obtained for elliptic equations in terms of fractional derivatives. Particularly, it is proven that the fractional elliptic operators generated by these equations are positive and also are generators of the analytic semigroups. Moreover, maximal regularity properties of the fractional abstract parabolic equation are established. As an application, the parameter-dependent anisotropic fractional differential equations and the system of fractional differential equations are studied.

中文翻译：

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分数抽象微分方程及其应用

研究了Banach空间中带参数的分数阶椭圆型方程的边值问题。根据分数导数，获得了椭圆方程的一致（L_ {p} \）-可分离性，并给出了清晰的分解估计。特别地，证明了由这些方程生成的分数椭圆算子是正的，并且也是解析半群的生成器。此外，建立了分数抽象抛物方程的最大正则性。作为应用，研究了参数相关的各向异性分数阶微分方程和分数阶微分方程组。