On some fractional differential equations in the Hilbert space.(English)Zbl 1160.47038

The equations considered in this paper are
${d^\alpha u_\alpha(t) \over d t^\alpha} = \lambda A u_\alpha(t) + (1 - \lambda) Bu_\alpha (t) \, , \qquad 0 < \alpha \leq 1 \, , \quad 0 \leq \lambda \leq 1,$
where $$A, B$$ are densely defined operators in a Hilbert space $$H$$ with $$D(AB)$$ dense; $$A$$ is symmetric, $$B$$ is skew symmetric, and $$A$$ and $$B$$ commute. This fractional differential equation is understood as the integral equation
$u_\alpha(t) = u_\alpha(0) + {1 \over \Gamma(\alpha)} \int_0^t (t - \tau)^{\alpha - 1}(\lambda A u_\alpha(\tau) + (1 - \lambda) B u_\alpha(\tau))\, d\tau \, .$
The results are on existence and uniqueness of solutions of the integral equation. There are applications to partial differential operators $$A, B.$$

MSC:

 47D09 Operator sine and cosine functions and higher-order Cauchy problems 34G10 Linear differential equations in abstract spaces 34G99 Differential equations in abstract spaces 35K90 Abstract parabolic equations