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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
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