ON BOUNDED SOLUTIONS OF SEMILINEAR FRACTIONAL ORDER DIFFERENTIAL INCLUSIONS IN HILBERT SPACES

We prove a result on a priori estimate for mild solutions to the initial value problem for a semilinear fractional-order differential inclusion in a separable Hilbert space. We assume that the linear part of the inclusion is presented by an unbounded strictly negatively defined operator and the multivalued nonlinearity satisfies an one-sided estimate. To prove this result, we use approximation methods based on, in particular, Yosida approximations of the linear part of the inclusion. The obtained result is applied to justify the existence of mild solutions to the initial value problems on finite intervals, and to prove the existence of mild solutions which are bounded on the semi-axis.