We consider the problem of determining the unknown term in the right-hand side of a second-order differential equation with unbounded operator generating a cosine operator function from the overspecified boundary data. We obtain necessary and sufficient conditions of the unique solvability of this problem in terms of location of the spectrum of the unbounded operator and properties of its resolvent.

In the class of scalar type spectral operators in a complex Banach space, a characterization of the generators of analytic $C_0$-semigroups in terms of the analytic vectors of the operators is found.

We let $\Omega \left(r\right)$ be the axially symmetric bounded domains which satisfy some suitable conditions, then the ground-state solutions of the semilinear elliptic equation in $\Omega \left(r\right)$ are nonaxially symmetric and concentrative on one side. Furthermore, we prove the necessary and sufficient condition for the symmetry of ground-state solutions.

Existence result for strongly nonlinear elliptic equation with a natural growth condition on the nonlinearity is proved.

Using variational arguments, we prove some nonexistence and multiplicity results for positive solutions of a system of $p$-Laplace equations of gradient form. Then we study a $p$-Laplace-type problem with nonlinear boundary conditions.

We study the existence of periodic trajectories for nonautonomous differential equations and inclusions remaining in a prescribed compact subset of an extended phase space. These sets of constraints are nonconvex right-continuous tubes not satisfying the viability tangential condition on the whole boundary. We find sufficient conditions for existence of viable periodic trajectories studying properties of the exit subset of the tube. A new approximation approach for continuous multivalued maps is presented.