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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.
We let be the axially symmetric bounded domains which satisfy some suitable conditions, then the ground-state solutions of the semilinear elliptic equation in are nonaxially symmetric and concentrative on one side. Furthermore, we prove the necessary and sufficient condition for the symmetry of ground-state solutions.
Using variational arguments, we prove some nonexistence and multiplicity results for positive solutions of a system of -Laplace equations of gradient form. Then we study a -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.