Local and Nonlocal Boundary Value Problems for Degenerating and Nondegenerating Pseudoparabolic Equations with a Riemann–Liouville Fractional Derivative

We study local and nonlocal boundary value problems for degenerating and nondegenerating third-order pseudoparabolic equations of the general form with variable coefficients and with a Riemann–Liouville fractional derivative. For their solutions, we obtain a priori estimates that imply the uniqueness of the solution and its stability with respect to the right-hand side and the initial data.     

Perturbation of the Hill operator by narrow potentials

We consider a perturbation of a periodic second order differential operator, defined on the real axis, which is a special case of the Hill operator. The perturbation is realized by a sum of two complex-valued potentials with compact supports. The potentials depend on two small parameters. One of them describes the lengths of the supports of the potentials and the reciprocal to the second one corresponds to the maximum values of the potentials. We obtain a sufficient condition, under fulfillment of which, the eigenvalues arise from the edges of non-degenerate lacunas of continuous spectrum, and construct their asymptotics. We also give a sufficient condition under which the eigenvalues do not arise.     

Simultaneous reduction to block triangular form and theorems on pairs of complex idempotents

It is shown that the properties of idempotents found in recent journal publications can be justified in an easier way if a classical theorem concerning the simultaneous reduction of a pair of complex idempotents to block triangular form is used.