Vol 64, No 3 (2018): Proceedings of the Crimean Autumn Mathematical School-Symposium

We consider the Sturm-Liouville operator perturbed by a convolution integral operator on a finite interval with Dirichlet boundary-value conditions and discontinuity conditions in the middle of the interval. We study the inverse problem of restoration of the convolution term by the spectrum. The problem is reduced to solution of the so-called main nonlinear integral equation with a singularity. To derive and investigate this equations, we do detailed analysis of kernels of transformation operators for the integrodifferential expression under consideration. We prove the global solvability of the main equation, this implies the uniqueness of solution of the inverse problem and leads to necessary and sufficient conditions for its solvability in terms of spectrum asymptotics. The proof is constructive and gives the algorithm of solution of the inverse problem.

In this paper, we review main directions and results of the theory of topological radicals. We consider applications to different problems in the theory of operators and Banach algebras.

We study the problem on small motions of an ideal stratified fluid with a free surface partially covered with crushed ice. The crushed ice is supposed to be ponderable particles of some matter floating on the free surface. These particles do not interact with each other during oscillations of the free boundary (or this interaction is neglible) and stay on the surface during these oscillations. Using the method of orthogonal projecting of boundary-value conditions on the free surface and introducing auxiliary problems, we reduce the original initial-boundary value problem to the equivalent Cauchy problem for a second-order differential equation in some Hilbert space. We obtain conditions under which there exists a strong with respect to time solution of the initial-boundary value problem describing the evolution of this hydraulic system.