Volume 4, Issue 2

Generalizations of the disconjugacy for a differential inclusion of neutral type are given. The relation between these generalizations and the existence of solutions of multipoint boundary value problems is discussed and the tests for disconjugacy are established. These tests are applicable to the linear differential equations of neutral type, too.

The Dirichlet problem for the equation y Δ W + αW y = – F is studied in the semi-circle x 2 + y 2 < 1, y > 0. The restrictions on F are established under which the problem is uniquely solvable in the definite generalized sense.

We establish necessary and sufficient conditions for imbeddings of weighted Orlicz–Lorentz spaces.

Sufficient conditions for oscillation and nonoscillation of second-order linear equations are established.

In the present paper we establish Lyapunov type integral inequalities related to the zeros of solutions of certain second-order differential equations by using elementary analysis. We also present some immediate applications of our results to study the asymptotic behavior of solutions of the corresponding differential equations.

For a class of Serre fibrations with a weak formal base X (or with a degenerated A ∞ -algebra structure on the integral cohomology H *( X )), obstructions are defined by means of spherical twisting cochains of ξ . In particular, for a given section s n : X n → E on n -skeleton of X , the problem of avoiding the ( n +1)th obstruction o ( s n ) ∈ H n +1 ( X ; π n ( F )) to the existence of a section on X n +1 reduces to solving a system of linear equations with respect to cohomology elements of the groups H i ( X ; π i ( F )), i < n . Homotopy classification theorems for sections as well as for weak formal maps are given, too.

A new class of non-isolated singularities called hyperplane singularities is introduced. Special deformations with simplest critical points are constructed and an algebraic expression for the number of Morse points is given. The topology of the Milnor fibre is completely studied.

Some functorial properties are studied for the Hörmander classes defined for symplectic bundles. The behavior of the Chern first form on a Lagrangian submanifold in an almost Hermitian manifold is also studied, and Fomenko's conjecture about the behavior of a Maslov class on minimal Lagrangian submanifolds is considered.