Volume 2, Issue 3

The aim of the paper is to study a global structure of solutions of four differential inequalities with respect to their zeros. The structure of an oscillatory solution is described, and the number of points with trivial Cauchy conditions is investigated.

Automorphisms of the family of all Sturm-Liouville equations y″ = qy are investigated. The classical Darboux transformation arises as a particular case of a general result.

It is proved that the even-order equation y (2 n ) + p ( t ) y = 0 is ( n, n ) oscillatory at ∞ if where .

In the first part [Duduchava, Natroshvili and Shargorodsky, Georgian Math. J. 2: 123–140, 1985] of the paper the basic boundary value problems of the mathematical theory of elasticity for three-dimensional anisotropic bodies with cuts were formulated. It is assumed that the two-dimensional surface of a cut is a smooth manifold of an arbitrary configuration with a smooth boundary. The existence and uniqueness theorems for boundary value problems were formulated in the Besov () and Bessel-potential () spaces. In the present part we give the proofs of the main results (Theorems 7 and 8) using the classical potential theory and the nonclassical theory of pseudodifferential equations on manifolds with a boundary.

Necessary and sufficient conditions are found to be imposed on a pair of weights, for which a weak type two-weighted reverse inequality holds, in the case of general maximal functions defined in homogeneous type spaces.

The Keldysh theorem for an elliptic equation with characteristic parabolic degeneration is generalized for the case of an elliptic equation of the second-order canonical form with order and type degeneration. The criteria under which the Dirichlet or Keldysh problems are correct are given in a one-sided neighborhood of the degeneration segment, enabling one to write the criteria in a single form. Moreover, some cases are pointed out in which it is even nessesary to give a criterion in the neighborhood because it is impossible to establish it on the segment of degeneracy of the equation.

The theorem of unique solvability of a spatial problem of Darboux type in Sobolev space is proved for a second-order hyperbolic equation.

The process of Fejér type trigonometric convolution and its discrete analogue have equivalent uniform residues. The situation changes under pointwise comparison. In this direction negative results have been obtained by different authors. One result of such a type is given in the present paper. In particular, a counter-example is constructed for which both comparisons diverge on the set of complete measure. The smoothness of the counter-example as well as some other problems are investigated.

The method of barriers is used to justify asymptotic representations of solutions of two-point boundary value problems for singularly perturbed quasilinear equations of the second and the third order. This paper is a continuation of [Rozov and Sushko, Georgian Math. J. 2: 99-110, 1995].