Vol 84, No 2 (2020)

We develop an approach which we used to deduce conditions of a new type for the existence of periodic solutions ofordinary differential equations and functional differential equations of point type. These conditions are based onthe use of asymptotic properties of solutions of differential equations which can be observed on shifts of solutionsand stated in terms of averages over the period on a distinguished sphere in the phase space. The development ofthis approach enables us to obtain conditions for the existence of bounded solutions for the same classes offunctional differential equations.

We consider a boundary-value problem for a system of two second-order ODE with distinct powers of asmall parameter at the second derivative in the first and second equations. Whenone of the two equations of the degenerate system has a double root, the asymptotic behaviour ofthe boundary-layer solution of the boundary-value problem turns out to be qualitatively different from the knownasymptotic behaviour in the case when those equations have simple roots. In particular,the scales of the boundary-layer variables and the very algorithm for constructing the boundary-layer seriesdepend on the type of the boundary conditions for the unknown functions. We construct and justify asymptoticexpansions of the boundary-layer solution for boundary conditions of a particular type. These expansions differfrom those for other boundary conditions.

We prove that weakly $o$-minimal theories of finite convexity rank having lessthan $2^{\omega}$ countable models are binary. Our main result is theconfirmation of Vaught's conjecture for weakly $o$-minimal theories of finiteconvexity rank.

We construct orthogonal trigonometric polynomials satisfying a new spectral conditionand such that their $L^{1}$-norms are bounded below and the uniform norm of their partial sums has extremely small order of growth. We obtain new results that relate the uniform norm and $\mathrm{QC}$-norm on subspaces of the vector space of trigonometric polynomials.