Volume 6, Issue 5

New oscillation and nonoscillation criteria are established for the equation 𝑢″ + 𝑝(𝑡)𝑢 = 0, where 𝑝 : ]1, + ∞[ → 𝑅 is the locally integrable function. These criteria generalize and complement the well known criteria of E. Hille, Z. Nehari, A. Wintner, and P. Hartman.

For the 𝑛th order nonlinear differential equation 𝑦 (𝑛) (𝑡) = 𝑓(𝑦(𝑡)), 𝑡 ∈ [0, 1], satisfying the multipoint conjugate boundary conditions, 𝑦 (𝑗) (𝑎 𝑖 ) = 0, 1 ≤ 𝑖 ≤ 𝑘, 0 ≤ 𝑗 ≤ 𝑛 𝑖 – 1, 0 = 𝑎 1 < 𝑎 2 < ⋯ < 𝑎 𝑘 = 1, and , where 𝑓 : ℝ → [0, ∞) is continuous, growth condtions are imposed on 𝑓 which yield the existence of at least three solutions that belong to a cone.

The solvability of the nonlocal boundary value problem 𝑢 𝑡 = 𝑎(𝑡, 𝑥, 𝑢, 𝑢 𝑥 )𝑢 𝑥𝑥 + 𝑏(𝑡, 𝑥, 𝑢, 𝑢 𝑥 ), 0 ≤ 𝑡 ≤ 𝑇, |𝑥| ≤ 𝑙, 𝑢(0, 𝑥) = 0, 𝑢(𝑡, –𝑙) = 𝑢(𝑡, 𝑙), 𝑢 𝑥 (𝑡, –𝑙) = 𝑢 𝑥 (𝑡, 𝑙) in a class of functions is investigated for a quasilinear parabolic equation. The solution uniqueness follows from the maximum principle.

Sufficient conditions of the existence and uniqueness of bounded on real axis solutions of systems of linear functional differential equations are established.

We consider some subsets of the Euclidean plane 𝐑 2 , having small linear sections (in all directions), and investigate those sets from the point of view of measurability with respect to certain invariant extensions of the classical Lebesgue measure on 𝐑 2 .

It is proved that the Dirichlet problem is correct in the characteristic rectangle 𝐷 𝑎𝑏 = [0, 𝑎] × [0, 𝑏] for the linear hyperbolic equation with the summable in 𝐷 𝑎𝑏 coefficients 𝑝 0 , 𝑝 1 , 𝑝 2 , 𝑝 3 and 𝑞 if and only if the corresponding homogeneous problem has only the trivial solution. The effective and optimal in some sense restrictions on 𝑝 0 , 𝑝 1 , 𝑝 2 and 𝑝 3 guaranteeing the correctness of the Dirichlet problem are established.

Four types of entire modular forms of weight are constructed for the congruence subgroup Γ 0 (4𝑁) when 𝑠 is even. One can find these forms helpful in revealing the arithmetical meaning of additional terms in the formulas for the number of representations of positive integers by positive quadratic forms with integral coefficients in an even number of variables.

A piecewise-homogeneous plane made up of twodifferent materials and reinforced by an elastic unclusion is considered on a semi-finite section where the different materials join. Vertical and horizontal forces are applied to the inclusion which haz a variable thichness and a variable elasticity modulus. Under certain conditions the problem is reduced to integrodifferential equations of third order. The solution is constructed effectively by applying the methods of theory of analytic functions to a boundary value problem of the Carleman type for a strip. Asymptotic estimates of normal contact stress are obtained.