Vol 55, No 8 (2019)

Necessary and sufficient conditions are obtained for a center as well as an isochronous center of holomorphic Newton equations with force function quadratic in velocities. These conditions do not rely on calculating focus quantities and isochronicity constants.

The equivalence problem for second-order equations with respect to point changes of variables is solved. Equations whose right-hand side is not a cubic polynomial in the first derivative are considered. A basis of differential invariants of these equations in both main and degenerate cases is constructed, as well as operators of invariant differentiation and “trivial” relations that hold for the invariants of any equation. The use of the invariants of an equation in constructing its integrals, symmetries, and representation in the form of Euler–Lagrange equations is discussed. A generalization of the derived formulas to the calculation of invariants of equations unsolved for the highest derivative is proposed.

We consider a discontinuous Dirac operator on the interval (0, 2π). It is assumed that its coefficient (potential) is a complex-valued matrix function integrable on (0, 2π). Criteria are established for the Riesz and unconditional basis properties of the system of root vector functions in L₂²(0, 2π). A theorem about the equivalent basis property in L_p²(0, 2π), 1 < p > ∞, is proved.

In a domain D ⊂ ℝⁿ divided by a hyperplane Σ into two parts D⁽¹⁾ and D⁽²⁾, we consider a p(x)-Laplace type equation with a small parameter and with exponent p(x) that has a logarithmic modulus of continuity in each part of the domain and undergoes a jump on Σ when passing from D⁽²⁾ to D⁽¹⁾. Under the assumption that the equation uniformly degenerates with respect to the small parameter in D⁽¹⁾, we establish the Hölder continuity of solutions with Hölder exponent independent of the parameter.

The dynamics of stochastic nonlinear parabolic equations is analyzed. Self-similar solutions of the Cauchy problem for a quasilinear stochastic equation of the parabolic type with power-law nonlinearities are constructed. The dynamics of the solutions and their supports is studied with the use of comparison theorems.

We study the solvability of a nonlinear boundary value problem for a partial differential equation with a small parameter multiplying the nonlinearity. The solvability conditions are first derived for the corresponding linear problem by the Fourier method and then used to state and prove theorems about the solvability of the nonlinear boundary value problem. If the corresponding homogeneous linear boundary value problem has nonzero solutions, then the solvability of the nonlinear boundary value problem is established using ideas of the Pon-tryagin method and the methods and means of the theory of rotation of completely continuous vector fields.

We construct a continuous family of continuous mappings of a compact metric space such that the topological entropy of these mappings as a function of the parameter is nowhere upper semicontinuous.