Vol 55, No 6 (2019)

We consider a class of nonlocal boundary value problems for linear systems of ordinary differential equations with nonlinear point loads. Conditions for the existence and uniqueness of a solution are obtained. The study uses a constructive method that can be applied to specific problems.

We study a bifurcation from the zero solution of the differential equation ẍ + x^p/^q = 0, where p > q > 1 are odd coprime numbers, under periodic (in particular, time-invariant) perturbations depending on a small positive parameter ε. The motion separation method is used to derive the bifurcation equation. To each positive root of this equation, there corresponds an invariant two-dimensional torus (a closed trajectory in the time-invariant case) shrinking to the equilibrium position x = 0 as ε → 0. The proofs use methods of the Krylov-Bogolyubov theory to study time-periodic perturbations and the implicit function theorem in the case of time-invari ant perturbations.

We establish new tests for the solvability and unique solvability of two-point boundary value problems for ordinary second-order differential equations with nonintegrable singularities in the time variable. In particular, we describe a set of functions f:]a, b[×ℝ → ℝ such that the condition \(\int\limits_a^b {{{\left( {t - a} \right)}^\ell }{{\left( {b - t} \right)}^\ell }|\left( {t,x} \right)|dt = + \infty } \) is satisfied for arbitrary x ∈ ℝ and ℓ > 0, but nevertheless, the boundary value problem \(u'' = f\left( {t,u} \right);\,\,u\left( {a + } \right) = 0,\,u\left( {b - } \right) = 0\) has a unique solution.

We consider the Cauchy problem for a second-order Petrovskii parabolic system with bounded continuous coefficients under the condition that the leading coefficients are Dini continuous in the spatial variables. We prove the uniqueness of the classical solution of this problem in the space of functions increasing with respect to the spatial variables, belonging to the Tikhonov class, and having derivatives that may be unbounded when approaching the initial data plane.

We study a nonlinear system of two elliptic differential equations with nonlinearities depending on a product of powers of the unknown functions. Sufficient conditions for the system to be reducible to a single equation are obtained. Some cases are singled out for which we find classes of exact solutions expressed via elementary and harmonic functions and solutions of the Liouville equation.

In a finite-dimensional Euclidean space, we consider the pursuit problem with one evader and a group of pursuers described by a system of the form D^(α)z_i = az_i + u_i - v, where D^(α)f is the Caputo derivative of order α ∈ (1, 2) of a function f. The set of admissible solutions u_i and v is a convex compact set, the objective set is the origin, and a is a real number. In addition, it is assumed that the evader does not leave a convex polyhedral cone with nonempty interior. We obtain sufficient conditions for the solvability of the pursuit problem in terms of the initial positions and the game parameters.

Upper and lower functions and fragmentation are considered for higher-order boundary value problems.

We obtain sufficient conditions for the nonexistence of time-global solutions of a fractional analog of the Burgers equation. Examples of solution blowup are considered for the Cauchy problem. The Mitidieri-Pokhozhaev nonlinear capacity method is used.