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We establish the conditions for the existence of one class of solutions of a nonautonomous binomial differential equation of the second order with rapidly varying nonlinearity and the asymptotic representations for these solutions and their first derivatives as t ↑ ω (ω ≤ +∞).
We construct a solution of the boundary-value problem for a system of linear inhomogeneous differential equations of the first order with rectangular matrices.
For systems of differential equations with pulse action at nonfixed times, we establish the conditions for the existence of optimal control. These conditions depend on the right-hand sides of the system and the quality criterion.
We establish constructive conditions of solvability and an algorithm for the construction of solutions of an autonomous periodic boundary-value problem for a matrix differential equation.
We study the linear inhomogeneous first order differential equation by′ + f(x) = y in the ring of formal power series with integer coefficients. Using the p-adic topology on the ring of integers, we construct a counterpart of the Hurwitz product of the Euler series and an arbitrary formal power series with integer coefficients. It is shown that the Euler series can be interpreted as the fundamental solution to the equation under consideration. Bibliography: 8 titles.
The asymptotic properties of the solutions of some third-order differential equation are examined. Sufficient conditions for the square integrability and oscillation of the solutions are established.
We study the existence of analytic solutions for some systems of ordinary differential equations partially solvable with respect to the derivatives. We establish sufficient conditions for the existence of analytic solutions of the Cauchy problem in the presence of a pole. An estimate for these solutions is obtained in a certain domain and the problem of the number of solutions is investigated.
We state solvability conditions for a multipoint boundary-value problem for linear second-order functional-differential equation.
For an autonomous differential equation with delay, we obtain a description of the space of solutions on the negative semi-axis.
In this paper, we obtain solvability conditions for the periodic boundary-value problem for a certain first-order differential equation unsolved with respect to the derivative. These condition were obtained by using the theorem on implicit operators.
For autonomous functional-differential equations with delays, we obtain an oscillation criterion, which allows one to reduce the oscillation problem to the calculation of a unique root of a real-valued function determined by the coefficients of the original equation. The criterion is illustrated by examples of equations with concentrated and distributed aftereffect, for which convenient oscillation tests are obtained.
We consider a class of autonomous functional-differential equations for which the properties of oscillation of all solutions and positiveness of the fundamental solution are mutually complementary.
We obtain new nonimprovable conditions of oscillation of all solutions to a linear differential equation with several variable delays and positive coefficients. These conditions have the form of the upper and lower limits of the sum of integrals of the coefficients over the sets that are determined only by the delay corresponding to this coefficient. These results differ from the well-known results in which a coarsening of the integration set is assumed.
We consider autonomous and periodic differential equations containing two scalar parameters and analyze the construction of boundaries of stability domains of equilibrium points for such equations in the plane of parameters. We indicate conditions under which a unique smooth boundary curve passes through a given point. We propose a scheme of approximate construction of this curve.
This paper is devoted to the theory of approximation of coefficient inverse problems for differential equations of parabolic, elliptic, and hyperbolic types in functional spaces. We present general statements of problems and their approximations and review results obtained earlier in the literature.
In this work, we study approximations of solutions of fractional differential equations of order 1/2. We present a new method of approximation and obtain the order of convergence. The presentation is given within the abstract framework of a semidiscrete approximation scheme, which includes finite-element methods, finite-difference schemes, and projection methods.
We consider the boundary value problem for a differential operator with an integrable potential and study the asymptotic behavior of solutions for large values of the spectral parameter. We derive an equation for eigenvalues and obtain asymptotics of eigenfunctions of the operator.
We prove an existence and uniqueness theorem for the linear differential equation u′(z) = Au(z)+e{sup γz}f(z), where A is a closed operator and f is an entgire vector-valued function of zero exponential type. Bibliography: 10 titles.