On the well-posedness of the Cauchy problem for quasilinear differential equations of neutral type

This paper proves the theorems on the continuous dependence of solutions when perturbations of the initial moment, the initial vector, the initial functions, the matrix coefficients, and the nonlinear summand on the right-hand side are small in the Euclidean and integral topologies, respectively. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 19, Optimal Control, 2006.     

Solvability of the Cauchy problem for higher-order linear functional differential equations

We obtain necessary and sufficient conditions for the unique solvability of the Cauchy problem for higher-order linear non-Volterra functional differential equations.     

On the Cauchy problem for some abstract nonlinear differential equations

In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A ₁(t)u,A ₂(t)u,?,A _?(t)u), (A _i (t),i = 1, 2, ?,?), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E ^? intoE and ?A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (A _i(t),i = 1, 2, ?,?), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations.     

On the Cauchy problem for hyperbolic differential equations with multiple characteristics

A priori estimates leading to existence results for the Cauchy problem are obtained for a class of linear variable coefficient hyperbolic operators with multiple characteristics. These estimates provide an extension of the energy inequalities known for strictly hyperbolic operators. Supported by NSF Grant No. GP-7234.     

On properties of solutions of a limit problem for systems of nonlinear functional differential equations of neutral type

For one class of systems of nonlinear functional differential equations, we study asymptotic properties of their solutions continuously differentiable and bounded for t ≥ T > 0 together with their first derivatives. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 217–224, February, 2008.     

On the Cauchy problem for a class of iterated Fuchsian partial differential equations

In this paper, we consider the Cauchy problem with ramified data for a class of iterated Fuchsian partial differential equations. We give an explicit representation of the solution in terms of Gauss hypergeometric functions. Our results are illustrated through some examples.     

On acyclicity of the set of solutions to the Cauchy problem for differential equations

The acyclicity of sets of solutions to the Cauchy problem is considered from the viewpoint of the axiomatic theory of solutions spaces of ordinary differential equations. We prove that the class of acyclic solution spaces is closed with respect to passing to the limit space. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 836–842, December, 1997 Translated by M. A. Shishkova     

On the approximate solution of the singular problem of Cauchy for ordinary differential equations

In the present paper an approximate solution of the singular problem of Cauchy for the ordinary differential equation of mth order is constructed and, by the method of finite differences, sufficient conditions are found for the convergence to the exact solution when the mesh width tends to zero.     

Investigation of the Cauchy problem for stochastic partial differential equations

It is shown that the solutions of stochastic linear parabolic equations with Poisson perturbations are stabilized in the mean square. The problem of determining the reserve of stability for a rod under random perturbations is studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1259–1265, September, 1993.     

Optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations

Nonimprovable, in a sense sufficient conditions guaranteeing the unique solvability of the problem

On the initial value problem for functional differential systems

For a system of functional differential equations of an arbitrary order the conditions are established for the initial value problem to be solvable on an infinite interval, and the structure of the set of solutions to this problem is studied.     

Parametrization of the Cauchy problem for systems of ordinary differential equations with limiting singular points

The method of solution continuation with respect to a parameter is used to solve an initial value problem for a system of ordinary differential equations with several limiting singular points. The solution is continued using an argument (called the best) measured along the integral curve of the problem. Additionally, a modified argument is introduced that is locally equivalent to the best one in the considered domain. Theoretical results are obtained concerning the conditioning of the Cauchy problem parametrized by the modified argument in a neighborhood of each point of its integral curve.     

Stability of the Cauchy problem for a Fedorov type matrix system of partial differential equations

Sufficient conditions on the coefficients are obtained for the stability of the nul solution for a Fedorov type matrix system of partial differential equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 583–590, June, 1991.