Involutive Distributions,Invariant Manifolds,and Defining Equations

We introduce the notion of an invariant solution relative to an involutive distribution. We give sufficient conditions for existence of such a solution to a system of differential equations. In the case of an evolution system of partial differential equations we describe how to construct auxiliary equations for functions determining differential constraints compatible with the original system. Using this theorem, we introduce linear and quasilinear defining equations which enable us to find some classes of involutive distributions, nonclassical symmetries, and differential constraints. We present examples of reductions and exact solutions to some partial differential equations stemming from applications.     

On One Class of Systems of Differential Equations and on Retarded Equations

We establish a connection between solutions to a broad class of large systems of ordinary differential equations and solutions to retarded differential equations. We prove that solving the Cauchy problem for systems of ordinary differential equations reduces to solving the initial value problem for a retarded differential equation as the number of equations increases unboundedly. In particular, the class of systems under consideration contains a system of differential equations which arises in modeling of multiphase synthesis.     

Linear measure functional differential equations with infinite delay

We use the theory of generalized linear ordinary differential equations in Banach spaces to study linear measure functional differential equations with infinite delay. We obtain new results concerning the existence, uniqueness, and continuous dependence of solutions. Even for equations with a finite delay, our results are stronger than the existing ones. Finally, we present an application to functional differential equations with impulses.     

On the symmetry classification and conservation laws for quasilinear differential equations of second order

We obtain a sufficient condition for the absence of tangent transformations admitted by quasilinear differential equations of second order and a sufficient condition for the linear autonomy of the operators of the Lie group of transformations admitted by weakly nonlinear differential equations of second order. We prove a theorem concerning the structure of conservation laws of first order for weakly nonlinear differential equations of second order. We carry out the classification by first-order conservation laws for linear differential equations of second order with two independent variables.     

On meromorphic solutions of linear partial differential equations of second order

This paper is concerned with entire and meromorphic solutions of linear partial differential equations of second order with polynomial coefficients. We will characterize entire solutions for a class of partial differential equations associated with the Jacobi differential equations, and give a uniqueness theorem for their meromorphic solutions in the sense of the value distribution theory, which also applies to general linear partial differential equations of second order. The results are complemented by various examples for completeness.     

A Branching Method for Studying Stability of a Solution to a Delay Differential Equation

We study stability of antisymmetric periodic solutions to delay differential equations. We introduce a one-parameter family of periodic solutions to a special system of ordinary differential equations with a variable period. Conditions for stability of an antisymmetric periodic solution to a delay differential equation are stated in terms of this period function.     

Impulsive nonlocal differential equations through differential equations on time scales

We propose a non-standard approach to impulsive differential equations in Banach spaces by embedding this type of problems into differential (dynamic) problems on time scales. We give an existence result for dynamic equations and, as a consequence, we obtain an existence result for impulsive differential equations.     

A class of stochastic differential equations with pathwise unique solutions

We propose a new method viz., using stochastic partial differential equations to study the pathwise uniqueness of stochastic (ordinary) differential equations. We prove the existence and pathwise uniqueness of a class of stochastic differential equations with coefficients in suitable Hermite-Sobolev class using our approach.     

Dichotomies for generalized ordinary differential equations and applications

In this work we establish the theory of dichotomies for generalized ordinary differential equations, introducing the concepts of dichotomies for these equations, investigating their properties and proposing new results. We establish conditions for the existence of exponential dichotomies and bounded solutions. Using the correspondences between generalized ordinary differential equations and other equations, we translate our results to measure differential equations and impulsive differential equations. The fact that we work in the framework of generalized ordinary differential equations allows us to manage functions with many discontinuities and of unbounded variation.     

Existence of solutions for a class of abstract differential equations with nonlocal conditions

We discuss the existence of mild, classical and strict solutions for a class of abstract differential equations with nonlocal conditions. Our technical approach allows the study of partial differential equations with nonlocal conditions involving partial derivatives or nonlinear expressions of the solution. Some concrete applications to partial differential equations are considered.     

On the stability of abstract monotone impulsive differential equations in terms of two measures

We consider differential equations in a Banach space subjected to an impulsive influence at fixed times. It is assumed that a partial ordering is introduced in the Banach space by using a normal cone and that the differential equations are monotone with respect to the initial data. We propose a new approach to the construction of comparison systems in finite-dimensional spaces without using auxiliary Lyapunov-type functions. On the basis of this approach, we establish sufficient conditions for the stability of this class of differential equations in terms of two measures. In this case, a Birkhoff measure is chosen as the measure of initial displacements, and the norm in the given Banach space is used as the measure of current displacements. We present some examples of investigations of the impulsive systems of differential equations in the critical cases and linear impulsive systems of partial differential equations.     

Multipoint boundary value problems of Neumann type for functional differential equations

We obtain the expression of the explicit solution to a class of multipoint boundary value problems of Neumann type for linear ordinary differential equations and apply these results to study sufficient conditions for the existence of solution to linear functional differential equations with multipoint boundary conditions, considering the particular cases of equations with delay and integro-differential equations.     

Stochastic center of systems of stochastic differential equations on the plane

We study a stochastic analogy of the famous center problem of Dulac for quadratic differential equations in the plane. We introduce the concept of center for systems of stochastic differential equations of It\^o''s type on the plane, called stochastic center. We derive a criterion for the existence of such a center. We apply it to obtain necessary and sufficient conditions for quadratic stochastic differential equations in dimension 2.     

Quasi-periodic solutions for perturbed autonomous delay differential equations

This work discusses the persistence of quasi-periodic solutions for delay differential equations. We prove that the perturbed system possesses a quasi-periodic solution under appropriate hypotheses if an unperturbed linear system has quasi-periodic solutions. We extend some well-known results on ordinary differential equations to delay differential equations.     

On the construction of non-affine jump-diffusion models

We describe a method for construction of jump analogues of certain one-dimensional diffusion processes satisfying solvable stochastic differential equations. The method is based on the reduction of the original stochastic differential equations to the ones with linear diffusion coefficients, which are reducible to the associated ordinary differential equations, by using the appropriate integrating factor processes. The analogues are constructed by means of adding the jump components linearly into the reduced stochastic differential equations. We illustrate the method by constructing jump analogues of several diffusion processes and expand the notion of market price of risk to the resulting non-affine jump-diffusion models.     

Solving homogeneous fractional differential equations of Euler type

We study linear homogeneous differential equations with three left Riemann-Liouville fractional derivatives; these equations are analogs of Euler ordinary differential equations. By using the direct and inverse Mellin transforms and residue theory, we obtain a complete system of linearly independent solutions. As a corollary, related results are proved for Euler ordinary differential equations.     

Theorems of comparison and stability with probability 1 for one-dimensional stochastic differential equations

We prove the comparison theorems for scalar stochastic differential equations in the case of different diffusion coefficients. Conditions are given of stability with probability 1 with respect to the trivial solution to stochastic differential equations with random coefficients. The results remain valid for deterministic analogs of stochastic differential equations with symmetric integrals.     

On zeros of solutions of linear differential equations

We consider linear differential equations with regular coefficients in $\text{¦ z ¦ < 1}$. We obtain sufficient conditions for all the solutions of these equations to vanish a given number of times at the most. First the results are obtained for differential equations of second order, then for differential equations of nth order, n > 2.     

Approximation for non-smooth functionals of stochastic differential equations with irregular drift

This paper aims at developing a systematic study for the weak rate of convergence of the Euler–Maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients. We apply our method to obtain the rates of approximation for the expectation of various non-smooth functionals of both stochastic differential equations and killed diffusion. We also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is Hölder continuous.     

Wazewski’s method for nonlinear evolution equations

We justify a method for reducing a wide class of nonlinear equations (including several partial differential equations) to ordinary differential equations in locally convex spaces. The possibilities of this method are demonstrated by an example of a class of nonlinear hyperbolic partial differential equations.