Solvability of a periodic boundary-value problem for systems of functional differential equations with cyclic matrices

We consider first-order systems of linear functional differential equations with regular operators. For families of systems of two equations we obtain the general necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem. For families of systems of n linear functional differential equations with cyclic matrices we obtain effective necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem.     

Degenerate integro-differential operators in Banach spaces and their applications

In this paper we consider linear integro-differential equations in Banach spaces with Fredholm operators at the highest-order derivatives and convolution-type Volterra integral parts. We obtain sufficient conditions for the unique solvability (in the classical sense) of the Cauchy problem for the mentioned equations and illustrate the abstract results with pithy examples. The studies are carried out in classes of distributions in Banach spaces with the help of the theory of fundamental operator functions of degenerate integro-differential operators. We propose a universal technique for proving theorems on the form of fundamental operator functions.     

On measure-compact operators

We obtain new sufficient solvability conditions for equations with measure-compact operators comparable with partially additive ones. We also prove new conditions under which these operators are locally condensing with respect to the Hausdorff measure of noncompactness. As an application of the obtained results we prove one property of bifurcation points which occur, in particular, in nonlinear mechanics.     

On the solvability of initial boundary-value problems for a class of operator-differential equations of third order

We obtain sufficient conditions for the regular solvability of initial boundary-value problems for a class of operator-differential equations of third order with variable coefficients on the semiaxis. These conditions are expressed only in terms of the operator coefficients of the equations under study. We obtain estimates of the norms of intermediate derivative operators via the discontinuous principal parts of the equations and also find relations between these estimates and the conditions for regular solvability.     

Constructive methods for obtaining the solution of the periodic boundary value problem for a system of matrix differential equations of Riccati type

In the nonsingular case, we obtain sufficient coefficient conditions for the unique solvability of the periodic boundary value problem for a system of matrix differential equations of Riccati type. We develop efficient algorithms for constructing the solution.     

On the unique solvability of a nonlocal boundary value problem with data on intersecting lines for systems of hyperbolic equations

We consider a nonlocal boundary value problem for a system of hyperbolic equations with two independent variables with data on intersecting lines one of which is a characteristic. In terms of the data of the nonlocal boundary value problem, we obtain sufficient coefficient conditions for its unique solvability.     

The solutions to some operator equations with corresponding operators not necessarily having closed ranges

In this paper, we study the solvability of the operator equations in the general setting of infinite-dimensional Hilbert space with corresponding operators no necessarily having closed range. We get the necessary and sufficient conditions for the existences of the solutions and obtain formulae in each case for the general selfadjoint, positive and real positive solutions to these operator systems.     

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.     

Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite-dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.     

Version of the elimination method for a system of partial differential equations

For a system of three first-order partial differential equations with three independent variables, we obtain sufficient conditions for one component of the solution to satisfy a third-order Bianchi equation. We also obtain conditions for the solvability of this system by quadratures.     

Estimates for the Cauchy function of linear singular differential equations and some applications

For higher-order linear singular equations, we find sharp estimates for the Cauchy function and its partial derivatives. We use these estimates to study the properties of solutions of singular differential inequalities and obtain optimal sufficient conditions for the unique solvability of the linear singular Cauchy problem.     

Existence and blow-up of solutions of a pseudoparabolic equation

We obtain sufficient blow-up conditions for the solution of a nonlinear differential problem with given initial and boundary conditions. We prove the solvability of this problem in any finite cylinder under some restrictions on the nonlinear operators.     

Nonlocal problem with integral conditions for a system of hyperbolic equations in characteristic rectangle

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations in rectangular domain. We investigate the questions of existence of unique classical solution to the problemunder consideration and approaches of its construction. Sufficient conditions of unique solvability to the investigated problem are established in the terms of initial data. The nonlocal problem with integral conditions is reduced to an equivalent problem consisting of the Goursat problem for the system of hyperbolic equations with functional parameters and functional relations. We propose algorithms for finding a solution to the equivalent problem with functional parameters on the characteristics and prove their convergence. We also obtain the conditions of unique solvability to the auxiliary boundary-value problem with an integral condition for the system of ordinary differential equations. As an example, we consider the nonlocal boundary-value problem with integral conditions for a two-dimensional system of hyperbolic equations.     

Construction and Structure Properties of Solutions of a Periodic Boundary Value Problem for a Generalization of the Matrix Lyapunov and Riccati Equations

We obtain constructive sufficient conditions for the unique solvability of a periodic boundary value problem for a matrix differential equation that generalizes the Lyapunov and Riccati equations, develop an algorithm for constructing the solution of this equation, estimate the domain where the solution is localized, and study the structural properties of the solution.     

Best mean-quasiconformal mappings

We look for best mean-quasiconformal mappings as extremals of the functional equal to the integral of the square of the functional of the conformality distortion multiplied by a special weight. The mapping inverse to an extremal is an extremal of the same functional. We obtain necessary and sufficient conditions for the Petrovskii ellipticity of the system of Euler equations for an extremal. We prove the local unique solvability of boundary values problems for this system in the 2-dimensional case. In the general case we prove the unique solvability of boundary value problems for the system linearized at the identity mapping.     

On the unique solvability of a family of two-point boundary-value problems for systems of ordinary differential equations

We consider a family of two-point boundary-value problems for systems of ordinary differential equations with functional parameters. This family is the result of the reduction of a boundary-value problem with nonlocal condition for a system of second-order, quasilinear hyperbolic equations by the introduction of additional functions. Using the parametrization method, we establish necessary and sufficient conditions of the unique solvability of the family of two-point boundary-value problems for a linear system in terms of the initial data. We also prove sufficient conditions of the unique solvability of the problem considered and propose an algorithm for its solution. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 21–39, 2006.     

On the “Destruction” of Solutions of Nonlinear Wave Equations of Sobolev Type with Cubic Sources

We consider model three-dimensional wave nonlinear equations of Sobolev type with cubic sources, and foremost, model three-dimensional equations of Benjamin-Bona-Mahony and Rosenau types with model cubic sources. An essentially three-dimensional nonlinear equation of spin waves with cubic source is also studied. For these equations, we investigate the first initial boundary-value problem in a bounded domain with smooth boundary. We prove local solvability in the strong generalized sense and, for an equation of Benjamin-Bona-Mahony type with source, we prove the unique solvability of a “weakened” solution. We obtain sufficient conditions for the “destruction” of the solutions of the problems under consideration. These conditions have the sense of a “large” value of the initial perturbation in the norms of certain Banach spaces. Finally, for an equation of Benjamin-Bona-Mahony type, we prove the “failure” of a “weakened” solution in finite time.     

Boundary value problem for abstract first order differential equation with integral condition

The present paper is devoted to the study of a boundary value problem for abstract first order linear differential equation with integral boundary conditions. We obtain necessary and sufficient conditions for the unique solvability and well-posedness. We also study the Fredholm solvability. Finally, we obtain a result of the stability of solution with respect to small perturbation.     

On the well-posedness of a boundary value problem for a class of fourth-order operator-differential equations

We find sufficient coefficient conditions for the well-posed solvability of a boundary value problem for a class of fourth-order operator-differential equations with multiple characteristics. Furthermore, we indicate the sharp values of norms of operators of intermediate derivatives in a Sobolev-type space. In addition, for the corresponding polynomial operator pencil, we prove the completeness of the part of its eigenvectors and associated vectors corresponding to the eigenvalues in the left half-plane.     

Matrix differential equations with separation of variables

We consider matrix differential equations with separation of variables and homogeneous equations reducible to them. For the mentioned equations we obtain sufficient conditions for the solvability in quadratures.