A boundary-value problem for parabolic equations with general nonlocal conditions

We study a boundary-value problem with general two-point conditions with respect to the time coordinate, and periodic conditions on the spatial coordinates for Shilov-parabolic equations with constant coefficients. We construct the solution in the form of a Fourier series. We establish conditions for existence and uniqueness of a classical solution of the problem. We prove quantitative theorems on a lower bound for the small denominators that arise in solving the problem. Translated fromMatematichni Methody i Fiziko-mekhanichni Polya, Vol. 38, 1995.     

Mixed Problem for an Ultraparabolic Equation in Unbounded Domain

We investigate a mixed problem for a nonlinear ultraparabolic equation in a certain domain Q unbounded in the space variables. This equation degenerates on a part of the lateral surface on which boundary conditions are given. We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation; these conditions do not depend on the behavior of the solution at infinity. The problem is investigated in generalized Lebesgue spaces.     

On the solvability of the Tricomi problem with a generalized Frankl transmission condition

We study the solvability of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions in the elliptic part of the domain. On the type change line of the equation, the solution gradient is subjected to a condition that is usually referred to as the generalized Frankl transmission condition. We show that the inhomogeneous Tricomi problem either has a unique solution or is conditionally solvable and the homogeneous problem has only the trivial solution. We write out an integral representation of the solution of this problem.     

Problem without initial conditions for linear and almost linear degenerate operator differential equations

We study the problem without initial conditions for linear and almost linear degenerate operator differential equations in Banach spaces. The uniqueness of a solution of this problem is proved in the classes of bounded functions and functions with exponential behavior as t → –∞. We also establish sufficient conditions for initial data under which there exists a solution of the considered problem in the class of functions with exponential behavior at infinity.     

On the existence and uniqueness of a generalized solution of the mixed problem for the wave equation with the second and third boundary conditions

We prove the uniqueness of a generalized solution of an initial-boundary value problem for the wave equation with boundary conditions of the third and second kind. In addition, we find a closed-form expression for the analytic solution of that problem with zero initial data. The result plays an important role in the investigation of the boundary control problem. We show how to use the obtained solution for the investigation of the boundary control problem in the case of subcritical time intervals for which the solution of the boundary control problem, if it exists at all, is unique. We obtain necessary and sufficient conditions for the existence of a unique solution in a class admitting the existence of finite energy.     

On a nonlocal generalization of the biharmonic Dirichlet problem

We consider a mixed problem with the Dirichlet boundary conditions and integral conditions for the biharmonic equation. We prove the existence and uniqueness of a generalized solution in the weighted Sobolev space W ₂². We show that the problem can be viewed as a generalization of the Dirichlet problem.     

Multipoint Problem with Multiple Nodes for Partial Differential Equations

We establish conditions for the existence and uniqueness of a solution of a problem with multipoint conditions with respect to a selected variable t (in the case of multiple nodes) and periodic conditions with respect to x ₁,..., x _p for a nonisotropic partial differential equation with constant complex coefficients. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of this problem.     

On the solvability of the tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions

We study the existence of a regular (classical) solution of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions. We find conditions under which the homogeneous problem has only the zero solution and give an example in which the homogeneous Tricomi problem has a nonzero solution. We also study the solvability of the inhomogeneous Tricomi problem.     

Nonlocal boundary-value problem for linear partial differential equations unsolved with respect to the higher time derivative

We study the well-posedness of the problem with general nonlocal boundary conditions in the time variable and conditions of periodicity in the space coordinates for partial differential equations unsolved with respect to the higher time derivative. We establish the conditions of existence and uniqueness of the solution of the considered problem. In the proof of existence of the solution, we use the method of divided differences. We also prove metric statements on the lower bounds of small denominators appearing in constructing the solution of the problem. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 370–381, March, 2007.     

Solution of the inverse quasiperiodic problem for the Dirac system

We present a complete solution of the inverse problem of spectral analysis for the Dirac operator with quasiperiodic boundary conditions. We prove a uniqueness theorem for the solution of the inverse problem and obtain necessary and sufficient conditions for a sequence of real numbers to be the spectrum of a quasiperiodic Dirac problem.     

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.     

A dynamic boundary-value problem without initial conditions for almost linear parabolic equations

We study a dynamic boundary-value problem without initial conditions for linear and almost linear parabolic equations. First, we establish conditions for the existence of a unique solution of a problem without initial conditions for a certain abstract implicit evolution equation in the class of functions with exponential behavior as t → −∞. Then, using these results, we prove the existence of a unique solution of the original problem in the class of functions with exponential behavior at infinity.     

Application of the finite integral transform method to solving a mixed problem with integro-differential conditions for a nonclassical equation

We consider a mixed problem with integro-differential boundary conditions for a nonclassical equation. Under certain conditions, we apply a finite integral transform to this problem and obtain a parametric problem. We introduce the notion of proper boundary conditions of the parametric problem, which is wider than the notion of regularity. By applying the inverse integral transform to the solution of the parametric problem, we obtain an analytic representation of the solution of the original mixed problem.     

On the existence of nonnegative solutions to the Dirichlet boundary value problem for the p-Laplace equation in the presence of exterior mass forces

We consider the Dirichlet problem for an inhomogeneous p-Laplace equation with nonlinear source in the presence of exterior mass forces.We obtain new sufficient conditions for the existence of a weak nonnegative bounded solution. The sufficient conditions are written in explicit form through the data of the problem.     

Nonlocal boundary-value problem for parabolic equations

We study the problem for Shilov parabolic equations of arbitrary order with constant coefficients with conditions nonlocal in time and periodic in space variables. We establish conditions for the existence and uniqueness of a classical solution of the problem and prove metric theorems on lower bounds of small denominators appearing in the construction of a solution of the problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 12, pp. 1621–1626, December, 1994.The work was supported by the Foundation for Fundamental Studies of Ukrainian State Committee on Science and Technology.     

Nonlocal boundary-value problem for parabolic equations with variable coefficients

We study the boundary-value problem for Petrovskii parabolic equations of arbitrary order with variable coefficients with conditions nonlocal in time. We establish conditions for the existence and uniqueness of a classical solution of this problem and prove metric theorems on lower bounds of small denominators appearing in the construction of a solution of the problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 7, pp. 915–921, July, 1995.This work was partially supported by the Foundation for Fundamental Research of the Ukrainian State Committee on Science and Technology.     

A Newton’s method for perturbed second-order cone programs

We develop optimality conditions for the second-order cone program. Our optimality conditions are well-defined and smooth everywhere. We then reformulate the optimality conditions into several systems of equations. Starting from a solution to the original problem, the sequence generated by Newton’s method converges Q-quadratically to a solution of the perturbed problem under some assumptions. We globalize the algorithm by (1) extending the gradient descent method for differentiable optimization to minimizing continuous functions that are almost everywhere differentiable; (2) finding a directional derivative of the equations. Numerical examples confirm that our algorithm is good for “warm starting” second-order cone programs—in some cases, the solution of a perturbed instance is hit in two iterations. In the progress of our algorithm development, we also generalize the nonlinear complementarity function approach for two variables to several variables.     

Application of nonclassical separation of variables to a nonlocal heat problem

We consider an initial-boundary value problem for the heat equation with a nonlocal two-point boundary condition containing a parameter. By separating the variables in an auxiliary function system, we construct a regular solution. We obtain sufficient conditions for the absolute and uniform convergence of the series in the auxiliary system. We prove conditions close to necessary ones for the existence of a regular solution of the initial-boundary value problem.     

Green’s functions for discrete mTH-order problems*

We investigate an mth-order discrete problem with additional conditions, described by m linearly independent linear functionals. We find the solution to this problem and present a formula and the existence condition of Green??s function if the general solution of a homogeneous equation is known. We obtain a relation between Green??s functions of two nonhomogeneous problems. It allows us to find Green??s function for the same equation, but with different additional conditions. The obtained results are applied to problems with nonlocal boundary conditions.     

A problem with formal initial conditions for differential equations with constant pseudodifferential coefficients

We establish conditions for the unique existence of a solution of a problem with formal initial conditions. We investigate the problem of its solvability in the case where a solution is not unique.