# Ptashnik B. I.

### Problem with integral conditions in the time variable for Sobolevtype system of equations with constant coefficients

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 530-549

In a domain obtained as a Cartesian product of an interval $[0, T]$ and the space $R^p, p \in N$, for a system of equations (with constant coefficients) unsolved with respect to the highest time derivative, we study a problem with integral conditions in the time variable in the class of functions almost periodic in the space variables. A criterion of uniqueness and sufficient conditions for the existence of the solution of this problem in different functional spaces are established. We use the metric approach to solve the problem of small denominators encountered in the construction of the solution.

### Boundary-value problem with mixed conditions for linear typeless partial differential equations

Ptashnik B. I., Repetylo S. M.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 665-682

In the domain obtained as the Cartesian product of a segment $0 \leq t \leq T$ by a $p$-dimensional torus in variables $x_1, ..., x_p$, $p \geq 1$, we study the problem with mixed boundary conditions in the variable $t$ for general (no restrictions are imposed on the type) linear partial differential equations of high order with constant coefficients isotropic with respect to the order of differentiation for all independent variables. We establish conditions for the unique solvability of the problem in various functional spaces and construct its solution in the form of a series with respect to systems of orthogonal functions of the variables $x_1, ..., x_p$.

### A Problem with Condition Containing an Integral Term for a Parabolic-Hyperbolic Equation

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 635-644

In a layer obtained as the Cartesian product of an interval $[−T_1 ,T_2], T_1 ,T_2 > 0$, and a space $ℝ_p, p ≥ 1$, we study a problem with nonlocal condition in the time variable containing an integral term for a mixed parabolic-hyperbolic equation in the class of functions almost periodic in the space variables. For this problem, we establish a criterion of uniqueness and sufficient conditions for the existence of solutions. To solve the problem of small denominators encountered in the construction of the solution, we use the metric approach.

### Multipoint Problem for *B*-Parabolic Equations

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 418-429

We establish conditions for the well-posedness of a problem for one class of parabolic equations with the Bessel operator in one of the space variables in a bounded domain with multipoint conditions in the time variable and some boundary conditions in the space coordinates. A solution of the problem is constructed in the form of a series in a system of orthogonal functions. We prove a metric theorem on lower bounds for the small denominators appearing in the solution of the problem.

### A problem with integral conditions with respect to time for Garding hyperbolic equations

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 252-265

In a domain that is the Cartesian product of an interval $[0,T]$ and the space $\mathbb{R}^p$, we investigate a problem for Garding hyperbolic equations having constant coefficients with integral conditions with respect to the time variable in a class of functions almost periodic in the space variables. A criterion for the uniqueness and sufficient conditions for the existence of a solution of the problem in different functional spaces are established. To solve the problem of small denominators that arises in the solution of the problem, the metric approach is used.

### Myroslav L’vovych Horbachuk (on his 70th birthday)

Adamyan V. M., Berezansky Yu. M., Khruslov E. Ya., Kochubei A. N., Kuzhel' S. A., Marchenko V. O., Mikhailets V. A., Nizhnik L. P., Ptashnik B. I., Rofe-Beketov F. S., Samoilenko A. M., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 439–442

### Well-posedness of boundary-value problems for multidimensional hyperbolic systems

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 192–203

By the method of characteristics, we investigate the well-posedness of local (the Cauchy problem, mixed problems) and nonlocal (with nonseparable and integral boundary conditions) problems for some multidimensional almost linear first-order hyperbolic systems. Reducing these problems to the systems of integral operator equations, we prove the existence and uniqueness of classical solutions.

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

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 370–381

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.

### Problems for partial differential equations with nonlocal conditions. Metric approach to the problem of small denominators

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1624–1650

A survey of works of the authors and their disciples devoted to the investigation of problems with nonlocal conditions with respect to a selected variable in cylindrical domains is presented. These problems are considered for linear equations and systems of partial differential equations that, in general, are ill posed in the Hadamard sense and whose solvability in certain scales of functional spaces is established for almost all (with respect to Lebesgue measure) vectors composed of the coefficients of the problem and the parameters of the domain.

### The Skorobogat'ko international mathematical conference

Ptashnik B. I., Samoilenko A. M.

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 143-144

### A Problem with Nonlocal Conditions for Partial Differential Equations Unsolved with Respect to the Leading Derivative

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1022-1034

In the domain that is the product of a segment and a *p*-dimensional torus, we investigate the well-posedness of a problem with nonlocal boundary conditions for a partial differential equation unsolved with respect to the leading derivative with respect to a selected variable. We establish conditions for the the classical well-posedness of the problem and prove metric theorems on the lower bounds of small denominators appearing in the course of its solution.

### Multipoint Problem with Multiple Nodes for Partial Differential Equations

Ptashnik B. I., Symotyuk M. M.

Ukr. Mat. Zh. - 2003. - 55, № 3. - pp. 400-413

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.

### Multipoint Problem for Nonisotropic Partial Differential Equations with Constant Coefficients

Ptashnik B. I., Symotyuk M. M.

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 241-254

We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable *t* and periodic conditions with respect to coordinates *x* _{1},..., *x* _{p} for a nonisotropic (concerning differentiation with respect to *t* and *x* _{1},..., *x* _{p}) partial differential equation with constant complex coefficients. We establish conditions for the existence and uniqueness of a solution of this problem and prove metric theorems on lower bounds for small denominators appearing in the course of the construction of its solution.

### A Multipoint Problem for Pseudodifferential Equations

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 22-29

We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable *t* and periodic conditions with respect to coordinates *x* _{1},...,*x* _{p} for equations unsolved with respect to the leading derivative with respect to *t* and containing pseudodifferential operators. We establish conditions for the unique solvability of this problem and prove metric assertions related to lower bounds for small denominators appearing in the course of its solution.

### Dirichlet-Type Problems for Systems of Partial Differential Equations Unresolved with Respect to the Highest Time Derivative

Bilusyak N.I., Komarnyts'ka L. I., Ptashnik B. I.

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1592-1602

We establish conditions for the correct solvability of problems for systems of partial differential equations unresolved with respect to the highest time derivative with Dirichlet-type conditions with respect to time and periodic conditions with respect to space variables. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of these problems.

### International scientific conference "New approaches to the solution of differential equations"

Ptashnik B. I., Samoilenko A. M.

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 575-576

### Some Pseudoparabolic Variational Inequalities with Higher Derivatives

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 94-105

We consider a pseudoparabolic variational inequality with higher derivatives. We prove the existence and uniqueness of a solution of this inequality with a zero initial condition.

### A Problem with Nonlocal Conditions for Partial Differential Equations with Variable Coefficients

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1328-1336

We establish conditions for the unique solvability of a problem for partial differential equations with coefficients dependent on variables *t* and *x* in a rectangular domain with nonlocal two-point conditions with respect to *t* and local boundary conditions with respect to *x*. We prove metric statements related to lower bounds of small denominators appearing in the course of solution of the problem.

### Boundary-Value Problem for Weakly Nonlinear Hyperbolic Equations with Variable Coefficients

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1281-1286

We establish conditions for the unique solvability of a boundary-value problem for a weakly nonlinear hyperbolic equation of order 2*n*, *n* > (3*p* + 1)/2, with coefficients dependent on the space coordinates and data given on the entire boundary of a cylindric domain \(D \subset \mathbb{R}^{p + 1}\) . The investigation of this problem is connected with the problem of small denominators.

### A Boundary-Value Problem for Weakly Nonlinear Hyperbolic Equations with Data on the Entire Boundary of a Domain

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 244-249

In a domain that is the Cartesian product of a segment and a *p*-dimensional torus, we investigate a boundary-value problem for weakly nonlinear hyperbolic equations of higher order. For almost all (with respect to Lebesgue measure) parameters of the domain, we establish conditions for the existence of a unique solution of the problem.

### A multipoint problem for partial differential equations unresolved with respect to the higher time derivative

Ukr. Mat. Zh. - 1999. - 51, № 12. - pp. 1604–1613

We investigate the well-posedness of problems for partial differential equations unresolved with respect to the higher time derivative with multipoint conditions with respect to time. By using the metric approach, we determine lower bounds for small denominators appearing in the course of the solution of the problems.

### A multipoint problem with multiple nodes for linear hyperbolic equations

Beresnevich V. V., Bernik V. I., Ptashnik B. I., Vasylyshyn P. B.

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1311–1316

We establish conditions for the unique solvability of a multipoint (with respect to the time coordinate) problem with multiple nodes for linear hyperbolic equations with constant coefficients in the class of functions periodic in the space variable. We prove metric statements concerning lower bounds of small denominators that appear in the course of construction of a solution of the problem.

### On certain nonlinear pseudoparabolic variational inequalities without initial conditions

Lavrenyuk S. P., Ptashnik B. I.

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 328–337

We consider a nonlinear pseudoparabolic variational inequality in a tube domain semibounded in variable*t*. Under certain conditions imposed on coefficients of the inequality, we prove the theorems of existence and uniqueness of a solution without any restriction on its behavior as*t*→−∞.

### A multipoint problem for partial integro-differential equations

Ptashnik B. I., Vasylyshyn P. B.

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1155–1168

We investigate a multipoint problem for a linear typeless partial differential operator with variable coefficients that is perturbed by a nonlinear integro-differential term. We establish conditions for the unique existence of a solution. We prove metric theorems on lower bounds of small denominators that arise in the course of investigation of the problem of solvability.

### All-Ukrainian scientific conference “New approaches to the solution of differential equations”

Ptashnik B. I., Samoilenko A. M.

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 454–455

### Nonlocal boundary-value problems for systems on linear partial differential equations

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1478–1487

We study the classical well-posedness of problems with nonlocal two-point conditions for typeless systems of linear partial differential equations with variable coefficients in a cylindrical domain. We prove metric theorems on lower bounds for small denominators that appear in the construction of solutions of such problems.

### Multipoint problem for typeless systems of differential equations with constant coefficients

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1236–1249

For typeless systems of differential equations with constant coefficients, we investigate the well-posedness of the problem with multipoint conditions for a selected variable and 2π-periodic conditions for the other coordinates. The conditions of univalent solvability are established and the metric theorems are proved for lower bounds of small denominators that appear in the construction of solutions of the problems.

### Problem with nonlocal conditions for weakly nonlinear hyperbolic equations

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 186–195

For weakly nonlinear hyperbolic equations of order *n, n*≥3, with constant coefficients in the linear part of the operator, we study a problem with nonlocal two-point conditions in time and periodic conditions in the space variable. Generally speaking, the solvability of this problem is connected with the problem of small denominators whose estimation from below is based on the application of the metric approach. For almost all (with respect to the Lebesgue measure) coefficients of the equation and almost all parameters of the domain, we establish conditions for the existence of a unique classical solution of the problem.

### Representation and investigation of solutions of a nonlocal boundary-value problem for a system of partial differential equations

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 184-194

We study the boundary-value problem for a system of partial differential equations with constant coefficients with conditions nonlocal in time. By using a metric approach, we prove the well-posedness of the problem in the scale of Sobolev spaces of functions periodic in space variables. By using matrix calculus, we construct an explicit representation of a solution.

### Multipoint problem for typeless factorized differential operators

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 66-79

We analyze the well-posedness of a problem with multipoint conditions in the time variable and periodic conditions in the spatial coordinates for differential operators decomposable into operators of the first order with complex coefficients. We establish conditions for the existence and uniqueness of the classical solution of the problem under consideration and prove metric theorems for the lower estimates of small denominators appearing in the process of construction of the solution.

### Boundary-value problems for hyperbolic equations with constant coefficients

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 795–802

By using a metric approach, we study the problem of well-posedness of boundary-value problems for hyperbolic equations of order*n* (*n*?2) with constant coefficients in a cylindrical domain. Conditions of existence and uniqueness of solutions are formulated in number-theoretic terms. We prove a metric theorem on lower estimates of small denominators that appear when constructing solutions.

### A nonlocal multipoint problem for pseudodifferential operators with analytic symbols

Il'kiv V. S., Polishchuk V. N., Ptashnik B. I., Salyga B. O.

Ukr. Mat. Zh. - 1986. - 38, № 5. - pp. 582–587

### An analog of the multipoint problem for partial differential equations with variable coefficients

Ukr. Mat. Zh. - 1983. - 35, № 6. - pp. 728-734

### Periodic solutions of a system of partial differential equations with constant coefficients

Polishchuk V. N., Ptashnik B. I.

Ukr. Mat. Zh. - 1980. - 32, № 2. - pp. 239 - 243

### The periodic boundary-value problem for a system of hyperbolic equations with constant coefficients

Polishchuk V. N., Ptashnik B. I.

Ukr. Mat. Zh. - 1978. - 30, № 3. - pp. 326–333

### Analog of the n-point problem for a linear hyperbolic equation

Ukr. Mat. Zh. - 1971. - 23, № 4. - pp. 472–481

### Dirichlet type problem for hyperbolic equations with constant coefficients

Ukr. Mat. Zh. - 1970. - 22, № 6. - pp. 841—848