2019
Том 71
№ 5

# Kmit I. Ya.

Articles: 3
Article (English)

### Fredholm solvability of a periodic Neumann problem for a linear telegraph equation

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 381-391

We investigate a periodic problem for the linear telegraph equation $$u_{tt} - u_{xx} + 2\mu u_t = f (x, t)$$ with Neumann boundary conditions. We prove that the operator of the problem is modeled by a Fredholm operator of index zero in the scale of Sobolev spaces of periodic functions. This result is stable under small perturbations of the equation where p becomes variable and discontinuous or an additional zero-order term appears. We also show that the solutions of this problem possess smoothing properties.

Article (English)

### Robustness of exponential dichotomies of boundary-value problems for general first-order hyperbolic systems

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 236-251

We examine the robustness of exponential dichotomies of boundary-value problems for general linear first-order one-dimensional hyperbolic systems. It is assumed that the boundary conditions guarantee an increase in the smoothness of solutions in a finite time interval, which includes reflection boundary conditions. We show that the dichotomy survives in the space of continuous functions under small perturbations of all coefficients in the differential equations.

Article (Ukrainian)

### 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.