# Kmit I. Ya.

### 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.

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

Kmit I. Ya., Recke L., Tkachenko V. I.

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.

### 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.

### On a nonlocal problem for a quasilinear first-order hyperbolic system with two independent variables

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1307–1311

A mixed problem with nonlocal conditions on a space variable is considered for a system of quasilinear first-order hyperbolic equations with two independent variables. The sufficient conditions of solvability of this system are given.