# Volume 70, № 3, 2018

### Irregular elliptic boundary-value problems and Hörmander spaces

Anop A. V., Kasirenko T. M., Murach A. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 299-317

We study nonregular elliptic problems with boundary conditions of higher orders and prove that these problems are Fredholm on appropriate pairs of the inner-product H¨ormander spaces that form a two-sided refined Sobolev scale. We prove a theorem on the regularity of generalized solutions to the problems in these spaces.

### Polynomial inequalities in regions with interior zero angles in the Bergman space

Abdullayev F. G., Balci S., Imash kyzy M.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 318-336

We investigate the order of growth of the moduli of arbitrary algebraic polynomials in the weighted Bergman space $A_p(G, h),\; p > 0$, in regions with interior zero angles at finitely many boundary points. We obtain estimations for algebraic polynomials in bounded regions with piecewise smooth boundary.

### A problem for one class of pseudodifferential evolutionary equations multipoint in the time variable

Gorodetskii V. V., Petryshyn R. I., Verezhak A. P.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 337-355

We establish the correct solvability of the multipoint (in the time variable) problem for the evolution equation with operator of differentiation of infinite order in generalized $S$-type spaces. The properties of the fundamental solution of this problem and the behavior of the solution $u(t, x)$ as $t \rightarrow +\infty$ are investigated.

### Criteria for the existence of an isolated solution of a nonlinear boundary-value problem

Dzhumabaev D. S., Temesheva S. M.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 356-365

A nonlinear two-point boundary-value problem for an ordinary differential equation is studied by the method of parametrization. We construct systems of nonlinear algebraic equations that enable us to find the initial approximation to the solution to the posed problem. In terms of the properties of constructed systems,we establish necessary and sufficient conditions for the existence of an isolated solution to the boundary-value problem under consideration.

### Bifurcation conditions for the solutions of weakly perturbed boundary-value problems for operator equations in Banach spaces

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 366-378

We obtain bifurcation conditions for the solutions of weakly perturbed boundary-value problems for operator equations in Banach spaces from the point $\varepsilon = 0$. A convergent iterative procedure is proposed for the construction of solutions as parts of series in powers of $\varepsilon$ with pole at the point $\varepsilon = 0$.

### On moduli of smoothness with Jacobi weights

Kopotun K. A., Leviatan D., Shevchuk I. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 379-403

We introduce the moduli of smoothness with Jacobi weights $(1 x)\alpha (1+x)\beta$ for functions in the Jacobi weighted spaces $L_p[ 1, 1],\; 0 < p \leq \infty $. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted spaces $L_p$. If $1 \leq p \leq \infty$, then these moduli are equivalent to certain weighted $K$-functionals (and so they are equivalent to certain weighted Ditzian – Totik moduli of smoothness for these $p$), while for $0 < p < 1$ they are equivalent to certain “Realization functionals”.

### Continuity in the parameter for the solutions of one-dimensional boundary-value problems for differential equations of higher orders in Slobodetsky spaces

Maslyuk H. O., Mikhailets V. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 404-411

We introduce the most general class of linear boundary-value problems for systems of ordinary differential equations of order $r \geq 2$ whose solutions belong to the Slobodetsky space $^{Ws+r}_p\bigl( (a, b),C_m\bigr),$ where $m \in N,\; s > 0$ and $p \in (1,\infty )$. We also establish sufficient conditions under which the solutions of these problems are continuous functions of the parameter in the Slobodetsky space $W^{s+r}_p\bigl( (a, b),C_m\bigr)$.

### Averaging of fuzzy systems

Perestyuk N. A., Skripnik N. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 412-428

We develop the ideas of the method of averaging for some classes of fuzzy systems (fuzzy differential equations with delay, fuzzy differential equations with pulsed action, fuzzy integral equations, fuzzy differential inclusions and differential inclusions with fuzzy right-hand sides without and with pulsed action).

### Approximate and information aspects of the numerical solution of unstable integral and pseudodifferential equations

Semenova E. V., Solodkii S. G.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 429-444

We present a review of the latest results obtained in the field of numerical solution of unstable integral and pseudodifferential equations. New versions of fully discrete projection and collocation methods are constructed and justified. It is shown that these versions are characterized by the optimal accuracy and cost efficiency, as far as the use of computational resources is concerned.