# Volume 65, № 2, 2013

### Normally solvable operator equations in a Banach space

Boichuk A. A., Pokutnyi A. A., Zhuravlev V. F.

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 163-174

On the basis of a generalization of the well-known Schmidt lemma to the case of linear, bounded, normally solvable operators in Banach spaces, we propose a procedure for the construction of a generalized inverse for a linear, bounded, normally solvable operator whose kernel and image are complementable in the indicated spaces. This construction allows one to obtain a solvability criterion for linear normally solvable operator equations and a formula for finding their general solutions.

### Hermite-Hadamard-type inequalities for *r*-convex functions using Riemann-Liouville fractional integrals

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 175-191

By using two fundamental fractional integral identities, we derive some new Hermite – Hadamard-type inequalities for differentiable *r*-convex functions and twice-differentiable *r*-convex functions involving Riemann – Liouville fractional integrals.

### Global weak solutions of the Navier?Stokes?Fokker?Planck system

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 192-225

We consider a coupled system of the Navier- Stokes and Fokker- Planck equations that describes the motion of a polydisperse suspension of solid particles in a viscous incompressible liquid. We prove the existence theorem and study some properties of global weak solutions of the initial boundary-value problem for this system.

### Oscillation of solutions of the second-order linear functional-difference equations

Karpenko O. V., Kravets V. I., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 226-235

We establish conditions for the oscillation of solutions of functional difference linear equations and discrete difference linear equations of the second order in the case where the corresponding solutions of their differential analogs are oscillating on a segment.

### 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.

### 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.

### Variations on some finite-dimensional fixed-point theorems

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 266-272

We give rather elementary topological proofs of some generalizations of fixed-point theorems in $\mathbb{R}^n$ due to Pireddu-Zanolin and Zgliczynski, which are useful in various questions related to ordinary differential equations.

### Existence, uniqueness, and estimation of solutions for a set of equations of perturbed motion

Martynyuk A. A., Martynyuk-Chernienko Yu. A.

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 273-295

We propose a regularization procedure for a set of equations of perturbed motion with uncertain values of parameters. Using the comparison principle, we establish conditions for the existence of solutions of the original system and the regularized system.

### On the structure of the general solution and conditions of solvability of the Cauchy problem for degenerate linear systems of higher-order differential equations

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 296-305

For a system of linear differential equations of order $p$ with identically degenerate coefficient matrix of the leading derivatives, we establish conditions under which it has a general solution of the Cauchy type. The structure of this solution is determined. Conditions for the existence and uniqueness of a solution of the corresponding initial-value problem are also found.

### Conditions for the existence of almost periodic solutions of nonlinear differential equations in Banach spaces

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 307-312

We obtain conditions for the existence of almost periodic solutions of nonlinear almost periodic differential equations in a Banach space without using the $\mathcal{H}$-classes of these equations.