Vol 235, No 1 (2018)

The conditions of solvability and the structure of a generalized Green operator of the Cauchy problem for a linear differential-algebraic system are found. The sufficient conditions of reducibility of a differential-algebraic equation to a sequence of systems joining differential and algebraic equations are constructed. An original classification and a single scheme of construction of the solutions of differentialalgebraic equations are proposed.

We have obtained the exact-by-order estimates of Kolmogorov, linear, and trigonometric widths of the classes \( {B}_{p,\theta}^{\varOmega } \) of periodic functions of many variables in the space B_1,1 the norm in which is stronger than the L₁-norm.

The asymptotic behavior of lower Q-homeomorphisms relative to a p-modulus in ℝⁿ, n ≥ 2, at a point is studied. A number of logarithmic estimates for the lower limits under various conditions imposed on the function Q are obtained. Some applications of these results to the Orlicz–Sobolev classes \( {W}_{\mathrm{loc}}^{1,\varphi } \) in ℝⁿ, n ≥ 3 under the Calderon-type condition imposed on the function φ and, in particular, to the Sobolev classes \( {W}_{\mathrm{loc}}^{1,p} \) for p > n – 1 are given. The example of a homeomorphism with finite distortion which shows the exactness of the found order of growth is constructed.

The results of this work are referred to the well-known trend of the geometric theory of functions of complex variable, namely, to the extreme problems on the classes of nonoverlapping domains. It was started by Lavrent’ev’s classical work [1], where, in particular, the problem of the product of conformal radii of two nonoverlapping domains was first solved. Now, this trend is intensively developed. The main results can be found in [2–8] and [9–13]. Our results present a generalization of some results in [7].

We obtain the exact-by-order estimates of the approximation of functions from the classes \( {S}_{p,\theta}^{\varOmega } \)B(ℝ^d) in the space L_q(ℝ^d), 1 < p < q < ∞, by entire functions of the exponential type with supports of their Fourier transforms in sets generated by the level surfaces of a function Ω.