2019
Том 71
№ 2

# Volume 48, № 7, 1996

Article (Ukrainian)

### Strongly nonlinear degenerate elliptic equations with discontinuous coefficients. I

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 867-875

This paper is concerned with the existence and uniqueness of variational solutions of the strongly nonlinear equation $$- \sum\limits_1^m {_i \frac{\partial }{{\partial x_i }}\left( {\sum\limits_1^m {_j a_{i,j} (x, u(x))\frac{{\partial u(x)}}{{\partial x_j }}} } \right) + g(x, u(x)) = f(x)}$$ with the coefficients a i,j (x, s) satisfying an eHipticity degenerate condition and hypotheses weaker than the continuity with respect to the variable s. Furthermore, we establish a condition on f under which the solution is bounded in a bounded open subset Ω of Rm.

Article (Russian)

### Inner turning point in the theory of singular Perturbations

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 876-890

We construct the uniform asymptotics of a solution of a singularly perturbed differential equation of Liouville type with an interior turning point.

Article (Russian)

### Chebyshev polynomial approximation on a closed subset with unique limit point and analytic extension of functions

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 891-900

We describe the domain of analyticity of a continuous function f in terms of the sequence of the best polynomial approximations of f on a compact set K(K ⊂ ℂ) and the sequence of norms of Chebyshev polynomials for K.

Article (Russian)

### On branches of small solutions of certain operator equations

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 901-909

In the case of double degeneration of a linearized problem, we study the points of bifurcation of the null solutions of nonlinear equations of special type.

Article (Ukrainian)

### Generalization of Berg-Dimovski convolution in spaces of analytic functions

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 910-919

In the space H(G) of functions analytic in a ρ-convex region G equipped with the topology of compact convergence, we construct a convolution for the operator J π+L where J ρ is the generalized Gel’fond-Leont’ev integration operator and L is a linear continuous functional on H(G). This convolution is a generalization of the well-known Berg-Dimovski convolution. We describe the commutant of the operator J π+L in ℋ(G) and obtain the representation of the coefficient multipliers of expansions of analytic functions in the system of Mittag-Leffler functions.

Article (Russian)

### On a Numerical-Analytic method for second-order difference equations

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 920-924

For second-order difference equations, we justify the scheme of the Samoilenko numerical-analytic method for finding periodic solutions.

Article (Russian)

### Solvability of nonlinear elliptic systems with measure

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 925-939

We prove the solvability of nonlinear elliptic systems in spaces dual to the Morrey spaces. As a main consequence, we establish that, under certain restrictions on the modulus of ellipticity of a system, systems with measure are solvable.

Article (Ukrainian)

### High-order asymptotics of a solution of one problem of optimal control over a distributed system with rapidly oscillating coefficients

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 940-948

High-order asymptotics is constructed and justified for optimal control over parabolic systems with rapidly oscillating coefficients in the principal part that describes high-intensity heat transfer processes in inhomogeneous and periodic media. The investigation is based on the use of methods of multiscale asymptotic decomposition and some results of the theory of averaging.

Article (Russian)

### On the solvability of physically nonlinear problems of Thermoelasticity

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 949-953

We study the problem of existence and uniqueness of generalized solutions of nonlinear vector boundary-value problems arising in the physically nonlinear theory of thermoelasticity. We prove the convergence of iteration processes in the space W 1 2.

Article (Russian)

### On the radii of convexity and starlikeness for some special classes of analytic functions in a disk

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 954-958

We introduce the class O α, 0≤α≤1, of functions w=ƒ(z), ƒ(0)=0, ƒ′(0)=0,..., ƒ (0) (n−1) =0, f (n)(0)=(n-l)! analytic in the disk |z|<1 and satisfying the condition $$\operatorname{Re} \left( {\frac{{1 - 2z^n \cos \Theta + z^{2n} }}{{z^{n - 1} }}f'(z)} \right) > \alpha , 0 \leqslant \Theta \leqslant \pi , n = 1,2,3,... .$$ We establish the radius of convexity in the class Oα and the radius of starlikeness in the class Uα of functions σ(z)=zƒ′(z), ƒ(z)⊂O α.

Article (Ukrainian)

### On the equivalence of the Euler-Pommier operators in spaces of analytic functions

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 958-971

In the space A (θ) of all one-valued functions f(z) analytic in an arbitrary region G ⊂ ℂ (0 ∈ G) with the topology of compact convergence, we establish necessary and sufficient conditions for the equivalence of the operators L 1 n z n Δ n + ... + α1 zΔ+α0 E and L 2= z n a n (z n + ... + za 1(z)Δ+a 0(z)E, where δ: (Δƒ)(z)=(f(z)-ƒ(0))/z is the Pommier operator in A(G), n ∈ ℕ, α n ∈ ℂ, a k (z) ∈ A(G), 0≤kn, and the following condition is satisfied: Σ j=s n−1 α j+1 ∈ 0, s=0,1,...,n−1. We also prove that the operators z s+1Δ+β(z)E, β(z) ∈ A R , s ∈ ℕ, and z s+1 are equivalent in the spaces A R, 0šRš-∞, if and only if β(z) = 0.

Article (Russian)

### Regularity of solutions of Degenerate Quasilinear Parabolic Equations (weighted case)

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 972-988

We establish the inner regularity of solutions and their derivatives with respect to spatial coordinates for a degenerate quasilinear parabolic equation of the second order.

Article (Russian)

### Two-sided estimates of a solution of the Neumann problem as $t \rightarrow \infty$ for a second-order Quasilinear Parabolic Equation

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 989-998

We establish exact upper and lower bounds as $t \rightarrow \infty$ for the norm ‖u(·, t)‖ L ∞(Ω) of a solution of the Neumann problem for a second-order quasilinear parabolic equation in the region D=Ω×{>0}, where Ω is a region with noncompact boundary.

Article (Russian)

### Well-posedness of the cauchy problem for complete second-order operator-differential equations

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 999-1006

For the equation y″(t)+Ay′(t)+By(t)=0, where A and B are arbitrary commuting normal operators in a Hilbert space H, we obtain a necessary and sufficient condition for well-posedness of the Cauchy problem in the space of initial data D(B)×(D(A)∩D(|B|1/2)) and for weak well-posedness of the Cauchy problem in H×H_(|A|+|B|1/2+1). This condition is expressed in terms of location of the joint spectrum of the operators A and B in C 2. In terms of location of the spectrum of the operator pencil z 2+Az+B in C 1, such a condition cannot be written.