# Volume 48, № 7, 1996

### 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 R^{m}.

### 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.

### 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*.

### 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.

### 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.

### 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.

### 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.

### 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.

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

Komarov G. N., Oshkhunov M. M.

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}.

### 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 *O*α and the radius of starlikeness in the class *U*α of functions σ(*z*)=*z*ƒ′(*z*), ƒ(*z*)⊂*O* _{α}.

### 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}=α_{ n } *z* ^{ n }Δ^{ n } + ... + α_{1} *z*Δ+α_{0} *E* and *L* _{2}= *z* ^{ n } *a* _{ n }(*z*)Δ^{ 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≤*k*≤*n*, 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+1}Δ+β(*z*)*E*, β(*z*) ∈ *A* _{ R }, *s* ∈ ℕ, and *z* ^{ s+1} are equivalent in the spaces *A* _{R}, 0š*R*š-∞, if and only if β(*z*) = 0.

### 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.

### 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.

### 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.