Vol 63, No 4 (2017): Differential and Functional Differential Equations

New Results

Maps Which Are Continuously Differentiable in the Sense of Michal and Bastiani but not of Fre´chet

Walther H.


We construct examples of nonlinear maps on function spaces which are continuously differentiable in the sense of Michal and Bastiani but not in the sense of Fre´chet. The search for such examples is motivated by studies of delay differential equations with the delay variable and not necessarily bounded.

Contemporary Mathematics. Fundamental Directions. 2017;63(4):543-556
pages 543-556 views

Existence of Weak Solution of the Aggregation Integro-Differential Equation

Vildanova V.F., Mukminov F.K.


In this work, we investigate the mixed problem for anisotropic integro-differential equation with variable nonlinearity indices. Using the discretization method with respect to time, we prove the existence of a weak solution in a bounded cylinder. We give an estimate of the lifetime of the solition.
Contemporary Mathematics. Fundamental Directions. 2017;63(4):557-572
pages 557-572 views

On Absence of Nonnegative Monotone Solutions for Some Coercive Inequalities in a Half-Space

Galakhov E.I., Salieva O.A.


Using the nonlinear capacity method, we investigate the problem of absence of nonnegative monotone solutions for a quasilinear elliptic inequality of type Δpu≥uq in a half-space in terms of parameters p and q.
Contemporary Mathematics. Fundamental Directions. 2017;63(4):573-585
pages 573-585 views

On the Stabilization Rate of Solutions of the Cauchy Problem for Nondivergent Parabolic Equations with Growing Lower-Order Term

Denisov V.N.


In the Cauchy problem L1u≡Lu+(b,∇u)+cu-ut=0,(x,t)∈D,u(x,0)=u0(x),x∈RN, for nondivergent parabolic equation with growing lower-order term in the half-space D=RN×[0,∞), N⩾3, we prove sufficient conditions for exponential stabilization rate of solution as t→+∞ uniformly with respect to x on any compact K in RN with any bounded and continuous in RN initial function u0(x).
Contemporary Mathematics. Fundamental Directions. 2017;63(4):586-598
pages 586-598 views

Differential Equation in a Banach Space Multiplicatively Perturbed by Random Noise

Zadorozhniy V.G., Konovalova M.A.


We consider the problem of finding the moment functions of the solution of the Cauchy problem for a first-order linear nonhomogeneous differential equation with random coefficients in a Banach space. The problem is reduced to the initial problem for a nonrandom differential equation with ordinary and variational derivatives. We obtain explicit formula for the mathematical expectation and the second-order mixed moment functions for the solution of the equation.
Contemporary Mathematics. Fundamental Directions. 2017;63(4):599-614
pages 599-614 views

G˚arding Cones and Bellman Equations in the Theory of Hessian Operators and Equations

Ivochkina N.M., Filimonenkova N.V.


In this work, we continue investigation of algebraic properties of G˚arding cones in the space of symmetric matrices. Based on this theory, we propose a new approach to study of fully nonlinear differential operators and second-order partial differential equations. We prove new-type comparison theorems for evolution Hessian operators and establish a relation between Hessian and Bellman equations.
Contemporary Mathematics. Fundamental Directions. 2017;63(4):615-626
pages 615-626 views

On Oscillations of Two Connected Pendulums Containing Cavities Partially Filled with Incompressible Fluid

Kopachevsky N.D., Voytitsky V.I., Sitshaeva Z.Z.


We consider the linearized problem on small oscillations of two pendulums connected to each other with a spherical hinge. Each pendulum has a cavity partially filled with incompressible fluid. We study the initial-boundary value problem as well as the corresponding spectral problem on normal motions of the hydromechanic system. We prove theorems on correct solvability of the problem on an arbitrary interval of time both in the case of ideal and viscous fluids in the cavities, and we study the corresponding spectral problems as well.
Contemporary Mathematics. Fundamental Directions. 2017;63(4):627-677
pages 627-677 views

Asymptotic Properties of Solutions of Two-Dimensional Differential-Difference Elliptic Problems

Muravnik A.B.


In the half-plane {<x<+}×{0<y<+}, the Dirichlet problem is considered for m differential-difference equations of the kind uxx+mk=1akuxx(x+hk,y)+uyy=0, where the amount of nonlocal terms of the equation is arbitrary and no commensurability conditions are imposed on their coefficients a1,..., am and the parameters h1,..., hm determining the translations of the independent variable x. The only condition imposed on the coefficients and parameters of the studied equation is the nonpositivity of the real part of the symbol of the operator acting with respect to the variable x. Earlier, it was proved that the specified condition (i. e., the strong ellipticity condition for the corresponding differential-difference operator) guarantees the solvability of the considered problem in the sense of generalized functions (according to the Gel’fand-Shilov definition), a Poisson integral representation of a solution was constructed, and it was proved that the constructed solution is smooth outside the boundary line. In the present paper, the behavior of the specified solution as y → +∞ is investigated. We prove the asymptotic closedness between the investigated solution and the classical Dirichlet problem for the differential elliptic equation (with the same boundary-value function as in the original nonlocal problem) determined as follows: all parameters h1,..., hm of the original differential-difference elliptic equation are assigned to be equal to zero. As a corollary, we prove that the investigated solutions obey the classical Repnikov-Eidel’man stabilization condition: the solution stabilizes as y → +∞ if and only if the mean value of the boundary-value function over the interval (-R, +R) has a limit as R → +∞.

Contemporary Mathematics. Fundamental Directions. 2017;63(4):678-688
pages 678-688 views

The Calderon-Zygmund Operator and Its Relation to Asymptotic Estimates for Ordinary Differential Operators

Savchuk A.M.


We consider the problem of estimating of expressions of the kind Υ(λ)=supx∈[0,1]∣∣∫x0f(t)eiλtdt∣∣. In particular, for the case f∈Lp[0,1], p∈(1,2], we prove the estimate ∥Υ(λ)∥Lq(R)≤C∥f∥Lp for any q>p′, where 1/p+1/p′=1. The same estimate is proved for the space Lq(dμ), where dμ is an arbitrary Carleson measure in the upper half-plane C+. Also, we estimate more complex expressions of the kind Υ(λ) arising in study of asymptotics of the fundamental system of solutions for systems of the kind y′=By+A(x)y+C(x,λ)y with dimension n as |λ|→∞ in suitable sectors of the complex plane.
Contemporary Mathematics. Fundamental Directions. 2017;63(4):689-702
pages 689-702 views

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