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

**Year:**2017**Articles:**9**URL:**https://journals.rudn.ru/CMFD/issue/view/1264**DOI:**https://doi.org/10.22363/2413-3639-2017-63-4

## Full Issue

## New Results

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

#### Abstract

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

### Existence of Weak Solution of the Aggregation Integro-Diﬀerential Equation

#### Abstract

**Contemporary Mathematics. Fundamental Directions**. 2017;63(4):557-572

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

#### Abstract

**Contemporary Mathematics. Fundamental Directions**. 2017;63(4):586-598

### Diﬀerential Equation in a Banach Space Multiplicatively Perturbed by Random Noise

#### Abstract

**Contemporary Mathematics. Fundamental Directions**. 2017;63(4):599-614

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

#### Abstract

**Contemporary Mathematics. Fundamental Directions**. 2017;63(4):615-626

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

#### Abstract

**Contemporary Mathematics. Fundamental Directions**. 2017;63(4):627-677

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

#### Abstract

In the half-plane $\{-\infty <x<+\infty \}\times \{0<y<+\infty \}$, the Dirichlet problem is considered for m diﬀerential-diﬀerence equations of the kind ${u}_{xx}+{{\sum}^{m}}_{k=1}{a}_{k}{u}_{xx}(x+{h}_{k},y)+{u}_{yy}=0$, 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

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

#### Abstract

**Contemporary Mathematics. Fundamental Directions**. 2017;63(4):689-702