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

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

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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

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

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**Contemporary Mathematics. Fundamental Directions**. 2017;63(4):573-585

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

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

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

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

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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

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