CC BY
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ISSN 1810-0198. Вестник ТГУ, т. 20, вып. 5, 2015
4. Hakl R., Lomtatidze A., Sremr J. Some boundary value problems for first order scalar functional differential equations. Brno: Masaryk University, 2002.
5. Kiguradze I., PuZa B. Boundary value problems for systems of linear functional differential equations. Brno: Masaryk University, 2003.
6. Бравый Е.И. Разрешимость краевых задач для линейных функционально-дифференциальных уравнений. Москва; Ижевск: Регулярная и хаотическая динамика, 2011.
7. Бравый Е.И. О разрешимости периодической краевой задачи для линейных функционально-дифференциальных уравнений // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2011. Т. 16. № 4. С. 1029-1032.
БЛАГОДАРНОСТИ: Работа выполнена в рамках госзадания Минобрнауки РФ (задание 2014/152, проект 1890) и поддержана РФФИ (проект 14-01-00338).
Поступила в редакцию 27 мая 2015 г.
Bravyi E.I. ON SOLVABILITY OF PERIODIC BOUNDARY VALUE PROBLEM AND DIRICHLET PROBLEM FOR SECOND ORDER FUNCTIONAL-DIFFERENTIAL EQUATIONS
The Dirichlet boundary value problem and the periodic boundary value problem for for some classes of linear second-order functional-differential equations are considered. Necessary and sufficient conditions of a unique solvability of the boundary value problem for all equations from these classes are obtained.
Key words: functional-differential equations; boundary value problems; periodic boundary value problem; solvability conditions; Dirichlet problem.
Бравый Евгений Ильич, Пермский национальный исследовательский политехнический университет, г. Пермь, Российская Федерация, кандидат физико-математических наук, старший научный сотрудник научно-исследовательского центра «Функционально-дифференциальные уравнения», email: bravyi@perm.ru
Bravyi Evgenii Ilich, Perm National Research Polytechnical University, Perm, the Russian Federation, Candidate of Physics and Mathematics, Senior Researcher of the Research Center «Functional-Differential Equations», e-mail: bravyi@perm.ru
УДК 517.968.4
ON CONNECTION BETWEEN CONTINUOUS AND DISCONTINUOUS HOMOGENIZED NEURAL FIELD EQUATIONS
© E. Burlakov, A. Ponosov, J. Wyller
Key words: discontinuous Hammerstein equations; solvability; continuous dependence. We study existence and continuous dependence of the solutions to the Hammerstein equation under the transition from continuous nonlinearities in the Hammerstein operator to the Heaviside nonlinearity in a vicinity of the solution, corresponding to the discontinuous nonlinearity case.
We consider the following generalization of the homogenized Amari neural field equation (see for example [1], [2])
dtu(t,x,x f ) = —u(t,x,x f)+ / u(x — y,x f — yf )fe (u(t,y))dyf dy,
i Y (D
t> 0, x € S С Rm, xf € У e Rk,
ISSN 1810-0198. Вестннк Try, t. 20, huo. 5, 2015
parameterized by 0 € [0, to) .
We assume that the functions involved in (1) satisfy the following assumptions:
(A1) For any xf € Y , the integration kernel w(-, xf) € C2(E, R) .
(A2) For any x € R , the integration kernel u(x, ■) € L(Y, ^, R) .
(A3) For 0 = 0 , the Hammerstein nonlinearity is represented by the Heaviside function
, , , i 0, u < 0, f0(u) = \ 1, u>0
with some threshold value 0 .
(A4) For 0 > 0, functions of the family fp : R ^ [0,1] are non-decreasing, continuous, and satisfying the following convergence conditions with respect to the parameter 0:
(i) fp ^ fp uniformly on R as 0 ^ 0, 0 € (0, to) ;
(ii) for any e > 0 , fp ^ f0 uniformly on R \ BR(0, e) as 0 ^ 0 .
If the stationary solution to (1) exists and does not depend on the fine-scale variable, it satisfies the following equation
u(x) = J(w)(x - y)fp(u(y))dy,
r 3 (2)
(w)(x) = u(x,xf)dxf, x € e C Rm,xf € Y•
Y
We are interested here in one particular type of solutions, which possesses the following properties.
Definition 1. Let 0 > 0 be fixed. We say that u € C R) satisfies the 0-condition if
N
(B1) there is a finite set of open bounded domains 0j C E such that u(x) > 0 on 0= |J 0j;
i= 1
N
(B2) for any point x of the boundary B = (J Bi of 0 , it holds true that u'(x) = 0;
i= 1
(B3) there exist a > 0 and r > 0 such that u(x) < 0 — a for all x € E \ BRm (0, r).
The following theorem provides conditions for convergence of the stationary solutions up to
(1), 0 > 0 , (if these solutions exist) to the stationary solution u0 to (1) at 0 = 0 .
Theorem 1. (Continuous dependence) Let the assumptions (A1) — (A4) hold true, 0 > 0 be fixed and u0 € C 1(Rm,R) satisfies the 0 -condition. Then there exists e > 0 such that for any (sufficiently large) closed Q C Rm, if we assume existence of solutions up € Bci(Q,R)(u0, e) to the equation (2) for any 0 € (0,1] (E = Q), then there exist a solution to (2) at 0 = 0 and it is a limit point of the set {up} . Moreover, if the solution of (2) at 0 = 0 (E = Q), say u0 , is unique then \\up — uo||ci(n,R) ^ 0 .
The next theorem provides a tool for proving existence of solutions to (2) for 0 € (0, to) using some knowledge about the solution to (2) at 0 = 0 .
Theorem 2. (Existence) Let the conditions of Theorem 1 be satisfied, the set Q and the constant e be taken from Theorem 1. Assume that there exists solution u0 € C 1(Rm,R) of
(2) at 0 = 0, which satisfies 0 -condition and which is unique in Bci(n;R)(u0, e1) (e1 < e), and deg(I—F0, Bci(n;R)(u0, e1), 0)=0 , where the operator F0 : Bci(n;R)(u0,e1) ^ C1 (Q,R) is given by
(F0u)(x) = (u)(x — y)f0(u(y))dy.
ISSN 1810-0198. Вестник ТГУ, т. 20, вып. 5, 2015
Then for any в € (0,1] , there exists solution € Bci(n,R)(uo,£i) to the equation (2) .
These results can be applied for justification of the usage of the Heaviside Hammerstein nonlinearities in the frameworks of [1] and [2], where it appreciably simplified both theoretical and numerical investigations.
REFERENCES
1. Svanstedt N., Wyller J., Malyutina E. A one population Amari model with periodic microstructure // Nonlinearity, 2014. № 27. P. 1391-1417.
2. Malyutina E., Wyller J., Ponosov A. Two bump solutions of a homogenized Wilson-Cowan model with periodic microstructure // Physica D. 2014. № 271. P. 19-31.
ACKNOWLEDGEMENTS: The present work is partially supported by RFBR (project № 14-01-97504).
Received 25 May 2015.
Бурлаков Е., Поносов А., Виллер Й. О СВЯЗИ НЕПРЕРЫВНЫХ И РАЗРЫВНЫХ УСРЕДНЁННЫХ УРАВНЕНИЙ НЕЙРОПОЛЕЙ
Изучаются существование и непрерывная зависимость решений интегральных уравнений Гам-мерштейна при переходе от непрерывной нелинейной части оператора Гаммерштейна к нелинейности типа Хевисайда в окрестности решения, соответствуюшего случаю разрывной нелинейной части.
Ключевые слова: разрывные операторы Гаммерштейна; разрешимость; непрерывная зависимость.
Burlakov Evgenii, Norwegian University of Life Sciences, As, Norway, Post-graduate Student, e-mail: evgenii.burlakov@nmbu.no
Бурлаков Евгений, Норвежский университет естественных наук, Аас, Норвегия, аспирант, e-mail: evgenii.burlakov@nmbu.no
Ponosov Arcady, Norwegian University of Life Sciences, As, Norway, Doctor of Physics and Mathematics, Professor, e-mail: arkadi.ponossov@nmbu.no
Поносов Аркадий, Норвежский университет естественных наук, Аас, Норвегия, доктор физико-математических наук, профессор, e-mail: arkadi.ponossov@nmbu.no
Wyller John, Norwegian University of Life Sciences, As, Norway, Doctor of Physics and Mathematics, Professor, e-mail: john.wyller@nmbu.no
Виллер Йон, Норвежский университет естественных наук, Аас, Норвегия, доктор физико-математических наук, профессор, e-mail: john.wyller@nmbu.no
