CC BY
26
применяя формулу Грина [6, с.541] и условия леммы, в пределе при е ^ 0, в силу равенства
2т — х)и(х, у)и(\/х2 + т2,у)(х(у =
!!н (х—т Ы^-*2 ^и(х>у
л/х2— т
(х(у,
будем иметь (16) и
12(х, у) = 2(1 + н(х — т) и2(х, у) + и2(\/х2 — т2, у) —
— (и(х, у) — иЫ х2 — т2, у))2] ^ 2(г + —=2=1) н (х — т)
и2 (х,у) +
■3 Т
+ ( / и2 — ( I \щ2 > о.
\/х2-Т 2 \/х2-Т 2
Вопрос существования решения задачи О в области О = Во и О\ и О2 и ■] связан с построением в О^ = О+ и О- и (к = 0,1, 2), на основании общих решений (14) функций и±(х,у), (х,у) € О± (к = 0,1, 2), удовлетворяющих условиям (2)-(7), (8), (15) в которых ф(х),фк(х)(к = 0,1, 2) заданы, а ш(х),и(х) подлежат определению. Для этого достаточно решить задачу О для уравнения (1), например, в области О2 = О+ и О- и 12 , то есть найти и±(а2(х),у), (х,у) € Во при условиях, согласно (2)-(7), (15):
и+(а2(х),Н) = ф(а2(х)), 0 < х < т, (17)
у+(а2(0),у) = и+(а2(т),у) = 0,0 < у < Н, (18)
и-(а2(х), —х) = ф2(а2(х)) =0, 0 < х < т/2, (19)
и-(а2(х), 0—) = и+(а2(х), 0+) = ш(а2(х)), 0 < х < т, (20)
и-у (о%(х), 0—) = и+у (а2(х), 0+) = V (а2(х)), 0 <х <т, (21) Ф(а2(0)) = ф(а2(т )),ш(а>2 (0)) = ш(а2(т)) = Ы^)) = 0.
Задача Коши. Найти в области О- решение и- (х,у) уравнения (1)из класса С (ВО--) П П С2(О-), удовлетворяющее условиям (20), (21), то есть
и-(а2(х), 0—) = ш(а2(х)), 0 < х < т,
и-у(®2(х), 0—) = V(а2(х)), 0 < х < т,
где ш(а2(х))^(а2(х)) — непрерывные достаточно гладкие функции, причем ш(^0^(0)) = = ш(а2(т ))=0.
Теорема 2. Если и(а%(х)) € С [0,т] П С2(0,т), v(a2 (х)) € С *(0,т), ш(а2(0))= ш(а^2(т))=0, то существует единственное решение задачи Коши и-(х,у) € С(О-) П С2(О-) вида
х+у
1
и2 (а2(х),у) = 1 У Мгл/Ых — у)(х + у — Щр'(Ь) + т(Ь)](М+ о
х-у
+2 I Мгл/Нх + у)(х — у — г))[р'(г) — т(г)Щ (х,у) € о-,
о
х
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где
X
х д
ш(а2(х)) + I о
р(х) = и(а$(х)) + У Ш(аКО) Х^дх Ыгу/^х — £М, (23)
о
X
г(х) = V (а2(х)) + У V )) дх (24)
Доказательство следует из (14), аналогично [7].
Функциональное соотношение между ш(а\(х)) и V(а\(х)), принесенное на линию изменения типа уравнения (1) у = 0, 0 <х<т, получим из (22), полагая у = —х и учитывая условие (19) задачи С :
2х
1
Ф2(а2(х)) = 2![р(г) — т(г)]сИ, 0 < х < т/2,
2
о
или, после замены х на х/2 и дифференцирования,
р'(г) = г(х) + 2(ф2(а2(х/2)))', 0 < х < т. (25)
Выражение (25) является искомым функциональным соотношением.
Задача Дирихле. В области Б+ найти решение и+ (х,у) € С(1)2+) П С2(Б2+) уравнения (1), удовлетворяющее условиям (17), (18), (20), то есть
и+(а2(х), К) = ф(а2(х)), 0 < х < т,
и+(а2(0),у) = и+(а2(т),у) = 0,0 < у < К,
и+(а2(х), 0+) = ш(а2(х)), 0 < х < т,
где ф(а2(х)),ш(а2(х)) — непрерывные достаточно гладкие функции, причем ш(^0^(0)) = = ш(а2(т)) = ф(а2(0)) = ф(а2(т)) = 0.
Теорема 3. Если ф(а%(х)), ш(а2(х)) € С [0,т] П С2(0,т) и ш(а^(0)) = ш(а2(т)) = = ф(а2(0))= ф(а^(т)) = 0, то существует единственное решение и++(х,у) € С(Б2+) П С2(Б+) задачи Дирихле вида
х +гу
и+ (а2(х),у)= ! 5о(г\/А2(х — гу)(х + гу — г))
р' (г)—
. х—гу
_^к2гкН(р'(г) — я* в'(г)) сг + М%у/А2(х + гу)(х — гу — г))х
к=0 * '
г
(26)
^я2гкн (р' (г) — я? в' (г))
к=о
сг, (х,у) € Б2,
где
в(х) = ф(а2(х)) + ых2+ К 0 МА2(х2 + К2)(1 — г))сг, (27)
а р(х) определяется равенством (23).
х
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Доказательство следует из (14), аналогично [7].
Найдем функциональное соотношение между ш(а^(х)) и V(а'2(х)), принесенное на линию у = 0, 0 < х < т.
Подставляя в равенство (14) условие (21), получаем уравнение
X
/г д
м+ (г, 0) ххг Мг\/А2х(х — г))сг, 0 <х<т,
о
обращая которое относительно М+у(х, 0) аналогично [1, с.49], придем к равенству г(х) = гр'(х) — я2хгкнр'(х) +2г^2 ЕХ^(2к+1)в'(х),
"X
к=0 к=0
представимому в виде
(1 — я2хгк)т(г) = —г(1 + К2хгк)р'(х) + 2гЕгхв'(х), 0 < х < т. (28)
Выражение (28) является искомым функциональным соотношением.
Вопрос существования решения задачи С в области Б2 = 0+ и Б- и 12 сводится к разрешимости системы функциональных соотношений (25), (28), то есть к разностному уравнению
(1 + )г(х) = р(х) = —(г + 1)(1 + Е^ )(^2(а2,(х/2)))'+ +(г + 1)Е^в'(х), 0 <х<т. Решение разностного уравнения (29) можно записать [8] в виде
(29)
\П
г(х) = ^(—г)ПЕТпр(х) = —(Ма2(х/2))У+
п=о
т т
+ 1'(Ф2(о2(Ф)))'Сх(х,№ + 1 в'(0С2(х,№, 0 <х<т,
(30)
где Сг(х,£)(г = 1,2) определяются аналогично [9].
Таким образом, учитывая (30) в (24), применяя формулы взаимного обращения получим
х
/г д ,_
г(г)х—Мгл/А2х(х — г))сг, 0 < х < т. (31)
о
На основании свойств функций ф(а^(х)), ф2(а^(х)), входящих в (29), (30), из (31) следует, что V(а2(х)) € С!(0,т).
Очевидно, интегрируя (25), подставляя р(х), г(х) из (23), (30), найдем ш(а^(х)) € С[0,т] П П С2(0,т).
Подстановка функций ш(а2(х)) и V(а2(х)) в формулы (22), (26) приводит к окончательному виду решения задачи Коши и задачи Дирихле в областях Б-- и 02+ , то есть в области
Б2 = Б+ и Б- и 12.
СПИСОК ЛИТЕРАТУРЫ 1. Векуа И.Н. Новые методы решения эллиптических уравнений. Москва; Ленинград; 1948.
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2. Прудников А.П., Брычков Ю.А., Маричев О.И. Интегралы и ряды. Специальные функции. М., 1983.
3. Векуа И.Н. Обращение одного интегрального преобразования и его некоторые приложения // Сообщения АН ГрузССР. 1945. Т. 6. № 3. С. 177-183.
4. Зарубин А.Н. Уравнения смешанного типа с запаздывающим аргументом. Орел, 1997.
5. Франкль Ф.И. Избранные труды по газовой динамике. М., 1973.
6. Тер-Крикоров А.М., Шабунин М.И. Курс математического анализа. М., 1988.
7. Зарубим А.Н. Задача Трикоми для опережающе-запаздывающего уравнения смешанного типа с замкнутой линией вырождения // Дифференциальные уравнения. 2015. Т. 51. № 10. С. 1315-1327.
8 . Зарубин А.Н. Краевая задача для уравнения смешанного типа с опережающе-запаздывающим аргументом // Дифференциальные уравнения. 2012. Т. 48. № 10. С. 1404-1411.
9 . Зарубин А.Н. Краевая задача для опережающе-запаздывающего уравнения смешанного типа с негладкой линией вырождения // Дифференциальные уравнения. 2014. Т. 50. № 10. С. 1362-1372.
Поступила в редакцию 20 октября 2016 г.
Чаплыгина Елена Викторовна, Орловский государственный университет им. И.С. Тургенева, г. Орел, Российская Федерация, кандидат физико-математических наук, доцент кафедры математического анализа и дифференциальных уравнений, e-mail: lena260581@yandex.ru
Зарубин Александр Николаевич, Орловский государственный университет им. И.С. Тургенева, г. Орел, Российская Федерация, доктор физико-математических наук, профессор, зав. кафедрой математического анализа и дифференциальных уравнений, e-mail: aleks_zarubin@mail.ru
UDC 517.95
DOI: 10.20310/1810-0198-2016-21-6-2098-2106
THE GELLERSTEDT PROBLEM FOR EQUATION OF THE MIXED TYPE WITH FUNCTIONAL DELAY AND ADVANCE
© E.V. Chaplygina, A.N. Zarubin
Orel State University named after I. S. Turgenev 95 Komsomolskaya St., Orel, Russian Federation, 302026 E-mail: aleks_zarubin@mail.ru
Investigates the task of Gellerstedt for the mixed type equation with the operator Lavrentiev-Bitsadze in the main part and a variable deviation of the argument. The General solution of the equation. Proved the uniqueness theorem without restrictions on the magnitude of the deviation. Found in the explicit integral representations of solutions in the field of ellipticity and hyperbolicity.
Key words: equation of mixed type; the Cauchy problem; the Dirichlet problem; difference equation; the Gellerstedt problem
REFERENCES
1. Vekua I.N. Novye metody resheniya ellipticheskih uravnenij. Moskva; Leningrad; 1948.
2. Prudnikov A.P., Brychkov YU.A., Marichev O.I. Integraly i ryady. Spetsial'nye funktsii. M., 1983.
3. Vekua I.N. Obrashchenie odnogo integral'nogo preobrazovaniya i ego nekotorye prilozheniya // Soobshcheniya AN GruzSSR. 1945. T. 6. № 3. S. 177-183.
4. Zarubin A.N. Uravneniya smeshannogo tipa s zapazdyvayushchim argumentom. Orel, 1997.
5. Frankl' F.I. Izbrannye trudy po gazovoj dinamike. M., 1973.
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6. Ter-Krikorov A.M., SHabunin M.I. Kurs matematicheskogo analiza. M., 1988.
7. Zarubin A.N. Zadacha Trikomi dlya operezhayushche-zapazdyvayushchego uravneniya smeshannogo tipa s zamknutoj liniej vyrozhdeniya // Differentsial'nye uravneniya. 2015. T. 51. № 10. S. 1315-1327.
8 . Zarubin A.N. Kraevaya zadacha dlya uravneniya smeshannogo tipa s operezhayushche-zapazdyvayushchim argumentom // Differentsial'nye uravneniya. 2012. T. 48. № 10. S. 1404-1411.
9 . Zarubin A.N. Kraevaya zadacha dlya operezhayushche-zapazdyvayushchego uravneniya smeshannogo tipa s negladkoj liniej vyrozhdeniya // Differentsial'nye uravneniya. 2014. T. 50. № 10. S. 1362-1372.
Received 20 October 2016
Chaplygina Elena Viktorovna, Orel State University named after I. S. Turgenev, Orel, the Russian Federation, Candidate of Physics and Mathematics, Associate Professor of the Mathematical Analysis and Differential Equations Department, e-mail: lena260581@yandex.ru
Zarubin Aleksander Nikolaevich, Orel State University named after I. S. Turgenev, Orel, the Russian Federation, Doctor of Physics and Mathematics, Professor, Head of Chair of Mathematical Analysis and Differential Equations, e-mail: aleks_zarubin@mail.ru
Информация для цитирования:
Чаплыгина Е.В., Зарубин А.Н. Задача Геллерстедта для уравнения смешанного типа с функциональным запаздыванием и опережением // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2016. Т. 21. Вып. 6. С. 2098-2106. DOI: 10.20310/1810-0198-2016-21-6-2098-2106
Chaplygina E.V., Zarubin A.N. Zadacha Gellerstedta dlya uravneniya smeshannogo tipa s funktsional'nym zapazdyvaniem i operezheniem [The Gellerstedt problem for equation of the mixed type with functional delay and advance]. Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Review. Series: Natural and Technical Sciences, 2016, vol. 21, no. 6, pp. 2098-2106. DOI: 10.20310/1810-0198-2016-21-6-2098-2106 (In Russian)
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UDC 517.922; 517.988.5
DOI: 10.20310/1810-0198-2016-21-6-2107-2112
COVERING MAPPINGS IN THE THEORY OF IMPLICIT SINGULAR
DIFFERENTIAL EQUATIONS
© A. I. Shindiapin ^ , E. S. Zhukovskiy 2)
Universidade Eduardo Mondlane 257 Praca 25 de Junho, Maputo, Mocambique, CP 257
E-mail: andrei.olga@tvcabo.co.mz 2) Tambov State University named after G.R. Derzhavin 33 Internatsionalnaya St., Tambov, Russian Federation, 392000 The Peoples' Friendship University of Russia 6 Miklukho-Maklay St., Moscow, Russian Federation, 117198 E-mail: zukovskys@mail.ru
We propose method of studying implicit singular differential equations based on the results of the covering mapping theory. The article consists of three sections. In the first section we give the necessary designations and definitions and formulate the theorem on Lipcshitz perturbations of covering mappings. In the second section we introduce special spaces of measurable functions making it possible to study singular equations by methods of functional analysis and formulate the results about the Nemytskii operator in those spaces. In the last section we provide conditions for the resolubility of the Cauchy problem for implicit singular differential equations.
Key words: Implicit singular differential equation; Cauchy problem; covering mapping; Lipschitz mapping; metric space
Some problems of physics, aerodynamics, and technological processes can be reduced to singular differential equations with non-summable coefficients [1], [2]. The images of corresponding mappings, used to model such cases, do not belong to Lebesgue's spaces, which result in certain difficulties in studying such processes. The paper [3] proposed a space of summable functions, in which singular mappings become regular ones. This makes it possible to apply methods of classical analysis to differential equations with non-summable singularities. In studying the nonholonomous mechanical systems (see, for example, [4]) it is necessity to deal with the implicit singular equations. The standard method to study implicit differential equations is to use theorems on implicit functions. However, such theorems cannot be applied if the function generating the differential equation is not smooth enough or it has degenerated derivative. Additional difficulties appear in the case of singular equations. In literature we could not find methods to study such equations.
Recent rapid developments of the theory of covering mappings povides new opportunities in studying singular implicit differential equations. The authors in [5] proved a theorem on non-linear Lipcshitz perturbations of covering mappings and a proposed a new approach to study the implicit equations based on this theorem. Those studies have been further developed in [6], [7].
In this article, we build on [3], [5]-[7] and propose a formalization of implicit singular differential equation in a form of an equation with covering mapping in a special space of measurable functions. As a result, new conditions for existence of solutions of the Cauchy problem for such equations and its estimations have been found.
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1. Covering mappings of metric spaces We will use the following definitions.
Let (X,pX), (Y,Py) be metric spaces. Denote the closed ball in space X with center in x and radius r> 0 by BX(x,r).
Definition 1 [8]. The mapping X ^ Y is an a-covering (covering with constant a> 0), if for any r> 0 and u € X there holds the inclusion
$>(Bx(u,r)) D By(&(u),ar). (1)
The mapping X ^ Y is a -covering if and only if
Vu € X Vy € Y 3x € X ^(x) = y, pX(x,u) ^ a-1 pY (y, V(u)). (2)
Note that the covering mapping is surjective. This follows from the inclusion (1), as any point y € Y belongs to the ball BYu),ar) for sufficiently big r. In the problems that follow, it is sufficient to consider the estimation (2) only for y € ^(X) and not for any y € Y. This allows to ease the definition of covering mapping in our case.
Definition 2 [5]. The mapping ^ : X ^ Y is called conditionally a -covering (conditionally covering with constant a> 0), if for any r> 0 and u € X there holds the
$>(Bx(u,r)) D By(&(u),ar) n ).
The mapping X ^ Y is conditionally a-covering if and only if
Vu € X Vy € V(X) 3x € X ty(x) = y, px(x,u) < a-1 py (y, V(u)). (3)
The following proposition, proved in [8], is generalization of A.A.Milyuting's well-known theorem on Lipschit's perturbations of covering mappings. We will formulate this result for a particular case in order not to complicate reading with the unnecessary details, which do not effect our results. Let $ : X2 ^ Y be a mapping and y € Y. Consider equation
F (x) = $(x,x)= y. (4)
Theorem 1 [5]. Let metric space X be complete and let the following conditions hold: • for all x € X the mapping $(^,x): X ^ Y is conditionally a -covering, closed and
y € $(X, x); (5)
• for all x € X the mapping $(x, •): X ^ Y is / -Lipschitz;
Then, if 3 < a, the equation (4) has a solution and for any u0 € X then there exists solution x = £ € X of this equation, which satisfy the inequality
PX(£,u0) < PY (y, $(u0,u0)). (6)
N o t e 1. If the mapping $(-,x) : X — Y is a covering one, then the inclusion (5) is trivial, and should be excluded from conditions of Theorem 1. In this case the equation (4) has a solution for any righthand part y € Y. Moreover, as the solution satisfying the condition (6) exists, that means that the mapping F : X — Y, F (x) = &(x,x) is (a — () -covering. Thus the theorem 1 gives the conditions of stability of the covering property in respect of Lipschitz perturbations.
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2. Conditions of covering for the Nemytskii singular operator
Let z0,v: [a, b] ^ R, v(t) > 0 for almost all t € [a, b] be two measurable functions. Following [3], we define the metric space L^([a,b],R, z0,v) of all measurable functions z :[a,b] ^R, for which the function t € [a, b] ^ is essentially bounded, with the distance
. |Z2(t) - z1(t)\ PLx (z2, zi) = vrai sup-—-.
t&[a,b] v(t)
The metric space L^([a,b],R, z0,v) is complete. In its definition we will drop the symbol zo, if z0(t) = 0, and the symbol v, if v(t) = 1. In particular, L^([a, b],R) is a "standard"Banach space of essentially bounded functions.
In order to apply the Theorem 1 to the implicit singular differential equations we will need covering conditions for the Nemytsky operator in the spaces L^([a,b],R, z0,v). Let us formulate the corresponding proposition.
Let z0,v :[a,b] ^R be measurable functions and the function g :[a,b] x R^R satisfy the Caratheodory conditions, i.e. measurable on the first and continuous on the second variable. Let y0(t) = g(t,z0(t)). We assume that for any r> 0 there exists such R> 0, such that for almost all t € [a, b] and for each u € [z0(t) — rv(t), z0(t) + rv(t)] we have \g(t, u) — y0(t)\ < R.
Let function g generate the Nemytskii operator
(Ngz)(t)=g(t,z(t)), t € [a, b]. (7)
Under the above conditions, this is a closed from L^([a,b],R, z0,v) to L^([a,b],R,y0).
Define Gv :[a,b] xR^ as
Gv(t,u) = g(t, u(t)v).
Theorem 2. Suppose there exists a> 0, such that for almost all t € [a,b] the mapping Gv(t, ■): R ^ R is an a -covering. Then Nemytskii operator (7) defined by Ng : L^([a, b], IR, z0,v) ^ ^ L^([a,b],R,y0) is also an a -covering. If the mapping Gv(t, ■) for almost all t € [a,b] is conditionally an a -covering, then the operator (7) is also conditionally an a -covering.
Let us illustrate the use Theorem 2 to study the Nemytskii operator generated by the following simple function.
Example 1. Let
g : [0,1] xR^R, g(t,z)= t(1 — t)z.
Let v(t) = t(1 — t), z0(t) = 0, y0(t) = 0. For such a function g the operator acts from the space L^([a,b],R,v) into space L^([a,b],R). As for any t € [0,1] function Gv (t,u)= u is 1 -covering by variable u , the operator Ng : L^([a, b],R,v) ^ L^([a, b], R) is also 1 -covering.
3. The Cauchy problem for the implicit singular differential equation
Now we will use Theorems 1,2 to study the resolubility of the Cauchy problem for the implicit singular differential equation.
Suppose z0, v :[a,b] ^ R are summable functions, with v(t) > 0 for almost all t € [a, b] . Suppose a summable function f : [a, b] x R2 ^ R, satisfies Caratheodory conditions, and y :[a,b] ^ R is a measurable function and y € IR.
Consider the Cauchy problem
f (t,x(t),x'(t)) = y(t), t € [a, b], x(a)= y- (8)
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The solution of the Cauchy problem does not have to be defined on all the [a, b], but must satisfy the given equation on [a, c], for some c € (a, b]. We will look for the solution of the Cauchy problem (8) in the class of AC^([a,c],R, z0,v), absolutely continuous functions x :[a,c] — R such that x' € L^([a,c],R, z0,v). We call the solution x^ € AC^([a,c\,R, z0,v) the continuation of the solution xc € AC^([a,c],R, z0,v), if c > c and x-^(t) = xc(t) for t € [a,c]. If there exists the continuation of the solution xc then we call xc continuable. We will call function x :[a,d) — R, maximally continued solution if the restriction of x onto [a, c] for any c € (a, d) is a solution of (8) and
• \x' (t) — zo(t)\ iim vrai sup----= œ.
c^ d-0 t£[a,c] v(t)
We now give conditions that imply the resolubility of the Cauchy problem (8). We define function
yo(t) = f (t, y +/ zo(s) ds,zo(t)).
a
We assume that for any r > 0 there exists R> 0, such that for almost all t € [a, b] for every x € [—r,r], u € [z0(t) — rv(t), z0(t) + rv(t)] the inequality \f(t,x,u) — y0(t)\ < R holds. Let Fv : [a, b] x R2 — R be defined as
Fv(t,x,u) = f{t, x, u(t)v).
Theorem 3. Suppose function Fv (t,x, ■) :R — R be conditionally covering for almost all t € [a, b], and for any x € R and u € R . Suppose further the function Fv(t, ^,u) : R — R is Lipschitz and y(t) € f(t,x,U). Then there exists c € (a,b] and exists defined on [a,c] solution xc € AC^([a, c],R, z0,v) of the problem (8). Any solution of (8) is continuable to the solution defined on the whole [a, b] or to the maximally continued solution.
Proof of this theorem is based on the representation of the problem (8) in a form of operator equation (4) in respect of the derivative of the unknown function. In our case the mapping $(-,u) : L^([a,c],R, z0,v) — L^([a, c],R,y0) is a Nemytskii operator, which due to Theorem 2 is conditionally an a -covering; and the mapping $(u, ■) : L^([a, c],R, z0,v) — L^([a,c],R,y0) is an integral operator, which is Lipschitz with the constant (, which is less than a when c is close to a. Thus, by Theorem ,1 problem (8) has a solution. The continuation of each solution is proved in a similar way.
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