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ISSN 1810-0198. Вестник Тамбовского университета. Серия Естественные и технические науки
Том 23, № 122
2018
DOI: 10.20310/1810-0198-2018-23-122-125-130
ON SEMIDISCRETIZATION METHODS FOR DIFFERENTIAL INCLUSIONS OF FRACTIONAL ORDER
£ M.I. Kamenskii1), V.V. Obukhovskii2), G. G. Petrosyan2'
Voronezh State University 1 University sq., Voronezh 394018, Russian Federation E-mail: mikhailkameiiski@mail.ru 2- Voronezh State Pedagogical University 86 Lenin St., Voronezh 394043. Russian Federation E-mail: valerio-ob2000@mail.ru, garikpetrosyan@yandex.ru
Abstract. The report provides semidiscretization diagram for semilinear differential inclusions of fractional order.
Keywords: fractional differential inclusion; semilinear differential inclusion; Cauchy problem; approximation; semidiscretization; fixed point; condensing map; measure of noncompactness
Introduction
Theories of differential inclusions and condensing mappings are of great importance in modern mathematics (see [1], [2]). In our work, we present further development of these theories for differential inclusions of fractional order.
For a semilinear fractional order differential inclusion in a separable Banach space E of the form
c№)i+£ Ar)i+0 F)t,x)t-^tg]l,Ti, (1)
consider the problem of existence of mild solutions to this inclusion satisfying the following periodic
x)l+[ x)T+ (2)
and anti-periodic
x)l+[ x)T+ (3)
boundary value conditions under the following basic assumptions.
The symbol c Dqx denotes the Caputo fractional derivative of order q Q )1,2-|r We suppose that the linear operator A satisfies condition
The work is supported by the Ministry of Education and Science of the Russian Federation in the frameworks of the project part of the state work quota (Project No 1.3464.2017/4.6).
),4+ A ; D)A+< E E E is a linear closed (not necessarily bounded) operator generating a C0-semigroup }£/)£-()-(>Q of bounded linear operators in E
We will assume that the multivalued nonlinearity F ; ] 1, Ti QE E Kv)EJr obeys the following conditions:
)F2+for each x Q E the multifunction F) ; ] 1, Ti e Kv )E+ admits a strongly continuous selection;
)F3+fora.e. £C?]l,Ti the multimap F)t1 E E Kv )F+is u.s.c.;
)F4+ there exists a function a Q 1, Ti+ such that
F)t,x+B c a)£-|}2 0 x)t+E+ for a.c. t Q ]l,Ti,
)F: + the -regularity condition: there exists a function ¡i Q L°°)]l,Ti+such that for each bounded set ! wc have:
X)F)t,\ -H-C fi)t~ix)\
for a.e. t Q ]1, Ti, where x is the Hausdorff MNC in E.
Along with inclusion (1), for a given sequence of positive numbers }hn\ converging to zero consider the inclusion
n*xh)t^Q Ahxh)t-H) Fh)t, xh)tM, t Q ] 1, Ti, (4)
where HQ H \ }hn\ is the semidiscretization parameter, Ah ; D)Ah~\—> Eh E Eh are closed linear operators in Banach spaccs Eh generating C0-semigroups }Uh)t-$t>0 . We assume that E0 [ E^A0 [ A. F0 [ F and continuous maps Fh ; ] 1, Ti O Eh Eh satisfying conditions )F2+ )F: +for each h Q H.
1. Basic concepts
Definition 1. A mild solution to the Cauchy problem for inclusion (1) with initial condition
x)l+[ x0 (5)
on an interval ] 1, ri < ] 1, Ti is a function x Q C)\ 1, T\=EJr which can be represented as
x)t+[ V)t^0 0 f )t )t t Q ]l,Ti,
Jo
where <j) Q
pea poo
V)t+[ / £q)6JJ)tWW, {)i+[ q / 0Zq)64J)tq6^M, Jo Jo
"E) )nq ° 2+ m)mrq^6 Q M+.
7T Z—' Tl(
n=l v
ON SEMIDISCRETIZATION METHODS FOR DIFFERENTIAL INCLUSIONS
127
By the symbol 1, ri we will denote the set of all mild solutions to the Cauchy problem (1), (5) on an interval 1, ri < ]l,Ti. In articles [3] and |4] we proved the following local existence result.
Theorem 1. Under conditions )F2+ )F: + there exists r Q )l,Ti such that
l,ri is a nonempty subset of the space C')] 1, ri=E-\r
Definition 2. We will say that problem (1), (5) satisfies condition )Q+provided: )Q2+ is a non-empty compact subset of C)] 1,Ti=E^
)QS+ the following extendability condition holds:
for every r Q )l,Ti.
We suppose that there exist linear operators Qh ; Eh € E. h Q H, Qq [ I and projection operators Eh, P0 [ I such that
PhQhi h, (6)
where J^ is the identity on Eh and
QhPhX e x (7)
as h e 1 for each x Q E. We suppose that the operators F\ and Qh are uniformly bounded
Ph C 2, Qh C2 (8)
for all h G H.
An initial condition for equation (4) will be given by the equality
xfc)l-K xh)T^ov xh)l+[ xh)T-tf (9)
2. Main results
Theorem 2. Under conditions )A-^)F2+ )F3-^)Q2+ )Q3+ let problem (1) - (2) )or (1) - (3)+ the solution x* on the interval 1, ai. Then, for a sufficiently small h > 1 problems (4), (9) have solutions Xh on the interval ]1, ai and
Qhxh € x*
as h £ 1.
Various applications of the theory of differential inclusions of fractional order can be found in papers [5], [6] and [7].
REFERENCES
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2. Borisovich Yu.G., Gelman B.D., Myshkis A.D., Obukhovskiy V.V. Vvedenie v teoriyu mnogo-znachnykh otobrazheniy i differentsial'nykh vklyucheniy [Introduction to the Theory of Many-Valued Separations and Differential Inclusions]. Moscow, Book House "Librokom" Publ., 2011. (In Russian).
3. Kamenskii M., Obukhovskii V., Petrosyan G., Yao J.-C. On semilinear fractional order differential inclusions in banach spaces. Fixed Point Theory, 2017, vol. 18, no. 1, pp. 269-292.
4. Kamenskii M., Obukhovskii V., Petrosyan G., Yao J.-C. Boundary value problems for semilinear differential inclusions of fractional order in a Banach space. Applicable Analysis, 2017, vol. 96, pp. 1-21.
5. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. Amsterdam, Elsevier Science, 2006, 541 p.
6. Petrosyan G.G., Afanasova M.S. O zadache koshi dlya differentsial'nogo vklyucheniya drobno-go poryadka s nelineynym granichnym usloviem [On the Cauchy problem for a differential inclusion of fractional order with nonlinear boundary conditions]. Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika - Proceedings of Voronezh State University. Series: Physics. Mathematics, 2017, no. 1, pp. 135-151. (In Russian).
7. Petrosyan G.G. On the structure of the solutions set of the Cauchy problem for a differential inclusions of fractional order in a Banach space. Nekotorye voprosy analiza, algebry, geometrii i matematicheskogo obrazovaniya [Some Questions of Analysis, Algebra, Geometry and Mathematical Education]. Voronezh, 2016, pp. 7-8.
Received 21 March 2018
Reviewed 26 April 2018
Accepted for press 5 June 2018
There is no conflict of interests.
Kamenskii Mikhail Igorevich, Voronezh State University, Voronezh, Russia, Doctor of Physical and Mathematical Sciences, Head of the Department of Functional Analysis and Operator Equations, e-mail: mikhailkamenski@mail.ru
Obukhovskii Valeri Vladimirovich, Voronezh State Pedagogical University, Voronezh, Russia, Doctor of Physical and Mathematical Sciences, Head of the Department of Higher Mathematics, e-mail: valerio-ob2000@mail.ru
Petrosyan Garik Gagikovich, Voronezh State Pedagogical University, Voronezh, Russia, Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Higher Mathematics, e-mail: garikpetrosyan@yandex.ru
For citation: Kamenskii M.I., Obukhovskii V.V., Petrosyan G.G. On semidiscretization methods for differential inclusions of fractional order. Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Reports. Series: Natural and Technical Sciences, 2018, vol. 23, no. 122, pp. 125-130. DOI: 10.20310/1810-0198-2018-23-122-125-130 (In Russian, Abstr. in Engl.).
ON SEMIDISCRETIZATION METHODS FOR DIFFERENTIAL INCLUSIONS
129
DOI: 10.20310/1810-0198-2018-23-122-125-130 УДК 517.92
О МЕТОДЕ ПОЛУДИСКРЕТИЗАЦИИ ДЛЯ ДИФФЕРЕНЦИАЛЬНЫХ ВКЛЮЧЕНИЙ ДРОБНОГО ПОРЯДКА
М. И. Каменский1^, В. В. Обуховский2), Г. Г. Петросян2)
ФГБОУ ВО «Воронежский государственный университет» 394018, Российская Федерация, г. Воронеж. Университетская пл., 1 E-mail: mikhailkaraenskiQmail. ru 2"' ФГБОУ ВО «Воронежский государственный педагогический университет» 394043, Российская Федерация, г. Воронеж, ул. Ленина, 86 E-mail: valerio-ob2000@mail.ru, garikpetrosyan@yandex.ru
Аннотация. В докладе приводится схема полудискретизации для полулинейных дифференциальных включений дробного порядка
Ключевые слова: дифференциальное включение дробного порядка; полулинейное дифференциальное включение; задача Коши; аппроксимация; полудискретизация: неподвижная точка: уплотняющее отображение; мера некомпактности
СПИСОК ЛИТЕРАТУРЫ
1. Ахмеров P.P., Каменский М.И., Потапов А.С., Родкина А.Е., Садовский Б.Н. Меры некомпактности и уплотняющие операторы. Новосибирск, Наука, 1986.
2. Борисович Ю.Г., Гелълшн Б.Д., Мышкис А.Д., Обуховский В.В. Введение в теорию многозначных отображений и дифференциальных включений. Издание 2-е, испр. и доп. М.: Книжный дом «Либроком», 2011.
3. Kamenskii М., Obukhovskii V., Petrosyan G., Yao J.-С. On semilinear fractional order differential inclusions in banach spaces // Fixed Point Theory. 2017. Vol. 18. № 1. P. 269-292.
4. Kamenskii M., Obukhovskii V., Petrosyan G., Yao J.-C. Boundary value problems for semilinear differential inclusions of fractional order in a Banach space // Applicable Analysis. 2017. Vol. 96. P. 1-21.
5. Kilbas A. A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier Science. 2006. 541 p.
6. Петросян P.P., Афанасова M.C. О задаче Коши для дифференциального включения дробного порядка с нелинейным граничным условием // Вестник ВГУ. Серия: Физика. Математика. 2017. № 1. С. 135-151.
7. Петросян Г. P. On the structure of the solutions set of the Cauchy problem for a differential inclusions of fractional order in a Banach space // Некоторые вопросы анализа, алгебры, геометрии и математического образования. Воронеж, 2016. С. 7-8.
Работа выполнена при поддержке Министерства образования и науки Российской Федерации в рамках проектной части государственного задания (проект № 1.3464.2017/4.6).
Поступила в редакцию 21 марта 2018 г. Прошла рецензирование 26 апреля 2018 г. Принята в печать 5 июня 2018 г. Конфликт интересов отсутствует.
Каменский Михаил Игоревич, Воронежский государственный университет, г. Воронеж, Российская Федерация, доктор физико-математических наук, зав. кафедрой функционального анализа и операторных уравнений, e-mail:mikhailkamenski@mail.ru
Обуховский Валерий Владимирович, Воронежский государственный педагогический университет, г. Воронеж, Российская Федерация, доктор физико-математических наук, зав. кафедрой высшей математики, e-mail: valerio-ob2000@mail.ru
Петросян Гарик Гагикович, Воронежский государственный педагогический университет, г. Воронеж, Российская Федерация, кандидат физико-математических наук, доцент кафедры высшей математики, e-mail: garikpetrosyan@yandex.ru
Для цитирования: Каменский М.И., Обуховский В.В., Петросян Г.Г. О методе полудискретизации для дифференциальных включений дробного порядка // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2018. Т. 23. № 122. С. 125-130. БОТ: 10.20310/1810-0198-2018-23-122-125-130
