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9
Том 23, № 122
2018
DOI: 10.20310/1810-0198-2018-23-122-136-144
ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR INCLUSIONS WITH CAUSAL MULTIOPERATORS AND THE METHOD OF INTEGRAL GUIDING POTENTIALS
ac S.V. Kornev, V.V. Obukhovskii
Voronezh State Pedagogical University 86 Lenina St., Voronezh 394043, Russian Federation E-mail: kornev _ [email protected], [email protected]
Abstract. In the present paper the method of integral guiding potentials is applied to study the problem of the asymptotic behavior of solutions for a differential inclusion with a causal multioperator. At first we consider the case when the multioperator is closed and convex-valued. Then the case of a non-convex-valued and lower semicon-tinuous right-hand part is considered.
Keywords: functional inclusion; causal multioperator; asymptotic behavior of solutions; integral guiding potential
Introduction
The study of systems governed by differential and functional equations with causal operators, which is due to Tonclli [1| and Tychonov [2], attracts the attention of many researchers. The term causal arises from the engineering and the notion of a causal operator turns out to be a powerful tool for unifying problems in ordinary differential equations, intcgro-differential equations, functional differential equations with finite or infinite delay, Volterra integral equations, neutral functional equations et al. (see the monograph [3]). In the present paper we apply the method of integral guiding potentials to the investigation of the asymptotic behavior of solutions for a differential inclusion with the multivalued causal operator.
The main ideas of the method of guiding functions were formulated by Krasnosel'skii and Pcrov in the fifties (see [4, 5]). Being geometrically clear, this method was originally applied to the study of periodic and bounded solutions of ordinary differential equations (see, e.g., [6-8]). Thereafter the method was extended to differential inclusions (see, e.g., [9. 10]), differential inclusion with the causal operator (see, e.g., |11, 12]) and other objects. The sphere of applications was extended to the study of qualitative behavior and bifurcations
The work is partially supported by the Ministry of Education and Science of the Russian Federation (project № 1.3464.2017/4.6) and the Russian Fund for Basic Research (projects №№ 17-51-52022, 16-01-00370, 16-01-00386).
of solutions (see, e.g., [13-16]) and asymptotics of solutions (see, e.g., [17-19]). These and other aspects of the method of guiding functions and its applications, as well as the additional bibliography, may be found in the recent monograph [20].
1. Main concepts
Let (X, dx) and (Y, dy) be metric spaces. By the symbols P(Y) and K(Y) we denote the collections of all nonempty and, respectively, nonempty and compact subsets of the space Y. If Y is a normed space, Cv ( K) and Kv(Y) denote the collections of all nonempty convex closed (and, respectively, compact) subsets of Y.
Definition 1. A multimap F : X oo P(Y) is called upper semicontinuous (u.s.c.) at the point x / X if for each open set V —>Y such that F(x) —> V there exists 5 > 0 such that dx(x, x') < ¿ implies F(x') —>V. A multimap F : X oo P(Y) is called u.s.c. if it is u.s.c. at each point x / X.
Definition 2. A multimap F : X oo P(Y) is called lower semicontinuous (l.s.c.) at a point x / X. if for each open set V —>Y such that F(x) { V 0= C there exists S > 0 such that dx(x,x') < ¿ implies F(x') { V 0= C. A multimap F : X oo P(Y) is called l.s.c. if it is l.s.c. at each point x / X.
Definition 3. Let I be a closed subset of M endowed with the Lebesgue measure. A multifunction F . I oo K(Y) is called measurable if, for each open subset W Y, its pre-image F~1(W) = }í / I : F(t) —>W\ is the measurable subset of J.
Remark 1. A u.s.c. multifunction is measurable. Each measurable multifunction F.I oo K(Y) has a measurable selection, i.e., there exists such measurable function / : / oo y, that f(t) / F(t) for a.e. t / F
Let T > 0 and a c 0 be given numbers. By the symbols C([kT <r, (k + 1)T]; R") and ¡/((kT, (fc-|-l)7T);KT1), where k / Z, we will denote the corresponding spaces of continuous and integrable functions with usual norms.
For any subset V -»■ L1 {(kT, (k + 1)T);K") and r / (kT,(k + l)T) we define the restriction of V on {kT, r) as
V 1^7»=}/ i[*r,T):f /v\.
Definition 4. We will say that l~l is a causal multioperator if for each k / % a multimap
n : C([kT <7, (k + 1)T]; Rn) L1^kT, (k + 1 )T);Mn) is defined in such a way that for each r / (kT. (k + 1)T) and for all
u(^v(>{/C([kT a7(k+l)T\;Wn)
the condition U \[kT-a,r]=V ItfeT-^r] implies n ('tí) ||;fcT,r)= n (v) ||fcT,r) •
Denote by V the Banach space C([ cr, 0]; R").
Example 1. Suppose that a multimap F : M. QfD00 Kv (R™) satisfies the following conditions:
(Fl) the multifunction F (>fc) : ffi 00 Kv (R") admits a measurable selection for every c/Vt
(F2) the multimap F(t,%: Voo Kv (Rn) is u.s.c. for a.e. t / M;
(F3) for every r > 0 there exists a locally integrable non-negative function r]r(^ / Ljoc (ffi) such that
\F (t, c)\ := sup}\y\ : y / F (t, c)\ > rjr (t) a.e. t / M, for all c / V, \c\ > r.
It is known (see, e.g., [9, 21]) that under conditions (Fl) - (F3) for each k / Z, the superposition multioperator QF : C ([kT a, (k + 1 )T]; R") L1 ((/kT, (k + 1)T);R"),
QF(u) = {f /L1((kT,(k+ 1)7]; R"): f (t) / F (t7ut) a.e. t / (kT, (k + 1)T) }
is well defined. Here ut / T> is defined as ut(9) = u(t + 0), 6 / [ <7,0]. It is easy to see that the multioperator Qf is causal.
Example 2. Let F : R 0'Poc Kv(M") be a multimap satisfying conditions (Fl) - (F3) of Example 1. Suppose that }K(t, s) : e < s > t < +€ | is a continuous (with respect to the norm) family of linear operators in M" and m / Ljoc(M\ R") is a given locally integrable function. Consider, for each k / Z, the Volterra type integral multioperator AT: C([kT a, (k + l)T];Mn) L1 ((kT, (k + 1 )T);Mn) defined as
A(u)(t) = m(t) + f K(t,s)F(s,us)ds, J kT
A{u) = }y / Ll ((kT, (k + 1)T); R") : y(t) = m(t) + [ K(t, s)f(s)ds : / / &(«)| .
J kT
It is also obvious that the multioperator N is causal.
Example 3. Suppose that a multimap F : R O'Coo iif(M™) satisfies the following condition of almost lower semicontinuity:
(Fl) there exists a sequence of disjoint closed sets }J,j| , Jn < M n = 1,2,... such that: (?') meas (M MJ.„ Jn) = 0; (ii) the restriction of F on each set Jn QT> is l.s.c.
Then (see, e.g., [9, 21]) under conditions (Fl), (F3), for each k / Z, the superposition multioperator Qf : C([kT a, (k + 1)7]; K") ^ L1 ({kT, (k + 1)T); E") is also well-defined and causal.
Now, suppose that ip / D is a given initial function. By the symbol D^ we will denote the set of all continuous functions x : [ a, +t ) oo R" such that x(t) = '0(i), t / [ a, 0] and the restriction of x to R+ = [0, +6 ) is absolutely continuous.
Considering the following abstract Cauchy problem for a functional inclusion with causal operator Q of the following form:
x' /Q(x), (1)
x(t) = i>(t), t/[ (7,0], (2)
where x( is an absolutely continuous function, we will study the problem of existence of solutions satisfying the estimate of the type
V(£)\>^, t / R+, (3)
where k > 0 and g is a. given function,
2. Main results 2.1. Convex-valued causal multioperators
By the symbols L1 and C we will denote the corresponding spaces of integrable functions
/ : R oo M" and continuous functions x : R oo R™ with the norm \x\c = sup \x(t)\.
te[o,T]
In this section we will assume that the causal multioperator D : Coo Cv(Ll) has convex values and satisfies the following conditions:
(l~l 1) for each bounded linear operator A \ Ll oo E, where E is a Banach space, the composition A Cn : C oo Cv(E) is closed;
(f)2) there exists a locally integrable non-negative function a( / Ljoc(BL+) such that for every x / C
\n(x)(i)\ > a(t)(l + \x(t)\) for a.e. t / R+.
To provide condition (fl 1) in Examples 1 and 2, it is sufficient to assume that the multimap F satisfies conditions (Fl) - (F3) (see, e.g., [9, Theorem 1.5.30]) and to fulfil condition (f)2), we can suppose, in Example 1, the following sublinear growth condition: for each x / C we have, for some non-negative integrable function 0(t) :
\F(t,xt)\ > /3(i)(l + \x(i)\) for a.e. t / R+, (4)
and, in Example 2, the global boundcdness condition \F(t, c)\ > 7(t) for some non-negative integrable function "f(t).
Denote by the collection of all C1 -functions V : M" 00 R satisfying the coercivity condition lim||T||_)._|_00 V(x) = ¡E . Notice that, given a function V / -Xi, for each r > 0 there exists k(r) > r such that if a,, := inf}V(:E), \x\ > r| then V(x) < ar, \x\ C k(r). Now, let g : R+ 00 M+ be a given C1 -function such that inf t / R| c 1.
Definition 5. A function V / is called an integral guiding potential for inclusion (1) along the function g if there exists ry > ¿r(0)(0)\ such that for every function x / Q ^ satisfying conditions
(i) there exists a largest finite number rf > 0 such that g(t)\x(t)\ > ry, t / [0, rf); (ii) there exists a (least) finite number if > rf such that ¿/(rf)\x(rf)\ = ky := fc {?>-); (Hi) \x'(t)\ > \Q(x)(t)\ for a.e. t / we have
r ( V^s^a)), g>(s)x(s) + g(s)f(s))ds c 0
Jt? t
for each summable selections / / Q(x), where rf := sup}r / [rf, r^), \g(T)x(r)\ = ry .
Now we are in position to formulate the main result of this paper.
Theorem 1. If V / 93 is an integral guiding potential for inclusion (1) along the function g then each solution of Cauchy problem (1), (2) satisfies the estimate
\x(t)\>kvx-L t/R+. (5)
2.2. Lower semicontinuous causal mult [operators
In this section we will consider the Cauchy problem for a class of functional inclusions with non-convex-valued lower semicontinuous causal multioperators. Namely, we will suppose that the causal multioperator D : C oo P{Ll) satisfies condition
(HL) H is l.s.c. and has closed decomposable values
and condition (H2).
As an example of a causal multioperator satisfying conditions (H and (H 2) we may consider the superposition multioperator Qp generated by a multimap F : M (jiD-oo /i (M"') satisfying conditions of almost lower semicontinuity (Fl) and the sublinear growth condition (4) (see, e.g., [9, 21]).
The following result holds true.
Theorem 2. Let D : C oo P(Ll) be a causal multioperator satisfying conditions (H ¿ ) and (H 2). If V \ MJ1 oo H is an integral guiding potential for inclusion (1) along the function g then each solution of Cauchy problem (1), (2) satisfies the estimate (5).
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20. Obukhovskii V., Zecca P., Loi N.V., Kornev S. Method of Guiding Functions in Problems of Nonlinear Analysis. Lecture Notes in Mathematics. Heidelberg, Springer, 2013.
21. Kamenskii M., Obukhovskii V., Zecca P. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. De Gruyter Series in Nonlinear Analysis and Applications. Berlin, New York, Walter de Gruyter, 2001.
Received 23 March 2018 Reviewed 24 April 2018 Accepted for press 5 June 2018 There is no conflict of interests.
Kornev Sergey Viktorovich, Voronezh State Pedagogical University, Voronezh, the Russian Federation, Doctor of Physics and Mathematics, Professor of the Chair of Higher Mathematics, e-mail: kornev _ [email protected]
Obukhovskii Valeri Vladimirovich, Voronezh State Pedagogical University, Voronezh, the Russian Federation, Doctor of Physics and Mathematics, Head of the Chair of Higher Mathematics, e-mail: [email protected]
For citation: Kornev S.V., Obukhovskii V.V. Asymptotic behavior of solutions for inclusions with causal multioperators and the method of integral guiding potentials. Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki -Tambov University Reports. Series: Natural and Technical Sciences, 2018, vol. 23, no. 122, pp. 136-144. DOI: 10.20310/18100198-2018-23-122-136-144 (In Engl., Abstr. in Russian).
DOI: 10.20310/1810-0198-2018-23-122-136-144 УДК 517.911
АСИМПТОТИЧЕСКОЕ ПОВЕДЕНИЕ РЕШЕНИЙ ВКЛЮЧЕНИЙ С КАУЗАЛЬНЫМИ МУЛЬТИОПЕРАТОРАМИ И МЕТОД ИНТЕГРАЛЬНЫХ НАПРАВЛЯЮЩИХ ПОТЕНЦИАЛОВ
С. В. Корнев, В. В. Обуховский
ФГБОУ ВО «Воронежский государственный педагогический университет» 394043, Российская Федерация, г. Воронеж, ул. Ленина, 86 E-mail: kornev _ [email protected]. [email protected]
Аннотация. В настоящей работе метод интегральных направляющих потенциалов применяется для изучения задачи об асимптотическом поведении решений дифференциального включения с каузальным мультиоператором. Сначала рассматривается случай, когда мультиоператор имеет замкнутые и выпуклые значения. Затем рассматривается случай невыпуклозначной полунепрерывной снизу правой части.
Ключевые слова: функциональное включение; каузальный мультиоператор; асимптотическое поведение решений: интегральный направляющий потенциал
СПИСОК ЛИТЕРАТУРЫ
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2. Тихонов A.H. О функциональных уравнениях типа Volterra и их применениях к некоторым задачам математической физики // Бюллетень Московского государственного университета. Секция А. Серия Математика и механика. 1938. Т. 1. Вып. 8. С. 1-25.
3. Corduneanu С. Functional Equations with Causal Operators. Stability and Control: Theory, Methods and Applications. London: Taylor and Francis, 2002.
4. Красносельский M.A. Оператор сдвига по траекториям дифференциальных уравнений. М.: Наука. 1966.
5. Красносельский М.А., Перов А.И. Об одном принципе существования ограниченных, периодических и почти-периодических решений у систем обыкновенных дифференциальных уравнений // ДАН СССР. 1958. Т. 123. № 2. С. 235-238.
6. Красносельский М.А., Забрейко П.П. Геометрические методы нелинейного анализа. М.: Наука, 1975.
7. Mawhin J. Topological Degree Methods in Nonlinear Boundary Value Problems // CBMS Regional Conference Series in Mathematics. Providence: American Mathematical Society, 1979.
8. Mawhin J., Ward J.R.Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations // Discrete and Continuous Dynamical Systems. 2002. Vol. 8. № 1. P. 39-54.
9. Борисович Ю.Г., Гельман Б.Д., Мышкис А.Д., Обуховский В.В. Введение в теорию многозначных отображений и дифференциальных включений. Изд. 2-е. М_: Либроком, 2011.
Работа выполнена при поддержке Министерства образования и науки РФ (проект № 1.3464.2017/4.6) и Российского фонда фундаментальных исследований (№№ 17-51-52022, 16-01-00370, 16-01-00386).
10. Gorniewicz L. Topological Fixed Point Theory of Multivalued Mappings. Second edition. Topological Fixed Point Theory and Its Applications. Dordrecht: Springer, 2006.
11. Корнев С.В., Обуховский В.В. Интегральные направляющие функции и периодические решения включений с каузальными операторами // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2016. Т. 21. Вып. 1. С. 55-65.
12. Kornev S., Obukhovskii V., Zecca P. Guiding functions and periodic solutions for inclusions with causal multioperators // Applicable Analysis. 2017. Vol. 96. Issue 3. P. 418-428.
13. Kryszewski W. Homotopy Properties of Set-Valued Mappings. Torun: Univ. N. Copernicus Publishing, 1997.
14. Obukhovskii V., Loi N.V., Kornev S. Existence and global bifurcation of solutions for a class of operator-differential inclusions // Differential Equations and Dynamical Systems. 2012. Vol. 20. P. 285-300.
15. Loi N.V., Obukhovskii V., Zecca P. On the global bifurcation of periodic solutions of differential inclusions in Hilbert spaces // Nonlinear Analysis. 2013. Vol. 76. P. 80-92.
16. Kornev S.V., Liou Y.-C. Multivalent guiding functions in the bifurcation problem of differential inclusions // Journal of Nonlinear Sciences and Applications. 2016. Vol. 9. Issue 8. P. 5259-5270.
17. Kornev S., Obukhovskii V., Yao J-C. On asymptotics of solutions for a class of functional differential inclusions // Discussiones Mathematicae Differential Inclusions, Control and Optimization. 2014. Vol. 34. № 2. P. 219-227.
18. Корнев С.В., Обуховский В.В. Асимптотическое поведение решений дифференциальных включений и метод направляющих функций // Дифференциальные уравнения. 2015. Т. 51. № 6. С. 700-705.
19. Obukhovskii V., Kamenskii M., Kornev S., Liou Y.-C. On asymptotics of solutions for a class of differential inclusions with a regular right-hand part // Journal of Nonlinear and Convex Analysis. 2017. Vol. 18. № 5. P. 967-975.
20. Obukhovskii V., Zecca P., Loi N.V., Kornev S. Method of Guiding Functions in Problems of Nonlinear Analysis. Lecture Notes in Mathematics. Heidelberg: Springer, 2013.
21. Kamenskii M, Obukhovskii V., Zecca P. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. De Gruyter Series in Nonlinear Analysis and Applications. Berlin; N. Y.: Walter de Gruyter, 2001.
Поступила в редакцию 23 марта 2018 г.
Прошла рецензирование 24 апреля 2018 г.
Принята в печать 5 июня 2018 г.
Конфликт интересов отсутствует.
Корнев Сергей Викторович, Воронежский государственный педагогический университет, г. Воронеж, Российская Федерация, доктор физико-математических наук, профессор кафедры высшей математики, e-mail: kornev _ [email protected]
Обуховский Валерий Владимирович, Воронежский государственный педагогический университет, г. Воронеж, Российская Федерация, доктор физико-математических наук, зав. кафедрой высшей математики, e-mail: [email protected]
Для цитирования: Корнев С.В., Обуховский В.В. Асимптотическое поведение решений включений с каузальными мультиоператорами и метод интегральных направляющих потенциалов // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2018. Т. 23. № 122. С. 136-144. DOI: 10.20310/1810-0198-2018-23-122-136-144
