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Определим множества
07 = {у Є В I (у,0) € и(7)}, 0* = 0^7 = {д Є В* | 'І у Є 0і_7 ду = 0}.
Зададим в сопряженном пространстве В* систему 0?* отношений эквивалентности, считая функционалы /, д V* ^-эквивалентными, если / — д Є 0*. Система 53* отношений эквивалентности элементов пространства В* обладает всеми перечисленными выше свойствами.
Т е о р е м а. Если линейный оператор F : В —»• В волътерров на системе 93, то сопряженный оператор F* : В* —> В* будет вольтерровым на системе 93*.
ЛИТЕРАТУРА
1. Жуковский Е.С. Непрерывная зависимость от параметров решений уравнений Вольтерра // Мат. сборник. 2006. Т. 197. № 10. С. 33-56.
БЛАГОДАРНОСТИ: Работа выполнена при поддержке Российского Фонда Фундаментальных Исследований (грант № 04-01-00324) и Норвежского комитета по развитию университетской науки и образования - NUFU (проект № PRO 06/02).
Поступила в редакцию 8 ноября 2006 г.
POSITIVE INVARIANCE AND PERIODIC SOLUTIONS FOR DIFFERENTIAL INCLUSION WITH NON-CONVEX RIGHT-HAND SIDE
© E. Panasenko
The talk is concerned with the non-autonomous ordinary differential inclusion in finite dimensional space with periodic, compact, but non necessarily convex valued right-hand side. It is shown that if for such an inclusion there exists a strongly positively invariant set then there exists a periodic solution of the inclusion which stays in SDt.
Let be a Euclidian space with the scalar product (x, y), x,y € Rn, usual norm |ar| — \J(x, y), and metric p(x, y) = |x — 2/|, and let comp(En) stand for a set of all compact subsets of Rn. By AC {{to, ¿i], Rn) we denote the space of all absolutely continuous functions x : [¿o? ¿l] Rn with the norm IklUc =
|a?(f0)| +
Consider an ordinary differential inclusion
x G F(t,x), (1)
where F : E x E" -) comp(Rn) satisfies Caratheodory conditions. As a solution of (1) on an interval I C E we suppose a function x 6 AC(I,Rn) satisfying inclusion (1) for a.e. t e /, so we deal with the Caratheodory type solutions.
Let a map M : E —t comp(Mn) be continuous and denote a set
971= {{t,x) <E E x En :x 6 (2)
which represents the graph of M. The set VJl is called strongly positively invariant under inclusion (1) if for every point z0 = (t0,x0) £ DJl any solution t —> x(t,z0) of the Cauchy problem for (1) with initial
condition x(io) = £o satisfies (t,x(t,zo)) £ 9Jt for every t ^ to.
The sufficient conditions for the set Ш to be strongly positively invariant under inclusion can be expressed in terms of so-called Lyapunov functions. Let r > 0 and let fflr = {(£, x) £ IK x Rn : x £ Mr(t)} denote a closed r-neighbor hood of the set 9Я. We say that continuous function V : WT -> R is a Lyapunov function with respect to the set Ш if V(i, я) = 0 for (t, x) £ дШ and V(t, x) > 0 for (t, x) 6 9JT\9Jt. If the Lyapunov function V in addition is locally lipschitz, we can consider the generalized Clarke derivative (see, e.g. [1]) of V at the point (t,x) in the direction (1, h) £ R x Rn which is defined as follows:
V»(t,*;/.) = limsup n<> + fi,y + Sh)-V(0,y)
<5 v0 -
The relation Vp(i, x) = max V"(t, x; h) we will call the derivative of function V with respect to inclusion
h£F(t,x)
(!)■
T h e o r e m 1. Let us have a Lyapunov function (t,x) -» V(t,x)i {t,x) £ ПЯ7-, which is locally lipschitz. If for some £ £ (0,r] the inequality V§-{t,x) ^ 0 holds for any {t,x) £ Ш1е\ЗЯ, then the set OR is strongly positively invariant.
We suppose now an ordinary differential inclusion
x £ F(t, ж), F(t + T,x) = F(t, z) (3)
under the following assumptions:
(PI) F : E x En —у comp(Rn) satisfies Caratheodory conditions;
(P2) there exists continuous, T-periodic map M : E —t comp(En) such that M(0) is convex and the
corresponding set Ш (see (2)) is strongly positively invariant under inclusion (3);
(P3) there exists an integrable function к : E —у M+ such that for a.e. t 6 E and each ж, y £ M(t)
dist(F(t, x),F(t, y)) ^ A:(i)|a: — y\.
Theorem 2. Let the maps F and M satisfy the conditions (PI) — (F3). Then there exists a periodic solution t —> x(t) for the problem (3) such that x(t) £ M(t) for all t.
REFERENCES
1. Clarke F.H. Optimization and Nonsmooth Analysis. Wiley Interscience, New York, 1983.
2. Benedetti IPanasenko E. Positive Invariance and Differential Inclusions with Periodic Right-Hand Side // Nonlinear Dynamics and Systems Theory, (to appear)
ACKNOWLEDGMENTS: The author is partially supported by RFBR Grant 04-01-00324.
Поступила в редакцию 8 ноября 2006 г.
К ПРОБЛЕМЕ ОБУЧЕНИЯ ДОКАЗАТЕЛЬСТВУ В КУРСЕ ГЕОМЕТРИИ ОСНОВНОЙ ШКОЛЫ © С.Н. Петрунина
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