О нелокальных дифференциальных уравнениях в гильбертовых пространствах Текст научной статьи по специальности «Математика»

УДК 517.927.4, 517.927.6 ON NONLOCAL DIFFERENTIAL EQUATIONS IN HILBERT SPACES

© N.V. Loi

Key words: nonlocal differential equation; bounding function.

In this paper by combining the continuity method, the bounding function method and the approximation method we prove the existence theorem for a class of nonlocal differential equations in Hilbert spaces.

Let H be a separable Hilbert space which is compactly embedded in a Banach space E. Let

{ei}T be the orthonormal basis of H. For every n € N denote by Hn = span{e1, ■■■ ,en} and

Pn - the projection onto Hn.

In this paper we consider a differential equation of the form:

x'(t) = f (t,x(t)) for a.e. t € [0,T], (1)

with a nonlocal condition

x(0) = Mx, (2)

where f: [0,T] x H ^ H and M: C([0,T]; H) ^ H satisfying the following conditions:

(f 1) f is (globally) measurable;

(f 2) for a.e. t € I the map f (t, ■): H ^ H is E — E continuous in the sense: for each w € H, for every e> 0 there exists 0 such that from w' € OE(w) it follows that f (t,w') € € Of (f (t,w)), where Of (z) denoted e— neighborhood of z in the space E;

(f 3) for every bounded subset Q С H there exists vq € L+[0,T] such that

llf(t, q)IIh < vQ(t)

for a.e. t € I;

(M) M is a linear bounded map such that ||M|| ^ 1.

(a) Condition (f 1) can be easily followed if f is a Carathe odory map, i.e., for every w € € H the function f (^,w): I ^ H is measurable and for a.e. t € I the map f (t, ■): H ^ H is continuous.

(b) The class of linear bounded maps satisfying condition (M) is large enough. In particular, it includes all the well-known problems:

(i) Mx = 0 (the general Cauchy condition x(0) = xo can be replaced by condition z(0) = 0 by a transformation z = x — x0 );

(ii) Mx = ±x(T) (periodic and anti-periodic problems);

(iii) Mx = 1 x(t)dt (mean value problem);

(iv) Mx = ^2i=Li aix(ti) with ai € R and ^i=Li \ai\ ^ 1, where

0 <t1 < ■ ■ ■ <tk0 ^ T (multi-points problem).

By a solution to problem (1)-(2) we mean an absolutely continuous function x: [0,T] ^ H that satisfies (1)-(2).

2578

The main result of this paper is the following statement.

Theorem. Let conditions (f 1) — (f3) and (M) hold. In addition, assume that there exist R0 >r0 > 0 such that

(w,f (t,w))H < 0. for all w € H: r0 < ||w||h < R0 and a.e. t € I.

Then problem (1)-(2) admits a solution satisfying ||x(t)||H ^ R0, Vt € [0,T].

Sketch of the proof. Let r* € (r0, R0) be an arbitrary number and denote Q = {x € C([0, T]; H): llx(t)lH < r*, t € [0,T]}.

For each n € N set Q(n = Q П C([0,T]; Hn) and consider the problem

x'(t)= Pnf (t,x(t)), for a.e. t € [0,T],

x(0) = PnMx, ()

whose solution is considered in the space C([0, T]; Hn).

It is shown that problem (3) has a solution xn € Q(n). The existence of a solution of problem (1)-(2) is obtained from the weak convergence of the set {xn}.

REFERENCES

1. J. Andres, L. Malaguti, V. Taddei, On boundary values problems in Banach spaces, Dyn. Sys. Appl., 18 (2009): 275-302.

2. Nguyen Van Loi, Method of guiding functions for differential inclusions in a Hilbert space, Differ. Uravn. 46 (2010), no. 10, 1433-1443 (in Russian); English tranl.: Differ. Equat. 46 (2010), no. 10, 1438-1447.

3. N.V. Loi, V. Obukhovskii, P. Zecca, Non-smooth guiding functions and periodic solutions of functional differential inclusions with infinite delay in Hilbert spaces, Fixed Point Theory, 13(2), 565-582, 2012.

Лой Н.В. О НЕЛОКАЛЬНЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЯХ В ГИЛЬБЕРТОВЫХ ПРОСТРАНСТВАХ

С помощью метода непрерывности, метода ограничивающей функции и метода приближений, доказывается теорема существования для класса нелокальных дифференциальных уравнений в гильбертовых пространствах. Ключевые слова: нелокальное дифференциальное уравнение; ограничивающая функция.

УДК 517.977

ОБ АППРОКСИМАЦИИ КОНФЛИКТНО-УПРАВЛЯЕМЫХ ФУНКЦИОНАЛЬНО-ДИФФЕРЕНЦИАЛЬНЫХ СИСТЕМ

© Н.Ю. Лукоянов, А.Р. Плаксин

Ключевые слова: теория управления; уравнения с запаздыванием; уравнения нейтрального типа.

Для конфликтно-управляемых систем функционально-дифференциальных уравнений рассматриваются процедуры управления с поводырем, движение которого описывается аппроксимирующей системой обыкновенных дифференциальных уравнений.

2579