CC BY
16
UDC 517.911, 517.968
On some problems for functional-differential inclusions with multi-valued impulses 1
We consider functional-differential inclusions with multi-valued impulses. We present an existence theorem for solutions and a theorem on the compactness of the set of all solutions. We establish some properties of a priory bounded sets of solutions
Keywords: functional-differential inclusions, multi-valued impulses, local solutions
Let E C [a, b] be a Lebesgue measurable set, Ln(E) a space of summable (in Lebesgue sense) functions x : E ^ Rn with the norm
| ■ | being the Euclidean norm in Rn, and the corresponding distance pLn(e)- Let comp [Rn] denote the set of all nonempty compacts of Rn,
We say that a set A C Ln[a,b] is convex with respect to switching, if for any functions x,y E A and any measurable set e C [a, b], the function X(e)X + X([a,b]\e)V belongs to A too. Here X(g) is the characteristic function of the set g, Let us denote by S(Ln[a, b]) the collection of sets in Ln[a,b] that are bounded, closed and convex with respect to switching, and by Q [S(Ln[a,b])] its part consisting of convex sets.
Fix m points ii,..., tm on [a, b] satisfving a < t\ < ... < tm < b. Let us denote by Cn[a, b] space of functions x(t) on [a, b] with values in Rn that are continuous
on each interval [a,t1^ (t1,t2^ .. ^ (tm, b] and have limits at points t1,... ,tm from
the right. Equip this space with the norm
For t G (a, b], let Cn[a, t] be the space of restrictions to [a, t] of functions in Cn[a, b], the norm is given by the same formula with b replaced by t.
1This work is supported by Russian Foundation for Basic Research (RFBR): grants 11-01-00645 and 11-01-00626
© A. I. Bulgakov, E. V. Malyutina
Derzhavin Tambov State University, Tambov, Russia
xllc"[o,6] = suP |x(t)|.
t£[a,b]
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We consider the problem
x G $(x),
Ax(tk) G Ik(x(ifc)), k = 1,...,m, x(a) = x0,
(l)
(2)
(3)
where a map $ : Cn[a, b] ^ S(Ln[a, b]) is continuous in sense of Hausdorff and meets the conditions: for any bounded set U C Cn[a,b] its mage $(U) is bounded by a summable function, maps Ik : Rn ^ comp [Rn] are continuous in sense of Hausdorff, and Ax(tk) is the jump x(tk + 0) — x(tk), k = l, 2,..., m.
Definition 8 A solution of problem (1)-(3) is a funct ion x G Cn [a, b] for that there exists a function q G $(x) such that for any t G [a,b] we have
From now on we suppose that $ is a Volterra operator,
[a, b]
we also call local solutions). First for t E (a,b] we define a continuous operator VT : Cn[a, t] ^ Cn[a,b] by:
For the closed interval [a, t], a solution of problem (l)-(3) is defined in the same wav as for [a, b] by Definition 1, Now x G Cn[a, t], the function q belongs to a set of restrictions to [a, t] of functions in $(VT(x)), and the ^^m is taken over k for that tk G [a, t]. Denote by H(x0,t) the set of solutions of (l)-(3) on [a, t].
For the half-open interval [a, c), a solution of problem (l)-(3) is a function x : [a, c) ^ Rn such that ^^s restriction to any interval [a, t], a < t < c, is a solution of problem (l)-(3) on this interval [a, t],
(l) (3)
the distance (in the space of summable functions) from any summable function to values of the map $ at solutions, if for any v G Ln[a, b] and each e > 0 there exists a solution x G Cn[a, b] of problem (1)-(3) such that for measurable set E C [a,b] we have
where q G $(x) is given by (4), If (5) holds for e = 0, we say that the set of solutions of (l)-(3) realizes the distance from any summable function to values of the map $ at solutions.
Definition 10 If there exists r > 0 such that for any t G (a, b] and any y G H(x0, t) we have ||y||cn[or] ^ r, then the set of all local solutions of problem (l)-(3) is called
■t m
(4)
q — v||Ln(£) ^ №"(e) [v, $(x)] + e^(E),
(5)
a priori bounded.
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Let 2Cn[a,b] denote the set of nonempty convex bounded subsets of Cn[a, b]. Define a continuous operator A : Ln[a,b] ^ Cn[a,b] bv
(Az)(t) = x0 + / z(s) ds, t E [a,b],
J a
and then an operator A : Cn[a, b] ^ 2Cn[a,b] bv
m
(Ax)(t) = A$(x) + ^ X(ifc,b](t)Ax(tfc). (6)
fc=i
Theorem 7 There exists t E (a, b] sucft that there exists a solution of (1)-(3) on [a,T ]•
(1) (3)
exists a convex compact K C Cn[a,b] sucft that H(x0,b) C K, and A(K) C K (for A, see (6)).
(1) (3)
(1) (3)
functions) from any summable function to values of the 'map $ at solutions. If moreover, $ maps to Q [S(£n[a,b])], then the set of solutions to problem (1)-(3) realizes the distance from any summable function to values of $ at solutions.
Proof. Since the map $ is continuous, then, according to [2], for any e > 0 and any
v E Ln[a, b] there exists a continuous map g : Cn[a,b] ^ Ln[a,b] such that for any y E Cn[a, b] we have g(y) E $(y) and
|g(y) - v||l"(E) ^ PLn(E) [v, $(y)] + eME) (7)
for any measurable set E C [a, b]. Keeping in mind (6), define a continuous operator G : Cn[a, b] ^ Cn[a, b] bv
m
G(y) = Ag(y) + ^ X(ii,6](t) i=1
with Ay(tk) E (y(tk)), k = 1,... ,m. Then we have an inclusion:
G(y) E A(y). (8)
Bv Theorem 2, there exists a convex compact K C C n[a,b] such th at A(K) C K.
G
of (l)-(3). Replacing g(x) in (7) bv q we get (5), Thus the first statement of the theorem is proved.
Now let us prove the second statement. By the statement proved above, for each i = 1, 2,... there exists a solution x* E H(x0, b) that satisfies (5) with e = 1/i, where q = q* is given by (4) with x = x*. As for any t E (a, b], the set of solutions
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H(x0, t) is closed, the sequence x* tends to a function x in Cn[a, b] when i ^ to. The sequence q* is bounded by a summable function, so it is feeblv compact in Ln[a,b], Without loss of generality, we can consider that q* tends to q weakly in Ln[a, b]. As the map $ is supposed to map to Q [S(Ln[a, b])], it is closed in the coarse topology of Ln[a,b], so that q E $(x), It follows from continuity of $ that x is a solution of
(l)-(3).
Now we show that x satisfies (5) with e = 0. Indeed, since q* ^ q weakly in Ln[a, b], then for each i = 1, 2,... there exists a finite collection of k(i) numbers Ajj ^ 0 j = 1, 2,..., k(i), such that Ai,j = 1 and the sequence
k(*)
A 'y ] qj+*
j=i
converges to q in Ln[a,b], Hence we obtain
k(*)
||q — v||Ln[a,b] ^ ||q — ||Ln[a,b] + ^ A*,j 1^'+* — v^"[a,6].
j=i
Bv virtue of choice of the sequence q* (q* E $(x*)) we have
k(*) k(*) a
|q - v||l>,6] ^ llq - Alll^M + ^2 A*)j Pl>,6] [v, $(Xj+j)] + (b - a) ^2 —j- .
j=1 j=1j +i
Passing to the limit when i ^ to we get
||q - v ||Ln[a,b] = ^*[a,i] [v, $(x)] .
The inclusion $(x) C Q [S(Ln[a, b])] implies, that for any measurable set E C [a,b] the latter equality with [a, b] replaced by E is true. □
Theorem 10 If problem (1)-(3) is a priori bounded, then th ere exists 5 > 0 such, that (1)-(3) is a priori bounded on the ball B[x0,5].
References
1, N, V, Azbelev, V, P. Maksimov, L, F, Rakhmatullina, Elements of theory of functional-differential equations, M.: Vvsshava Shkola, 1987,
2, A, I, Bulgakov, Continuous branches of multi-valued maps and integral inclusions with non-eonvex images and their applications, Diff. Eq,, 1992, vol. 28, No, 3, 371-379.
3, S, T, Zavalishin, A, N, Sesekin, Impulsive processes. Models and applications, M,: Nauka, 1991,
4, A, N, Tikhonov, Functional equations of Volterra type and their applications to some problems in mathematical physics, Bull, Moscow Univ., Section A, 1938, vol. 68, No. 4, 1-25.
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