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UDC 517.911, 517.968
Functional-differential inclusions with impulses without switching convexity assumption 1
We consider functional-differential inclusions with multi-valued impulses. We do not suppose that the right hand side is convex with respect to switching. We introduce the notion of a generalized solution and prove that for a Cauchy problem with a Volterra operator (in sense of A. N. Tikhonov) a local generalized solution does exist and can be continued to a «m^imal> interval. Finally we give a density principle for generalized solutions
Keywords: functional-differential inclusions with impulses, generalized solutions, convexity with respect to switching, Volterra operators, a priori boundness
Functional-differential inclusions with impulses, see, for example, [1], [2], [3], have numerous and useful applications in mathematics and techniques. In the present work we consider inclusions whose right hand side is not supposed to be convex with respect to switching. We introduce notions of generalized solutions and generalized quasi-solutions of the Cauchy problem when the right hand side is a Volterra (in the sense of A. N. Tikhonov [4]) operator. We study the solvability and the extendabilitv of generalized solutions and study connections between different families of solutions. Finally we give a density principle for generalized 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 distanee pLn(e) ■ Denote bv Q(Ln[a, b]) the family of nonempty closed sets of Ln[a,b] bounded bv summable functions.
Fix m points t\,..., tm on [a, b] satisfvi ng a < t\ < ... <tm < b. Let us denote by Cn[a, b] the space of functions x(t) on [a, b] with values in Rn that are continuous
1This work is supported by Russian Foundation for Basic Research (RFBR): grants 11-01-00645 and 11-01-00626
© A. I. Bulgakov, O. V. Filippova
Derzhavin Tambov State University, Tambov, Russia
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on each interval [a,ti], (ti,t2], ..., (tm, b] and have limits at points ti,... ,tm from the right. Equip this space with the norm
INIo-kb] = suP |x(t)|.
te[a,b]
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.
Definition 1 A set A C Ln[a,b] is convex with respect to switching, if for any functions x, y G A and any measurable set e C [a, b], the function X(e) x + X([«,b]\e) y belongs to A too. Here X(g) is the characteristic function of the set g, Let us denote by n(Ln[a, b]) the collection of sets in Ln[a,b] that are bounded, closed and convex with respect to switching, and by Q [n(Ln[a,b])] its part consisting of convex sets.
For A C Ln[a,b], let swA (the “convex switching” hull of A) denote the set of functions X(e) x + X([«,b]\e) y with x, y G ^^d swA the closure of swA in Ln[a, b].
Consider the problem
x G $(x), (1)
Ax(tfc) = Ik(x(tk)), k = 1,..., m, (2)
x(a) = x0, (3)
where a map $ : Cn[a,b] ^ Q(Ln[a,b]) meets the condition: for any bounded set U C Cn[a,b] the image $(U) is bounded by a summable function, the maps Ik : Rn ^ Rn are continuous, Ax(tk) = x(tk + 0) — x(tk), k = 1, 2, ...,m.
Let us define the map $ : Cn[a,b] ^ n(Ln[a,b]) («convex with respect to switching») as
$ (x) = sw$(x).
Definition 2 A generalized solution of problem (l)-(3) is a function x G Cn[a,b] for that there exists q G $ (x) such that
t m
q(s) ds + E X(tfc,6](t)Ax(tk), t G [a,b]. (4)
k=i
Let t g (a,b]. For a set A of functions z on [a,b], we denote bv A|T the set of restrictions z|T functions z to [a, t].
Definition 3 A map $ : Cn[a,b] ^ Q(Ln[a,b]) is a Volterra operator in sense of Tikhonov [4], if the condition x|T = y|T implies $(x)|T = $(y)|T.
x(t) = x0
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[a, b]
First for t G (a, b] we define a continuous operator VT : Cn[a,r] ^ Cn[a,b] by:
For the dosed interval [a,T], a generalized solution of problem (l)-(3) is defined in the same wav as for [a, b] by Definition 2, Now x G Cn[a, t], the function q belongs to a set $(VT(x))|T and the sum is taken over k for that tk G [a, t].
Denote by H(x0,t) the set of all generalized solutions of (l)-(3) on [a, t].
For the half-open interval [a,c), a generalized solution of problem (l)-(3) is a function x : [a, c) ^ Rn such that its restriction to any interval [a, t], a < t < c, is a generalized solution of problem (l)-(3) on this interval [a,T],
Definition 4 A generalized solution x : [a, c) ^ Rn of problem (l)-(3) is called
y
segment [a, t], t G [c, b], such that x(t) = y(t) for t G [a, c).
Theorem 1 There exists t G (a, b] such that a solution of problem (1)-(3) exists on a segment [a, t].
Theorem 2 The solution x : [a,c) ^ Rn of (1)-(3) is continuable to a segment [a, t], t G [c, b], if and only if limt^c-0 |x(t)| < ro.
Theorem 3 Any generalized solution y of problem (1)-(3) on a segment [a, t] can be continued to a noncontinuable solution x on [a, c), c G (t, b], or on [a,b].
Definition 5 If there exists r > 0 such that for any t G (a, b] and anv y G H(x0, t) we have ||y||g„[aT] ^ r, then the set of all local generalized solutions of problem (l)-(3) is called a priori bounded.
(1) (3)
priori bounded. Then H(x0,T) = 0 for any t G (a, b].
Definition 6 Let a function x G Cn[a,b] have representation (4) with x and q replaced by x and q respectively. Assume that this function x is the limit in Cn[a, b] of a sequence of functions x*, i = 1, 2,..., having representation (4) with x and q replaced by x* and q* respectively such that q* G $ (x^d Ax*(tk) k = 1,..., m,
xx
Let H(x0) be the set of all generalized quasi-solutions of problem (l)-(3).
(1) (3)
Define a map $co : Cn[a,b] ^ Q [n(Ln[a, b])^v $co(x) = co$(x)), the closure of the convex hull of $ (x).
Consider the problem
x G $co, Ax(tk) = Ik(x(tk)), x(a) = xo.
Let Ho(x0,T) denote the set of all solutions of (5) on [a,T], t G (a,b].
(5)
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Theorem 5 H(x0) = Hco(x0,b).
Definition 7 We say that impulses Ik : Rn ^ Rn k = 1,..., m, and a map $ : Cn[a,b] ^ Q(Ln[a, b]) possess the property A, if:
1) for each k = 1,... ,m there exists a continuous non-decreasing function Ik : R+ ^ R+ such that Ik(0) = 0 and for any x, y G Rn we have
|Ik(x) — Ik(y)| ^ $k(|x — y|);
2) there exists an isotonic continuous Volterra operator r : C+[a,b] ^ L + [a,b], satisfying the following conditions:
(i) r(0) = 0.
(ii) for all x, y G Cn[a, b] and any measurable set E C [a, b] we have
hL"(E)[ $(x); $(y)] ^ |r(|x — y|)^ 1(E)
(h denotes the Hausdorff distance),
(iii) the set of all solutions of
y = r(y^ Ay(tk) = $k(y(tk)), y(a) = 0
is a priori bounded,
(1) — (3)
priori bounded and the condition A is satisfied. Then H(x0,b) = 0 and the closure of H(x0,b) m Cn[a, b] is precisely Hco(x0,b).
References
1, A, I, Bulgakov, O, P. Belyaeva, A, N, Maehina, Functional-differential inclusions with map, non-necessarv eonvexed with respect to switching, Vestnik Udmurt, Univ., Math,, meeh,, 2005, No, 1, 3-20,
2, A, I, Bulgakov, A, I, Korobko, O, V, Filippova, To theory of functional-differential inclusions with impulses, Vestnik Udmurt, Univ., Math,, meeh,, 2008, No. 2, 24-27.
3. A. Bressan, G. Colombo. Extensions and selections of maps with decomposable values, Studia. math., 1988, vol. 90, No. 1, 69-86.
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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