ON UNIQUE SOLVABILITY OF THE INITIAL VALUE PROBLEM FOR ...

48 N. Dilna
(5) L 1 ([a, b], R n ) is the Banach space of Lebesgue integrable vector functions
u : [a, b] → R n with the standard norm
L 1 ([a, b], R n ) ∋ u ↦−→
∫ b
a
‖u(s)‖ ds.
The notion of a solution of the problem under consideration is defined in
the standard way (see, e.g., [1]).
Definition 1. We say that a vector function u = (u k ) n k=1
: [a, b] → Rn
is a solution of the problem (1), (2) if it satisfies the system (1) almost everywhere
on the interval [a, b] and possesses the property (2) at the point a.
We will use in the sequel the natural notion of positivity of a linear
operator.
Definition 2. A linear operator l =(l k ) n k=1 : D([a, b], Rn )→L 1 ([a, b], R n )
is said to be positive if
vrai min (l k u)(t) ≥ 0, k = 1, 2, . . . , n,
t∈[a,b]
for any u = (u k ) n k=1 from D([a, b], Rn ) with non-negative components.
Consider the linear semihomogeneous problem for the functional differential
equation
u ′ k = (l k u)(t) + q k (t), t ∈ [a, b], k = 1, 2 . . . , n, (3)
with the initial value condition
u k (a) = 0, k = 1, 2, . . . , n, (4)
where l k : D([a, b], R n ) → L 1 ([a, b], R), k = 1, 2, . . . , n, are linear operators,
{q k | k = 1, 2, . . . , n} ⊂ L 1 ([a, b], R). The following definition is motivated
by a notion used, in particular, in [5], [11].
Definition 3. A linear operator l =(l k ) n k=1 : D([a, b], Rn )→L 1 ([a, b], R n )
is said to belong to the set S a ([a, b], R n ) if the semihomogeneous initial
value problem (3), (4) has a unique solution u = (u k ) n k=1 for any {q k | k =
1, 2, . . . , n} ⊂ L 1 ([a, b], R) and, moreover, the solution of (3), (4) possesses
the property
min u k(t) ≥ 0, k = 1, 2, . . . , n, (5)
t∈[a,b]
whenever the functions q k , k = 1, 2, . . . , n, appearing in (3) are non-negative
almost everywhere on [a, b].
3. Main Results
The following statements are true.