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

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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.
Unique Solvability of the Non-Linear Cauchy Problem 49 Theorem 1. Let there exist linear operators φ 0 , φ 1 : D([a, b], R n ) → L 1 ([a, b], R n ), φ i = (φ ik ) n k=1 , i = 0, 1, satisfying the inclusions φ 0 ∈ S a ([a, b], R n ), φ 0 + φ 1 ∈ S a ([a, b], R n ) (6) and such that the inequalities ∣ (fk u)(t) − (f k v)(t) − φ 0k (u − v)(t) ∣ ∣ ≤ φ1k (u − v)(t), k = 1, 2, . . . , n, (7) are true for arbitrary absolutely continuous functions u = (u k ) n k=1 and v = (v k ) n k=1 from [a, b] to Rn possessing the properties u k (a) = v k (a), u k (t) ≥ v k (t) for t ∈ [a, b], k = 1, 2, . . . , n. (8) Then the Cauchy problem (1), (2) is uniquely solvable for arbitrary real c k , k = 1, 2, . . . , n. Theorem 2. Assume that there exist positive linear operators g i = (g ik ) n i=1 : D([a, b], Rn ) → L 1 ([a, b], R n ), i = 0, 1, such that the inequalities ∣ (fk u)(t) − (f k v)(t) + g 1k (u − v)(t) ∣ ∣ ≤ g0k (u − v)(t), k = 1, 2, . . . , n, (9) hold on [a, b] for any vector functions u = (u k ) n k=1 and v = (v k) n k=1 from D([a, b], R n ) with the properties (8). Moreover, let the inclusions g 0 + (1 − 2θ)g 1 ∈ S a ([a, b], R n ), −θg 1 ∈ S a ([a, b], R n ) (10) be true. Then the initial value problem (1), (2) has a unique solution for all c k , k = 1, 2, . . . , n. Corollary 1. Let there exist positive linear operators g i = (g ik ) n k=1 : D([a, b], R n ) → L 1 ([a, b], R n ), i = 0, 1, satisfying the condition (9) for arbitrary absolutely continuous functions u = (u k ) n k=1 and v = (v k) n k=1 with the properties (8) and, moreover, such that the inclusions g 0 ∈ S a ([a, b], R n ), − 1 2 g 1 ∈ S a ([a, b], R n ) (11) hold. Then the initial value problem (1), (2) has a unique solution for any c k , k = 1, 2, . . . , n. Corollary 2. Let there exist positive linear operators g i = (g ik ) n k=1 : D([a, b], R n ) → L 1 ([a, b], R n ), i = 0, 1, satisfying the condition (9) for arbitrary absolutely continuous functions u = (u k ) n k=1 and v = (v k) n k=1 with the properties (8) and, moreover, such that the inclusions g 0 + 1 2 g 1 ∈ S a ([a, b], R n ), − 1 4 g 1 ∈ S a ([a, b], R n ) (12) are true. Then the problem (1), (2) has a unique solution for any c k , k = 1, 2, . . . , n.
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