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

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52 N. Dilna 6.1. Existence of a solution of (18), (19). To show the existence of a solution of (18), (19), it is sufficient to impose some conditions on µ 1 and ω 1 only. More precisely, the following statement is true. Proposition 1. The problem (18), (19) is solvable for any c ∈ R if at least one of the following conditions is satisfied: ∫ b a [µ 1 (s)] − ds < 1, ω 1 (t) ≤ t for a.e. t ∈ [a, b]; ∫ b √ [µ 1 (s)] + ds < 1 + 2√1 − a ∫ b a (21a) [µ 1 (s)] − ds . (21b) Here, by definition, [u(t)] + :=max {u(t), 0} and [u(t)] − :=max {−u(t), 0} for t ∈ [a, b] and u : [a, b] → R. To prove Proposition 1, the following lemmata can be used. Let f : D([a, b], R) → L 1 ([a, b], R) be an operator and h : D([a, b], R) → R be a continuous linear functional. Lemma 1 ([6, Lemma 3.1]). Let there exist a linear operator l :C([a, b], R) → L 1 ([a, b], R) such that the problem u ′ (t) + (lu)(t) = 0, u(a) = 0 (22) has only the trivial solution and for all v ∈ C([a, b], R) |(lv)(t)| ≤ η(t) max |v(s)|, t ∈ [a, b], (23) s∈[a,b] where η ∈ L 1 ([a, b], R) does not depend on v. Moreover, assume that there exists a positive number ρ such that for every δ ∈ (0, 1) and for an arbitrary solution u ∈ D([a, b], R) of the problem the estimate u ′ (t) + (lu)(t) = δ[(fu)(t) + (lu)(t)], u(a) = δh(u) (24) holds. Then the problem has at least one solution. max |u(t)| ≤ ρ (25) t∈[a,b] u ′ (t) = (fu)(t), t ∈ [a, b], u(a) = h(u) Lemma 1 is, in fact, [6, Theorem 1] in the formulation of [5]. Lemma 2 (Lemma 3.4 from [5]). Let there exist a linear operator l such that the condition (23) is satisfied and the homogeneous problem (22) has only the trivial solution. Then there exists a positive number r such that for any q ∈ L([a, b], R) and real c every solution u of the problem u ′ (t) = (lu)(t) + q(t), u(a) = c (26)
Unique Solvability of the Non-Linear Cauchy Problem 53 admits the estimate ( ∫ b max |u(t)| ≤ r |c| + t∈[a,b] Lemma 3. Problem (18), (19) is solvable if the problem has no non-trivial solution. a ) |q(s)| ds . (27) u ′ (t) = −µ 1 (t)u(ω 1 (t)), t ∈ [a, b], (28) u(a) = 0 (29) Proof. Indeed, assume that the problem (28), (29) has no non-trivial solution. Let u be a solution of the problem (26). Then where u ′ (t) = (lu)(t) + Q(t), u(a) = c, (30) Q(t) := µ 0 (t) √ 1 + λ(t) sin u(ω 0 (t)), t ∈ [a, b]. (31) Using the estimate ∣ µ0 (t) √ 1 + λ(t) sin u(ω 0 (t)) ∣ ∣ ≤ |µ0 (t)| √ 1 + λ(t) valid for a.e. t ∈ [a, b] and taking Lemma 2 into account, we conclude that an arbitrary solution u of the problem (26) satisfies the estimate and Let us put ( ∫ b max |u(t)| ≤ r |c| + t∈[a,b] ( ∫ b ρ := r |c| + a a |µ 0 (s)| √ ) 1 + λ(s) ds . |µ 0 (s)| √ ) 1 + λ(s) ds (32) (lu)(t) := −µ 1 (t)u(ω 1 (t)), t ∈ [a, b], (33) and return to the problem (24). In this notation, all solutions of the problem (24) satisfy the estimate (25). So, using the assumption that the problem (28), (29) has only the trivial solution and applying Lemmas 1 and 2, we prove Lemma 3. □ Lemma 4. Let at least one of the conditions (21a) and (21b) be satisfied. Then the homogeneous Cauchy problem (28), (29) has no non-trivial solution. Proof. This statement follows from [11, Corollary 3.3] and [2, Theorem 1.3] with the operator l defined by the formula (33). □
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ON UNIQUE SOLVABILITY OF THE INITIAL VALUE PROBLEM FOR ...

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