Asymptotic Formula for Oscillatory Solutions of Some Singular Nonlinear Differential Equation

and Applied Analysis 3 Then for each B ∈ B, L problem 1.1 , 2.1 has a unique solution u. If B / 0, then the solution u is damped and oscillatory with decreasing amplitudes and lim t→∞ u t 0. 2.4 Proof. The assertion follows from Theorems 2.3, 2.10 and 3.1 in 19 . Example 2.3. The functions i p t t, p t t ln t 1 , k, ∈ 0,∞ , ii p t t α sin t, α ∈ −1, 1 , iii p t t/ 1 t , k, ∈ 0,∞ , < k satisfy 1.5 , 1.6 , and 2.2 . The functions i p t ln t 1 , p t arctan t, p t t/ 1 t , k ∈ 0,∞ satisfy 1.5 , 1.6 , but not 2.2 the third condition . The function i p t t α sin t, α ∈ −1, 1 , k ∈ 1,∞ , satisfy 1.5 , 1.6 but not 2.2 the second and third conditions . Example 2.4. Let k ∈ 0,∞ . i The function f x ⎧ ⎨ ⎩ −kx, for x ≤ 0, x x − 1 , for x > 0, 2.5 satisfies 1.2 with L 1, 1.3 , 1.4 with B − 3k −1/2 and 2.3 . ii The function f x ⎧ ⎨ ⎩ kx2, for x ≤ 0, x x − 1 , for x > 0, 2.6 satisfies 1.2 with L 1, 1.3 , 1.4 with B − 2k −1/3 but not 2.3 the second condition . In the next section, the generalized Matell’s theorem which can be found as Theorem 6.5 in the monograph by Kiguradze will be useful. For our purpose, we provide its following special case. Consider an interval J ⊂ R. We write AC J for the set of functions absolutely continuous on J and ACloc J for the set of functions belonging to AC I for each compact 4 Abstract and Applied Analysis interval I ⊂ J . Choose t0 > 0 and a function matrix A t ai,j t i,j≤2 which is defined on t0,∞ . Denote by λ t and μ t eigenvalues of A t , t ∈ t0,∞ . Further, suppose λ lim t→∞ λ t , μ lim t→∞ μ t 2.7 be different eigenvalues of the matrix A limt→∞A t , and let l and m be eigenvectors of A corresponding to λ and μ, respectively. Theorem 2.5 see 20 . Assume that ai,j ∈ ACloc t0,∞ , ∣ ∣ ∣ ∣ ∣ ∫∞ t0 ai,j t dt ∣ ∣ ∣ ∣ ∣ < ∞, i, j 1, 2, 2.8 and that there exists c0 > 0 such that ∫ t s Re ( λ τ − μ τ dτ ≤ c0, t0 ≤ s < t, 2.9 or ∫∞ t0 Re ( λ τ − μ τ dτ ∞, ∫ t s Re ( λ τ − μ τ dτ ≥ −c0, t0 ≤ s < t. 2.10 Then the differential system x′ t A t x t 2.11 has a fundamental system of solutions x t , y t such that lim t→∞ x t e− ∫ t t0 λ τ dτ l, lim t→∞ y t e− ∫ t t0 μ τ dτ m. 2.12 3. Asymptotic Formula In order to derive an asymptotic formula for a damped oscillatory solution u of problem 1.1 , 2.1 , we need a little stronger assumption than 2.3 . In particular, the function f x /x should have a negative derivative at x 0. Theorem 3.1. Assume that 1.2 – 1.6 , and 2.2 hold. Assume, moreover, that there exist η > 0 and c > 0 such that f x x ∈ AC−η, η, lim x→ 0 f x x −c. 3.1 Abstract and Applied Analysis 5 Then for each B ∈ B, L problem 1.1 , 2.1 has a unique solution u. If B / 0, then the solution u is damped and oscillatory with decreasing amplitudes such thatand Applied Analysis 5 Then for each B ∈ B, L problem 1.1 , 2.1 has a unique solution u. If B / 0, then the solution u is damped and oscillatory with decreasing amplitudes such that lim sup t→∞ √ p t |u t | < ∞. 3.2 Proof. We have the following steps: Step 1 construction of an auxiliary linear differential system . Choose B ∈ B, L , B / 0. By Theorem 2.2, problem 1.1 , 2.1 has a unique oscillatory solution u with decreasing amplitudes and satisfying 2.4 . Having this solution u, define a linear differential equation v′′ p′ t p t v′ f u t u t v, 3.3 and the corresponding linear differential system x′ 1 x2, x ′ 2 f u t u t x1 − p′ t p t x2. 3.4