TY - JOUR

T1 - Davies-Harrell representations, Otelbaev's inequalities and properties of solutions of Riccati equations

AU - Chernyavskaya, N. A.

AU - Shuster, L. A.

PY - 2007/10/15

Y1 - 2007/10/15

N2 - We consider an equation(1)y″ (x) = q (x) y (x), x ∈ R, under the following assumptions on q:(2)0 ≤ q ∈ L1loc (R), underover(∫, - ∞, x) q (t) d t > 0, underover(∫, x, ∞) q (t) d t > 0 for all x ∈ R . Let v (respectively u) be a positive non-decreasing (respectively non-increasing) solution of (1) such thatv′ (x) u (x) - u′ (x) v (x) = 1, x ∈ R . These properties determine u and v up to mutually inverse positive constant factors, and the function ρ (x) = u (x) v (x), x ∈ R, is uniquely determined by q. In the present paper, we obtain an asymptotic formula for computing ρ (x) as | x | → ∞. As an application, under conditions (2), we study the behavior at infinity of solution of the Riccati equationz′ (x) + z (x)2 = q (x), x ∈ R .

AB - We consider an equation(1)y″ (x) = q (x) y (x), x ∈ R, under the following assumptions on q:(2)0 ≤ q ∈ L1loc (R), underover(∫, - ∞, x) q (t) d t > 0, underover(∫, x, ∞) q (t) d t > 0 for all x ∈ R . Let v (respectively u) be a positive non-decreasing (respectively non-increasing) solution of (1) such thatv′ (x) u (x) - u′ (x) v (x) = 1, x ∈ R . These properties determine u and v up to mutually inverse positive constant factors, and the function ρ (x) = u (x) v (x), x ∈ R, is uniquely determined by q. In the present paper, we obtain an asymptotic formula for computing ρ (x) as | x | → ∞. As an application, under conditions (2), we study the behavior at infinity of solution of the Riccati equationz′ (x) + z (x)2 = q (x), x ∈ R .

KW - Asymptotics on the diagonal

KW - Green function

KW - Riccati equations

KW - Sturm-Liouville operator

UR - http://www.scopus.com/inward/record.url?scp=34250624555&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2006.10.092

DO - 10.1016/j.jmaa.2006.10.092

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AN - SCOPUS:34250624555

SN - 0022-247X

VL - 334

SP - 998

EP - 1021

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -