TY - JOUR

T1 - Necessary and sufficient conditions for the solvability of a problem of hartman and wintner

AU - Chernyavskaya, N.

AU - Shuster, L.

PY - 1997/1/1

Y1 - 1997/1/1

N2 - The equation (1) (r(x)y′(x))′ = q(x)y(x) is regarded as a perturbation of (2) (r(x)z′(x))′ = qi(x)z(x), where the latter is nonoscillatory at infinity. The functions r(x), q1(x) are assumed to be continuous real-valued, r(x) > 0, whereas q(x) is continuous complex-valued. A problem of Hartinan and Wintner regarding the asymptotic integration of (1) for large x by means of solutions of (2) is studied. A new statement of this problem is proposed, which is equivalent to the original one if q(x) is real-valued. In the general case of (x) being complex-valued a criterion for the solvability of the HartmanWintner problem in the new formulation is obtained. The result improves upon the related theorems of Hartman and Wintner, Trench, Śimśa and some results of Chen.

AB - The equation (1) (r(x)y′(x))′ = q(x)y(x) is regarded as a perturbation of (2) (r(x)z′(x))′ = qi(x)z(x), where the latter is nonoscillatory at infinity. The functions r(x), q1(x) are assumed to be continuous real-valued, r(x) > 0, whereas q(x) is continuous complex-valued. A problem of Hartinan and Wintner regarding the asymptotic integration of (1) for large x by means of solutions of (2) is studied. A new statement of this problem is proposed, which is equivalent to the original one if q(x) is real-valued. In the general case of (x) being complex-valued a criterion for the solvability of the HartmanWintner problem in the new formulation is obtained. The result improves upon the related theorems of Hartman and Wintner, Trench, Śimśa and some results of Chen.

UR - http://www.scopus.com/inward/record.url?scp=21944448551&partnerID=8YFLogxK

U2 - 10.1090/s0002-9939-97-04186-5

DO - 10.1090/s0002-9939-97-04186-5

M3 - Article

AN - SCOPUS:21944448551

VL - 125

SP - 3213

EP - 3228

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 11

ER -