CONDITIONS FOR SOLVABILITY OF THE HARTMAN–WINTNER PROBLEM IN TERMS OF COEFFICIENTS

[摘要] The Equation (1) $(r(x)y')'=q(x)y(x)$ is regarded as a perturbation of (2) $(r(x)z'(x))'=q_1(x)z(x)$. The functions $r(x)$, $q_1(x)$ are assumed to be continuous real valued, $r(x)>0$, $q_1(x)ge0$, whereas $q(x)$ is continuous complex valued. A problem of Hartman and Wintner regarding the asymptotic integration of (1) for large $x$ by means of solutions of (2) is studied. Sufficiency conditions for solvability of this problem expressed by means of coefficients $r(x)$, $q(x)$, $q_1(x)$ of Equations (1) and (2) are obtained.AMS 2000 Mathematics subject classification: Primary 34E20

[学科分类] 数学（综合）

[关键词] asymptotics of solutions;differential equations of second order;Hartman–Wintner problem