Some asymptotic properties of solutions of \((a(t)x')'-q(t)f(x)=r(t)\). (English) Zbl 0622.34060
Differential equations: qualitative theory, 2nd Colloq., Szeged/Hung. 1984, Colloq. Math. Soc. János Bolyai 47, 347-359 (1987).
[For the entire collection see Zbl 0607.00010.]
The authors study the nonlinear limit-point / limit-circle problem for forced second order nonlinear differential equations of the type \((E_ 1)\) \((a(t)x')'-q(t)f(x)=r(t)\) under the assumptions that \(a(t)>0\), \(q(t)>0\) and xf(x)\(\geq 0\) for all x. A solution x(t) of \((E_ 1)\) is said to be of nonlinear limit-circle type if \(\int^{\infty}_{t_ 0}x(s)f(x(s))ds<\infty\) and of nonlinear limit-point type otherwise, i.e., \(\int^{\infty}_{t_ 0}x(s)f(x(s))ds=\infty\).
The authors study the nonlinear limit-point / limit-circle problem for forced second order nonlinear differential equations of the type \((E_ 1)\) \((a(t)x')'-q(t)f(x)=r(t)\) under the assumptions that \(a(t)>0\), \(q(t)>0\) and xf(x)\(\geq 0\) for all x. A solution x(t) of \((E_ 1)\) is said to be of nonlinear limit-circle type if \(\int^{\infty}_{t_ 0}x(s)f(x(s))ds<\infty\) and of nonlinear limit-point type otherwise, i.e., \(\int^{\infty}_{t_ 0}x(s)f(x(s))ds=\infty\).
Reviewer: P.N.Bajaj
MSC:
| 34E05 | Asymptotic expansions of solutions to ordinary differential equations |
