Curves of positive solutions of boundary value problems on time-scales |
| |
| Authors: | Fordyce A Davidson Bryan P Rynne |
| |
| Institution: | aDivision of Mathematics, Dundee University, Dundee DD1 4HN, Scotland, UK;bDepartment of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK |
| |
| Abstract: | Let TR be a time-scale, with a=infT, b=supT. We consider the nonlinear boundary value problem | where λR+:=0,∞), and satisfies the conditions We prove a strong maximum principle for the linear operator defined by the left-hand side of (1), and use this to show that for every solution (λ,u) of (1)–(2), u is positive on T a,b . In addition, we show that there exists λmax>0 (possibly λmax=∞), such that, if 0λ<λmax then (1)–(2) has a unique solution u(λ), while if λλmax then (1)–(2) has no solution. The value of λmax is characterised as the principal eigenvalue of an associated weighted eigenvalue problem (in this regard, we prove a general existence result for such eigenvalues for problems with general, nonnegative weights).
| |
| Keywords: | Time-scales Nonlinear boundary value problem Positive solutions Strong maximum principle Weighted eigenvalue problem |
| 本文献已被 ScienceDirect 等数据库收录！ |
|
