Curves of positive solutions of boundary value problems on time-scales

[摘要] Let T subset of R be a time-scale, with alpha = inf T, b = sup T. We consider the nonlinear boundary value problem -[p(t)u(Delta)(t)](Delta) + q(t)u(sigma)(t) = gimelf(t, u(sigma) (t)), onT, u(a) = u(b) = 0, where gimel is an element ofR(+) := [0, infinity), and f : T x R --> R satisfies the conditions f(t,xi)>0, (t,xi) is an element of T x R, f (t,xi) > f(xi)(t,xi)xi, (t,xi) is an element of T x R+. 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 (gimel,u) of (1)-(2), u is positive on T \ {a, b}. In addition, we show that there exists gimel(max) > 0 (possibly gimelmax = infinity), such that, if 0 less than or equal to gimel < gimel(max) then (1)-(2) has a unique solution u (gimel), while if gimel greater than or equal to gimel(max) then (1)-(2) has no solution. The value gimel(max) 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). (C) 2004 Elsevier Inc. All rights reserved.

[发布日期] 2004-12-15

[关键词] time-scales;nonlinear boundary value problem;positive solutions;strong maximum principle;weighted eigenvalue problem