ON EXACT LIMITS FOR THE NEGATIVITY OF THE GREEN FUNCTION OF A TWO-POINT BOUNDARY VALUE PROBLEM

The question of the positivity of the Green function G(x, s, lambda) of the two-point boundary value problem (-1)(k)u((n)) - lambda integral(1)(0) u(s)d(s)r(x, s) - f(x), x is an element of [0, l], B(k)u = 0, ((*)), where, B(m)u : = (u(0), u'(0), . . . , u((n - m - 1))(0), u(l),-u '(l), u ''(l), . . . , (-1)(m - 1)u((m - 1))(l)), with non-decreasing r(x, <middle dot> ) for almost all x is an element of [0, l] is reduced to estimating the eigenvalues of auxiliary boundary value problems. The Green function G(x, s, lambda) is positive if and only if -min {lambda(k - 1), lambda(k+1)} <= lambda < lambda(k) (there are also small clarifying details that are not needed for the ordinary differential equation (-1)(k+1) u ((n) )- lambda p(x)u = f(x)). Here, lambda(m) is the smallest positive eigenvalue of the boundary value problem (-1)(m)u((n) )- lambda integral(1 )(0)u(s)d(s)r(x, s) = 0, B(m)u = 0 (m is an element of {0, . . . , n}).