Solvability for two-point boundary value problems

Suppose that h is an element of L-1(0, pi), g is an element of C(R,R), and lim(\zeta\ --> infinity)(g(t)/t) = 0. With the Saddle Point Theorem, the solvability is proved for the two-point boundary value problem -u " = u + g(u) - h(x), u(0) = u(pi) = 0, under the condition that <(F(-infinity))over bar> integral(0)(pi) sin xdx < integral(0)(pi) h(x)sin xdx < <(F(+infinity)under bar> integral(0)(pi) sin xdx, where <(F-infinity)over bar> = lim sup(t --> -x)F(t), <(F(+infinity)under bar> = lim inf(t --> + infinity) F(t), and [GRAPHICS] (C) 1997 Academic Press.