Solvability of a nonlinear two-point boundary value problem at resonance II

In this article we apply the Leray-Schauder continuation method to obtain existence theorems of solutions for (i) u" + u + g(x, u) = h in (0, pi), u(0) = u(pi) = 0 in which the non-linearity g grow superlinearly in u in one of directions u --> infinity and u --> -infinity, and may grow sublinearly in the other, and for (ii) -u" - u + g(x, u) = h in (0, pi), u(0) = u(71) = 0 in which the nonlinearity g has no growth restriction in u as \u\ --> infinity. The L-1(0,pi) function h may satisfy integral(0)(pi) g(beta)(-)(x)sin x dx < integral(0)(pi) h(x) sin x dx = O < integral(0)(pi) g(alpha)(+)(x) sin x dx, where alpha, beta greater than or equal to 0, g(alpha)(+)(x) = lim (u-->infinity) inf g(x,u)\u\(alpha), and g(beta)(-) (x) = lim(u-->-infinity) sup g (x,u) \u\(beta). (C) 2003 Elsevier Science Ltd. All rights reserved.