Existence of Solutions to u″ + u + g(t, u, u′) = p(t), u(0) = u(π) = 0

Abstract

Existence and multiplicity results for the boundary value problem[formula]are presented. The proofs are based on the alternative method, a connectedness result, the contraction mapping principle, and a detailed analysis of the bifurcation equation utilizing, e.g., a generalization of the mean value theorem for integrals. We shall obtain results with g bounded or unbounded, having finite limits at ±∞ or without limits, thus extending some recent results in the literature. The proofs offer a constructive way to find the bounds for p̄ and to find numerically the number of solutions and the approximative solutions.