Resonance and nonresonance in terms of average values (Q1817259)

The problem \(u''(t)+k^2u(t)+g(u(x))+h(x)=0\), \(x\in(0,\pi)\), \(u(0)=x(\pi)=0\) is considered where \(k\) is a positive integer, \(g:\mathbb{R}\to\mathbb{R}\) is a continuous function with at most linear growth, and \(h\in L^2[0,\pi]\). Existence results are proved. The proofs rely on the Leray-Schauder degree argument and apply proved convergence results. Three illustrative examples are considered.