Nonresonance Conditions on the Potential for a Second-Order Periodic Boundary Value Problem

Abstract

We consider the periodic problem \[ \begin {array}{*{20}{c}} { - u'' = f(u) + h(t),} \\ {u(0) = u(2\pi ),\qquad u’(0) = u’(2\pi ),} \\ \end {array} \] and prove its solvability for any given $h$, under new assumptions on the asymptotic behaviour of the potential of the nonlinearity $f$, with respect to two consecutive eigenvalues of the associated linear problem.