With a view towards defining a pseudo Fuik spectrum (PFS) we study, using a time-mapping approach, the eigenvalue-like problem {(φ(u′))′ + Aφ(u⁺) − Bφ(u⁻) = 0, u(a) = 0 = u(b), where A > 0, B > 0.

We show here that this PFS, which we denote by , consists of a set of curves in (⁺)² resembling the classical Fuik spectrum, i.e., when φ(s) = s.

Our main existence result, which deals with nonresonance with respect to the PFS, can be roughly stated as follows: if for s > 0 sufficiently large and for almost every t [a, b], the pair (f(t, − s)/φ(− s), f(t, s)/φ(s)) lies in a compact rectangle contained in an open component of (⁺)²\ which intersects the diagonal, then problem (P) has at least a solution.