Double resonance for one-sided superlinear or singular nonlinearities

Abstract

We deal with the problem of existence of periodic solutions for the scalar differential equation x + f (t, x) = 0 when the asymmetric nonlinearity satisfies a one-sided superlinear growth at infinity. The nonlinearity is asked to be next to resonance and a Landesman-Lazer type of condition will be introduced in order to obtain a positive answer. Moreover we provide also the corresponding result for equations with a singularity and asymptotically linear growth at infinity, showing a further application to radially symmetric systems.

A systematic approach to nonresonance conditions for periodically forced planar Hamiltonian systems

“…Indeed, if k ≠ m 2 ∕4 for any integer m, Theorem 3.1 applies. Remark 6.5 Similar situations have been considered in [4,5,15,31] for more general nonlinearities. Since problems "near resonance" are concerned, some restrictions must be imposed on the nonlinearity (in our approach, condition (6.10)).…”

Double resonance in Sturm-Liouville planar boundary value problems

“…In presence of resonance we need to add additional assumptions, e.g. of Landesman-Lazer type, as suggested in [13] (see also [6,7,8,23,24] for related results).…”

On a singular periodic Ambrosetti–Prodi problem

“…Equations of the type (E s ) have already been considered in the literature, taking into account several situations. E.g., systems with an attractive singularity of Keplerian type have been studied in [9,12,14]; the case of repulsive singularity has been treated in [10,11,13,26]; bouncing solutions were found in [27]. See also the interesting monograph [29].…”