Resonant Sturm–Liouville Boundary Value Problems for Differential Systems in the Plane

Abstract

We study the Sturm-Liouville boundary value problem associated with the planar differential system Jz = ∇V (z) + R(t, z), where V (z) is positive and positively 2-homogeneous and R(t, z) is bounded. Assuming Landesman-Lazer type conditions, we obtain the existence of a solution in the resonant case. The proofs are performed via a shooting argument. Some applications to boundary value problems associated with scalar second order asymmetric equations are discussed.MSC 2010 Classification 34B15.

Double resonance in Sturm-Liouville planar boundary value problems

“…In this section we recall some notations and contents from [1,14]. Let us consider the planar system…”

“…For this reason, boundary value problems related to (1.4) present a particular interest in literature, see e.g. [1,11,14,25] and the references therein.…”

“…Recalling that the unperturbed system (1.4) has an isochronous center of minimal period τ V , and borrowing the definition from [1], we say that the unperturbed problem…”

“…Conversely, if the unperturbed problem (1.7) is resonant, then the existence of a solution to problems as in (1.6) is ensured assuming an additional condition: in [1] the introduction of a Landesman-Lazer type of assumption provides an existence result. In these notes we continue the study performed in [1,14] by Boscaggin, Fonda and Garrione. In particular, we are going to consider the wider class of problems…”

“…The paper is organized as follows. In Section 2 we present some preliminary results: in Section 2.1 some properties of the autonomous system (1.4) are listed borrowing some notations from [1,14], then in Section 2.2 we add a first perturbation presenting some properties of solutions of systems (1.8) in the the semi-autonomous case r ≡ 0. We present the main Theorems 3.2 (nonresonance), 3.7 (simple resonance) and 3.8 (double resonance) in the successive Sections 3.1, 3.2 and 3.3.…”