Continuation Theorems for Ambrosetti-Prodi Type Periodic Problems

Abstract

We study the existence of periodic solutions u(·) for a class of nonlinear ordinary differential equations depending on a real parameter s and obtain the existence of closed connected branches of solution pairs (u, s) to various classes of problems, including some cases, like the superlinear one, where there is a lack of a priori bounds. The results are obtained as a consequence of a new continuation theorem for the coincidence equation Lu = N (u, s) in normed spaces. Among the applications, we discuss also an example of existence of global branches of periodic solutions for the Ambrosetti-Prodi type problem u + g(u) = s + p(t), with g satisfying some asymmetric conditions.

Existence and multiplicity results for some nonlinear problems with singular ϕ-Laplacian

“…For each λ ∈ [0, 1], the nonlinear operator M # on C 1 # associated to (15) by Proposition 2 is the operator…”

Multiplicity Results for Asymptotically Linear Equations, Using the Rotation Number Approach

“…The existence of such a connected branch of solutions can be alternatively guaranteed by theorems, which rely on the Leray-Schauder continuation principle [31]. In this framework, we refer, among others, to [10], [11], [35] and, moreover, to the more recent contribution of Mawhin-Rebelo-Zanolin [36]. A result from [36] is used here to achieve the abstract topological theorem given in Section 2.…”

“…In this framework, we refer, among others, to [10], [11], [35] and, moreover, to the more recent contribution of Mawhin-Rebelo-Zanolin [36]. A result from [36] is used here to achieve the abstract topological theorem given in Section 2. We also mention a work by M. Struwe [47,Theorem 1] for analogous arguments (under some stronger assumptions).…”

Periodic solutions to parameter-dependent equations with a $$\phi $$-Laplacian type operator

“…However, when the nonlinearity is bounded, one could achieve the existence of at least a T -periodic solution for some ranges of the parameter s (cf. [16,18]).…”

“…The key ingredient for the proofs is Theorem 2.2 in Section 2, combined with arguments inspired from [5,7,15]. In the same section, following [16,18], we also recall a result of Amann, Ambrosetti and Mancini type on bounded nonlinearities (cf. [1]).…”