On a singular periodic Ambrosetti–Prodi problem

Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations

“…In particular, we focus our attention on the classical result of Lazer and McKenna [23] dated 1987 and its generalizations due to Del Pino et al [6] in 1992 and Fonda and Ghirardelli [9] in 2010. We also refer to [3,10,17,27] for related results and to [28,30] and references therein for a comprehensive introduction to this topic.…”

“…Lemma 2.5. It is possible to find δ, R 1 , s 0 , with 0 < δ < R 1 < 1 2 c 0 and s 0 > s, where c 0 and s are given by Lemma 2.3, with the following property: for every s ≥ s 0 , if y is a solution of (17), with (α,…”

“…Lemma 2.6. For every s ≥ s 0 , any solution to the Cauchy problem (17) associated to an initial datum (α, β) such that…”

“…In this case, by ( 16), we get the estimate |ρ i | ≤ (1 + C)ρ i + C, then again by a Gronwall argument the proof easily follows. Now, by a standard non-resonance argument, we can prove that there exists χ 0 > 0 large enough to guarantee that every solution to (17), such that ρ i (t) > χ 0 for every t ∈ [0, T ] and a certain index i, satisfies n i < rot i (y) < n i + 1.…”

Existence and multiplicity of periodic solutions to differential equations with attractive singularities

“…The idea how to obtain multiplicity results is similar to the previous works, usually we use homotopy invariance and additivity property of Leray-Schauder degree (see, e.g. [2,6,7,15,19,23,30]). However, the nature of this kind of problems request quite new approach in how to construct the strict lower and upper functions.…”