1. On the positive periodic solutions of a class of Liénard equations with repulsive singularities in degenerate case.
- Author
-
Yu, Xingchen, Song, Yongli, Lu, Shiping, and Godoy, José
- Subjects
- *
DEGENERATE differential equations , *DIOPHANTINE equations , *DIFFERENTIAL equations , *EQUATIONS , *TOPOLOGICAL degree - Abstract
In this paper, we study the existence, multiplicity and dynamics of positive periodic solutions to a generalized Liénard equation with repulsive singularities. The Ambrosetti-Prodi type result is proved in the absence of the so-called anticoercivity condition. Furthermore, with s as a parameter, under some conditions on the function h , it has been shown that for any M > 1 there exists s M ∈ R such that the equation x ″ + f (x) x ′ + h (t , x) = s has two positive T -periodic solutions u 1 (⋅ ; s) and u 2 (⋅ ; s) satisfying min { u 1 (t ; s) : t ∈ [ 0 , T ] } > M and min { u 2 (t ; s) : t ∈ [ 0 , T ] } < 1 / M for every s < s M. As a by-product of the property, we obtain sufficient conditions to guarantee the existence of positive T -periodic solutions of indefinite differential equations. • Establish a refinement result of the Ambrosetti-Prodi type in the absence of anti-coercivity condition. • Investigate the dynamic behaviors of positive periodic solutions to a generalized Liénard equation. • Provide a new approach to study the existence of positive periodic solutions for indefinite differential equations. • Find a relationship between the strict lower functions and Leray-Schauder degree. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF