Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type

• Main Author: Yu Zhu
• Format: Article
• Language: English
• Published: SpringerOpen 2020-10-01
• Series: Boundary Value Problems
• Subjects: Periodic solution; Neutral equation; Mawhin’s continuation theorem; Attractive singularity
• Online Access: http://link.springer.com/article/10.1186/s13661-020-01462-w

Abstract In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type ( x ( t ) − c x ( t − σ ) ) ″ + f ( x ( t ) ) x ′ ( t ) − φ ( t ) x μ ( t ) + α ( t ) x γ ( t ) = e ( t ) , $$ \bigl(x(t)-cx(t-\sigma)\bigr)''+f\bigl(x(t) \bigr)x'(t)-\varphi(t)x^{\mu}(t)+ \frac{\alpha(t)}{x^{\gamma}(t)}=e(t), $$ where f : ( 0 , + ∞ ) → R $f:(0,+\infty)\rightarrow R$ , φ ( t ) > 0 $\varphi(t)>0$ and α ( t ) > 0 $\alpha(t)>0$ are continuous functions with T-periodicity in the t variable, c, γ are constants with | c | < 1 $|c|<1$ , γ ≥ 1 $\gamma\geq1$ . Many authors obtained the existence of periodic solutions under the condition 0 < μ ≤ 1 $0<\mu\leq1$ , and we extend the result to μ > 1 $\mu>1$ by using Mawhin’s continuation theorem as well as the techniques of a priori estimates. At last, an example is given to show applications of the theorem.