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

Abstract

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