English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

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    ,    $\varphi(t)&gt;0$   φ  (  t  )  &gt;  0     and    $\alpha(t)&gt;0$   α  (  t  )  &gt;  0     are continuous functions with  T -periodicity in the  t  variable,  c ,  γ  are constants with    $|c|&lt;1$   |  c  |  &lt;  1    ,    $\gamma\geq1$   γ  ≥  1    . Many authors obtained the existence of periodic solutions under the condition    $0&lt;\mu\leq1$   0  &lt;  μ  ≤  1    , and we extend the result to    $\mu&gt;1$   μ  &gt;  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.

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