New results on the existence of periodic solutions for Rayleigh equations with state-dependent delay

Abstract

We consider a Rayleigh-type equation with state–dependent delay ℘″(ϑ)+𝒱1(ϑ,℘(ϑ))+A(℘(ϑ−𝒱2(ϑ,℘(ϑ))))=B(ϑ).\wp ''\left( \vartheta \right) + \mathcal{V}{_1}\left( {\vartheta ,\wp \left( \vartheta \right)} \right) + A\left( {\wp \left( {\vartheta - \mathcal{V}{_2}\left( {\vartheta ,\wp \left( \vartheta \right)} \right)} \right)} \right) = B\left( \vartheta \right). We establish a set of new su˚cient conditions on the existence of at least one positive periodic solution by using the continuation theorem of coincidence degree theory. Our results not only provide a new approach but also generalize previous results. An example with graphical representations are presented to illustrate the results.

Keywords

• rayleigh equation
• mawhin’s continuation theorem
• state-dependent delay
• periodic solutions
• 34l30
• 47h11
• 54h25