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

Abstract

Abstract We consider a Rayleigh-type equation with state–dependent delay ℘ ″ ( ϑ ) + 𝒱 1 ( ϑ , ℘ ( ϑ ) ) + A ( ℘ ( ϑ − 𝒱 2 ( 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.