We study some properties of the range of the relativistic pendulum operator P, that is, the set of possible continuous T -periodic forcing terms p for which the equation Px = p admits a T -periodic solution over a Tperiodic time scale T. Writing p(t) = p0(t) + p, we prove the existence of a nonempty compact interval I(p0), depending continuously on p0, such that the problem has a solution if and only if p ∈ I(p0) and at least two different solutions when p is an interior point. Furthermore, we give sufficient conditions for nondegeneracy; specifically, we prove that if T is small then I(p0) is a neighbourhood of 0 for arbitrary p0. The results in the present paper improve the smallness condition obtained in previous works for the continuous case T = R.
<p style='text-indent:20px;'>Using a Lyapunov-Krasovskii functional, new results concerning the global stability, boundedness of solutions, existence and non-existence of <inline-formula><tex-math id="M3">\begin{document}$ T $\end{document}</tex-math></inline-formula>-periodic solutions for a kind of delayed equation for a <inline-formula><tex-math id="M4">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula>-Laplacian operator are obtained. An application is given for the well known sunflower equation.</p>
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