ENG 
ESP Session S12 - Delay and functional differential equations and applications
Tuesday, July 13, 14:15 ~ 14:50 UTC-3
Multiple periodic solutions for dynamic Liénard equations with delay and singular $\varphi$-laplacian of relativistic type
Mariel Paula Kuna
Departamento de Matemática, FCEyN, Universidad de Buenos Aires e IMAS-CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work, we study the existence and multiplicity of $T$-periodic solutions $x:\mathbb{T} \rightarrow \mathbb{R}$ to the following equation with delay on time scales
\[(\varphi(x^{\Delta}(t)))^{\Delta} +h(x(t))x^{\Delta}(t)+ g(x(t-r)) =p(t) \ \ \ t\in \mathbb{T},\]
where ${\mathbb{T}}$ is an arbitrary $T$-periodic nonempty closed subset of $\mathbb{R}$ (\textit{time scale}), $\varphi:(-a,a)\rightarrow \mathbb{R}$ is an
increasing homeomorphism with $0 \smallskip Under appropriate assumptions, multiple solutions are obtained as fixed points of an operator arising on a nonlinear Lyapunov-Schmidt decomposition.
\smallskip A special case of interest is the sunflower equation with relativistic effects on time scales, namely
\[ \left( \frac {x^\Delta(t)}
{\sqrt {1- \frac{x^\Delta(t)^2}{c^{2}} }}\right)^\Delta+ ax^\Delta(t) + b \sin( x(t-r)) = p(t).\]
We prove that if $T$ is small, then the equation has a $T$-periodic solution. The results improve the smallness condition obtained in previous works for the continuous case $\mathbb T=\mathbb R$ and without delay. The bound shall be expressed in terms of $k(\mathbb T)$, the optimal constant of the Sobolev inequality
$$\|x-\overline x\|_{\infty}\leq k\|x^\Delta\|_{\infty}, \;\; x\in C^1_T.$$ \noindent {\bf References} \medskip \noindent [1] P. Amster, M. P. Kuna and D. P. Santos, {\em Existence and multiplicity of periodic solutions for dynamic equations with delay and singular $\varphi$-Laplacian of relativistic type}, submitted. \medskip \noindent [2] P. Amster, M. P. Kuna and D. P. Santos, {\em On the solvability of the periodically forced
relativistic pendulum equation on time scales}, Electron. J. Qual. Theory Differ. Equ. 2020, No. 62, 1-11. Joint work with Pablo Amster (Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IMAS-CONICET, Argentina) and Dionicio Pastor Santos (Universidad Nacional del Centro de la Provincia de Buenos Aires, Argentina).