On the Stability and the Existence and Non-existence of T Periodic Solutions for Nonlinear Delayed Differential Equations with Friction and phi-Laplacian

AMSTER, PABLO; KUNA, MARIEL PAULA; SANTOS, DIONICIO PASTOR

Evento Virtual, Montevideo

Congreso; IV Congreso Latinoamericano de Matemáticos; 2021

Let us consider the following problem$$(\varphi(x'(t) ))' + h(x(t),x'(t))x'(t)+ g(x(t-r)) =p(t), \ \ \ t\in [0,\infty),$$where $\varphi:\mathbb{R} \rightarrow \mathbb{R}$ is an increasing homeomorphism such that $\varphi(0)=0$, $r$ is a positive constant, $g:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous differentiable function, and $h:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ and $p:[0,\infty) \rightarrow \mathbb{R}$ are continuous functions such that $p$ is $T$-periodic in $t$ for $T$ a positive constant. \medskipUsing Lyapunov-Krasovskii functional and under appropriate assumptions we obtain new results on the global stability, boundedness of solutions, existence and non-existence of $T-$periodic solutions.