arXiv subjects
Publications and source records attributed to Mariel P. Kuna.
We study the existence and multiplicity of periodic solutions for singular $φ$-laplacian equations with delay on time scales. We prove the existence of multiple solutions using topological methods based on the Leray-Schauder degree. A special case is the $T$-periodic problem for the forced pendulum equation with relativistic effects.
We study some properties of the range of the relativistic pendulum operator $\mathcal P$, that is, the set of possible continuous $T$-periodic forcing terms $p$ for which the equation $\mathcal P x=p$ admits a $T$-periodic solution over a $T$-periodic time scale $\mathbb T$. Writing $p(t)=p_0(t)+\overline p$, we prove the existence of a nonempty compact interval $\mathcal I(p_0)$, depending continuously on $p_0$, such that the problem has a solution if and only if $\overline p\in \mathcal I(p_0)$ and at least two different solutions when $\overline p$ is an interior point. Furthermore, we give sufficient conditions for nondegeneracy; specifically, we prove that if $T$ is small then $\mathcal I(p_0)$ is a neighbourhood of $0$ for arbitrary $p_0$. The results in the present paper improve the smallness condition obtained in previous works for the continuous case $\mathbb T=\mathbb R$.
A second order ordinary differential equation with a superlinear term $g(x,u)$ under radiation boundary conditions is studied. Using a shooting argument, all the results obtained in a previous work for a Painlevé II equation are extended. It is proved that the uniqueness or multiplicity of solutions depend on the interaction between the mapping $\frac {\partial g}{\partial u}(\cdot,0)$ and the first eigenvalue of the associated linear operator. Furthermore, two open problems regarding, on the one hand, the existence of sign-changing solutions and, on the other hand, exact multiplicity are solved.
Small non-autonomous perturbations around an equilibrium of a nonlinear delayed system are studied. Under appropriate assumptions, it is shown that the number of $T$-periodic solutions lying inside a bounded domain $Ω\subset \R^N$ is, generically, at least $|χ\pm 1|+1$, where $χ$ denotes the Euler characteristic of $Ω$. Moreover, some connections between the associated fixed point operator and the Poincaré operator are explored.