Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity

Pablo Amster; Manuel Zamora

doi:10.3934/dcds.2018211

Article Contents

2018, Volume 38, Issue 10: 4819-4835. Doi: 10.3934/dcds.2018211

This issue Previous Article Next Article On continuity equations in space-time domains

Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity

Pablo Amster^1, and
Manuel Zamora^2, ,

1.
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, 1428, Buenos Aires, Argentina
2.
Departamento de Matemática, Grupo de Investigación en Sistemas Dinámicos y Aplicaciones (GISDA), Universidad de Oviedo, C/ Federico García Lorca, n18, Oviedo, Spain

^* Corresponding author: Manuel Zamora
^* Corresponding author: Manuel Zamora

Received: January 31, 2017

Revised: November 30, 2017

Published: September 2018

The first author is supported by projects UBACyT 20020120100029BA and CONICET PIP11220130100006CO, and the second author was supported by FONDECYT, project no. 11140203.

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Abstract

TWe prove the existence of $T$-periodic solutions for the second order non-linear equation

$ {\left( {\frac{{u'}}{{\sqrt {1 - {{u'}^2}} }}} \right)^\prime } = h(t)g(u), $

where the non-linear term $g$ has two singularities and the weight function $h$ changes sign. We find a relation between the degeneracy of the zeroes of the weight function and the order of one of the singularities of the non-linear term. The proof is based on the classical Leray-Schauder continuation theorem. Some applications to important mathematical models are presented.

Keywords:

Mathematics Subject Classification: Primary: 34C25, 34B18; Secondary: 34B30.

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Figure 1. The figure illustrates a possible behaviour of any $T-$periodic solution $u$ of (20) when $t_*$ is included on $[\bar{\ell}_i, \bar{\ell}_i')$ (Case Ⅰ.) or if $t_*\in (\bar{\ell_i}, \bar{\ell}_i']$ (Case Ⅱ.).

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Figure 2. The figure illustrates a possible behaviour of the $T-$periodic solution $u$ of (16) when $t_*$ is included on $[a_i, a_i']$, distinguishing if $t_*\in [a_i, \bar{\ell}_i)$ (Case Ⅰ. a)) or $t_*\in (\bar{\ell}_i', a_i']$ (Case Ⅰ. b)).

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Figure 3. The figure illustrates a possible behaviour of the $T-$periodic solution $u$ of (16) on the interval $[\bar{\ell}_i', \ell_i']$, assuming that $u(a_i')<\alpha+2\delta$.

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Figure 4. The figure illustrates a possible behaviour of the $T-$periodic solution $u$ of (16) on the interval $[\bar{\ell}_i', \ell_i']$, assuming that $u(a_i')\geq \alpha+2\delta$.

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