A dynamical approach to lower and upper solutions for planar systems 'To the memory of Massimo Tarallo'

Alessandro Fonda; Rodica Toader

doi:10.3934/dcds.2021012

Article Contents

2021, Volume 41, Issue 8: 3683-3708. Doi: 10.3934/dcds.2021012

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A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo"

Alessandro Fonda^1, , and
Rodica Toader^2,

1.
Dipartimento di Matematica e Geoscienze, Università di Trieste, P.le Europa 1, Trieste, Italy
2.
Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, Via delle Scienze 206, Udine, Italy

^* Corresponding author: Alessandro Fonda
^* Corresponding author: Alessandro Fonda

Received: October 31, 2020

Early access: January 2021

Published: August 2021

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Abstract

We prove the existence of bounded and periodic solutions for planar systems by introducing a notion of lower and upper solutions which generalizes the classical one for scalar second order equations. The proof relies on phase plane analysis; after suitably modifying the nonlinearities, the Ważewski theory provides a solution which remains bounded in the future. For the periodic problem, the Massera Theorem applies, yielding the existence result. We then show how our result generalizes some well known theorems on the existence of bounded and of periodic solutions. Finally, we provide some corollaries on the existence of almost periodic solutions for scalar second order equations.

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Mathematics Subject Classification: Primary: 34C11, 34C25; Secondary: 34C27.

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Figure 1. In the red region $ f(t,x,y)>\alpha'(t) $, in the green one $ f(t,x,y)<\alpha'(t) $

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Figure 2. A section of the set $ V $ at time $ t $ with its egress points

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Figure 3. The grid made of quadrilaterals near the point $ (\alpha(t),y_\alpha(t)) $

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