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

  • * Corresponding author: Alessandro Fonda

    * Corresponding author: Alessandro Fonda 
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  • 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.

    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) $

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

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

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