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An asymptotic method for calculating limits of maximal means

Authors:

O P Filatov

Institution:

(1) Samara State University, USSR

Abstract:

For a continuous almost periodic function $f: \mathbb{R}^{m_\tau } \times \mathbb{R}^{m_\gamma } \to \mathbb{R}$ , we show that the function

$M_f \left( \mu \right) = \mathop {\lim }\limits_{\Delta \to \infty } \mathop {\sup }\limits_{\tau , \gamma } \frac{1}{\Delta }\smallint _0^\Delta f\left( {\tau \left( {\mu t} \right),\gamma \left( t \right)} \right)dt,$

where the supremum is taken over all solutions of the system of differential inclusion $\dot \tau \in \mu G_\tau ,\tau \left( 0 \right) = \tau _0$ , $\dot \gamma \in G_{_\gamma } ,\gamma \left( 0 \right) = \gamma _0$ , has the following limit (as μ→+0):

$\Psi _f = \mathop {\lim }\limits_{\Delta \to \infty } \mathop {\sup }\limits_\tau \frac{1}{\Delta }\int_0^\Delta {\Phi (\tau (\mu t))dt,} where \Phi (\tau _0 ) = \mathop {\lim }\limits_{\Delta \to \infty } \mathop {\sup }\limits_\gamma \frac{1}{\Delta }\int_0^\Delta {f(\tau _0 ,\gamma (t))dt} .$

, Thus if the parameter μ is small, then $\Psi _f = \lim _{\mu \to + 0} M_f \left( \mu \right)$ and the limit of the maximal mean can approximately be determined by solving problems of smaller dimensionality. Moreover, if the compact sets $G_\tau \subset \mathbb{R}^{m_\tau }$ and $G_\gamma \subset \mathbb{R}^{m_\gamma }$ are nondegenerate, then Ψ_f is independent of initial data. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 431–438, September, 1999.

Keywords:

almost periodic functions asymptotic method limit of maximal means generalized differential equation Bochner almost periodic function

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