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ON THE PHASE-VOLUME METHOD FOR NONLINEAR DIFFERENCE EQUATIONS}

Janos Karsai, John G. Graef, Michael Li

Abstract


The Milloux-Hartman theorem states that if thezero solution of the differential equation $x'=A(t)x$ is stable, where$x\in \cset^m$ and $A(t)$ is a continuous $m\times m$ matrix function,then there exists a nontrivialsolution that tends to zero as $t\rightarrow \infty$ if and only if$\int_{t_0}^{\infty}\trace A(s)\, ds =-\infty$. Several variations of thistheorem have been proved for linear systems. This theorem was extended tolinear difference equations by Peil and Peterson and toimpulsive differential equations by Graef and Karsai. Karsai investigated this problem for nonlinear ordinarydifferential equations, and pointed out that thegeneralization is far from immediate.  Here, the authorsprove a Milloux-Hartman type theorem for nonlineardifference equations, and formulate analogous results forthe existence of unbounded solutions.

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