Necessary and sufficient conditions for the oscillations of a multiplicative delay logistic equation

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Abstract
Necessary and sufficient conditions are presented for the oscillation of all positive solutions of the multiplicative delay logistic equation\[dN(t)dt=r(t)N(t)(1−∏j=1m(N(gj(t)K))\frac {{dN\left ( t \right )}}{{dt}} = r\left ( t \right )N\left ( t \right ) \left ( {1 - \prod \limits _{j = 1}^m { \left ( {\frac {{N\left ( {g_j}\left ( t \right ) \right .}}{K}} \right ) } } \right )\]about the positive equilibriumKK. The cases whenr(t)=rr\left ( t \right ) = randgj(t)=t−τj{g_j}\left ( t \right ) = t - {\tau _j}orgj(t)=[t−kj]{g_j}\left ( t \right ) = \left [ {t - {k_j}} \right ], [•] denoting the greatest integer function, wherer,τjr, {\tau _j}, andkj{k_j}are positive constants,j=1,2,...,mj = 1, 2,...,m, are also included.

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