Periodic Solutions of Certain Nonlinear Integral Equations with a Time Lag
Bernard D. Coleman, George H. Renninger · SIAM Journal on Applied Mathematics · 1976
Solutions y of equations of the type \[ ( 1 )\qquad y( t ) = m\left( {1 - \int_0^\infty {e^{ - s} g( {y( {t - \gamma - s} )} )ds} } \right) \] with $\gamma $ a given positive number, with g a given function obeying $g( 0 ) = 0$ and $g( x ) > 0$ for $x > 0$, and with m the “positive part” function defined by $m( x ) = \frac{1}{2}( {x + | x |} )$, describe the response of certain neural networks to constant excitation. It is here shown that if (1) has a periodic solution of the form \[ ( 2 )\qquad y( t ) = \{ \begin{gathered} G( t ) > 0,\quad ,0 0, \], with $G( \zeta ) = 0$ for $\zeta $ in $[ { - \gamma ,0} ]$. Conditions on $\gamma $ and g sufficient to insure the existence of a periodic solution of the form (2) are given, and graphs of such solutions are presented for the special case in which g is linear.