In this paper, we consider a discrete delay problem with negative feedback
x(
t)=
f(
x(
t),
x(
t−1)) along with a certain family of time discretizations with stepsize 1/
n. In the original problem, the attractor admits a nice Morse decomposition. We proved in (T. Gedeon and G. Hines, 1999,
J. Differential Equations151, 36-78) that the discretized problems have global attractors. It was proved in (T. Gedeon and K. Mischaikow, 1995,
J. Dynam. Differential Equations7, 141-190) that such attractors also admit Morse decompositions. In (T. Gedeon and G. Hines, 1999,
J. Differential Equations151, 36-78) we proved certain continuity results about the individual Morse sets, including that if
f(
x,
y)=
f(
y), then the individual Morse sets are upper semicontinuous at
n=∞. In this paper we extend this result to the general case; that is, we prove for general
f(
x,
y) with negative feedback that the Morse sets are upper semicontinuous.
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