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Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. II
In this paper we continue the research started in a previous paper, where we proved that the linear differential equation
(1) x0 = A(t)x + f(t)
with Levitan almost periodic coefficients has a unique Levitan almost periodic solution, if it has at least one bounded solution and the bounded solutions of the homogeneous equation
(2) x0 = A(t)x
are homoclinic to zero (i.e. lim jtj!+1 j'(t)j = 0 for all bounded solution ' of (2)).
If the coefficients of (1) are Bohr almost periodic and all bounded solutions
of equation (2) are homoclinic to zero, then the equation (1) admits a unique
almost automorphic solution.
In this second part we first generalise this result for linear functional differential
equations (FDEs) of the form
(3) x0 = A(t)xt + f(t);
as well as for neutral FDEs.
Analogous results for functional difference equations with finite delay and some classes of partial differential equations are also given.
We study the problem of existence of Bohr/Levitan almost periodic solutions
of differential equations of type (3) in the context of general semi-group
non-autonomous dynamical systems (cocycles), in contrast with the group
non-autonomous dynamical systems framework considered in the first part
Almost periodic solutions of retarded SICNNs with functional response on piecewise constant argument
We consider a new model for shunting inhibitory cellular neural networks,
retarded functional differential equations with piecewise constant argument.
The existence and exponential stability of almost periodic solutions are
investigated. An illustrative example is provided.Comment: 24 pages, 1 figur
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