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Integral manifolds of differential equations with piecewise constant argument of generalized type. (English) Zbl 1122.34054

Equations with piecewise constant delayed argument are studied, like

x ˙(t)=f(t,x(t),x([t]))·

One main assumption is that there is a linear part x ˙(t)=A(t)x(t) of the equation which has an exponential dichotomy. Manifolds of solutions converging to zero in forward/backward time are constructed using a Perron-type approach. Existence and uniqueness of bounded/periodic solutions is obtained (as a consequence of the exponential dichotomy). The author uses successive approximations instead of the contraction theorem.

34K19Invariant manifolds (functional-differential equations)
34K12Growth, boundedness, comparison of solutions of functional-differential equations
34K13Periodic solutions of functional differential equations