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Abstract This paper studies the existence of affine-periodic solutions which have the form of x ( t + T ) = Q x ( t ) $x(t+T)=Qx(t)$ with some nonsingular matrix Q. Depending on the structure of Q, they can be periodic, anti-periodic, quasi-periodic or even unbounded. Krasnosel’skii–Perov type existence theorem, asymptotic and homotopy equivalence approaches are given.