On nonautonomous linear systems of differential and difference equations with R-symmetric coefficient matrices

Let Cn×n(I) denote the set of continuous n×n matrices on an interval I. We say that R∈Cn×n(I) is a nontrivial k-involution if where ζ=e-2πi/k, d0+d1+⋯+dk-1=n, and with Uℓ∈Cdℓ×dℓ(I). We say that A∈Cn×n(I) is R-symmetric if R(t)A(t)R-1(t)=A(t), t∈I, and we show that if A is R-symmetric then solving x′=A(t)x or x′=A(t)x+f(t) reduces to solving k independent dℓ×dℓ systems, 0⩽ℓ⩽k-1. We consider the asymptotic behavior of the solutions in the case where I=[t0,∞). Finally, we sketch analogous results for linear systems of difference equations.