A Floquet Decomposition for Volterra Equations with Periodic Kernel and a Transform Approach to Linear Recursion Equations.

Abstract The primary aim of this paper is to show that a functional equation in E n , x ( t ) = ∝ −∞ t W ( t , s ) x ( s ) ds with piecewise continuous m × m kernel W ( t , s ) satisfying, for T , Ω, μ all positive, W ( t + T , s + T ) = W ( t , s ), t , s real, ∥W(t, s)∥⩽ Ωe −μ(t − s) , t ⩾ s , admits, for each β x ( t ) = x F ( t ) + X β ( t ), applicable to a wide class of solutions x ( t ) for t ⩾ 0, where, for some B = B(β), ∥x β (t)¦⩽ Be −βt , t ⩾ 0, and x F (t) is a linear combination of “Floquet-type” solutions having the form t q e λt p ( t ), q integer ⩾ 0, λ ϵ C , Re(λ) > −β, p ( t ) being a continuous n -vector function such that p ( t + T ) = p ( t ). The theorem is proved by converting the above equation to a linear recursion equation of convolution type ∑ k = −∞ 0 Q k x k + j = 0 in L m 0 [0, T ] and studying this equation by transform methods. In the process a secondary objective of this paper, examination of some general properties of equations of the form ∑ k = −∞ ∞ Q k x k + j = 0 within the same transform framework, is realized.