Periodic solutions of linear neutral integro-differential equations

Abstract

Abstract Consider the linear neutral FDEddt[x(t)+ Ax(t− τ)] = Z R [dL(s)]x(t+ s) + f(t)where x and f are n-dimensional vectors; A is an n×n constant matrix and L(s) is an n×nmatrix function with bounded total variation. Some necessary and sufficient conditionsare given which guarantee the existence and uniqueness of periodic solutions to the aboveequation.Key words Linear neutral FDE, bounded total variation, periodic solution2000 MR Subject Classification 34K15 1 Introduction Let C n be the space of n-dimensional complex vectors with the norm | ·| defined by|z| = max 1≤j≤n |z j |, for all z = (z 1 ,···,z n ) ∈ C n . (1)Let Z = (z ij ) ∈ C n 2 and k · k denotes the matrix norm induced by | · | , then it is easy to seethatkZk = max 1≤i≤n Xn j=1 |z ij |. (2)LetUC b (R,C n ) = {ϕ : R → C n is uniformly continuous on R andkϕk = sup s∈R |ϕ(s)| 0 , letC T = {ϕ : R → C n is continuous and ϕ(t +T) ≡ ϕ(t)}. 1 Received July 5, 2001. The Project Supported by NSFC (19801014,10171065,19971026)