Schaefer's fixed point theorem is used to study the controllability in an infinite delay system x ′ ( t ) = G ( t , x t ) + ( B u ) ( t ) . A compact map or homotopy is constructed enabling us to show that if there is an a priori bound on all possible solutions of the companion control system x ′ ( t ) = λ [ G ( t , x t ) + ( B u ) ( t ) ] , 0 < λ < 1 , then there exists a solution for λ = 1 . The a priori bound is established by means of a Liapunov functional or applying an integral inequality. Applications to integral control systems are given to illustrate the approach.