Zeitschrift f¨ ur Analysis und ihre Anwendungen c  European Mathematical Society Journal for Analysis and its Applications Volume 26 (2007), 363–375 Asymptotically Almost Periodic and Almost Periodic Solutions for Partial Neutral Integrodifferential Equations Hern´ an Henr´ ıquez, Eduardo Hern´andez M. and Jos´ e C. dos Santos Abstract. In this paper we study the existence of almost periodic and asymptotically almost periodic solutions for a class of partial neutral functional integro-differential equation with unbounded delay. Keywords. Almost periodic solutions, integro-differential equations, neutral equa- tions Mathematics Subject Classification (2000). Primary 34K14, 45N05, 35R10, 34K40, secondary 45D05, 34K30. 1. Introduction In this work we study the existence of asymptotically almost periodic and almost periodic solutions for a class of partial neutral integro-differential equation with unbounded delay modelled in the form d dt D(t, u t )= AD(t, u t )+  t 0 B(t − s)D(s, u s )ds + g(t, u t ), (1) where A : D(A) ⊂ X → X, B(t): D(B(t)) ⊂ X → X , t ≥ 0, are linear closed and densely defined operators on a Banach space X ; D(B(t)) ⊃ D(A) for every t ≥ 0; the history x t :(−∞, 0] → X , x t (θ)= x(t + θ), belongs to some H. Henr´ ıquez: Departamento de Matem´atica, Universidad de Santiago de Chile, Casilla 307, Correo-2, Santiago, Chile; hhenriqu@lauca.usach.cl E. Hern´andez M.: Instituto de Ciˆ encias Matem´aticas e de Computac˜ ao, Universidade de S˜ ao Paulo, Caixa Postal 668, CP. 13560-970, S˜ ao Carlos SP, Brazil; lalohm@icmc.usp.br J. P. C. dos Santos: Departamento de Ciˆ encias Exatas Universidade Federal de Alfe- nas, 3713-000 Alfenas; MG, Brazil; zepaulo@unifal-mg.edu.br The work of the professor Henr´ ıquez was supported by FONDECYT, Grant 1050314.