  In the present work we give some sufficient conditions to obtain a unique
almost automorphic solution to abstract nonlinear integral equations which are
simultaneously of advanced and delayed type and also a unique asymptotically
almost automorphic mild solution to abstract integro-differential equations
with nonlocal initial conditions, both situations are posed on Banach spaces.
Also, we develop a Bohr-Neugebauer type result for the abstract integral
equations. Before that, we introduce the notion of $\lambda$-bounded functions,
develop the appropriate abstract theory and discuss the almost periodic
situation. As applications, we study the existence of an asymptotically almost
automorphic solution to integro-differential equations modeling heat conduction
in materials with memory and also the existence of the almost automorphic
solution to semilinear parabolic evolution equations with finite delay.
