ASYMPTOTICALLY ALMOST PERIODIC SOLUTIONS TO NONLOCAL DIFFERENTIAL EQUATIONS

A BSTRACT . In this work, we consider the asymptotic behavior of solutions on the half line to nonlocal fractional differential equations of the form k ∗ u (cid:48) = Au ( t )+ f ( t , u ( t )) , where k ∗ u (cid:48) stands for the nonlocal derivative of u in Caputo’s sense corresponding to a singular kernel k , A is a positively deﬁnite, selfadjoint operator; the nonlinearity f is either locally Lipschitz continuous with respect to the second variable or of sublinear growth for small time and Lipschitz continuous for large time. Our main results show that if f is an asymptotically almost periodic function on t then the problem possesses asymptotically almost periodic solutions.