ASYMPTOTICALLY ALMOST PERIODIC SOLUTIONS TO NONLOCAL DIFFERENTIAL EQUATIONS

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We consider the asymptotic behavior of solutions on the half-line to nonlocal fractional differential equations of the form k∗u′=Au(t)+f(t,u(t)), where k∗u′ 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.