ASYMPTOTICALLY ALMOST PERIODIC SOLUTIONS TO NONLOCAL DIFFERENTIAL EQUATIONS

Abstract

We consider the asymptotic behavior of solutions on the half-line to nonlocal fractional differential equations of the form $k\ast u^{\prime }=Au(t)+f(t,u(t))$, where $k\ast u^{\prime }$ 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.