The Asymptotic Behavior of Solutions of a Fractional Integro-differential Equation

In this paper, we study the asymptotic behavior of solutions for an initial value problem with a nonlinear fractional integro-differential equation. Most of the existing results in the literature assume the continuity of the involved kernel. We consider here a kernel that is not necessarily continuous, namely, the kernel of the Riemann-Liouville fractional integral operator that might be singular. We determine certain sufficient conditions under which the solutions, in an appropriate underlying space, behave eventually like power functions. For this purpose, we establish and generalize some well-known integral inequalities with some crucial estimates. Our findings are supported by examples and numerical calculations.