Asymptotic Behavior for Fractional Systems with Lower-Order Fractional Derivatives

The asymptotic behavior, the decay and the boundedness of solutions are discussed for the system of fractional differential equations including two types of fractional derivatives the Caputo fractional derivative (CFD) and the Riemann-Liouville fractional derivative (RLFD). Reasonable sufficient conditions are determined ensuring that solutions for the system with nonlinear right hand sides approach a power type function, power type decay and boundedness as time goes to infinity. Our approach is based on appropriate desingularization techniques and generalized the inequality of the Gronwall-Bellman. Convenient assessments and lemmas such as a fractional version of L’Hopital’s rule are used.