Characterizing the asymptotic behavior (limiting behavior for large values of time) and stability of solutions of differential equations in general is of great importance in numerous applications. This task is not always easy and becomes even more challenging when dealing with nonlinear differential equations of non-integer order. Their investigation requires special techniques and new tools. We propose to investigate the asymptotic behavior and stability of a class of nonlinear fractional differential equations and systems, where the equations involve fractional derivatives of Riemann-Liouville type or Caputo type. The source terms may be nonlinear and involve themselves lower order fractional derivatives. In applications, these derivatives may account for fractional damping. Our objective is to characterize or at least seek reasonable conditions under which solutions decay to zero or to the equilibrium at infinity. In particular, we will identify sufficient conditions for power-type decay behavior and boundedness in general. To bypass the difficulties caused by the nonlocality nature of the fractional derivatives and the singularity of the kernels, we plan to use appropriate modified and generalized inequalities, estimation techniques, and asymptotic theories. Our results could be utilized to predict the behavior of many physical systems modeled by linear or nonlinear fractional differential equations and systems for which explicit analytical solutions may not be available. They will help also developing some numerical treatments for approximate solutions. In addition, our results will enrich the limited results available currently for fractional order problems.
|Effective start/end date
|11/04/17 → 11/04/19
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