A fractional Gronwall inequality and the asymptotic behaviour of global solutions of Caputo fractional problems

[摘要] We study the asymptotic behaviour of global solutions of some nonlinear integral equations related to some Caputo fractional initial value problems. We consider problems of fractional order between 0 and 1 and of order between 1 and 2, each in two cases: when the nonlinearity depends only on the function, and when the nonlinearity also depends on fractional derivatives of lower order. Our main tool is a new Gronwall inequality for integrals with singular kernels, which we prove here, and a related boundedness property of a fractional integral of anfunction.

[学科分类] 数学（综合）

[关键词] Fractional derivatives;asymptotic behaviour;Gronwall inequality;weakly singular kernel.