Fractional operators with exponential kernels and a Lyapunov type inequality

[摘要] In this article, we extend fractional calculus with nonsingular exponential kernels, initiated recently by Caputo and Fabrizio, to higher order. The extension is given to both left and right fractional derivatives and integrals. We prove existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial value problems by using Banach contraction theorem. Then we prove Lyapunov type inequality for the Riemann type fractional boundary value problems within the exponential kernels. Illustrative examples are analyzed and an application about Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.

[学科分类] 数学（综合）

[关键词] CFC fractional derivative;CFR fractional derivative;Lyapunov inequality;boundary value problem;higher order;exponential kernel