Linear higher-order fractional differential and integral equations

Date
2023-01-04
Authors
Lan, Kunquan
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
We study the equivalences and the implications between linear (or homogeneous) nth order fractional differential equations (FDEs) and integral equations in the spaces L1(a,b) and C[a,b] when n≥ 2. We establish the equivalence in C[a,b] of the IVP of the nth order FDE subject to the initial condition u(i)(a)=ui for i in {0,1,...,n-1} when n≥2. The difficulty is that the known conditions for such equivalence for the first order FDEs are not sufficient for equivalence in the nth order FDEs with n≥2. In this article we provide additional conditions to ensure the equivalence for the nth order FDEs with n≥2. In particular, we obtain conditions under which solutions of the integral equations are solutions of the linear nth order FDEs. These results are keys for further studying the existence of solutions and nonnegative solutions to initial and boundary value problems of nonlinear nth order FDEs. This is done via the corresponding integral equations by topological methods such as the Banach contraction principle, fixed point index theory, and degree theory.
Description
Keywords
Fractional differential equation, Integral equation, Equivalence, Initial value problem
Citation
Lan, K. (2023). Linear higher-order fractional differential and integral equations. <i>Electronic Journal of Differential Equations, 2023</i>(01), pp. 1-20.