Boundary value problems of singular multi-term fractional differential equations with impulse effects

Abstract

We study a class of boundary value problems for nonlinear impulsive fractional differential equations. A weighted function Banach space and a completely continuous nonlinear operator are constructed firstly. Some existence results for solutions of these problems are established continuously. Our analysis relies on the well known Schauder's fixed point theorem. Examples are given to illustrate the main results finally.

A New Variational Iteration Method for a Class of Fractional Convection-Diffusion Equations in Large Domains

“…Also, by this definition we can see that the singularity is removed [53]. It should be noted that some other methods for fractional calculus are introduced in [54,55] such as He's fractional derivative and Xiao-Jun Yang's definition.…”

Boundary value problems of multi‐term fractional differential equations with impulse effects

“…Boundary value problems for impulsive fractional differential equations have many applications in modeling of physical and chemical processes . Recent studies on the solvability of boundary value problems for impulsive fractional differential equations may be seen in and reference therein. In , Liu and Jia studied the existence of solutions of the following boundary value problem for multi‐term impulsive fractional differential equation (BVP for short) with impulse effects where …”

Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative

“…Nowadays, several theoretical results on Hadamard fractional differential equations have been obtained. Two surveys [2,27] proved the existence of solutions and weak solutions for some classes of Hadamard type fractional differential equations with and without impulse effects and presented the analytical solutions in terms of the Mittag-Leffler function. The authors in [4] investigated the initial and boundary value problems of Hadamard fractional differential equations and inclusions, and obtained some new results on them.…”

“…For α = 1, we have E 1 (z) = e z , which is the exponential function. Based on the Theorem 3.6 and 3.8 in [27], we present two important lemmas as follows.…”

“…For the abstract operators A and B involved in system (1), the last formula is given in terms of the operator S α (t) and K α (t), because of the uniqueness of the solution for a linear system (see e.g. [22] or the Theorem 3.8 of [27]). Moreover, the uniqueness of the solution shows that when A is a real number, S α (t) = E α (At α ), K α (t) = E α,α (At α ) and the solution coincides with that in [27].…”