Abstract
A class of periodic boundary value problems for higher order fractional differential equations with impulse effects is considered. We first convert the problem to an equivalent integral equation. Then, using a fixed-point theorem in Banach space, we establish existence results of solutions for this kind of boundary value problem for impulsive singular higher order fractional differential equations. Two examples are presented to illustrate the efficiency of the results obtained.
Highlights
Fractional differential equation is a generalization of ordinary differential equation to arbitrary non-integer orders
The reader may refer to the books and monographs [1,2,7,9] for fractional calculus and developments on fractional differential and fractional integrodifferential equations with applications
In [18], the authors studied the existence of positive solutions of the following non-linear impulsive fractional differential equation with generalized periodic boundary value conditions:
Summary
Fractional differential equation is a generalization of ordinary differential equation to arbitrary non-integer orders. The reason is that it is difficult to transform a boundary value problem for higher order impulsive fractional differential equations to integral equations. In [18], the authors studied the existence of positive solutions of the following non-linear impulsive fractional differential equation with generalized periodic boundary value conditions:. In [14], the existence of solutions of a high-order impulsive boundary value problem for quasi-linear fractional differential equations was studied. Motivated by [14], in this paper, we discuss the following periodic boundary value problems for non-linear impulsive singular fractional differential equation:. The first purpose of this paper is to present a new method for converting BVPs for impulsive fractional differential equation to integral equations, see Lemma 2.11 in Sect.
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