Abstract
We present a new general method for converting an impulsive fractional differential equation to an equivalent integral equation. Using this method and employing a fixed point theorem in Banach space, we establish existence results of solutions for a boundary value problem of impulsive singular higher order fractional differential equation. An example is presented to illustrate the efficiency of the results obtained. A conclusion section is given at the end of the paper.
Highlights
Fractional differential equation is a generalization of ordinary differential equation to arbitrary non-integer orders
Solvability of boundary value problems for higher order ordinary differential equations were investigated by many authors
There has been no papers concerned with the solvability of boundary value problems for higher order impulsive fractional differential equations since it is difficult to convert an impulsive fractional differential equation to an equivalent integral equation
Summary
Fractional differential equation is a generalization of ordinary differential equation to arbitrary non-integer orders. It is interesting to generalize results on boundary value problems for higher order ordinary differential equations; in mentioned papers, in [21], authors studied existence of solutions of the following boundary value problem for higher order fractional differential equation. There has been no papers concerned with the solvability of boundary value problems for higher order impulsive fractional differential equations since it is difficult to convert an impulsive fractional differential equation to an equivalent integral equation. In [28], a general method for converting an impulsive fractional differential equation to an equivalent integral equation was presented. Ð2:1Þ is a solution of cDa0þ xðtÞ 1⁄4 hðtÞ; a:e:; t 2 ðti; tiþ1ði 2 INm0 Þ ð2:2Þ if and only if there exist constants cv0 IR such that. For x 2 X, denote fxðtÞ 1⁄4 f ðt; xðtÞÞ and IjxðtsÞ 1⁄4
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