Some Existence Results for High Order Fractional Impulsive Differential Equation on Infinite Interval

In this paper, we consider the high order impulsive differential equation on infinite interval D 0 + α u t + f t , u t , J 0 + β u t , D 0 + α − 1 u t = 0 , t ∈ 0 , ∞ ∖ t k k = 1 m △ u t k = I k u t k , t = t k , k = 1 , … , m u 0 = u ′ 0 = ⋯ = u n − 2 0 = 0 , D 0 + α − 1 u ∞ = u 0 By applying Schauder fixed points and Altman fixed points, we obtain some new results on the existence of solutions. The nonlinear term of the equation contains fractional integral operator J β u t and lower order derivative operator D 0 + α − 1 u t . An example is presented to illustrate our results.

Ambiguous notation of Grünwald-Letnikov differintegral

The paper discussed the problem of Grünwald-Letnikov differintegral notation in which non-integer order can be incorrectly interpreted as a higher or lower order derivative. Taking the problem into consideration the author’s proposal is new notation of differintegrals.