The Right Equivalent Integral Equation of Impulsive Caputo Fractional-Order System of Order ϵ∈(1,2)

Abstract

For the impulsive fractional-order system (IFrOS) of order ϵ∈(1,2), there have appeared some conflicting equivalent integral equations in existing studies. However, we find two fractional-order properties of piecewise function and use them to verify that these given equivalent integral equations have some defects to not be the equivalent integral equation of the IFrOS. For the IFrOS, its limit property shows the linear additivity of the impulsive effects. For the IFrOS, we use the limit analysis and the linear additivity of the impulsive effects to find its correct equivalent integral equation, which is a combination of some piecewise functions with two arbitrary constants; that is, the solution of the IFrOS is a general solution. Finally, a numerical example is given to show the equivalent integral equation and the non-uniqueness of the solution of the IFrOS.

Impulsive fractional partial differential system and its correct equivalent integral equation

“…Similarity to the proof of Theorem 1 in [45], we also assume that $$$ e\left(s,t\right) $$$ is equal to $$$$ g\left({U}_j\left(\mathrm{Y}\left({s}_j^{-},t\right)\right)\right)\underset{U_j\left(\mathrm{Y}\left({s}_j^{-},t\right)\right)\to 0}{\lim }e\left(s,t\right), $$$$ (where $$$ g\left(\cdotp \right) $$$ is an undetermined function) with …”

“…Impulsive fractional order system (IFrOS) has become a focus of research and several hundreds of articles are found by searching the topic of IFrOS from Web of Science. For the IFrOS, its equivalent integral equation is key in studying numerical solution [18, 19], existence of solution [20–28], oscillation behavior [29, 30], periodic motion [31], solvability [32], asymptotic behavior of solution [33], stability [34–36], integral solution [37–45], and so on. Furthermore, the impulsive fractional partial differential order system (IFrPDOS) was firstly introduced in [46] by …”

On the equivalent integral equality of impulsive Hadamard fractional order system

“…The impulsive fractional differential equations (IFrDE) have recently been a research highlight that there have appeared many articles regarding the subject of the IFrDE on the Web of science. For the system of IFrDE, the equivalent integral equality (EIE) serves as an key tool in study of these properties (such as numerical solution [3,4], oscillation behavior [5,6], stability [7][8][9], periodic motion [10], solvability [11], asymptotic behavior of solution [12], existence of solution [13][14][15][16][17][18][19][20][21][22][23][24][25] and nonuniqueness of solution [26] etc. )…”