Solvability for Impulsive Fractional Langevin Equation

Xu, Mengrui; Sun, Shurong; Han, Zhenlai

doi:10.11948/20180170

Cited by 3 publications

(4 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then, by Definition 2, we have 0 D α t y(t) = 0. Therefore, combining (13) with the Fractional Comparison Principle (Lemma 4), we can obtain V(x(t)) ≤ V(x(0)) for all t > 0.…”

Section: Boundedness Resultsmentioning

confidence: 96%

“…In recent years, fractional calculus has attracted much attention from researchers [9][10][11][12][13][14]. The fractional derivative at time t is not defined locally and depends on the total effects of the classical integer-order derivatives on the interval [0, t], so it can be used to describe the variation of a system in which the instantaneous change rate depends on the past state, which is called the memory effect in a visualized manner [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Influences of the Order of Derivative on the Dynamical Behavior of Fractional-Order Antisymmetric Lotka–Volterra Systems

2023

Fractal Fract

View full text Add to dashboard Cite

This paper studies the dynamic behavior of a class of fractional-order antisymmetric Lotka–Volterra systems. The influences of the order of derivative on the boundedness and stability are characterized by analyzing the first-order and 0<α<1-order antisymmetric Lotka–Volterra systems separately. We show that the order does not affect the boundedness but affects the stability. All solutions of the first-order system are periodic, while the 0<α<1-order system has no non-trivial periodic solution. Furthermore, the 0<α<1-order system can be reduced on a two-dimensional space and the reduced system is asymptotically stable, regardless of how close to zero the order of the derivative used is. Some numerical simulations are presented to better verify the theoretical analysis.

show abstract

“…Then, by Definition 2, we have 0 D α t y(t) = 0. Therefore, combining (13) with the Fractional Comparison Principle (Lemma 4), we can obtain V(x(t)) ≤ V(x(0)) for all t > 0.…”

Section: Boundedness Resultsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Influences of the Order of Derivative on the Dynamical Behavior of Fractional-Order Antisymmetric Lotka–Volterra Systems

2023

Fractal Fract

View full text Add to dashboard Cite

show abstract

“…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

({, \begin{matrix} left left \\ array_{(0 +, 0 +)}^{C} D_{(s, t)}^{e} Υ (s, t) = F (s, t, Υ (s, t)), & array (s, t) \in Θ, s \neq s_{j} (j = 1,2, \dots, J), \\ array Υ (s_{j}^{+}, t) - Υ (s_{j}^{-}, t) = U_{j} (Υ (s_{j}^{-}, t)), & array t \in [0, c], j = 1,2, \dots, J, \\ array Υ (s, 0) = η (s), Υ (0, t) = ξ \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

Impulsive fractional partial differential system and its correct equivalent integral equation

Zhang,

Liu,

Peng

et al. 2024

Math Methods in App Sciences

View full text Add to dashboard Cite

It is found that there are two piecewise functions to satisfy the conditions in the impulsive fractional partial differential system (IFrPDS), which deduce that the three different equivalent integral equations of the IFrPDS given in the cited papers are inappropriate. Next, by applying two limit properties of the IFrPDS and the properties of piecewise function, the new formula of equivalent integral equation of the IFrPDS is discovered that is the integral equation with an arbitrary continuously differentiable function of on to reveal the non‐uniqueness of the IFrPDE's solution. Finally, an example is provided to expound the computation of the equivalent integral equation of the IFrPDS.

show abstract

“…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. )…”

Section: Introductionmentioning

confidence: 99%

On the equivalent integral equality of impulsive Hadamard fractional order system

Zhang

2024

Preprint

View full text Add to dashboard Cite

For the impulsive Hadamard fractional order system (IHFrOS), we apply the fractional property of segment function to construct the equivalent integral equality (EIE) of a special case of the IHFrOS to deduce that the proposed EIE of the IHFrOS in cited paper is incomplete. And next, we combine the limit analysis with the linear additivity of impulse effects to find the correct IHFrOS’s EIE that is an integral equation of the combination of ψ( t) and Ψ j ( t ) ( j=1 ,2 ,…,N) with an arbitrary constant, which reveal the nonuniqueness of the IHFrOS’s solution. Furthermore, we also give another IHFrOS’s EIE by the connection between two IHFrOSs. Finally, we show the calculation process of the EIE and the nonuniqueness of solution for two IHFrOSs by applying two numerical examples.

show abstract

Solvability for Impulsive Fractional Langevin Equation

Cited by 3 publications

References 21 publications

Influences of the Order of Derivative on the Dynamical Behavior of Fractional-Order Antisymmetric Lotka–Volterra Systems

Influences of the Order of Derivative on the Dynamical Behavior of Fractional-Order Antisymmetric Lotka–Volterra Systems

Impulsive fractional partial differential system and its correct equivalent integral equation

On the equivalent integral equality of impulsive Hadamard fractional order system

Contact Info

Product

Resources

About