Functional and Impulsive Differential Equations of Fractional Order

Stamova, Ivanka; Stamov, Gani

doi:10.1201/9781315367453

Cited by 86 publications

(71 citation statements)

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“…by x(t) = x(t; t 0 , x 0 ). Note that, according to the second (impulsive) condition in (1) [36,37], the solutions x(t) of type (1) systems are piecewise continuous functions that have points of discontinuity of the first kind t k and are left continuous at these moments. For such functions, the following identities are satisfied:…”

Section: Definition 2 Systemmentioning

confidence: 99%

“…. , we will introduce the class V q t k of piecewise continuous auxiliary Lyapunov-type functions that will be used in our research (see, for example, [36,41] and the references therein).…”

Section: Definition 2 Systemmentioning

confidence: 99%

“…Furthermore, we will use a comparison result [29,36], and, for this reason together with (1), we consider the comparison equation:…”

Section: Definition 2 Systemmentioning

confidence: 99%

“…For example, in [29], impulsive perturbations are not considered. It is well known that impulses can significantly affect the stability behavior of a system, and, therefore, considering impulses into systems of differential equations is a very worthwhile research project with potential applications [34,36,37]. Impulsive systems with conformable derivatives have been considered only in [38,39], where some oscillation criteria and inequalities are proposed.…”

mentioning

confidence: 99%

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Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds

2019

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In this paper, an impulsive fractional-like system of differential equations is introduced. The notions of practical stability and boundedness with respect to h-manifolds for fractional-like differential equations are generalized to the impulsive case. For the first time in the literature, Lyapunov-like functions and their derivatives with respect to impulsive fractional-like systems are defined. As an application, an impulsive fractional-like system of Lotka–Volterra equations is considered and new criteria for practical exponential stability are proposed. In addition, the uncertain case is also investigated.

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Section: Definition 2 Systemmentioning

confidence: 99%

“…. , we will introduce the class V q t k of piecewise continuous auxiliary Lyapunov-type functions that will be used in our research (see, for example, [36,41] and the references therein).…”

Section: Definition 2 Systemmentioning

confidence: 99%

“…Furthermore, we will use a comparison result [29,36], and, for this reason together with (1), we consider the comparison equation:…”

Section: Definition 2 Systemmentioning

confidence: 99%

mentioning

confidence: 99%

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Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds

2019

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“…Some authors use the derivative defined by (2.3) to study stability properties of delay fractional differential equations ( [22,24,26]) and delayed reaction-diffusion cellular neural networks of fractional order ( [25]). The proofs are based on the following comparison result (we will give it with appropriate technical corrections).…”

Section: Remarks On Some Applications Of Derivatives Of Lyapunov Funcmentioning

confidence: 99%

Lyapunov functions to Caputo reaction-diffusion fractional neural networks with time-varying delays

Agarwal¹,

Hristova²,

O’Regan³

2018

J. Math. Computer Sci.

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A reaction diffusion equation with a Caputo fractional derivative in time and with time-varying delays is considered. Stability properties of the solutions are studied via the direct Lyapunov method and arbitrary Lyapunov functions (usually quadratic Lyapunov functions are used). In this paper we give a brief overview of the most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. These derivatives are applied to various types of reaction-diffusion fractional neural network with variable coefficients and time-varying delays. We show the quadratic Lyapunov functions and their Caputo fractional derivatives are not applicable in some cases when one studies stability properties. Some sufficient conditions for stability are obtained and we illustrate our theory on a particular nonlinear Caputo reaction-diffusion fractional neural network with time dependent delays.

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Persistence of traveling waves to the time fractional Keller‐Segel system with a small parameter

Chen

Alsaedi

Stamova

2023

Math Methods in App Sciences

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This paper aims to investigate the time fractional Keller‐Segel system with a small parameter. After the fractional order traveling wave transformation, the heteroclinic orbit of the degenerate time fractional Keller‐Segel system is demonstrated through the method of constructing a suitable invariant region. Moreover, the persistence of traveling waves in the system with a small parameter can be further illustrated. The results are mainly reliance on the application of geometric singular perturbation theory and Fredholm theorem, which are fundamental theoretical frameworks for dealing with problems of complexity and high dimensionality. Eventually, the asymptotic behavior is depicted by the asymptotic theory to illustrate the rate of decay for traveling waves.

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Functional and Impulsive Differential Equations of Fractional Order

Cited by 86 publications

References 0 publications

Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds

Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds

Lyapunov functions to Caputo reaction-diffusion fractional neural networks with time-varying delays

Persistence of traveling waves to the time fractional Keller‐Segel system with a small parameter

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