On stability of a class of second alpha-order fractal differential equations

Cemil Tunç; Alireza Khalili Golmankhaneh; Cemil Tunç; Alireza Khalili Golmankhaneh

doi:10.3934/math.2020141

AIMS Mathematics

2020, Volume 5, Issue 3: 2126-2142. doi: 10.3934/math.2020141

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Research article

On stability of a class of second alpha-order fractal differential equations

Cemil Tunç ¹,
Alireza Khalili Golmankhaneh ^{2
,
,}

1.
Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080, Turkey
2.
Young Researchers and Elite Club, Urmia Branch, Islamic Azad University, Urmia, Iran

Received: 03 November 2019 Accepted: 11 February 2020 Published: 26 February 2020
MSC : 28A78, 28A80, 35B35, 35B40, 81Q35

In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions that are not differentiable or integrable on totally disconnected fractal sets such as middle-μ Cantor sets. Analogues of the Lyapunov functions and their features are given for asymptotic behaviors of fractal differential equations. The stability of fractal differentials in the sense of Lyapunov is defined. For the suggested fractal differential equations, sufficient conditions for the stability and uniform boundedness and convergence of the solutions are presented and proved. We present examples and graphs for more details of the results.
- fractal calculus,
- staircase function,
- Cantor-like sets,
- fractal stability,
- fractal convergence
Citation: Cemil Tunç, Alireza Khalili Golmankhaneh. On stability of a class of second alpha-order fractal differential equations[J]. AIMS Mathematics, 2020, 5(3): 2126-2142. doi: 10.3934/math.2020141

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Abstract

In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions that are not differentiable or integrable on totally disconnected fractal sets such as middle-μ Cantor sets. Analogues of the Lyapunov functions and their features are given for asymptotic behaviors of fractal differential equations. The stability of fractal differentials in the sense of Lyapunov is defined. For the suggested fractal differential equations, sufficient conditions for the stability and uniform boundedness and convergence of the solutions are presented and proved. We present examples and graphs for more details of the results.

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