Comparison theorems and distributions of solutions to uncertain fractional difference equations

Lu, Qinyun; Zhu, Yuanguo

doi:10.1016/j.cam.2020.112884

Cited by 22 publications

(5 citation statements)

References 16 publications

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“…Lemma 2.4 (see [29]) Suppose that ψ is a given real-valued function. The difference equation (1.2) has one solution if and only if y is a solution of the following fractional Taylor difference equation:…”

Section: Preliminariesmentioning

confidence: 99%

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Existence of solutions for a class of nonlinear fractional difference equations of the Riemann–Liouville type

et al. 2022

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Nonlinear fractional difference equations are studied deeply and extensively by many scientists by using fixed-point theorems on different types of function spaces. In this study, we combine fixed-point theory with a set of falling fractional functions in a Banach space to prove the existence and uniqueness of solutions of a class of fractional difference equations. The most important part of this article is devoted to correcting a significant mistake made in the literature in using the power rule by providing further conditions for its validity. Also, we provide specific conditions under which difference equations have attractive solutions and the solutions are also asymptotically stable. Furthermore, we construct some fractional difference examples in order to illustrate the validity of the observed results.

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Section: Preliminariesmentioning

confidence: 99%

“…Next, in [29] and [30], the problem of the power rule was solved by considering a new difference equation as follows:…”

Section: Introductionmentioning

confidence: 99%

Existence of solutions for a class of nonlinear fractional difference equations of the Riemann–Liouville type

et al. 2022

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“…Following this, discussions on distribution stability [12], almost sure stability [13] and exponential stability [14] of UDEs were conducted. Subsequently, Lu and Zhu [15] investigated uncertain fractional differential equation, while Wang [16] researched uncertain catenary differential equation. Recently, some examples in the financial field have verified that UDEs are more suitable for fitting data, such as Alibaba stock price [17], currency exchange rate [18], and interest rate [19].…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of uncertain simple pendulum equation ^*

Qiu,

Liu

2024

J. Phys. A: Math. Theor.

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The law of motion of a simple pendulum system is described by an uncertain simple pendulum equation which is a second-order uncertain differential equation driven by Liu process. The stability of a simple pendulum system refers to whether the system tends to the equilibrium state under small perturbation. In order to discuss the sensitivity of the uncertain simple pendulum equation to the perturbation in the initial state, we give the concept of many kinds of stability of the uncertain simple pendulum equation, including almost deterministic stability, distributional stability and exponential stability. And, the sufficient conditions of almost deterministic stability, distributional stability and exponential stability of the uncertain simple pendulum equation are proved respectively.

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“…Furthermore, the concept of discrete fractional calculus was rooted in the few decades; it gets attention of many researchers recently because of their inquisitive thinking (see [5][6][7] and references therein). The main reason is that these problems and operators have a wide range of practical applications, such as mathematical analysis [8,9], stability analysis [10][11][12], probability and statistics [13][14][15], geometry [16,17], ecology [18,19], and topology [20][21][22].…”

Section: Introductionmentioning

confidence: 99%

On convexity analysis for discrete delta Riemann–Liouville fractional differences analytically and numerically

et al. 2023

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In this paper, we focus on the analytical and numerical convexity analysis of discrete delta Riemann–Liouville fractional differences. In the analytical part of this paper, we give a new formula for the discrete delta Riemann-Liouville fractional difference as an alternative definition. We establish a formula for the $\Delta ^{2}$ Δ 2 , which will be useful to obtain the convexity results. We examine the correlation between the positivity of $({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})$ ( w 0 RL Δ α f ) ( t ) and convexity of the function. In view of the basic lemmas, we define two decreasing subsets of $(2,3)$ ( 2 , 3 ) , $\mathscr{H}_{\mathrm{k},\epsilon}$ H k , ϵ and $\mathscr{M}_{\mathrm{k},\epsilon}$ M k , ϵ . The decrease of these sets allows us to obtain the relationship between the negative lower bound of $({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})$ ( w 0 RL Δ α f ) ( t ) and convexity of the function on a finite time set $\mathrm{N}_{w_{0}}^{\mathrm{P}}:=\{w_{0}, w_{0}+1, w_{0}+2,\dots , \mathrm{P}\}$ N w 0 P : = { w 0 , w 0 + 1 , w 0 + 2 , … , P } for some $\mathrm{P}\in \mathrm{N}_{w_{0}}:=\{w_{0}, w_{0}+1, w_{0}+2,\dots \}$ P ∈ N w 0 : = { w 0 , w 0 + 1 , w 0 + 2 , … } . The numerical part of the paper is dedicated to examinin the validity of the sets $\mathscr{H}_{\mathrm{k},\epsilon}$ H k , ϵ and $\mathscr{M}_{\mathrm{k},\epsilon}$ M k , ϵ for different values of k and ϵ. For this reason, we illustrate the domain of solutions via several figures explaining the validity of the main theorem.

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Comparison theorems and distributions of solutions to uncertain fractional difference equations

Cited by 22 publications

References 16 publications

Existence of solutions for a class of nonlinear fractional difference equations of the Riemann–Liouville type

Existence of solutions for a class of nonlinear fractional difference equations of the Riemann–Liouville type

Stability analysis of uncertain simple pendulum equation ^*

On convexity analysis for discrete delta Riemann–Liouville fractional differences analytically and numerically

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Comparison theorems and distributions of solutions to uncertain fractional difference equations

Cited by 22 publications

References 16 publications

Existence of solutions for a class of nonlinear fractional difference equations of the Riemann–Liouville type

Existence of solutions for a class of nonlinear fractional difference equations of the Riemann–Liouville type

Stability analysis of uncertain simple pendulum equation *

On convexity analysis for discrete delta Riemann–Liouville fractional differences analytically and numerically

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Product

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Stability analysis of uncertain simple pendulum equation ^*