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17 pages, 751 KiB

Open AccessArticle

Finite-Time Stability of a Class of Nonstationary Nonlinear Fractional Order Time Delay Systems: New Gronwall–Bellman Inequality Approach

by Mihailo P. Lazarević, Stjepko Pišl and Darko Radojević

Mathematics 2025, 13(9), 1490; https://doi.org/10.3390/math13091490 - 30 Apr 2025

Viewed by 273

Abstract

This paper aims to analyze finite-time stability (FTS) for a class of nonstationary nonlinear two-term fractional-order time-delay systems with

α, β \in (0, 2)

. Using a new type of generalized Gronwall–Bellman inequality, we derive new FTS stability criteria for these [...] Read more.

This paper aims to analyze finite-time stability (FTS) for a class of nonstationary nonlinear two-term fractional-order time-delay systems with

α, β \in (0, 2)

. Using a new type of generalized Gronwall–Bellman inequality, we derive new FTS stability criteria for these systems in terms of the Mittag–Leffler function. We demonstrate that our theoretical results are less conservative than those presented in the existing literature. Finally, we provide three numerical examples using a modified Adams–Bashforth–Moulton algorithm to illustrate the applicability of the proposed stability conditions. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing for Applied Mathematics)

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20 pages, 345 KiB

Open AccessArticle

A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives

by Abdelhamid Mohammed Djaouti, Zareen A. Khan, Muhammad Imran Liaqat and Ashraf Al-Quran

Mathematics 2024, 12(11), 1654; https://doi.org/10.3390/math12111654 - 24 May 2024

Cited by 8 | Viewed by 1471

Abstract

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We [...] Read more.

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for

p = 2

to results in the

L^{p}

space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)

21 pages, 396 KiB

Open AccessArticle

Existence, Uniqueness, and Averaging Principle of Fractional Neutral Stochastic Differential Equations in the

L^{p}

Space with the Framework of the Ψ-Caputo Derivative

by Abdelhamid Mohammed Djaouti, Zareen A. Khan, Muhammad Imran Liaqat and Ashraf Al-Quran

Mathematics 2024, 12(7), 1037; https://doi.org/10.3390/math12071037 - 30 Mar 2024

Cited by 9 | Viewed by 1249

Abstract

In this research work, we use the concepts of contraction mapping to establish the existence and uniqueness results and also study the averaging principle in

L^{p}

space by using Jensen’s, Grönwall–Bellman’s, Hölder’s, and Burkholder–Davis–Gundy’s inequalities, and the interval translation technique for a [...] Read more.

In this research work, we use the concepts of contraction mapping to establish the existence and uniqueness results and also study the averaging principle in

L^{p}

space by using Jensen’s, Grönwall–Bellman’s, Hölder’s, and Burkholder–Davis–Gundy’s inequalities, and the interval translation technique for a class of fractional neutral stochastic differential equations. We establish the results within the framework of the

Ψ

-Caputo derivative. We generalize the two situations of

p = 2

and the Caputo derivative with the findings that we obtain. To help with the understanding of the theoretical results, we provide two applied examples at the end. Full article

(This article belongs to the Special Issue Recent Research on Fractional Calculus: Theory and Applications)

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16 pages, 322 KiB

Open AccessArticle

A Result Regarding Finite-Time Stability for Hilfer Fractional Stochastic Differential Equations with Delay

by Man Li, Yujun Niu and Jing Zou

Fractal Fract. 2023, 7(8), 622; https://doi.org/10.3390/fractalfract7080622 - 15 Aug 2023

Cited by 5 | Viewed by 1509

Abstract

Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the [...] Read more.

Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the existence and uniqueness of understanding, respectively. Further, finite-time stability is obtained using the generalized Grönwall–Bellman inequality. As verification, an example is provided to support the theoretical results. Full article

(This article belongs to the Special Issue Analysis of Caputo-Type Fractional Derivatives and Differential Equations)

16 pages, 1192 KiB

Open AccessArticle

Robust Stabilization for Uncertain Non-Minimum Phase Switched Nonlinear System under Arbitrary Switchings

by Khalil Jouili and Walid Belhadj

Symmetry 2023, 15(3), 596; https://doi.org/10.3390/sym15030596 - 25 Feb 2023

Cited by 4 | Viewed by 1455

Abstract

This paper addresses the problem of stabilization of non-minimum phase switched nonlinear systems where the internal dynamics with symmetries or non-symmetries of each mode may be unstable. The authors initially build a stabilizing Lyapunov controller for each mode in order to stabilize its [...] Read more.

This paper addresses the problem of stabilization of non-minimum phase switched nonlinear systems where the internal dynamics with symmetries or non-symmetries of each mode may be unstable. The authors initially build a stabilizing Lyapunov controller for each mode in order to stabilize its own unstable internal dynamics. The proposed approach is based on the exact input-output feedback linearization technique and the Lyapunov stability theory. The stability results for non-minimum phase switched nonlinear systems with arbitrary switching rules are then obtained using generalized Gronwall–Bellman inequalities. Finally, numerical examples are provided to demonstrate the efficacy of the achieved results. Full article

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16 pages, 312 KiB

Open AccessArticle

On Some Generalizations of Integral Inequalities in n Independent Variables and Their Applications

by Waleed Abuelela, Ahmed A. El-Deeb and Dumitru Baleanu

Symmetry 2022, 14(11), 2257; https://doi.org/10.3390/sym14112257 - 27 Oct 2022

Cited by 1 | Viewed by 1302

Abstract

Throughout this article, generalizations of some Grónwall–Bellman integral inequalities for two real-valued unknown functions in n independent variables are introduced. We are looking at some novel explicit bounds of a particular class of Young and Pachpatte integral inequalities. The results in this paper [...] Read more.

Throughout this article, generalizations of some Grónwall–Bellman integral inequalities for two real-valued unknown functions in n independent variables are introduced. We are looking at some novel explicit bounds of a particular class of Young and Pachpatte integral inequalities. The results in this paper can be utilized as a useful way to investigate the uniqueness, boundedness, continuousness, dependence and stability of nonlinear hyperbolic partial integro-differential equations. To highlight our research advantages, several implementations of these findings will be presented. Young’s method, which depends on a Riemann method, will follow to prove the key results. Symmetry plays an essential role in determining the correct methods for solving dynamic inequalities. Full article

(This article belongs to the Section Mathematics)

14 pages, 318 KiB

Open AccessArticle

On Some Dynamic (ΔΔ)^∇- Gronwall–Bellman–Pachpatte-Type Inequalities on Time Scales and Its Applications

by Ahmed A. El-Deeb, Alaa A. El-Bary and Jan Awrejcewicz

Symmetry 2022, 14(9), 1902; https://doi.org/10.3390/sym14091902 - 11 Sep 2022

Cited by 3 | Viewed by 1632

Abstract

In the present paper, some new generalizations of dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales are established. Some integral and discrete Gronwall–Bellman–Pachpatte-type inequalities that are given as special cases of main results are original. The main results are proved by using the dynamic [...] Read more.

In the present paper, some new generalizations of dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales are established. Some integral and discrete Gronwall–Bellman–Pachpatte-type inequalities that are given as special cases of main results are original. The main results are proved by using the dynamic Leibniz integral rule on time scales. To highlight our research advantages, several implementations of these findings are presented. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. Full article

(This article belongs to the Section Mathematics)

14 pages, 314 KiB

Open AccessArticle

Some Generalizations of (∇∇)^Δ–Gronwall–Bellman–Pachpatte Dynamic Inequalities on Time Scales with Application

by Ahmed A. El-Deeb, Alaa A. El-Bary and Jan Awrejcewicz

Symmetry 2022, 14(9), 1823; https://doi.org/10.3390/sym14091823 - 2 Sep 2022

Cited by 1 | Viewed by 1578

Abstract

As a new usage of Leibniz integral rule on time scales, we proved some new extensions of dynamic Gronwall–Pachpatte-type inequalities on time scales. Our results extend some existing results in the literature. Some integral and discrete inequalities are obtained as special cases of [...] Read more.

As a new usage of Leibniz integral rule on time scales, we proved some new extensions of dynamic Gronwall–Pachpatte-type inequalities on time scales. Our results extend some existing results in the literature. Some integral and discrete inequalities are obtained as special cases of the main results. The inequalities proved here can be used in the analysis as handy tools to study the stability, boundedness, existence, uniqueness and oscillation behavior for some kinds of partial dynamic equations on time scales. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. Full article

(This article belongs to the Section Mathematics)

15 pages, 300 KiB

Open AccessArticle

Some Gronwall–Bellman Inequalities on Time Scales and Their Continuous Forms: A Survey

by Francesca Barich

Symmetry 2021, 13(2), 198; https://doi.org/10.3390/sym13020198 - 26 Jan 2021

Cited by 6 | Viewed by 2589

Abstract

Some generalizations of the Gronwall–Bellman (G–B) inequality are presented in this paper in continuous form and on time scales. After S. Hilger introduced the time scales theory in 1988, over the years many mathematicians have studied new versions of this inequality according to [...] Read more.

Some generalizations of the Gronwall–Bellman (G–B) inequality are presented in this paper in continuous form and on time scales. After S. Hilger introduced the time scales theory in 1988, over the years many mathematicians have studied new versions of this inequality according to new results; the purpose of this paper is to present some of them. Therefore, in the Introduction, some generalizations of G–B inequality in continuous forms, linear and nonlinear are presented. In the second section, some important and interesting results on time scales theory are given. In the third and main part of our paper, G–B inequalities on time scales and their possible connection with G–B inequalities presented in the introduction are investigated. In particular, in the third section of this work, more attention is given to G–B type inequalities on time scales discussed in the last four years. Full article

(This article belongs to the Special Issue Symmetry in Abstract Differential Equations)

20 pages, 351 KiB

Open AccessArticle

Dissipativity of Fractional Navier–Stokes Equations with Variable Delay

by Lin F. Liu and Juan J. Nieto

Mathematics 2020, 8(11), 2037; https://doi.org/10.3390/math8112037 - 16 Nov 2020

Cited by 2 | Viewed by 2145

Abstract

We use classical Galerkin approximations, the generalized Aubin–Lions Lemma as well as the Bellman–Gronwall Lemma to study the asymptotical behavior of a two-dimensional fractional Navier–Stokes equation with variable delay. By modifying the fractional Halanay inequality and the comparison principle, we investigate the dissipativity [...] Read more.

We use classical Galerkin approximations, the generalized Aubin–Lions Lemma as well as the Bellman–Gronwall Lemma to study the asymptotical behavior of a two-dimensional fractional Navier–Stokes equation with variable delay. By modifying the fractional Halanay inequality and the comparison principle, we investigate the dissipativity of the corresponding system, namely, we obtain the existence of global absorbing set. Besides, some available results are improved in this work. The existence of a global attracting set is still an open problem. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)

17 pages, 701 KiB

Open AccessArticle

Finite-Time Mittag–Leffler Synchronization of Neutral-Type Fractional-Order Neural Networks with Leakage Delay and Time-Varying Delays

by Călin-Adrian Popa and Eva Kaslik

Mathematics 2020, 8(7), 1146; https://doi.org/10.3390/math8071146 - 13 Jul 2020

Cited by 13 | Viewed by 3222

Abstract

This paper studies fractional-order neural networks with neutral-type delay, leakage delay, and time-varying delays. A sufficient condition which ensures the finite-time synchronization of these networks based on a state feedback control scheme is deduced using the generalized Gronwall–Bellman inequality. Then, a different state [...] Read more.

This paper studies fractional-order neural networks with neutral-type delay, leakage delay, and time-varying delays. A sufficient condition which ensures the finite-time synchronization of these networks based on a state feedback control scheme is deduced using the generalized Gronwall–Bellman inequality. Then, a different state feedback control scheme is employed to realize the finite-time Mittag–Leffler synchronization of these networks by using the fractional-order extension of the Lyapunov direct method for Mittag–Leffler stability. Two numerical examples illustrate the feasibility and the effectiveness of the deduced sufficient criteria. Full article

(This article belongs to the Special Issue Advances and New Trends in Modeling and Control of Neural Network Models)

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15 pages, 310 KiB

Open AccessArticle

A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative

by Jehad Alzabut, Weerawat Sudsutad, Zeynep Kayar and Hamid Baghani

Mathematics 2019, 7(8), 747; https://doi.org/10.3390/math7080747 - 15 Aug 2019

Cited by 11 | Viewed by 3766

Abstract

New versions of a Gronwall–Bellman inequality in the frame of the generalized (Riemann–Liouville and Caputo) proportional fractional derivative are provided. Before proceeding to the main results, we define the generalized Riemann–Liouville and Caputo proportional fractional derivatives and integrals and expose some of their [...] Read more.

New versions of a Gronwall–Bellman inequality in the frame of the generalized (Riemann–Liouville and Caputo) proportional fractional derivative are provided. Before proceeding to the main results, we define the generalized Riemann–Liouville and Caputo proportional fractional derivatives and integrals and expose some of their features. We prove our main result in light of some efficient comparison analyses. The Gronwall–Bellman inequality in the case of weighted function is also obtained. By the help of the new proposed inequalities, examples of Riemann–Liouville and Caputo proportional fractional initial value problems are presented to emphasize the solution dependence on the initial data and on the right-hand side. Full article

(This article belongs to the Special Issue Inequalities)

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