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

^{1}

^{2}

^{3}

^{*}

^{†}

^{‡}

*Fractal Fract’s*Editorial Board Members)

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Lemma**

**1.**

**Lemma**

**2.**

**Remark**

**1.**

**Definition**

**3.**

**Definition**

**4.**

- V is defined on G, V has nonnegative values and $V(t,0)=0$ for $t\ge {t}_{k}$;
- V is continuous in G, $q$—differentiable in t and locally Lipschitz continuous with respect to its second argument on each of the sets ${G}_{k}$;
- For each $k=0,1,2,\dots $ and $x\in {\mathbb{R}}^{n}$, there exist the finite limits$$V({t}_{k}^{-},x)=\underset{\stackrel{t\to {t}_{k}}{t<{t}_{k}}}{lim}V(t,x),\phantom{\rule{1.em}{0ex}}V({t}_{k}^{+},x)=\underset{\stackrel{t\to {t}_{k}}{t>{t}_{k}}}{lim}V(t,x),$$

**Definition**

**5.**

**Lemma**

**3.**

**Lemma**

**4.**

## 3. Main Results

#### 3.1. Practical Stability Criteria

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 3.2. Boundedness Results

**Definition**

**6.**

**Definition**

**7.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

## 4. Applications

**Theorem**

**6.**

**Proof.**

**Remark**

**2.**

**Definition**

**8.**

**Theorem**

**7.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations, 1st ed.; Elsevier Science B.V: Amsterdam, The Netherlands, 2006; ISBN 0444518320. [Google Scholar]
- Podlubny, I. Fractional Differential Equations, 1st ed.; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Cattani, C.; Srivastava, H.M.; Yang, X.-J. (Eds.) Fractional Dynamics, 1st ed.; De Gryuter: Berlin, Germany, 2015; ISBN 978-3-11-047209-7. [Google Scholar]
- Ahmad, B.; Alsaedi, A.; Ntouyas, S.K.; Tariboon, J. Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, 1st ed.; Springer: Cham, Switzerland, 2017; ISBN 978-3-319-84831-0, 978-3-319-52141-1. [Google Scholar]
- Al-Ghafri, K.S.; Rezazadeh, H. Solitons and other solutions of (3 + 1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation. Appl. Math. Nonlinear Sci.
**2019**, 4, 289–304. [Google Scholar] [CrossRef] - Atangana, A.; Gómez–Aguilar, J.F. Numerical approximation of Riemann–Liouville definition of fractional derivative: From Riemann–Liouville to Atangana–Baleanu. Numer. Methods Partial. Differ. Equ.
**2018**, 34, 1502–1523. [Google Scholar] [CrossRef] - Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J.J. Fractional Calculus: Models and Numerical Methods, 2nd ed.; World Scientific: Singapore, 2016; ISBN 9813140038. [Google Scholar]
- Bayın, S.S. Definition of the Riesz derivative and its application to space fractional quantum mechanics. J. Math. Phys.
**2016**, 57, 123501. [Google Scholar] [CrossRef] [Green Version] - Gao, W.; Ghanbari, B.; Baskonus, H.M. New numerical simulations for some real world problems with Atangana–Baleanu fractional derivative. Chaos Solitons Fractals
**2019**, 128, 34–43. [Google Scholar] [CrossRef] - Jarad, F.; Abdeljawad, T.; Hammouch, Z. On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative. Chaos Solitons Fractals
**2018**, 117, 16–20. [Google Scholar] [CrossRef] - Morales-Delgado, V.F.; Gómez-Aguilar, J.F.; Saad, K.M.; Khan, M.A.; Agarwal, P. Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach. Physica A
**2019**, 523, 48–65. [Google Scholar] [CrossRef] - Saqib, M.; Khan, I.; Shafie, S. Application of Atangana–Baleanu fractional derivative to MHD channel flow of CMC-based-CNT’s nanofluid through a porous medium. Chaos Solitons Fractals
**2018**, 116, 79–85. [Google Scholar] [CrossRef] - Taneco–Heránndez, M.A.; Morales–Delgado, V.F.; Gómez-Aguilar, J.F. Fractional Kuramoto–Sivashinsky equation with power law and stretched Mittag-Leffler kernel. Physica A
**2019**, 527, 121085. [Google Scholar] [CrossRef] - De Oliveira, E.C.; Tenreiro Machado, J.A. A review of definitions for fractional derivatives and integral. Math. Probl. Eng.
**2014**, 2014, 238459. [Google Scholar] [CrossRef] - Ortigueira, M.D.; Tenreiro Machado, J.A. What is a fractional derivative? J. Comput. Phys.
**2015**, 293, 4–13. [Google Scholar] [CrossRef] - Ortigueira, M.; Machado, J. Which Derivative? Fractal Fract.
**2017**, 1, 3. [Google Scholar] [CrossRef] - Abdeljawad, T. On conformable fractional calculus. J. Comput. Appl. Math.
**2015**, 279, 57–66. [Google Scholar] [CrossRef] - Anderson, D.R.; Ulness, D.J. Properties of the Katugampola fractional derivative with potential application in quantum mechanics. J. Math. Phys.
**2015**, 56, 063502. [Google Scholar] [CrossRef] [Green Version] - Eslami, M.; Rezazadeh, H. The first integral method for Wu-Zhang system with conformable time-fractional derivative. Calcolo
**2016**, 53, 475–485. [Google Scholar] [CrossRef] - Katugampola, U. A new fractional derivative with classical properties. arXiv
**2014**, arXiv:1410.6535. [Google Scholar] - Khalil, R.; Al Horani, M.; Yousef, A.; Sababheh, M. A new definition of fractional derivative. J. Comput. Appl. Math.
**2014**, 264, 65–70. [Google Scholar] [CrossRef] - Pospíšil, M.; Pospíšilova Škripkova, L. Sturm’s theorems for conformable fractional differential equation. Math. Commun.
**2016**, 21, 273–281. [Google Scholar] - Souahi, A.; Ben Makhlouf, A.; Hammami, M.A. Stability analysis of conformable fractional-order nonlinear systems. Indag. Math.
**2017**, 28, 1265–1274. [Google Scholar] [CrossRef] - Yel, G.; Baskonus, H.M. Solitons in conformable time-fractional Wu–Zhang system arising in coastal design. Pramana J. Phys.
**2019**, 93, 57. [Google Scholar] [CrossRef] - Martynyuk, A.A.; Stamova, I.M. Fractional-like derivative of Lyapunov-type functions and applications to the stability analysis of motion. Electron. J. Differ. Equ.
**2018**, 2018, 1–12. [Google Scholar] - Kiskinov, H.; Petkova, M.; Zahariev, A. Remarks about the existence of conformable derivatives and some consequences. arXiv
**2019**, arXiv:1907.03486. [Google Scholar] - Martynyuk, A.A. On the stability of the solutions of fractional-like equations of perturbed motion. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki
**2018**, 6, 9–16. (In Russian) [Google Scholar] [CrossRef] - Martynyuk, A.A.; Stamov, G.; Stamova, I. Integral estimates of the solutions of fractional-like equations of perturbed motion. Nonlinear Anal. Model. Control
**2019**, 24, 138–149. [Google Scholar] [CrossRef] - Martynyuk, A.A.; Stamov, G.; Stamova, I. Practical stability analysis with respect to manifolds and boundedness of differential equations with fractional-like derivatives. Rocky Mt. J. Math.
**2019**, 49, 211–233. [Google Scholar] [CrossRef] - Ballinger, G.; Liu, X. Practical stability of impulsive delay differential equations and applications to control problems. In Optimization Methods and Applications. Applied Optimization; Yang, X., Teo, K.L., Caccetta, L., Eds.; Kluwer: Dordrecht, The Netherlands, 2001; Volume 52, pp. 3–21. [Google Scholar]
- Bernfeld, S.R.; Lakshmikantham, V. Practical stability and Lyapunov functions. Thoku Math. J.
**1980**, 32, 607–613. [Google Scholar] [CrossRef] - Lakshmikantham, V.; Leela, S.; Martynyuk, A.A. Practical Stability of Nonlinear Systems; World Scientific: Teaneck, NJ, USA, 1990; ISBN 981-02-0351-9. [Google Scholar]
- Martynyuk, A.A. (Ed.) Advances in Stability Theory at the End of the 20th Century. Stability and Control: Theory, Methods and Applications, 1st ed.; Taylor and Francis: New York, NY, USA, 2002; ISBN 0-203-16657-4. [Google Scholar]
- Stamova, I.M.; Stamov, G.T. Applied Impulsive Mathematical Models, 1st ed.; Springer: Cham, Switzerland, 2016; ISBN 978-3-319-28060-8. [Google Scholar]
- Yang, C.; Zhang, Q.; Zhou, L. Practical stabilization and controllability of descriptor systems. Int. J. Inf. Syst. Sci.
**2005**, 1, 455–465. [Google Scholar] - Stamova, I.M.; Stamov, G.T. Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications, 1st ed.; CRC Press, Taylor and Francis Group: Boca Raton, FL, USA, 2017; ISBN 9781498764834. [Google Scholar]
- Wang, J.; Feckan, M.; Zhou, Y. A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal.
**2016**, 19, 806–831. [Google Scholar] [CrossRef] - Sitho, S.; Ntouyas, S.K.; Agarwal, P.; Tariboon, J. Noninstantaneous impulsive inequalities via conformable fractional calculus. J. Inequal. Appl.
**2018**, 2018, 261. [Google Scholar] [CrossRef] - Tariboon, J.; Ntouyas, S.K. Oscillation of impulsive conformable fractional differential equations. Open Math.
**2016**, 14, 497–508. [Google Scholar] [CrossRef] [Green Version] - Cicek, M.; Yaker, C.; Gücen, M.B. Practical stability in terms of two measures for fractional order systems in Caputo’s sense with initial time difference. J. Frankl. Inst.
**2014**, 351, 732–742. [Google Scholar] [CrossRef] - Stamova, I.M.; Henderson, J. Practical stability analysis of fractional-order impulsive control systems. ISA Trans.
**2016**, 64, 77–85. [Google Scholar] [CrossRef] [PubMed] - Bernfeld, S.R.; Corduneanu, C.; Ignatyev, A.O. On the stability of invariant sets of functional differential equations. Nonlinear Anal.
**2003**, 55, 641–656. [Google Scholar] [CrossRef] - Bohner, M.; Stamova, I.; Stamov, G. Impulsive control functional differential systems of fractional order: Stability with respect to manifolds. Eur. Phys. J. Spec. Top.
**2017**, 226, 3591–3607. [Google Scholar] [CrossRef] - Smale, S. Stable manifolds for differential equations and diffeomorphisms. Ann. Scuola Norm. Sup. Pisa
**1963**, 3, 97–116. [Google Scholar] - Stamov, G. Lyapunov’s functions and existence of integral manifolds for impulsive differential systems with time-varying delay. Methods Appl. Anal.
**2009**, 16, 291–298. [Google Scholar] - Liu, B.; Liu, X.; Liao, X. Robust stability of uncertain impulsive dynamical systems. J. Math. Anal. Appl.
**2004**, 290, 519–533. [Google Scholar] [CrossRef] [Green Version] - Stamov, G.T.; Alzabut, J.O. Almost periodic solutions in the PC-space for uncertain impulsive dynamical systems. Nonlinear Anal.
**2011**, 74, 4653–4659. [Google Scholar] [CrossRef] - Stamov, G.T.; Simeonov, S.; Stamova, I.M. Uncertain impulsive Lotka–Volterra competitive systems: Robust stability of almost periodic solutions. Chaos Solitons Fractals
**2018**, 110, 178–184. [Google Scholar] [CrossRef] - Aguila-Camacho, N.; Duarte-Mermoud, M.A. Boundedness of the solutions for certain classes of fractional differential equations with application to adaptive systems. ISA Trans.
**2016**, 60, 82–88. [Google Scholar] [CrossRef] - Ahmad, S.; Stamova, I.M. (Eds.) Lotka–Volterra and Related Systems: Recent Developments in Population Dynamics, 1st ed.; Walter de Gruyter: Berlin, Germany, 2013; ISBN 978-3-11-026984-0. [Google Scholar]
- Gopalsamy, K. Stability and Oscillation in Delay Differential Equations of Population Dynamics, 1st ed.; Springer: Dordrecht, The Netherlands, 1992; ISBN 978-0-7923-1594-0. [Google Scholar]
- Takeuchi, Y. Global Dynamical Properties of Lotka–Volterra Systems; World Scientific: Singapore, 1996; ISBN 978-981-02-2471-4, 978-981-4499-63-7. [Google Scholar]
- Li, M.; Duan, Y.; Zhang, W.; Wang, M. The existence of positive periodic solutions of a class of Lotka–Volterra type impulsive systems with infinitely distributed delay. Comput. Math. Appl.
**2005**, 49, 1037–1044. [Google Scholar] [CrossRef] - Liu, X.; Rohlf, K. Impulsive control of a Lotka–Volterra system. IMA J. Math. Control Inform.
**1998**, 15, 269–284. [Google Scholar] [CrossRef] - Liu, Z.; Wu, J.; Tang, R. Permanence and extinction of an impulsive delay competitive Lotka–Volterra model with periodic coefficients. IMA J. Appl. Math.
**2009**, 74, 559–573. [Google Scholar] [CrossRef] - Agrawal, S.K.; Srivastava, M.; Das, S. Synchronization between fractional-order Ravinovich–Fabrikant and Lotka–Volterra systems. Nonlinear Dyn.
**2012**, 69, 2277–2288. [Google Scholar] [CrossRef] - Stamov, G.; Stamova, I.M. On almost periodic processes in impulsive fractional-order competitive systems. J. Math. Chem.
**2018**, 56, 583–596. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Stamov, G.; Martynyuk, A.; Stamova, I.
Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to *h*-Manifolds. *Fractal Fract.* **2019**, *3*, 50.
https://doi.org/10.3390/fractalfract3040050

**AMA Style**

Stamov G, Martynyuk A, Stamova I.
Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to *h*-Manifolds. *Fractal and Fractional*. 2019; 3(4):50.
https://doi.org/10.3390/fractalfract3040050

**Chicago/Turabian Style**

Stamov, Gani, Anatoliy Martynyuk, and Ivanka Stamova.
2019. "Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to *h*-Manifolds" *Fractal and Fractional* 3, no. 4: 50.
https://doi.org/10.3390/fractalfract3040050