STABILITY RESULTS AND EXISTENCE THEOREMS FOR NONLINEAR DELAY-FRACTIONAL DIFFERENTIAL EQUATIONS WITH $ \varphi^*_P $-OPERATOR

Hasib Khan; Cemil Tunç; Aziz Khan

doi:10.11948/20180322

2020 Volume 10 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2020 > 10(2): 584-597

Next Article Previous Article

Hasib Khan, Cemil Tunç, Aziz Khan. STABILITY RESULTS AND EXISTENCE THEOREMS FOR NONLINEAR DELAY-FRACTIONAL DIFFERENTIAL EQUATIONS WITH $ \varphi^*_P $-OPERATOR[J]. Journal of Applied Analysis & Computation, 2020, 10(2): 584-597. doi: 10.11948/20180322

Citation:

Hasib Khan, Cemil Tunç, Aziz Khan. STABILITY RESULTS AND EXISTENCE THEOREMS FOR NONLINEAR DELAY-FRACTIONAL DIFFERENTIAL EQUATIONS WITH $ \varphi^*_P $-OPERATOR[J]. Journal of Applied Analysis & Computation, 2020, 10(2): 584-597. doi: 10.11948/20180322

STABILITY RESULTS AND EXISTENCE THEOREMS FOR NONLINEAR DELAY-FRACTIONAL DIFFERENTIAL EQUATIONS WITH $ \varphi^*_P $-OPERATOR

1.
Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal, Dir Upper, P. O. Box 18000, Khybar Pakhtunkhwa, Pakistan
2.
Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080 Van, Turkey
3.
Department of Mathematics and General Sciences, Prince Sultan University, P.O.Box66833, Riyadh11586, Saudi Arabia

Corresponding author: Email address: cemtunc@yahoo.com(C. Tunc)

Abstract

Abstract

The study of delay-fractional differential equations (fractional DEs) have recently attracted a lot of attention from scientists working on many different subjects dealing with mathematically modeling. In the study of fractional DEs the first question one might raise is whether the problem has a solution or not. Also, whether the problem is stable or not? In order to ensure the answer to these questions, we discuss the existence and uniqueness of solutions (EUS) and Hyers-Ulam stability (HUS) for our proposed problem, a nonlinear fractional DE with $ p $-Laplacian operator and a non zero delay $ \tau>0 $ of order $ n-1<\nu^*,\,\epsilon<n $, for $ n\geq 3 $ in Banach space $ \mathcal{A} $. We use the Caputo's definition for the fractional differential operators $ \mathcal{D}^{\nu^*},\,\, \mathcal{D}^{\epsilon} $. The assumed fractional DE with $ p $-Laplacian operator is more general and complex than that studied by Khan et al. Eur Phys J Plus, (2018);133:26.
- Hybrid fractional differential equations /
- Hyers-Ulam stability /
- Caputo's fractional derivative /
- existence and uniqueness /
- topological degree theory
MSC: 26A33, 34B82, 45N05

FullText(HTML)

References

[1]	A. Ali, B. Samet, K. Shah and R. A. Khan, Existence and stability of solution to a toppled systems of differential equations of non-integer order, Bound Value Prob., 2017, 16, 13 pages. Google Scholar
[2]	D. Baleanu, R.P. Agarwal, H. Mohammadi and S. Rezapour, Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound Value Probl., 2013, 2013(112), 1-8. Google Scholar
[3]	A. Cabada, S. Dimitrijevic, T. Tomovic and S. Aleksic, The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions, Math. Meth. Appl. Sci., 2016. DOI: 10.1002/mma.4105. Google Scholar
[4]	J. Caballero, M.A. Darwish and K. Sadarangani, Positive solutions in the space of Lipschitz functions for fractional boundary value problems with integral boundary conditions, Mediterr J Math., 2017, 14(5), 201-215. doi: 10.1007/s00009-017-1001-z CrossRef Google Scholar
[5]	D. Chen and W. Liu, Chaotic behavior and its control in a fractional-order energy demand-supply system, J Comput Nonlinear Dyn., 2016. DOI: 10.1115/1.4034048. Google Scholar
[6]	N. Cong and H. Tuan, Existence, uniqueness, and exponential boundedness of Global solutions to delay fractional differential equations, Mediterr. J. Math., 2017. DOI: 10.1007/s00009-017-0997-41660-5446/17/050001-12. Google Scholar
[7]	K. Deimling, Nonlinear functional analysis, Springer, New York, 1985. Google Scholar
[8]	B. Dhage and N. Jadhav, Basic results in the theory of hybrid differential equations with linear perturbations of second type, Tamkang Journal of Mathematics, 2013, 44(2), 171-186. Google Scholar
[9]	W. Deng, C. Li and J. Lu, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dyn., 2007, 48(4), 409-416. doi: 10.1007/s11071-006-9094-0 CrossRef Google Scholar
[10]	D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988. Google Scholar
[11]	L. Guo, L. Liu and Y. Wu, Iterative unique positive solutions for singular $p$-Laplacian fractional differential equation system with several parameters, Nonlinear Analysis: Modelling and Control, 2018, 23, 182-203. doi: 10.15388/NA.2018.2.3 CrossRef $p$-Laplacian fractional differential equation system with several parameters" target="_blank">Google Scholar
[12]	L. Hu and S. Zhang, Existence results for a coupled system of fractional differential equations with $p$-Laplacian operator and infinite-point boundary conditions, Bound Value Prob., 2017, 88, 16 pages. Google Scholar
[13]	R. Hilfer(Ed), Application of fractional calculus in physics World scientific publishing Co, Singapore, 2000. Google Scholar
[14]	S. Haristova and C. Tunc, Stability of nonlinear Volterra integro-differential equations with Caputo fractional derivative and bounded delays, Electron. J. Differential Equations., 2019, 21-82, 1-11. Google Scholar
[15]	X. Han and X. Yang, Existence and multiplicity of positive solutions for a system of fractional differential equation with parameters, Bound Value Prob., 2017, 78, 12 pages. Google Scholar
[16]	Z. Hu, W. Liu and J. Liu, Existence of solutions for a coupled system of fractional $p$-Laplacian equations at resonance, Adv Difference Equ., 2013, 312, 14 pages. Google Scholar
[17]	R. W. Ibrahim and H. A. Jalab, Existence of Ulam stability for iterative fractional differential equations based on fractional entropy, Entropy, 2015, 17(5), 3172-3181. doi: 10.3390/e17053172 CrossRef Google Scholar
[18]	A. Iannizzotto and N. S. Papageorgiou, Existence and multiplicity results for resonant fractional boundary value problems, Am Institute Math Sci., 2018. DOI: 10.3934/dcdss.2018028. Google Scholar
[19]	F. Isaia, On a nonlinear integral equation without compactness, Acta Maths. Univ. Comenianae., 2006, LXXV(2), 233-240. Google Scholar
[20]	H. Jafari, D. Baleanu, H. Khan, R. A. Khan and A. Khan, Existence criterion for the solution of fractional order $p$-Laplacian boundary value problem, Bound Value Probl, 2015, 164, 10 pages. Google Scholar
[21]	W. Jiang, J. Qiu and C. Yang, The existence of solutions for fractional differential equations with p-Laplacian at resonance, Chaos, 2017, 27(3), 032102. doi: 10.1063/1.4979367 CrossRef Google Scholar
[22]	H. Jafari, H. K. Jassim, M, Qurashi and D. Baleanu, On the existence and uniqueness of solutions for local fractional differential equations, Entropy, 2016. DOI: 10.3390/e18110420. Google Scholar
[23]	H. Khan, Y. Li, W. Chen, D. Baleanu and A. Khan, Existence theorems and Hyers-Ulam stability for a coupled system of fractional differential equations with p-Laplacian operator, Boundary Value Problems, 2017. DOI: 10.1186/s13661-017-0878-6. Google Scholar
[24]	A. A. Kilbas and H. M. Srivastava, Trujillo JJ. Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Amsterdam, 2006. Google Scholar
[25]	A. Khan, Y. Li, K. Shah and T. S. Khan, On coupled $p$-Laplacian fractional differential equations with nonlinear boundary conditions, \textitComplexity, 2017, 9 pages, Article ID 8197610. Google Scholar
[26]	H. Khan, J. F. G$\breve{0}$mez-Aguilar and T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos, Solitons & Fractals, 2019, 127, 422-427. $\breve{0}$mez-Aguilar and Abdeljawad T., Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos" target="_blank">Google Scholar
[27]	H. Khan, C. Tunc, W. Chen and A. Khan, Existence theorems and Hyers-Ulam stability for a class of hybrid fractional differential equations with $p$-Laplacian operator, J. Appl. Anal. Comput., 2018, 8(4), 1211-1226. $p$-Laplacian operator" target="_blank">Google Scholar
[28]	H. Khan, F. Jarad, T. Abdeljawad and A. Khan, A singular ABC-fractional differential equation with p-Laplacian operator, Chaos, Solitons & Fractals., 2019, 129, 56-61. Google Scholar
[29]	A. Khan, M. I. Syam, A. Zada and H. Khan, Stability analysis of nonlinear fractional differential equations with Caputo and Riemann-Liouville derivatives, Eur Phys J Plus, 2018. DOI: 10.1140/epjp/i2018-12119-6. Google Scholar
[30]	W. Liu and K. Chen, Chaotic behavior in a new fractional-order love triangle system with competition, J Appl Anal Comput., 2015, 5(1), 103-113. Google Scholar
[31]	X. Liu, L. Liu and Y. Wu, Existence of positive solutions for a singular nonlinear fractional differential equation with integral boundary conditions involving fractional derivatives, Boundary Value Problems, 2018, 24. https://doi.org/10.1186/s13661-018-0943-9. Google Scholar
[32]	A. G. Lakoud and A. Ashyralyev, Positive solutions for a system of fractional differential equations with nonlocal integral boundary conditions, Differ Equ Dyn Syst., 2017, 25(4), 519-526. doi: 10.1007/s12591-015-0255-9 CrossRef Google Scholar
[33]	B. Liang, X. Peng and C. Qu, Existence of solutions to a nonlinear parabolic equation of fourth-order in variable exponent spaces, Entropy, 2016. DOI: 10.3390/e18110413. Google Scholar
[34]	Y. Li, Existence of positive solutions for fractional differential equations involving integral boundary conditions with $p$-Laplacian operator, Adv Difference Equ., 2017, 2017: 135, 11 pages. Google Scholar
[35]	N. I. Mahmudov and S. Unul, Existence of solutions of $\epsilon\in (2, 3]$ order fractional three-point boundary value problems with integral conditions, Abstr Appl Anal., 2014. DOI: 10.1155/2014/198632. Google Scholar
[36]	N. I. Mahmudov and S. Unul, Existence of solutions of fractional boundary value problems with $p$-Laplacian operator, Bound Value Prob., 2015, 2015: 99, 16 pages. Google Scholar
[37]	N. I. Mahmudov and S. Unul, On existence of BVP'S for impulsive fractional differential equations, Adv Difference Equ., 2017. DOI: 10.1186/s13662-016-1063-4. Google Scholar
[38]	I. Podlubny, Fractional differential equations, Academic Press, New York, 1999. Google Scholar
[39]	Y. Qiao and Z. Zhou, Existence of positive solutions of singular fractional differential equations with infinite-point boundary conditions, Adv Difference Equ, 2017, 2017(8), 9 pages. Google Scholar
[40]	S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. Google Scholar
[41]	S. G. Samko and A. A. Kilbas, Marichev OI. Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. Google Scholar
[42]	T. Shen, W. Liu and X. Shen, Existence and uniqueness of solutions for several BVPs of fractional differential equations with $p$-Laplacian operator, Mediterr J Math., 2016. DOI: 10.1007/s00009-016-0766-9. Google Scholar
[43]	N. T. Thanh, H. Trinh and V. N. Phat, Stability analysis of fractional differential time-delay equations, IET Control Theory Appl., 2017, 11(7), 1006-1015. doi: 10.1049/iet-cta.2016.1107 CrossRef Google Scholar
[44]	C. Urs, Coupled fixed point theorems and applications to perodic boundary value problems, Miskolc Math Notes, 2013, 14(1), 323-333. doi: 10.18514/MMN.2013.598 CrossRef Google Scholar
[45]	S. Vong, Positive solutions of singular fractional differential equations with integral boundary conditions, Mathematical and Computer Modelling, 2013, 57, 1053-1059. doi: 10.1016/j.mcm.2012.06.024 CrossRef Google Scholar
[46]	S. Xie and Y. Xie, Positive solutions of higher-order nonlinear fractional differential system with nonlocal boundary conditions, J Appl Anal Comput., 2016, 6(4), 1211-1227. Google Scholar
[47]	W. Zhang, W. Liu and T. Chen, Solvability for a fractional $p$-Laplacian multipoint boundary value problem at resonance on infinite interval, Adv Differ Equ., 2016. DOI: https://doi.org/10.1186/s13661-018-0954-6. Google Scholar
[48]	W. Zhang and W. Liu, Existence of solutions for fractional differential equations with infinite point boundary conditions at resonance, Boundary Value Problems, 2018. https://doi.org/10.1186/s13661-018-0954-6. Google Scholar
[49]	X. Zhang, P. Agarwal, Z. Liu, H. Peng, F. You and Y. Zhu, Existence and uniqueness of solutions for stochastic differential equations of fractional-order $q>1$ with finite delays, Adv Difference Equ., 2017. DOI: 10.1186/s13662-017-1169-3. Google Scholar