# Existence and Stability of the Solution to a Coupled System of Fractional-order Differential with a $p$-Laplacian Operator under Boundary Conditions

## DOI:

https://doi.org/10.5556/j.tkjm.53.2022.3335## Keywords:

topological degree theorem, Hyers-Ulam stability, Caputo's fractional derivative; existence and uniqueness; coupled system of FDEs## Abstract

This paper is devoted to studying the uniqueness and existence of the solution to a nonlinear coupled system of (FODEs) with p-Laplacian operator under integral boundary conditions (IBCs). Our problem is based on Caputo fractional derivative of orders $ \sigma,\lambda $, where $ k-1\leq\sigma,\lambda<k, k\geq3$. For these aims, the nonlinear coupled system will be converted into an equivalent integral equations system by the help of Green function. After that, we use Leray-Schauder's and topological degree theorems to prove the existence and uniqueness of the solution. Further, we study certain conditions for the Hyers-Ulam stability of the solution to the suggested problem. We give a suitable and illustrative example as an application of the results.

## References

X. Han and X. Yang. Existence and multiplicity of positive solutions for a system of fractional differential equation with parameters. Bound. Value Probl. 2017, 1-12. (2017).

R. A. Khan, A. Khan, A. Samad and H. Khan. On existence of solutions for fractional differential equations with p-Laplacian operator. J. Fract. Calc. Appl. 5(2), 28-37 (2014).

L. Diening, P. Lindqvist and B. Kawohl. Mini-Workshop : the p-Laplacian operator and applications. Oberwolfach Rep. 10(1), 433-482 (2013).

H. Jafari, D. Baleanu, H. Khan, R. A. Khan and A. Khan. Existence criterion for the solutions of fractional orderp-Laplacian boundary value problems. Bound. Value Probl.2015, 1-10, (2015).

L. Zhang, W. Zhang, X. Liu and M. Jia. Existence of positive solutions for integral boundary value problems of fractional differential equations with p-Laplacian, Advances in Difference Equations, Paper No. 1-19 pages, 2017.

A. Khan, T. S. Khan, M. I. Syam and H. Khan. Analytical solutions of time-fractional wave equation by double Laplace transform method, Eur. Phys. J. Plus (2019) 134: 163 DOI 10.1140/epjp/i2019-12499-y.

H. Khan, Y. Li, A. Khan and A. Khan. Existence of solution for a fractional-order Lotka-Volterra reaction-diffusion model with Mittag-Leffler kernel. Mathematical Methods in the Applied Sciences. 2019 Jun;42(9):3377-87. DOI: 10.1002/mma.5590

R. Agarwal, S. Hristova and D. O’Regan. Stability of solutions to impulsive Caputo fractional differential equations. Electron.J.Differ. Equ.2016,Article 1-22,(2016).

E. Cetin and F. S. Topal. Existence of solutions for fractional four point boundary value problems with p-Laplacian operator, Journal of Computational Analysis and Applications, vol. 19, no. 5, pp. 892-903, 2015.

A. Alkhazzan, P. Jiang, D. Baleanu, H. Khan and A. Khan. (2018). Stability and existence results for a class of nonlinear fractional differential equations with singularity. Mathematical Methods in the Applied Sciences. 9321-9334.

R. Hilfer. Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

K. S. Miller and B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.

N. D. Cong, T. S Doan, S. Siegmund and H. T. Tuan. Linearized asymptotic stability for fractional differential equations.Electron. J. Qual. Theory Differ. Equ. 2016, Article ID39(2016).

S. Kumar, D. Kumar and J. Singh. Fractional modelling arising in unidirectional propagation of long waves in dispersive media. Adv. Nonlinear Anal. 5(4), 383-394 (2016).

I. Area, J. Losada and J. J. Nieto. A note on fractional logistic equation. Physica A 444,182-187 (2016).

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo. Theory and Applications of Fractional Differential Equations, New York, NY, USA, Elsevier, 2006.

I. Bachar, H. Maagli and V. Radulescu. Fractional Navier boundary value problems. Bound. Value Probl. 2016, Article ID 79 (2016).

H. Khan, T. Abdeljawad, C. Tunç, A. Alkhazzan and A. Khan. Minkowski’s inequality for the Journal of Inequalities and Applications. (2019) 2019:96 AB-fractional integral operator,

https://doi.org/10.1186/s13660-019-2045-3,1-12.

H. Khan, A. Khan, T. Abdeljawad and A. Alkhazzan. Existence results in Banach space for a nonlinear impulsive system. Advances in Difference Equations. 2019 Dec 1;2019(1):18,1-16.

S. Peng and J. R. Wang. Existence and Ulam-Hyers stability of ODEs involving two Caputo fractional derivatives.Electron. J. Qual. Theory Differ. Equ.2015, Article ID 52 (2015).1-16.

I. Bachar, H. Maagli and V. Radulescu. Fractional Navier boundary value problems. Bound. Value Probl.2016, Article ID79 (2016).

R. Agarwal, S. Hristova and D. O’Regan. Stability of solutions to impulsive Caputo fractional differential equations. Electron.J.Differ.Equ.2016, Article ID58(2016),1-22.

H. Khan, C. Tunç, D. Baleanu, A. Khan and A. Alkhazzan. Inequalities for n-class of functions using the Saigo fractional integral operator. Revista de La Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. 113, no. 3 (2019): 2407-2420

Doi:10.1007/s13398-019-00624-5 .

H. Khan, C. Tunç, A. Alkhazan, B. Ameen and A. Khan. A generalization of Minkowski’s inequality by Hahn integral operator. Journal of Taibah University for Science, 12(5), 506-513 (2018).

B. Ahmad and J. J. Nieto. Existence Results for a couple System of Nonlinear Fractional Differential Equations with Three Point Boundary Conditions, Compur. Math. Appl. 58 (2009), 1838-1843.

C. Z. Bai and J. X. Fang. The existence of a positive solution for a singular coupled systems of nonlinear fractional differential equations, Appl. Math. Comput. 150 (2004), 611-621.

M. A. Abdellaoui and Z. Dahmani. Solvability for a coupled system of nonlinear fractional integro-differential equations, Note di Matematica 35 (2015), 95-107.

H. Khan, Y. J. Li, H. G. Sun and A. Khan. Existence of solution and Hyers-Ulam stability for a coupled system of fractional differential equations with p-Laplacian operator, J. Nonlinear Sci. Appl. 10 (2017), 5219-5229.

X. Su. Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett. 22 (2009),64-69.

K. Shah and C. Tunc. Existence theory and stability analysis to a system of boundary value problem, J. Taibah Univ. Sci. 11 (2017), 1330-1342.

M. Iqbal, Y. Li, K. Shah and R. A. Khan. Application of topological degree method for solutions of coupled systems of multi points boundary value problems of fractional order hybrid differential equations.Complexity 2017, Article ID 7676814 (2017).

S. Samina, K. Shah and R. A. Khan. Existence of positive solutions to a coupled system with three-point boundary conditions via degree theory. Commun. Nonlinear Anal. 3, 34-43 (2017).

K. Shah, P. Kumam and I. Ullah. On Ulam Stability and Multiplicity Results to a Nonlinear Coupled System with Integral Boundary Conditions. Mathematics,7(3), 223. (2019). Doi:10.3390/math7030223

L. L. Cheng, W.-B. Liu and Q.-Q. Ye. Boundary value problem for a coupled system of fractional differential equations with p-Laplacian operator at resonance,Electron. J. Differential Equations, 2014 (2014), 12 pages. 1.

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 Probl.1-16,88(2017).

J. Tariboon and S. K. Ntouyas and W. Sudsutad. Coupled systems of Riemann-Liouville fractional differential equations with Hadamard fractional integral boundary conditions. J. Nonlinear Sci. Appl. 9 (2016), 295-308.

DOI: 10.1007/978-3-319-52141-1-6.

J. Mawhin. Topological degree methods in nonlinear boundary value problems (No. 40). American Mathematical Soc, 1979.

F. Isaia. On a nonlinear integral equation without compactness. Acta Math. Univ. Comen.75,233-240 (2006).

J. Wang, Y. Zhou and W. Wei. Study in fractional differential equations by means of topological degree methods.Numer. Funct. Anal. Optim. 33(2), 216-238 (2012).

K. Shah and R. A. Khan. Existence and uniqueness results in a coupled system of fractional order boundary value problems by topological degree theory. Numer. Funct. Anal. Optim. 37, 887-899(2016).

K. Shah, A. Ali and R. A. Khan. Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems. Bound. Value Probl. 2016 (1),1-12.

K. Deimling. Nonlinear Functional Analysis. Springer, New York (1985).

H. Khan, W. Chen, A. Khan, T. S. Khan TS and Q. M. Al-Madlal. Hyers-Ulam stability and existence criteria for coupled fractional differential equations involving p-Laplacian operator. Advances in Difference Equations. 2018 Dec 1;2018(1):455,1-16.

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.13,4623-4637(2016).

I. Stamova. Mittag-Leffler stability of impulsive differential equations of fractional order. Quart. Appl. Math. 73(3), 525-535 (2015).

D. Baleanu and O. G. Mustafa. On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59(5), 1835-1841 (2010).

A. Zada, S. Faisal and Y. Li. Hyers-Ulam-Rassias stability of non-linear delay differential equations.J.Nonlinear Sci.Appl.10, 504-510(2017).

D. Baleanu, R. P. Agarwal, H. Mohammadi and S. Rezapor. Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces. Bound. Value Probl. 2013, 112 (2013), 1-8.

D. Baleanu, R. P. Agarwal, H. Khan, R. A. Khan and H Jafari. On the existence of solution for fractional differential equations of order 3 < δ ≤ 4 . Adv. Differ. Equ. 2015, 362 (2015), 1-9.

H. L. Royden and P. Fitzpatrick. Real Analysis. New York:Macmillan;1988.

D. Baleanu, O. G. Mustafa and R. P. Agarwal. On the solution set for a class of sequential fractional differential equations. J. Phys. A, Math. Theor. 43.38 (2010): 385209.

D. Baleanua, O. G. Mustafa and R. P. Agarwal. An existence result for a superlinear fractional differential equation. Appl. Math. Lett. 23(9), 1129-1132 (2010).

W. Al-Sadi, H. Zhenyou and A. Alkhazzan. Existence and stability of a positive solution for nonlinear hybrid fractional differential equations with singularity. Journal of Taibah University for Science. 2019;13:951-60.

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*53*(1), 37-58. https://doi.org/10.5556/j.tkjm.53.2022.3335

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