# Existence of solution and Hyers-Ulam stability for a coupled system of fractional differential equations with $p$-Laplacian operator

Volume 10, Issue 10, pp 5219--5229
Publication Date: October 14, 2017 Submission Date: July 15, 2017
• 1375 Views

### Authors

Hasib Khan - State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, 210098, Nanjing, P. R. China. - Shaheed Benazir Bhutto University Sheringal, Dir Upper, 18000, Khyber Pakhtunkhwa, Pakistan. Yongjin Li - Department of Mathematics, Sun Yat-sen University, 510275, Guangzhou, P. R. China. Hongguang Sun - State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, 210098, Nanjing, P. R. China. Aziz Khan - Department of Mathematics, University of Peshawar, 25000, Khyber Pakhtunkhwa, Pakistan.

### Abstract

Models with $p$-Laplacian operator are common in different scientific fields including; plasma physics, chemical reactions design, physics, biophysics, and many others. In this paper, we investigate existence and uniqueness of solution and Hyers-Ulam stability for a coupled system of fractional differential equations with $p$-Laplacian operator. The Hyers-Ulam stability means that a differential equation has a close exact solution which is generated by the approximate solution of the differential equation and the error in the approximation can be estimated. We use topological degree method and provide an expressive example as an application of the work.

### Keywords

• Existence and uniqueness of solution
• Hyers-Ulam stability
• topological degree method
• $p$-Laplacian operator

•  47H10
•  54H25

### References

• [1] A. Ali, B. Samet, K. Shah, R. A. Khan , Existence and stability of solution to a toppled systems of differential equations of non-integer order, Bound. Value Probl., 2017 (2017), 13 pages.

• [2] G. A. Anastassiou , On right fractional calculus, Chaos Solitons Fractals, 42 (2009), 365–376.

• [3] D. Baleanu, R. P. Agarwal, H. Mohammadi, S. Rezapour , Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces , Bound. Value Probl., 2013 (2013), 8 pages.

• [4] D. Băleanu, O. G. Mustafa , On the global existence of solutions to a class of fractional differential equations, Comput. Math. Appl., 59 (2010), 1835–1841.

• [5] D. Băleanu, O. G. Mustafa, R. P. Agarwal , An existence result for a superlinear fractional differential equation, Appl. Math. Lett., 23 (2010), 1129–1132.

• [6] D. Băleanu, O. G. Mustafa, R. P. Agarwal , On the solution set for a class of sequential fractional differential equations, J. Phys. A, 43 (2010), 7 pages.

• [7] J. Brzdęk, L. Cădariu, K. Ciepliński, A. Fošner, Z. Leśniak, Survey on recent Ulam stability results concerning derivations, J. Funct. Spaces, 2016 (2016), 9 pages.

• [8] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, II, Reprinted from Geophys. J. R. Astr. Soc., 13 (1967), 529–539, Fract. Calc. Appl. Anal., 11 (2008), 4–14.

• [9] L.-L. Cheng, W.-B. Liu, 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.

• [10] P. Găvruţa, S.-M. Jung, Y.-J. Li, Hyers-Ulam stability for second-order linear differential equations with boundary conditions, Electron. J. Differential Equations, 2011 (2011), 5 pages.

• [11] A. Granas, J. Dugundji , Fixed point theory , Springer Monographs in Mathematics, Springer-Verlag, New York (2003)

• [12] R. Hilfer (Ed.) , Applications of fractional calculus in physics , World Scientific Publishing Co., Inc., River Edge, NJ (2000)

• [13] Z.-G. Hu, W.-B. Liu, J.-Y. Liu, Existence of solutions for a coupled system of fractional p-Laplacian equations at resonance, Adv. Difference Equ., 2013 (2013), 14 pages.

• [14] F. Isaia , On a nonlinear integral equation without compactness, Acta Math. Univ. Comenian. (N.S.), 233–240. (2006)

• [15] H. Jafari, D. Baleanu, H. Khan, R. A. Khan, A. Khan , Existence criterion for the solutions of fractional order p-Laplacian boundary value problems, Bound. Value Probl., 2015 (2015), 10 pages.

• [16] R. A. Khan, A. Khan , Existence and uniqueness of solutions for p-Laplacian fractional order boundary value problems, Comput. Methods Differ. Equ., 205–215. (2014)

• [17] R. A. Khan, A. Khan, A. Samad, H. Khan , On existence of solutions for fractional differential equation with p-Laplacian operator , J. Fract. Calc. Appl., 5 (2014), 28–37.

• [18] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006)

• [19] P. Kumam, A. Ali, K. Shah, R. A. Khan , Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations, J. Nonlinear Sci. Appl., 10 (2017), 2986–2997.

• [20] N. I. Mahmudov, S. Unul , Existence of solutions of $\alpha\in (2, 3]$ order fractional three-point boundary value problems with integral conditions, Abstr. Appl. Anal., 2014 (2014), 12 pages.

• [21] N. I. Mahmudov, S. Unul , Existence of solutions of fractional boundary value problems with p-Laplacian operator, Bound. Value Probl., 2015 (2015), 16 pages.

• [22] N. I. Mahmudov, S. Unul , On existence of BVP’s for impulsive fractional differential equations, Adv. Difference Equ., 2017 (2017), 16 pages.

• [23] K. S. Miller, B. Ross , An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993)

• [24] K. B. Oldham, J. Spainer, The fractional calculus, Theory and applications of differentiation and integration to arbitrary order, With an annotated chronological bibliography by Bertram Ross, Mathematics in Science and Engineering, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1974)

• [25] I. Podlubny , Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)

• [26] K. R. Prasad, B. M. B. Krushna, Multiple positive solutions for a coupled system of p-Laplacian fractional order two-point boundary value problems, Int. J. Differ. Equ., 2014 (2014), 10 pages.

• [27] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Nikol’skiı, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993)

• [28] T.-F. Shen,W.-B. Liu, X.-H. Shen , Existence and uniqueness of solutions for several BVPs of fractional differential equations with p-Laplacian operator , Mediterr. J. Math., 13 (2016), 4623–4637.

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

• [30] H.-G. Sun, Z.-P. Li, Y. Zhang, W. Chen , Fractional and fractal derivative models for transient anomalous diffusion: model comparison, Chaos Solitons Fractals, 102 (2017), 346–353.

• [31] H.-G. Sun, Y. Zhang, W. Chen, D. M. Reeves, Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media, J. Contam. Hydrol., 157 (2014), 47–58.

• [32] C. Urs, Coupled fixed point theorems and applications to periodic boundary value problems, Miskolc Math. Notes, 14 (2013), 323–333.