Geodesic
Stable monotone iterative solutions to a class of boundary value problems of nonlinear fractional order differential equations
Ali, Sajjad1 ; Arif, Muhammad 1 ; Lateef, Durdana 2 ; Akram, Mohammad 3
1 Department of Mathematics, Abdul Wali Khan University of Mardan, Khyber Pakhtunkhwa, Pakistan
2 Department of Mathematics, College of Science, Taibah University, Madinah, KSA
3 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, KSA
Journal of nonlinear sciences and its applications, Tome 12 (2019) no. 6, p. 376-386.

Voir la notice de l'article provenant de la source International Scientific Research Publications

We construct sufficient conditions for existence of extremal solutions to boundary value problem (BVP) of nonlinear fractional order differential equations (NFDEs). By combing the method of lower and upper solution with the monotone iterative technique, we construct sufficient conditions for the iterative solutions to the problem under consideration. Some proper results related to Hyers-Ulam type stability are investigated. Base on the proposed method, we construct minimal and maximal solutions for the proposed problem. We also construct and provide maximum error estimates and test the obtain results by two examples.
DOI : 10.22436/jnsa.012.06.04
Classification : 34A08, 34A45
Keywords: Nonlinear fractional differential equations, iterative technique, upper and lower solutions, uniqueness and existence

Ali, Sajjad 1 ; Arif, Muhammad  1 ; Lateef, Durdana  2 ; Akram, Mohammad  3

1 Department of Mathematics, Abdul Wali Khan University of Mardan, Khyber Pakhtunkhwa, Pakistan
2 Department of Mathematics, College of Science, Taibah University, Madinah, KSA
3 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, KSA 
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Ali, Sajjad; Arif, Muhammad ; Lateef, Durdana ; Akram, Mohammad . Stable monotone iterative solutions to a class of boundary value problems of nonlinear fractional order differential equations. Journal of nonlinear sciences and its applications, Tome 12 (2019) no. 6, p. 376-386. doi : 10.22436/jnsa.012.06.04. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.012.06.04/

[1] Agarwal, R. P.; Belmekki, M.; Benchohra, M. A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Difference Equ., Volume 2009 (2009), pp. 1-47 | DOI | Zbl

[2] Agarwal, R. P.; Benchohra, M.; Hamani, S. A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., Volume 109 (2012), pp. 973-1033 | Zbl | DOI

[3] Ahmad, N.; Ali, Z.; Shah, K.; Zada, A. Analysis of implicit type nonlinear dynamical problem of impulsive fractional differential equation, Complexity, Volume 2018 (2018), pp. 1-15 | DOI

[4] Ahmad, B.; Nieto, J. J. Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl., Volume 58 (2009), pp. 1838-1843 | Zbl | DOI

[5] Ali, N.; Fatima, B. B.; Shah, K.; Khan, R. A. Hyers-Ulam stability of a class of nonlocal boundary value problem having triple solutions, Int. J. Appl. Comput. Math., Volume 4 (2018), pp. 1-12 | Zbl | DOI

[6] Al-Refai, M.; Hajji, M. Ali Monotone iterative sequences for nonlinear boundary value problems of fractional order, Nonlinear Anal., Volume 74 (2011), pp. 3531-3539 | Zbl | DOI

[7] Bai, C.-Z.; Fang, J.-X. The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations, Appl. Math. Comput., Volume 150 (2004), pp. 611-621 | DOI

[8] Benchohra, M.; Graef, J. R.; Hamani, S. Existence results for boundary value problems with nonlinear fractional differential equations, Applicable Anal., Volume 87 (2008), pp. 851-863

[9] Bushnaq, S.; Hussain, W.; K. Shah On nonlinear implicit fractional differential equations without compactness, J. Nonlinear Sci. Appl., Volume 10 (2017), pp. 5528-5539 | Zbl

[10] Cabadal, A.; Wang, G. T. Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., Volume 389 (2012), pp. 403-411 | DOI

[11] Coster, C. De; Habets, P. Two-point Boundary Value Problems: Lower and Upper Solutions, Elsevier B. V., Amsterdam, 2006 | Zbl

[12] Gafiychuk, V.; Datsko, B.; Meleshko, V.; Blackmore, D. Analysis of the solutions of coupled nonlinear fractional reactiondiffusion equations, Chaos Solitons Fractals, Volume 41 (2009), pp. 1095-1104 | DOI

[13] He, Z. M.; He, X. M. Monotone iterative technique for impulsive integro-differential equations with periodic boundary conditions, Comput. Math. Appl., Volume 48 (2004), pp. 73-84 | DOI

[14] Hilfer, R. Applications of Fractional Calculus in Physics, World Scientific Publishing Co., River Edge, 2000 | DOI

[15] Khalil, H.; Shah, K.; Khan, R. A. Upper and lower solutions to a coupled system of nonlinear fractional differential equations, Progress in Fractional Differential Equations Applications, Volume 2016 (2016), pp. 1-10

[16] Khan, R. A. Existence and approximation of solutions to three-point boundary value problems for fractional differential equations, Electron. J. Qual. Theory Differ. Equ., Volume 2011 (2011), pp. 1-8 | DOI

[17] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006

[18] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S. Monotone iterative technique for nonlinear differential equations, Pitman Publishing Inc., Boston, 1985

[19] Li, F.; Jia, M.; Liu, X.; Li, C.; Li, G. Existence and uniqueness of solutions of secend-order three-point boundary value problems with upper and lower solutions in the reverse order , Nonlinear Anal., Volume 68 (2008), pp. 2381-2388 | DOI

[20] Li, F. F.; Sun, J. T.; Jia, M. Monotone iterative method for the second-order three-point boundary value problem with upper and lower solutions in the reversed order, Comput. Math. Appl., Volume 217 (2011), pp. 4840-4847 | DOI | Zbl

[21] Li, M.; Wang, J. R.; O’Regan, D. Existence and Ulam’s stability for conformable fractional differential equations with constant coefficients, Bull. Malays. Math. Sci. Soc., Volume 2017 (2017), pp. 1-22 | DOI

[22] Liu, X. P.; Jia, M. Multiple solutions for fractional differential equations with nonlinear boundary conditions, Comput. Math. Appl., Volume 59 (2010), pp. 2880-2886 | DOI

[23] McRae, F. A. Monotone iterative technique and existence results for fractional differential equations, Nonlinear Anal., Volume 71 (2009), pp. 6093-6096 | DOI

[24] Miller, K. S.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993

[25] I. Podlubny Fractional Differential Equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, San Diego, 1999

[26] Rassias, T. M. On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., Volume 72 (1978), pp. 297-300 | DOI

[27] Rassias, T. M. On the stability of functional equations and a problem of Ulam, Acta Appl. Math., Volume 62 (2000), pp. 23-130 | DOI

[28] I. A. Rus Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., Volume 26 (2010), pp. 103-107 | Zbl

[29] Samko, S. G.; Kilbas, A. A.; Marichev, O. I. Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993

[30] Shah, K.; Khan, R. A. Iterative solutions to a coupled system of nonlinear fractional differential equations, J. Fract. Cal. Appl., Volume 7 (2016), pp. 40-50

[31] Shah, K.; Khan, R. A. Iterative scheme for a coupled system of fractional-order differential equations with three-point boundary conditions , Math. Methods Appl. Sci., Volume 41 (2018), pp. 1047-1053 | Zbl | DOI

[32] Su, X. Boundary value problem for a coupled systemof nonlinear fractional differential equations, Appl. Math. Lett., Volume 22 (2009), pp. 64-69 | DOI

[33] Wang, G. T.; Agarwal, R. P.; A. Cabada Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations, Appl. Math. Lett., Volume 25 (2012), pp. 1019-1024 | DOI

[34] Wang, J. R.; Feckan, M.; Zhou, Y. Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions, Bull. Sci. math., Volume 141 (2017), pp. 727-746 | DOI | Zbl

[35] Wang, J. R.; Shah, K.; Ali, A. Existence and Hyers-Ulam stability of fractional nonlinear impulsive switched coupled evolution equations, Math. Methods Appl. Sci., Volume 41 (2018), pp. 2392-2402 | DOI | Zbl

[36] Wang, J. H.; Xiang, H. J.; Liu, Z. G. Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations, Int. J. Differ. Equ., Volume 2010 (2010), pp. 1-12 | Zbl

[37] Xu, N.; Liu, W. B. Iterative solutions for a coupled system of fractional differential-integral equations with two-point boundary conditions, Appl. Math. Comput., Volume 244 (2014), pp. 903-911 | DOI | Zbl

[38] Yang, W. G. Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions, Comput. Math. Appl., Volume 63 (2012), pp. 288-297 | DOI

[39] E. Zeidler Nonlinear Functional Analysis and its Applications: II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990 | Zbl | DOI

[40] Zhang, S. Q. Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives, Nonlinear Anal., Volume 71 (2009), pp. 2087-2093 | Zbl | DOI

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