Stable monotone iterative solutions to a class of boundary value problems of nonlinear fractional order differential equations
Authors
Sajjad Ali
 Department of Mathematics, Abdul Wali Khan University of Mardan, Khyber Pakhtunkhwa, Pakistan.
Muhammad Arif
 Department of Mathematics, Abdul Wali Khan University of Mardan, Khyber Pakhtunkhwa, Pakistan.
Durdana Lateef
 Department of Mathematics, College of Science, Taibah University, Madinah, KSA.
Mohammad Akram
 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, KSA.
Abstract
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 HyersUlam 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.
Keywords
 Nonlinear fractional differential equations
 iterative technique
 upper and lower solutions
 uniqueness and existence
MSC
References

[1]
R. P. Agarwal, M. Belmekki, M. Benchohra, A survey on semilinear differential equations and inclusions involving RiemannLiouville fractional derivative, Adv. Difference Equ., 2009 (2009), 47 pages.

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

[3]
N. Ahmad, Z. Ali, K. Shah, A. Zada, Analysis of implicit type nonlinear dynamical problem of impulsive fractional differential equation, Complexity, 2018 (2018), 15 pages.

[4]
B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with threepoint boundary conditions, Comput. Math. Appl., 58 (2009), 1838–1843.

[5]
N. Ali, B. B. Fatima, K. Shah, R. A. Khan, HyersUlam stability of a class of nonlocal boundary value problem having triple solutions, Int. J. Appl. Comput. Math., 4 (2018), 12 pages.

[6]
M. AlRefai, M. Ali Hajji, Monotone iterative sequences for nonlinear boundary value problems of fractional order, Nonlinear Anal., 74 (2011), 3531–3539.

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

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

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

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

[11]
C. De Coster, P. Habets, Twopoint Boundary Value Problems: Lower and Upper Solutions, Elsevier B. V., Amsterdam (2006)

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

[13]
Z. M. He, X. M. He, Monotone iterative technique for impulsive integrodifferential equations with periodic boundary conditions, Comput. Math. Appl., 48 (2004), 73–84.

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

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

[16]
R. A. Khan, Existence and approximation of solutions to threepoint boundary value problems for fractional differential equations, Electron. J. Qual. Theory Differ. Equ., 2011 (2011), 8 pages.

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

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

[19]
F. Li, M. Jia, X. Liu, C. Li, G. Li, Existence and uniqueness of solutions of secendorder threepoint boundary value problems with upper and lower solutions in the reverse order , Nonlinear Anal., 68 (2008), 2381–2388.

[20]
F. F. Li, J. T. Sun, M. Jia, Monotone iterative method for the secondorder threepoint boundary value problem with upper and lower solutions in the reversed order, Comput. Math. Appl., 217 (2011), 4840–4847.

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

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

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

[24]
K. S. Miller, B. Ross, 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]
T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300.

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

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

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

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

[31]
K. Shah, R. A. Khan, Iterative scheme for a coupled system of fractionalorder differential equations with threepoint boundary conditions , Math. Methods Appl. Sci., 41 (2018), 1047–1053.

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

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

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

[35]
J. R. Wang, K. Shah, A. Ali, Existence and HyersUlam stability of fractional nonlinear impulsive switched coupled evolution equations, Math. Methods Appl. Sci., 41 (2018), 2392–2402.

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

[37]
N. Xu, W. B. Liu, Iterative solutions for a coupled system of fractional differentialintegral equations with twopoint boundary conditions, Appl. Math. Comput., 244 (2014), 903–911.

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

[39]
E. Zeidler , Nonlinear Functional Analysis and its Applications: II/B: Nonlinear Monotone Operators, SpringerVerlag, New York (1990)

[40]
S. Q. Zhang, Monotone iterative method for initial value problem involving RiemannLiouville fractional derivatives, Nonlinear Anal., 71 (2009), 2087–2093.