**Volume 10, Issue 10, pp 5174--5181**

**Publication Date**: 2017-10-12

http://dx.doi.org/10.22436/jnsa.010.10.05

Mohammed Al-Refai - Department of Mathematical Sciences, United Arab Emirates University, P. O. Box 15551, Al Ain, UAE

Dumitru Baleanu - Department of Mathematics and Computer Science, Cankaya University, 06530 Ankara, Turkey

We extend the results concerning the fractional derivatives of a function at its extreme points to fractional derivatives of arbitrary order. We also give an estimate of the error and present two examples to illustrate the validity of the results. The presented results are valid for both Caputo and Riemann-Liouville fractional derivatives.

Extreme points, higher order fractional derivatives, Caputo derivative, Riemann-Liouville derivative

[1] A. B. Abdulla, M. Al-Refai, A. Al-Rawashdeh, On the existence and uniqueness of solutions for a class of non-linear fractional boundary value problems, J. King Saud Univ. Sci., 28 (2016), 103–110.

[2] R. P. Agarwal, M. Benchohra, S. Hamani, A survey on existing results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973–1033.

[3] M. Al-Refai, Basic results on nonlinear eigenvalue problems with fractional order, Electron. J. Differential Equations, 2012 (2012), 12 pages.

[4] M. Al-Refai, On the fractional derivative at extreme points, Electron. J. Qual. Theory Differ. Equ., 2012 (2012), 5 pages.

[5] M. Al-Refai, Yu. Luchko, Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications, Fract. Calc. Appl. Anal., 17 (2014), 483–498.

[6] M. Al-Refai, Yu. Luchko, Maximum principle for the multi-term time-fractional diffusion equations with the Riemann- Liouville fractional derivatives, Appl. Math. Comput., 257 (2015), 40–51.

[7] M. Al-Refai, Yu. Luchko, Analysis of fractional diffusion equations of distributed order: Maximum principles and its applications, Analysis, 36 (2016), 123–133.

[8] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, World Scientific Publishing, Hackensack, (2012).

[9] D. Baleanu, O. Mustafa, Asymptotic Integration and Stability for Differential Equations of Fractional Order, World Scientific Publishing, Hackensack, (2015).

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

[11] Y. Luchko, Fractional diffusion and wave propagation, Springer, 2014 (2014), 36 pages.

[12] J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140–1153.

[13] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, (1999).

[14] J. Sabatier, O. P. Agarwal, J. A. Tenreiro Machado, Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering, Springer, Netherlands, (2007).

[15] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Switzerland, (1993).

[16] M. Stynes, J. L. Gracia, A finite difference method for a two-point boundary value problem with Caputo fractional derivative, IMA Journal of Numerical Analysis, 35 (2014), 698–721.

[17] M. Syam, M. Al-Refai, Positive solutions and monotone iterative sequences for a class of higher order boundary value problems of fractional order, J. Fract. Calc. Appl., 4 (2013), 147–159.

[18] W. Xie, J. Xiao, Z. Luo, Existence of solutions for Riemann-Liouville fractional boundary value problem, Abstract and Applied Analysis, Abstr. Appl. Anal., 2014 (2014), 9 pages.

[19] X.-J. Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, Thermal Science, 2016 (2016), 13 pages.

[20] X. J. Yang, A new fractional operator of variablr order: application in the description of anomalous diffusion, Physica A: Statis. Mechanics Appl., 481 (2017), 276–283.

[21] X. J. Yang, F. Gao, H. M. Srivastava, New rheological models within local fractional derivative, Rom. Rep. Phys., 2017 (2017), 12 pages.

[22] H. Ye, F. Liu, V. Anh, I. Turner, Maximum principle and numerical method for the multi-term time-space Riesz-Caputo fractional differential equations, Appl. Math. Comput., 227 (2014), 531–540.