# American Institute of Mathematical Sciences

June  2017, 10(3): 505-521. doi: 10.3934/dcdss.2017025

## Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative

 1 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China 2 Department of Mathematics, Changzhi University, Changzhi Shanxi 046011, China

* Corresponding author

Received  June 2016 Revised  January 2017 Published  February 2017

In this paper, using the weighted space method and a fixed point theorem, we investigate the Hyers-Ulam-Rassias stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville derivative on the continuous function space. We obtain some sufficient conditions in order that the nonlinear fractional differential equations are stable on the continuous function space. The results improve and extend some recent results. Finally, we construct some examples to illustrate the theoretical results.

Citation: Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025
##### References:
 [1] S. András and A. R. Mészáros, Ulam--Hyers stability of dynamic equations on time scales via Picard operators, Appl. Math. Comput., 219 (2013), 4853-4864. doi: 10.1016/j.amc.2012.10.115. Google Scholar [2] S. András and A. R. Mészáros, Ulam--Hyers stability of elliptic partial differential equations in Sobolev spaces, Appl. Math. Comput., 229 (2014), 131-138. doi: 10.1016/j.amc.2013.12.021. Google Scholar [3] S. András and J. J. Kolumbán, On the Ulam--Hyers stability of first order differential systems with nonlocal initial conditions, Nonlinear Analysis, 82 (2013), 1-11. doi: 10.1016/j.na.2012.12.008. Google Scholar [4] L. Cădariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math., 4 (2003), 1-7. Google Scholar [5] L. Cădariu, L. Găvruţa and P. Găvruţa, Weighted space method for the stability of some nonlinear equations, Appl. Anal. Discrete Math., 6 (2012), 126-139. doi: 10.2298/AADM120309007C. Google Scholar [6] P. Găvruţa and L. Găvruţa, A new method for the generalized Hyers-Ulam-Rassias stability, Int. J. Nonlinear Anal. Appl., 1 (2010), 11-18. Google Scholar [7] M. E. Gordji, Y. Cho, M. Ghaemi and B. Alizadeh, Stability of the second order partial differential equations, J. Inequal. Appl., 2011 (2011), 1-10. doi: 10.1186/1029-242X-2011-81. Google Scholar [8] B. Hegyi and S.-M. Jung, On the stability of Laplace's equation, Appl. Math. Lett., 26 (2013), 549-552. doi: 10.1016/j.aml.2012.12.014. Google Scholar [9] R. W. Ibrahim, Ulam stability of boundary value problem, Kragujevac J. Math., 37 (2013), 287-297. Google Scholar [10] S.-M. Jung, A fixed point approach to the stability of differential equations $y'=F(x, y)$, Bull. Malays. Math. Sci. Soc., 33 (2010), 47-56. Google Scholar [11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations Elsevier, Amsterdam, 2006. Google Scholar [12] Y. N. Li, H. R. Sun and Z. Feng, Fractional abstract Cauchy problem with order $α ∈ (1, 2)$, Dyn. Partial Diff. Equations, 13 (2016), 155-177. doi: 10.4310/DPDE.2016.v13.n2.a4. Google Scholar [13] N. Lungu and D. Popa, Hyers-Ulam stability of a first order partial differential equation, J. Math. Anal. Appl., 385 (2012), 86-91. doi: 10.1016/j.jmaa.2011.06.025. Google Scholar [14] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations John Wiley, New York, 1993.Google Scholar [15] D. Popa and I. Raşa, On the Hyers--Ulam stability of the linear differential equation, J. Math. Anal. Appl., 381 (2011), 530-537. doi: 10.1016/j.jmaa.2011.02.051. Google Scholar [16] I. Podlubny, Fractional Differential Equations Academic Press, San Diego, 1999. Google Scholar [17] I. A. Rus, Ulam stability of ordinary differential equations, Stud. Univ. Babes-Bolyai, Math., 54 (2009), 125-133. Google Scholar [18] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91-96. Google Scholar [19] H. Rezaei, S.-M. Jung and Th. M. Rassias, Laplace transform and Hyers--Ulam stability of linear differential equations, J. Math. Anal. Appl., 403 (2013), 244-251. doi: 10.1016/j.jmaa.2013.02.034. Google Scholar [20] Y. H. Su and Z. Feng, Existence theory for an arbitrary order fractional differential equation with deviating argument, Acta Appl. Math., 118 (2012), 81-105. doi: 10.1007/s10440-012-9679-1. Google Scholar [21] J. Wang and Y. Zhou, Mittag--Leffler--Ulam stabilities of fractional evolution equations, Appl. Math. Lett., 25 (2012), 723-728. doi: 10.1016/j.aml.2011.10.009. Google Scholar [22] J. Wang, L. Lv and Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2530-2538. doi: 10.1016/j.cnsns.2011.09.030. Google Scholar [23] J. Wang and Y. Zhang, A class of nonlinear differential equations with fractional integrable impulses, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3001-3010. doi: 10.1016/j.cnsns.2014.01.016. Google Scholar [24] J. Wang and X. Li, A uniform method to Ulam--Hyers stability for some linear fractional equations, Mediterr. J. Math., 13 (2016), 625-635. doi: 10.1007/s00009-015-0523-5. Google Scholar

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##### References:
 [1] S. András and A. R. Mészáros, Ulam--Hyers stability of dynamic equations on time scales via Picard operators, Appl. Math. Comput., 219 (2013), 4853-4864. doi: 10.1016/j.amc.2012.10.115. Google Scholar [2] S. András and A. R. Mészáros, Ulam--Hyers stability of elliptic partial differential equations in Sobolev spaces, Appl. Math. Comput., 229 (2014), 131-138. doi: 10.1016/j.amc.2013.12.021. Google Scholar [3] S. András and J. J. Kolumbán, On the Ulam--Hyers stability of first order differential systems with nonlocal initial conditions, Nonlinear Analysis, 82 (2013), 1-11. doi: 10.1016/j.na.2012.12.008. Google Scholar [4] L. Cădariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math., 4 (2003), 1-7. Google Scholar [5] L. Cădariu, L. Găvruţa and P. Găvruţa, Weighted space method for the stability of some nonlinear equations, Appl. Anal. Discrete Math., 6 (2012), 126-139. doi: 10.2298/AADM120309007C. Google Scholar [6] P. Găvruţa and L. Găvruţa, A new method for the generalized Hyers-Ulam-Rassias stability, Int. J. Nonlinear Anal. Appl., 1 (2010), 11-18. Google Scholar [7] M. E. Gordji, Y. Cho, M. Ghaemi and B. Alizadeh, Stability of the second order partial differential equations, J. Inequal. Appl., 2011 (2011), 1-10. doi: 10.1186/1029-242X-2011-81. Google Scholar [8] B. Hegyi and S.-M. Jung, On the stability of Laplace's equation, Appl. Math. Lett., 26 (2013), 549-552. doi: 10.1016/j.aml.2012.12.014. Google Scholar [9] R. W. Ibrahim, Ulam stability of boundary value problem, Kragujevac J. Math., 37 (2013), 287-297. Google Scholar [10] S.-M. Jung, A fixed point approach to the stability of differential equations $y'=F(x, y)$, Bull. Malays. Math. Sci. Soc., 33 (2010), 47-56. Google Scholar [11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations Elsevier, Amsterdam, 2006. Google Scholar [12] Y. N. Li, H. R. Sun and Z. Feng, Fractional abstract Cauchy problem with order $α ∈ (1, 2)$, Dyn. Partial Diff. Equations, 13 (2016), 155-177. doi: 10.4310/DPDE.2016.v13.n2.a4. Google Scholar [13] N. Lungu and D. Popa, Hyers-Ulam stability of a first order partial differential equation, J. Math. Anal. Appl., 385 (2012), 86-91. doi: 10.1016/j.jmaa.2011.06.025. Google Scholar [14] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations John Wiley, New York, 1993.Google Scholar [15] D. Popa and I. Raşa, On the Hyers--Ulam stability of the linear differential equation, J. Math. Anal. Appl., 381 (2011), 530-537. doi: 10.1016/j.jmaa.2011.02.051. Google Scholar [16] I. Podlubny, Fractional Differential Equations Academic Press, San Diego, 1999. Google Scholar [17] I. A. Rus, Ulam stability of ordinary differential equations, Stud. Univ. Babes-Bolyai, Math., 54 (2009), 125-133. Google Scholar [18] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91-96. Google Scholar [19] H. Rezaei, S.-M. Jung and Th. M. Rassias, Laplace transform and Hyers--Ulam stability of linear differential equations, J. Math. Anal. Appl., 403 (2013), 244-251. doi: 10.1016/j.jmaa.2013.02.034. Google Scholar [20] Y. H. Su and Z. Feng, Existence theory for an arbitrary order fractional differential equation with deviating argument, Acta Appl. Math., 118 (2012), 81-105. doi: 10.1007/s10440-012-9679-1. Google Scholar [21] J. Wang and Y. Zhou, Mittag--Leffler--Ulam stabilities of fractional evolution equations, Appl. Math. Lett., 25 (2012), 723-728. doi: 10.1016/j.aml.2011.10.009. Google Scholar [22] J. Wang, L. Lv and Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2530-2538. doi: 10.1016/j.cnsns.2011.09.030. Google Scholar [23] J. Wang and Y. Zhang, A class of nonlinear differential equations with fractional integrable impulses, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3001-3010. doi: 10.1016/j.cnsns.2014.01.016. Google Scholar [24] J. Wang and X. Li, A uniform method to Ulam--Hyers stability for some linear fractional equations, Mediterr. J. Math., 13 (2016), 625-635. doi: 10.1007/s00009-015-0523-5. Google Scholar
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