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.

[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.

[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.

[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.

[5]

L. CădariuL. 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.

[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.

[7]

M. E. GordjiY. ChoM. 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.

[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.

[9]

R. W. Ibrahim, Ulam stability of boundary value problem, Kragujevac J. Math., 37 (2013), 287-297.

[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.

[11]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations Elsevier, Amsterdam, 2006.

[12]

Y. N. LiH. 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.

[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.

[14]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations John Wiley, New York, 1993.

[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.

[16]

I. Podlubny, Fractional Differential Equations Academic Press, San Diego, 1999.

[17]

I. A. Rus, Ulam stability of ordinary differential equations, Stud. Univ. Babes-Bolyai, Math., 54 (2009), 125-133.

[18]

V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91-96.

[19]

H. RezaeiS.-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.

[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.

[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.

[22]

J. WangL. 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.

[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.

[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.

show all references

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.

[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.

[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.

[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.

[5]

L. CădariuL. 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.

[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.

[7]

M. E. GordjiY. ChoM. 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.

[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.

[9]

R. W. Ibrahim, Ulam stability of boundary value problem, Kragujevac J. Math., 37 (2013), 287-297.

[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.

[11]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations Elsevier, Amsterdam, 2006.

[12]

Y. N. LiH. 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.

[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.

[14]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations John Wiley, New York, 1993.

[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.

[16]

I. Podlubny, Fractional Differential Equations Academic Press, San Diego, 1999.

[17]

I. A. Rus, Ulam stability of ordinary differential equations, Stud. Univ. Babes-Bolyai, Math., 54 (2009), 125-133.

[18]

V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91-96.

[19]

H. RezaeiS.-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.

[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.

[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.

[22]

J. WangL. 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.

[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.

[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.

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