Hyers-Ulam Stability of a Generalized Second-Order Nonlinear Differential Equation

ABSTRACT

In this paper we have established the stability of a generalized nonlinear second-order differential equation in the sense of Hyers and Ulam. We also have proved the Hyers-Ulam stability of Emden-Fowler type equation with initial conditions.

Cite this paper

M. Qarawani, "Hyers-Ulam Stability of a Generalized Second-Order Nonlinear Differential Equation,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 1857-1861. doi: 10.4236/am.2012.312252.

M. Qarawani, "Hyers-Ulam Stability of a Generalized Second-Order Nonlinear Differential Equation,"

References

[1] S. M. Ulam, “Problems in Modern Mathematics,” John Wiley & Sons, New York, 1964.

[2] D. H. Hyers, “On the Stability of the Linear Functional Equation,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 27, No. 4, 1941, pp. 222-224. doi:10.1073/pnas.27.4.222

[3] T. M. R assias, “On the Stability of the Linear Mapping in Banach Spaces,” Proceedings of the American Mathemaical Society, Vol. 72, No. 2, 1978, pp. 297-300. doi:10.1090/S0002-9939-1978-0507327-1

[4] T. Miura, S.-E. Takahasi and H. Choda, “On the Hyers- Ulam Stability of Real Continuous Function Valued Dif- ferentiable Map,” Tokyo Journal of Mathematics, Vol. 24, No. 2, 2001, pp. 467-476. doi:10.3836/tjm/1255958187

[5] S. M. Jung, “On the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings,” Journal of Mathematics Analysis and Application, Vol. 204, No. 1, 1996, pp. 221-226. doi:10.1006/jmaa.1996.0433

[6] C. G. Park, “On the Stability of the Linear Mapping in Banach Modules,” Journal of Mathematics Analysis and Application, Vol. 275, No. 2, 2002, pp. 711-720. doi:10.1016/S0022-247X(02)00386-4

[7] C. Alsina and R. Ger, “On Some Inequalities and Stability Results Related to the Exponential Function,” Journal of Inequalities and Application, Vol. 2, No. 4, 1998, pp. 373-380.

[8] E. Takahasi, T. Miura and S. Miyajima, “On the HyersUlam Stability of the Banach Space-Valued Differential Equation ,” Bulletin of the Korean Mathematical Society, Vol. 39, No. 2, 2002, pp 309-315. doi:10.4134/BKMS.2002.39.2.309

[9] T. Miura, S. Miyajima and S.-E. Takahasi, “A Characterization of Hyers-Ulam Stability of First Order Linear Differential Operators,” Journal of Mathematics Analysis and Application, Vol. 286, No. 1, 2003, pp. 136-146.

[10] S. M. Jung, “Hyers-Ulam Stability of Linear Differential Equations of First Order,” Journal of Mathematics Analysis and Application, Vol. 311, No. 1, 2005, pp. 139-146. doi:10.1016/j.jmaa.2005.02.025

[11] G. Wang, M. Zhou and L. Sun, “Hyers-Ulam Stability of Linear Differential Equations of First Order,” Applied Mathematics Letters, Vol. 21, No. 10, 2008, pp 1024-1028. doi:10.1016/j.aml.2007.10.020

[12] Y. Li, “Hyers-Ulam Stability of Linear Differential Equations,” Thai Journal of Mathematics, Vol. 8, No 2, 2010, pp. 215-219.

[13] Y. Li and Y. Shen, “Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order,” International Journal of Mathematics and Mathematical Sciences, Vol. 2009, 2009, Article ID: 576852, p 7.

[14] P. Gavruta, S. Jung and Y. Li, “Hyers-Ulam Stability for Second-Order Linear Differential Equations With Boundary Conditions,” Electronic Journal of Differential Equations, Vol. 2011, No. 80, 2011, pp. 1-7.

[15] M. N. Qarawani, “Hyers-Ulam Stability of Linear and Nonlinear Differential Equations of Second Order,” International Journal of Applied Mathematics, Vol. 1, No. 4, 2012, pp. 422-432.

[16] I. A. Rus, “Ulam Stability of Ordinary Differential Equations,” Studia Universitatis Babe?-Bolyai—Series Mathematica, Vol. LIV, No. 4, 2009, pp. 125-133.

[1] S. M. Ulam, “Problems in Modern Mathematics,” John Wiley & Sons, New York, 1964.

[2] D. H. Hyers, “On the Stability of the Linear Functional Equation,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 27, No. 4, 1941, pp. 222-224. doi:10.1073/pnas.27.4.222

[3] T. M. R assias, “On the Stability of the Linear Mapping in Banach Spaces,” Proceedings of the American Mathemaical Society, Vol. 72, No. 2, 1978, pp. 297-300. doi:10.1090/S0002-9939-1978-0507327-1

[4] T. Miura, S.-E. Takahasi and H. Choda, “On the Hyers- Ulam Stability of Real Continuous Function Valued Dif- ferentiable Map,” Tokyo Journal of Mathematics, Vol. 24, No. 2, 2001, pp. 467-476. doi:10.3836/tjm/1255958187

[5] S. M. Jung, “On the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings,” Journal of Mathematics Analysis and Application, Vol. 204, No. 1, 1996, pp. 221-226. doi:10.1006/jmaa.1996.0433

[6] C. G. Park, “On the Stability of the Linear Mapping in Banach Modules,” Journal of Mathematics Analysis and Application, Vol. 275, No. 2, 2002, pp. 711-720. doi:10.1016/S0022-247X(02)00386-4

[7] C. Alsina and R. Ger, “On Some Inequalities and Stability Results Related to the Exponential Function,” Journal of Inequalities and Application, Vol. 2, No. 4, 1998, pp. 373-380.

[8] E. Takahasi, T. Miura and S. Miyajima, “On the HyersUlam Stability of the Banach Space-Valued Differential Equation ,” Bulletin of the Korean Mathematical Society, Vol. 39, No. 2, 2002, pp 309-315. doi:10.4134/BKMS.2002.39.2.309

[9] T. Miura, S. Miyajima and S.-E. Takahasi, “A Characterization of Hyers-Ulam Stability of First Order Linear Differential Operators,” Journal of Mathematics Analysis and Application, Vol. 286, No. 1, 2003, pp. 136-146.

[10] S. M. Jung, “Hyers-Ulam Stability of Linear Differential Equations of First Order,” Journal of Mathematics Analysis and Application, Vol. 311, No. 1, 2005, pp. 139-146. doi:10.1016/j.jmaa.2005.02.025

[11] G. Wang, M. Zhou and L. Sun, “Hyers-Ulam Stability of Linear Differential Equations of First Order,” Applied Mathematics Letters, Vol. 21, No. 10, 2008, pp 1024-1028. doi:10.1016/j.aml.2007.10.020

[12] Y. Li, “Hyers-Ulam Stability of Linear Differential Equations,” Thai Journal of Mathematics, Vol. 8, No 2, 2010, pp. 215-219.

[13] Y. Li and Y. Shen, “Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order,” International Journal of Mathematics and Mathematical Sciences, Vol. 2009, 2009, Article ID: 576852, p 7.

[14] P. Gavruta, S. Jung and Y. Li, “Hyers-Ulam Stability for Second-Order Linear Differential Equations With Boundary Conditions,” Electronic Journal of Differential Equations, Vol. 2011, No. 80, 2011, pp. 1-7.

[15] M. N. Qarawani, “Hyers-Ulam Stability of Linear and Nonlinear Differential Equations of Second Order,” International Journal of Applied Mathematics, Vol. 1, No. 4, 2012, pp. 422-432.

[16] I. A. Rus, “Ulam Stability of Ordinary Differential Equations,” Studia Universitatis Babe?-Bolyai—Series Mathematica, Vol. LIV, No. 4, 2009, pp. 125-133.