DOI QR코드

DOI QR Code

ON HYERS-ULAM STABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS

  • Huang, Jinghao (Department of Mathematics Sun Yat-Sen University) ;
  • Jung, Soon-Mo (Mathematics Section College of Science and Technology) ;
  • Li, Yongjin (Department of Mathematics Sun Yat-Sen University)
  • Received : 2014.05.27
  • Published : 2015.03.31

Abstract

We investigate the stability of nonlinear differential equations of the form $y^{(n)}(x)=F(x,y(x),y^{\prime}(x),{\cdots},y^{(n-1)}(x))$ with a Lipschitz condition by using a fixed point method. Moreover, a Hyers-Ulam constant of this differential equation is obtained.

Keywords

References

  1. C. Alsina and R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998), no. 4, 373-380.
  2. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66. https://doi.org/10.2969/jmsj/00210064
  3. J. Baker, J. Lawrence, and F. Zorzitto, The stability of the equation f(x+y) = f(x)f(y), Proc. Amer. Math. Soc. 74 (1979), no. 2, 242-246. https://doi.org/10.1090/S0002-9939-1979-0524294-6
  4. D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223-237. https://doi.org/10.1090/S0002-9904-1951-09511-7
  5. M. Burger, N. Ozawa, and A. Thom, On Ulam stabiltity, Israel J. Math (2012), 1-21.
  6. K. Cieplinski, Applications of fixed point theorems to the Hyers-Ulam stability of functional equations - a survey, Ann. Func. Anal. 3 (2012), no. 1, 151-164. https://doi.org/10.15352/afa/1399900032
  7. J. B. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309. https://doi.org/10.1090/S0002-9904-1968-11933-0
  8. V. A. Faiziev, Th. M. Rassias, and P. K. Sahoo, The space of (${\psi},{\gamma}$)-additive mappings on semigroups, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4455-4472. https://doi.org/10.1090/S0002-9947-02-03036-2
  9. G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), no. 1-2, 143-190. https://doi.org/10.1007/BF01831117
  10. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431-434. https://doi.org/10.1155/S016117129100056X
  11. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436. https://doi.org/10.1006/jmaa.1994.1211
  12. P. Gavruta, S.-M. Jung, and Y. Li, Hyers-Ulam stability for second-order linear differential equations with boundary conditions, Electron. J. Diff. Equ. 2011 (2011), no. 80, 1-5.
  13. O. Hatori, K. Kobayasi, T. Miura, H. Takagi, and S. E. Takahasi, On the best constant of Hyers-Ulam stability, J. Nonlinear Convex Anal. 5 (2004), no. 3, 387-393.
  14. T. Huuskonen and J. Vaisala, Hyers-Ulam constants of Hilbert spaces, Studia Math. 153 (2002), no. 1, 31-40. https://doi.org/10.4064/sm153-1-3
  15. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  16. D. H. Hyers and S. M. Ulam, On approximate isometries, Bull. Amer. Math. Soc. 51 (1945), 288-292. https://doi.org/10.1090/S0002-9904-1945-08337-2
  17. D. H. Hyers and S. M. Ulam, Approximate isometries of the space of continuous functions, Ann. Math. 48 (1947), 285-289. https://doi.org/10.2307/1969171
  18. D. H. Hyers and S. M. Ulam, On the stability of differential expressions, Math. Mag. 28 (1954), 59-64. https://doi.org/10.2307/3029365
  19. K.-W. Jun and Y.-H. Lee, A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation, J. Math. Anal. Appl. 238 (1999), no. 1, 305-315. https://doi.org/10.1006/jmaa.1999.6546
  20. S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (2004), no. 10, 1135-1140. https://doi.org/10.1016/j.aml.2003.11.004
  21. S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order. III, J. Math. Anal. Appl. 311 (2005), no. 1, 139-146. https://doi.org/10.1016/j.jmaa.2005.02.025
  22. S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order. II, Appl. Math. Lett. 19 (2006), no. 9, 854-858. https://doi.org/10.1016/j.aml.2005.11.004
  23. S.-M. Jung, A fixed point approach to the stability of differential equations y' = F(x, y), Bull. Malays. Math. Sci. Soc. (2) 33 (2010), no. 1, 47-56.
  24. S.-M. Jung, D. Popa, and M. Th. Rassias, On the stability of the linear functional equation in a single variable on complete metric groups, J. Global Optimi. 59 (2014), no. 1, 165-171. https://doi.org/10.1007/s10898-013-0083-9
  25. Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, New York, 2009.
  26. Y.-H. Lee, S.-M. Jung, and M. Th. Rassias, On an n-dimensional mixed type additive and quadratic functional equation, Appl. Math. Comput. 228 (2014), 13-16. https://doi.org/10.1016/j.amc.2013.11.091
  27. Y. Li, Hyers-Ulam stability of linear differential equations $y''={\lambda}^2y$, Thai J. Math. 8 (2010), no. 2, 215-219.
  28. Y. Li and Y. Shen, Hyers-Ulam stability of nonhomogeneous linear differential equations of second order, Internat. J. Math. Math. Sci 2009 (2009), Article ID 576852, 7 pp.
  29. Y. Li and Y. Shen, Hyers-Ulam stability of linear differential equations of second order, Appl. Math. Lett. 23 (2010), no. 3, 306-309. https://doi.org/10.1016/j.aml.2009.09.020
  30. T. Miura, On the Hyers-Ulam stability of a differentiable map, Sci. Math. Jpn. 55 (2002), no. 1, 17-24.
  31. T. Miura, S.-M. Jung, and S.-E. Takahasi, Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations $y'={\lambda}y$, J. Korean Math. Soc. 41 (2004), no. 6, 995-1005. https://doi.org/10.4134/JKMS.2004.41.6.995
  32. T. Miura, M. Miyajima, and S.-E. Takahasi, Hyers-Ulam stability of linear differential operator with constant coefficients, Math. Nachr. 258 (2003), 90-96. https://doi.org/10.1002/mana.200310088
  33. T. Miura, H. Oka, S.-E. Takahasi, and N. Niwa, Hyers-Ulam stability of the first order linear differential equation for Banach space-valued holomorphic mappings, J. Math. Inequal. 3 (2007), no. 3, 377-385.
  34. M. Ob loza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat. 13 (1993), 259-270.
  35. M. Ob loza, Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Mat. 14 (1997), 141-146.
  36. M. Omladic and P. Semrl, On non linear perturbations of isometries, Math. Ann. 303 (1995), no. 1, 617-628. https://doi.org/10.1007/BF01461008
  37. C.-G. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), no. 2, 711-720. https://doi.org/10.1016/S0022-247X(02)00386-4
  38. D. Popa and I. Rasa, Hyers-Ulam stability of the linear differential operator with non-constant coefficients, Appl. Math. Comput. 219 (2012), no. 4, 1562-1568. https://doi.org/10.1016/j.amc.2012.07.056
  39. V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), no. 1, 91-96.
  40. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  41. J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), no. 1, 126-130. https://doi.org/10.1016/0022-1236(82)90048-9
  42. Th. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), no. 4, 989-993. https://doi.org/10.1090/S0002-9939-1992-1059634-1
  43. Th. M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), no. 2, 325-338. https://doi.org/10.1006/jmaa.1993.1070
  44. H. Rezaei and S.-M. Jung, A fixed point approach to the stability of linear differential equations, Bull. Malays. Math. Sci. Soc. (2), in press.
  45. 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), no. 1, 244-251. https://doi.org/10.1016/j.jmaa.2013.02.034
  46. P. K. Sahoo and Pl. Kannappan, Intoduction to Functional Equations, Chapman and Hall CRC, Boca Raton, Florida, 2011.
  47. P. Semrl and J. Vaisala, Nonsurjective nearisometries of Banach spaces, J. Funct. Anal. 198 (2003), no. 1, 268-278. https://doi.org/10.1016/S0022-1236(02)00049-6
  48. S.-E. Takahasi, T. Miura, and S. Miyajima, On the Hyers-Ulam stability of the Banach space-valued differential equation $y'={\lambda}y$, Bull. Korean Math. Soc. 39 (2002), no. 2, 309-315. https://doi.org/10.4134/BKMS.2002.39.2.309
  49. S. M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960.

Cited by

  1. On the Hyers-Ulam Stability of First-Order Impulsive Delay Differential Equations vol.2016, 2016, https://doi.org/10.1155/2016/8164978
  2. Invariance of Hyers-Ulam stability of linear differential equations and its applications vol.2015, pp.1, 2015, https://doi.org/10.1186/s13662-015-0617-1
  3. Hyers-Ulam stability of delay differential equations of first order vol.289, pp.1, 2016, https://doi.org/10.1002/mana.201400298
  4. Hyers-Ulam stability of nonlinear differential equations with fractional integrable impulses vol.40, pp.15, 2017, https://doi.org/10.1002/mma.4405
  5. Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems vol.2017, pp.1, 2017, https://doi.org/10.1186/s13662-017-1248-5
  6. Hyers–Ulam stability of linear functional differential equations vol.426, pp.2, 2015, https://doi.org/10.1016/j.jmaa.2015.02.018
  7. Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations vol.2018, pp.1099-0526, 2018, https://doi.org/10.1155/2018/6423974
  8. Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays pp.1744-2516, 2018, https://doi.org/10.1080/17442508.2018.1551400