ON HYERS-ULAM STABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 52, Issue 2, 2015, pp.685-697
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2015.52.2.685

Title & Authors

ON HYERS-ULAM STABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS

Huang, Jinghao; Jung, Soon-Mo; Li, Yongjin;

Huang, Jinghao; Jung, Soon-Mo; Li, Yongjin;

Abstract

We investigate the stability of nonlinear differential equations of the form $y^{(n)}(x)

Keywords

Hyers-Ulam stability;generalized Hyers-Ulam stability;nonlinear differential equations;fixed point theorem;

Language

English

Cited by

1.

2.

3.

4.

5.

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.

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.

4.

D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223-237.

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.

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.

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.

9.

G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), no. 1-2, 143-190.

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.

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.

15.

D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.

16.

17.

D. H. Hyers and S. M. Ulam, Approximate isometries of the space of continuous functions, Ann. Math. 48 (1947), 285-289.

18.

D. H. Hyers and S. M. Ulam, On the stability of differential expressions, Math. Mag. 28 (1954), 59-64.

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.

20.

S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (2004), no. 10, 1135-1140.

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.

22.

S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order. II, Appl. Math. Lett. 19 (2006), no. 9, 854-858.

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.

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.

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.

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.

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.

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.

37.

C.-G. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), no. 2, 711-720.

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.

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.

41.

J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), no. 1, 126-130.

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.

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.

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.

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.

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.

49.

S. M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960.