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ON HYERS-ULAM STABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS
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 Title & Authors
ON HYERS-ULAM STABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS
Huang, Jinghao; Jung, Soon-Mo; Li, Yongjin;
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 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
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On the Hyers-Ulam Stability of First-Order Impulsive Delay Differential Equations, Journal of Function Spaces, 2016, 2016, 1  crossref(new windwow)
2.
Invariance of Hyers-Ulam stability of linear differential equations and its applications, Advances in Difference Equations, 2015, 2015, 1  crossref(new windwow)
3.
Hyers-Ulam stability of delay differential equations of first order, Mathematische Nachrichten, 2016, 289, 1, 60  crossref(new windwow)
4.
Hyers-Ulam stability of nonlinear differential equations with fractional integrable impulses, Mathematical Methods in the Applied Sciences, 2017, 40, 15, 5502  crossref(new windwow)
5.
Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems, Advances in Difference Equations, 2017, 2017, 1  crossref(new windwow)
6.
Hyers–Ulam stability of linear functional differential equations, Journal of Mathematical Analysis and Applications, 2015, 426, 2, 1192  crossref(new windwow)
 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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

9.
G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), no. 1-2, 143-190. crossref(new window)

10.
Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431-434. crossref(new window)

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. crossref(new window)

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. crossref(new window)

15.
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. crossref(new window)

16.
D. H. Hyers and S. M. Ulam, On approximate isometries, Bull. Amer. Math. Soc. 51 (1945), 288-292. crossref(new window)

17.
D. H. Hyers and S. M. Ulam, Approximate isometries of the space of continuous functions, Ann. Math. 48 (1947), 285-289. crossref(new window)

18.
D. H. Hyers and S. M. Ulam, On the stability of differential expressions, Math. Mag. 28 (1954), 59-64. crossref(new window)

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. crossref(new window)

20.
S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (2004), no. 10, 1135-1140. crossref(new window)

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. crossref(new window)

22.
S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order. II, Appl. Math. Lett. 19 (2006), no. 9, 854-858. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

37.
C.-G. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), no. 2, 711-720. crossref(new window)

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. crossref(new window)

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. crossref(new window)

41.
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), no. 1, 126-130. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

49.
S. M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960.