HYERS-ULAM-RASSIAS STABILITY OF THE BANACH SPACE VALUED LINEAR DIFFERENTIAL EQUATIONS y′ = λy

- Journal title : Journal of the Korean Mathematical Society
- Volume 41, Issue 6, 2004, pp.995-1005
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2004.41.6.995

Title & Authors

HYERS-ULAM-RASSIAS STABILITY OF THE BANACH SPACE VALUED LINEAR DIFFERENTIAL EQUATIONS y′ = λy

Miura, Takeshi Miura; Jung, Soon-Mo; Takahasi, Sin-Ei;

Miura, Takeshi Miura; Jung, Soon-Mo; Takahasi, Sin-Ei;

Abstract

The aim of this paper is to prove the stability in the sense of Hyers-Ulam- Rassias of the Banach space valued differentialequation y' = λy, where λ is a complex constant. That is, suppose f is a Banach space valued strongly differentiable function on an open interval. If f is an approximate solution of the equation y' = λy, then there exists an exact solution of the equation near to f.

Keywords

Hyers-Ulam-Rassias stability;differential equation;

Language

English

Cited by

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

17.

18.

19.

20.

21.

22.

24.

25.

26.

27.

References

1.

C. Alsina and R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998), 373–380.

3.

D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224.

4.

S.-M. Jung and K. Lee, Hyers-Ulam-Rassias stability of linear differential equations, to appear.

5.

T. Miura, S. Miyajima and S.-E. Takahasi, Hyers-Ulam stability of linear differential operator with constant coefficients, Math. Nachr. 258 (2003), 90–96.

6.

T. Miura, S. Miyajima and S.-E. Takahasi, A characterization of Hyers-Ulam stability of first order linear differential operators, J. Math. Anal. Appl. 286 (2003), 136–146.

7.

Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.

8.

Th. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993.

9.

W. Rudin, Real and Complex Analysis (3rd Edition), McGraw-Hill, 1987.

10.

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), 309–315.

11.

S. M. Ulam, Problems in Modern Mathematics, Chap. VI, Science Editions, Wiley, New York, 1964.

12.

S. M. Ulam, Sets, Numbers and Universes Selected Works, Part III, MIT Press, Cambridge, MA, 1974.