Ulam's type stability of impulsive ordinary differential equations

Journal of Mathematical Analysis and Applications

Volumn 395, Issue 1, 2012, Pages 258-264

  Wang, JinRong ^a   Fečkan, Michal ^b,c   Zhou, Yong ^d  

^a GUIZHOU UNIVERSITY   (China)

^b Faculty of Natural Sciences   (Slovakia)

^c INSTITUTE OF CHEMISTRY   (Slovakia)

^d XIANGTAN UNIVERSITY   (China)

Author keywords

Impulsive ordinary differential equations; Ulam's type stability

Indexed keywords

EID: 84862984768     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2012.05.040     Document Type: Article

References (30)

1. Ulam S.M. A Collection of Mathematical Problems 1968, Interscience Publishers, New York.
2. Hyers D.H. On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 1941, 27:222-224.
3. Rassias Th.M. On the stability of linear mappings in Banach spaces. Proc. Amer. Math. Soc. 1978, 72:297-300.
4. L. Cǎdariu, Stabilitatea Ulam-Hyers-Bourgin pentru ecuatii functionale, Ed. Univ. Vest Timişara, Timişara, 2007.
5. Hyers D.H., Isac G., Rassias Th.M. Stability of Functional Equations in Several Variables 1998, Birkhäuser.
6. Jung S.-M. Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis 2001, Hadronic Press, Palm Harbor.
7. Jung S.-M. Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 2004, 17:1135-1140.
8. Jung S.-M. Hyers-Ulam stability of linear differential equations of first order, III. J. Math. Anal. Appl. 2005, 311:139-146.
9. Jung S.-M., Rassias Th.M. Generalized Hyers-Ulam stability of Riccati differential equation. Math. Inequal. Appl. 2008, 11:777-782.
10. '(t)=λy(t-τ). Abstr. Appl. Anal. 2010, 2010:10. Article ID 372176.
11. Jung S.-M. Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis 2011, Springer, New York.
12. Brzdȩk J., Jung S.-M. A note on stability of an operator linear equation of the second order. Abstr. Appl. Anal. 2011, 2011:15. Article ID 602713.
13. Li Y., Shen Y. Hyers-Ulam stability of nonhomogeneous linear differential equations of second order. Int. J. Math. Math. Sci. 2009, 2009:7. Article ID 576852.
14. Miura T., Miyajima S., Takahasi S.-E. A characterization of Hyers-Ulam stability of first order linear differential operators. J. Math. Anal. Appl. 2003, 286:136-146.
15. Miura T., Miyajima S., Takahasi S.-E. Hyers-Ulam stability of linear differential operator with constant coefficients. Math. Nachr. 2003, 258:90-96.
16. Rus I.A. Ulam stability of ordinary differential equations. Studia Univ. Babeş-Bolyai Math. 2009, 54:125-133.
17. Obłoza M. Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat. 1993, 13:259-270.
18. Obłoza M. Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk.-Dydakt. Prace Mat. 1997, 14:141-146.
19. '=λy. Bull. Korean Math. Soc. 2002, 39:309-315.
20. Cimpean D.S., Popa D. Hyers-Ulam stability of Euler's equation. Appl. Math. Lett. 2011, 24:1539-1543.
21. Popa D., Raşa I. On the Hyers-Ulam stability of the linear differential equation. J. Math. Anal. Appl. 2011, 381:530-537.
22. Lungu N., Popa D. Hyers-Ulam stability of a first order partial differential equation. J. Math. Anal. Appl. 2012, 385:86-91.
23. Ibrahim R.W. Approximate solutions for fractional differential equation in the unit disk. Electron. J. Qual. Theory Differ. Equ. 2011, 64:1-11.
24. Wang J., Lv L., Zhou Y. Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electron. J. Qual. Theory Differ. Equ. 2011, 63:1-10.
25. Wang J., Lv L., Zhou Y. New concepts and results in stability of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 2012, 17:2530-2538.
26. Wang J., Zhou Y. Mittag-Leffler-Ulam stabilities of fractional evolution equations. Appl. Math. Lett. 2012, 25:723-728.
27. Samoilenko A.M., Perestyuk N.A. Stability of solutions of differential equations with impulse effect. Differ. Equ. 1977, 13:1981-1992.
28. Bainov D.D., Hristova S.G. Integral inequalities of Gronwall type for piecewise continuous functions. J. Appl. Math. Stoch. Anal. 1997, 10:89-94.
29. Tatar N.-E. An impulsive nonlinear singular version of the Gronwall-Bihari inequality. J. Inequal. Appl. 2006, 2006:12. Article ID 84561.
30. Butler G., Rogers T. A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations. J. Math. Anal. Appl. 1971, 33:77-81.