Stability in linear neutral difference equations with variable delays

Abdelouaheb Ardjouni; Ahcene Djoudi

Mathematica Bohemica (2013)

  • Volume: 138, Issue: 3, page 245-258
  • ISSN: 0862-7959

Abstract

top
In this paper we use the fixed point method to prove asymptotic stability results of the zero solution of a generalized linear neutral difference equation with variable delays. An asymptotic stability theorem with a sufficient condition is proved, which improves and generalizes some results due to Y. N. Raffoul (2006), E. Yankson (2009), M. Islam and E. Yankson (2005).

How to cite

top

Ardjouni, Abdelouaheb, and Djoudi, Ahcene. "Stability in linear neutral difference equations with variable delays." Mathematica Bohemica 138.3 (2013): 245-258. <http://eudml.org/doc/260713>.

@article{Ardjouni2013,
abstract = {In this paper we use the fixed point method to prove asymptotic stability results of the zero solution of a generalized linear neutral difference equation with variable delays. An asymptotic stability theorem with a sufficient condition is proved, which improves and generalizes some results due to Y. N. Raffoul (2006), E. Yankson (2009), M. Islam and E. Yankson (2005).},
author = {Ardjouni, Abdelouaheb, Djoudi, Ahcene},
journal = {Mathematica Bohemica},
keywords = {fixed point; stability; neutral difference equation; variable delay; neutral linear difference equation; stability; fixed point; contraction operator; variable delay},
language = {eng},
number = {3},
pages = {245-258},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Stability in linear neutral difference equations with variable delays},
url = {http://eudml.org/doc/260713},
volume = {138},
year = {2013},
}

TY - JOUR
AU - Ardjouni, Abdelouaheb
AU - Djoudi, Ahcene
TI - Stability in linear neutral difference equations with variable delays
JO - Mathematica Bohemica
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 138
IS - 3
SP - 245
EP - 258
AB - In this paper we use the fixed point method to prove asymptotic stability results of the zero solution of a generalized linear neutral difference equation with variable delays. An asymptotic stability theorem with a sufficient condition is proved, which improves and generalizes some results due to Y. N. Raffoul (2006), E. Yankson (2009), M. Islam and E. Yankson (2005).
LA - eng
KW - fixed point; stability; neutral difference equation; variable delay; neutral linear difference equation; stability; fixed point; contraction operator; variable delay
UR - http://eudml.org/doc/260713
ER -

References

top
  1. Ardjouni, A., Djoudi, A., 10.1016/j.na.2010.10.050, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 2062-2070. (2011) Zbl1216.34069MR2781737DOI10.1016/j.na.2010.10.050
  2. Berezansky, L., Braverman, E., 10.1016/j.jmaa.2004.09.042, J. Math. Anal. Appl. 304 (2005), 511-530. (2005) Zbl1068.39004MR2126547DOI10.1016/j.jmaa.2004.09.042
  3. Berezansky, L., Braverman, E., Liz, E., 10.1080/10236190500141050, J. Difference Equ. Appl. 11 (2005), 785-798. (2005) Zbl1078.39005MR2159797DOI10.1080/10236190500141050
  4. Burton, T. A., Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, Mineola (2006). (2006) Zbl1160.34001MR2281958
  5. Burton, T. A., Furumochi, T., Fixed points and problems in stability theory for ordinary and functional differential equations, Dyn. Syst. Appl. 10 (2001), 89-116. (2001) Zbl1021.34042MR1844329
  6. Chatzarakis, G. E., Miliaras, G. N., Asymptotic behavior in neutral difference equations with variable coefficients and more than one delay arguments, J. Math. Comput. Sci. 1 (2011), 32-52. (2011) MR2913376
  7. Elaydi, S., An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York (1999). (1999) Zbl0930.39001MR1711587
  8. Elaydi, S., 10.1006/jmaa.1994.1037, J. Math. Anal. Appl. 181 (1994), 483-492. (1994) Zbl0796.39004MR1260872DOI10.1006/jmaa.1994.1037
  9. Elaydi, S., Murakami, S., 10.1080/10236199808808097, J. Difference Equ. Appl. 3 (1998), 203-218. (1998) Zbl0891.39013MR1616085DOI10.1080/10236199808808097
  10. Eloe, P., Islam, M., Raffoul, Y. N., 10.1016/S0898-1221(03)00081-6, Comput. Math. Appl. 45 (2003), 1033-1039. (2003) Zbl1051.39003MR2000576DOI10.1016/S0898-1221(03)00081-6
  11. Gyori, I., Hartung, F., Stability in delay perturbed differential and difference equations, T. Faria Topics in Functional Differential and Difference Equations Papers of the conference on functional differential and difference equations, Lisbon, Portugal, July 26-30, 1999 AMS, Providence. Fields Inst. Commun. (2001), 181-194. (2001) Zbl0990.34066MR1821781
  12. Islam, M., Raffoul, Y. N., 10.1080/1023619031000101516, J. Difference Equ. Appl. 9 (2003), 819-825. (2003) Zbl1055.39011MR1995220DOI10.1080/1023619031000101516
  13. Islam, M., Yankson, E., Boundedness and stability in nonlinear delay difference equations employing fixed point theory, Electron. J. Qual. Theory Differ. Equ. 2005, electronic only (2005) 18 p. Zbl1103.39009MR2191670
  14. Kelly, W. G., Peterson, A. C., Difference Equations: An Introduction with Applications, Academic Press, San Diego (2001). (2001) MR1765695
  15. Liz, E., 10.1080/10236198.2010.549007, J. Difference Equ. Appl. 17 (2011), 203-220. (2011) Zbl1216.39021MR2783344DOI10.1080/10236198.2010.549007
  16. Liz, E., 10.1016/j.jmaa.2004.08.048, J. Math. Anal. Appl. 303 (2005), 492-498. (2005) Zbl1068.39017MR2122232DOI10.1016/j.jmaa.2004.08.048
  17. Malygina, V. V., Kulikov, A. Y., On precision of constants in some theorems on stability of difference equations, Func. Differ. Equ. 15 (2008), 239-248. (2008) Zbl1153.39003MR2384923
  18. Pituk, M., 10.1016/j.aml.2004.06.005, Appl. Math. Lett. 17 (2004), 779-783. (2004) Zbl1068.39019MR2072834DOI10.1016/j.aml.2004.06.005
  19. Raffoul, Y. N., 10.1016/j.jmaa.2006.01.044, J. Math. Anal. Appl. 324 (2006), 1356-1362. (2006) Zbl1115.39015MR2266564DOI10.1016/j.jmaa.2006.01.044
  20. Raffoul, Y. N., 10.1080/10236190410001652847, J. Difference Equ. Appl. 10 (2004), 1229-1242. (2004) Zbl1068.39021MR2100724DOI10.1080/10236190410001652847
  21. Raffoul, Y. N., 10.1016/S0022-247X(03)00051-9, J. Math. Anal. Appl. 279 (2003), 639-650. (2003) Zbl1022.39004MR1974051DOI10.1016/S0022-247X(03)00051-9
  22. Smart, D. R., Fixed Point Theorems, Cambridge Tracts in Mathematics 66. Cambridge University Press, London (1974). (1974) Zbl0297.47042MR0467717
  23. Yankson, E., Stability in discrete equations with variable delays Electronic J. Qual. Theory Differ. Equ. 2009, electronic only 2009, 7 p, . MR2480418
  24. Yankson, E., Stability of Volterra difference delay equations, Electronic J. Qual. Theory Differ. Equ. 2006, electronic only (2006), 14 p. Zbl1113.39013MR2263079
  25. Zhang, B., 10.1016/j.na.2005.02.081, Nonlinear Anal., Theory Methods Appl. 63 (2005), e233--e242. (2005) Zbl1159.34348DOI10.1016/j.na.2005.02.081
  26. Zhang, B. G., Tian, C. J., Wong, P. J. Y., Global attractivity of difference equations with variable delay, Dyn. Contin. Discrete Impulsive Syst. 6 (1999), 307-317. (1999) Zbl0938.39016MR1708451

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.