Stability in linear neutral difference equations with variable delays

Ardjouni, Abdelouaheb ; Djoudi, Ahcene

Mathematica Bohemica, Tome 138 (2013) no. 3, pp. 245-258

Abstract (VO)

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).

Détail

DOI : 10.21136/MB.2013.143436

Classification : 39A30, 39A70
Keywords: fixed point; stability; neutral difference equation; variable delay

Bibliographie

[1] Ardjouni, A., Djoudi, A.: Fixed points and stability in linear neutral differential equations with variable delays. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 2062-2070. | DOI | MR | Zbl

[2] Berezansky, L., Braverman, E.: On exponential dichotomy, Bohl-Perron type theorems and stability of difference equations. J. Math. Anal. Appl. 304 (2005), 511-530. | DOI | MR | Zbl

[3] Berezansky, L., Braverman, E., Liz, E.: Sufficient conditions for the global stability of nonautonomous higher order difference equations. J. Difference Equ. Appl. 11 (2005), 785-798. | DOI | MR | Zbl

[4] Burton, T. A.: Stability by Fixed Point Theory for Functional Differential Equations. Dover Publications, Mineola (2006). | MR | Zbl

[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. | MR | Zbl

[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. | MR

[7] Elaydi, S.: An Introduction to Difference Equations. Undergraduate Texts in Mathematics. Springer, New York (1999). | MR | Zbl

[8] Elaydi, S.: Periodicity and stability of linear Volterra difference systems. J. Math. Anal. Appl. 181 (1994), 483-492. | DOI | MR | Zbl

[9] Elaydi, S., Murakami, S.: Uniform asymptotic stability in linear Volterra difference equations. J. Difference Equ. Appl. 3 (1998), 203-218. | DOI | MR | Zbl

[10] Eloe, P., Islam, M., Raffoul, Y. N.: Uniform asymptotic stability in nonlinear Volterra discrete systems. Comput. Math. Appl. 45 (2003), 1033-1039. | DOI | MR | Zbl

[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. {\it 29} (2001), 181-194. | MR | Zbl

[12] Islam, M., Raffoul, Y. N.: Exponential stability in nonlinear difference equations. J. Difference Equ. Appl. 9 (2003), 819-825. | DOI | MR | Zbl

[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. | MR | Zbl

[14] Kelly, W. G., Peterson, A. C.: Difference Equations: An Introduction with Applications. Academic Press, San Diego (2001). | MR

[15] Liz, E.: Stability of non-autonomous difference equations: simple ideas leading to useful results. J. Difference Equ. Appl. 17 (2011), 203-220. | DOI | MR | Zbl

[16] Liz, E.: On explicit conditions for the asymptotic stability of linear higher order difference equations. J. Math. Anal. Appl. 303 (2005), 492-498. | DOI | MR | Zbl

[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. | MR | Zbl

[18] Pituk, M.: A criterion for the exponential stability of linear difference equations. Appl. Math. Lett. 17 (2004), 779-783. | DOI | MR | Zbl

[19] Raffoul, Y. N.: Stability and periodicity in discrete delay equations. J. Math. Anal. Appl. 324 (2006), 1356-1362. | DOI | MR | Zbl

[20] Raffoul, Y. N.: Periodicity in general delay nonlinear difference equations using fixed point theory. J. Difference Equ. Appl. 10 (2004), 1229-1242. | DOI | MR | Zbl

[21] Raffoul, Y. N.: General theorems for stability and boundedness for nonlinear functional discrete systems. J. Math. Anal. Appl. 279 (2003), 639-650. | DOI | MR | Zbl

[22] Smart, D. R.: Fixed Point Theorems. Cambridge Tracts in Mathematics 66. Cambridge University Press, London (1974). | MR | Zbl

[23] Yankson, E.: Stability in discrete equations with variable delays Electronic J. Qual. Theory Differ. Equ. 2009, electronic only 2009, 7 p. | MR

[24] Yankson, E.: Stability of Volterra difference delay equations. Electronic J. Qual. Theory Differ. Equ. 2006, electronic only (2006), 14 p. | MR | Zbl

[25] Zhang, B.: Fixed points and stability in differential equations with variable delays. Nonlinear Anal., Theory Methods Appl. 63 (2005), e233--e242. | DOI | Zbl

[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. | MR | Zbl