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Keywords:
fixed point; stability; neutral difference equation; variable delay
Summary:
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).
References:
[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 10.1016/j.na.2010.10.050 | MR 2781737 | Zbl 1216.34069
[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 10.1016/j.jmaa.2004.09.042 | MR 2126547 | Zbl 1068.39004
[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 10.1080/10236190500141050 | MR 2159797 | Zbl 1078.39005
[4] Burton, T. A.: Stability by Fixed Point Theory for Functional Differential Equations. Dover Publications, Mineola (2006). MR 2281958 | Zbl 1160.34001
[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 1844329 | Zbl 1021.34042
[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 2913376
[7] Elaydi, S.: An Introduction to Difference Equations. Undergraduate Texts in Mathematics. Springer, New York (1999). MR 1711587 | Zbl 0930.39001
[8] Elaydi, S.: Periodicity and stability of linear Volterra difference systems. J. Math. Anal. Appl. 181 (1994), 483-492. DOI 10.1006/jmaa.1994.1037 | MR 1260872 | Zbl 0796.39004
[9] Elaydi, S., Murakami, S.: Uniform asymptotic stability in linear Volterra difference equations. J. Difference Equ. Appl. 3 (1998), 203-218. DOI 10.1080/10236199808808097 | MR 1616085 | Zbl 0891.39013
[10] Eloe, P., Islam, M., Raffoul, Y. N.: Uniform asymptotic stability in nonlinear Volterra discrete systems. Comput. Math. Appl. 45 (2003), 1033-1039. DOI 10.1016/S0898-1221(03)00081-6 | MR 2000576 | Zbl 1051.39003
[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 1821781 | Zbl 0990.34066
[12] Islam, M., Raffoul, Y. N.: Exponential stability in nonlinear difference equations. J. Difference Equ. Appl. 9 (2003), 819-825. DOI 10.1080/1023619031000101516 | MR 1995220 | Zbl 1055.39011
[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 2191670 | Zbl 1103.39009
[14] Kelly, W. G., Peterson, A. C.: Difference Equations: An Introduction with Applications. Academic Press, San Diego (2001). MR 1765695
[15] Liz, E.: Stability of non-autonomous difference equations: simple ideas leading to useful results. J. Difference Equ. Appl. 17 (2011), 203-220. DOI 10.1080/10236198.2010.549007 | MR 2783344 | Zbl 1216.39021
[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 10.1016/j.jmaa.2004.08.048 | MR 2122232 | Zbl 1068.39017
[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 2384923 | Zbl 1153.39003
[18] Pituk, M.: A criterion for the exponential stability of linear difference equations. Appl. Math. Lett. 17 (2004), 779-783. DOI 10.1016/j.aml.2004.06.005 | MR 2072834 | Zbl 1068.39019
[19] Raffoul, Y. N.: Stability and periodicity in discrete delay equations. J. Math. Anal. Appl. 324 (2006), 1356-1362. DOI 10.1016/j.jmaa.2006.01.044 | MR 2266564 | Zbl 1115.39015
[20] Raffoul, Y. N.: Periodicity in general delay nonlinear difference equations using fixed point theory. J. Difference Equ. Appl. 10 (2004), 1229-1242. DOI 10.1080/10236190410001652847 | MR 2100724 | Zbl 1068.39021
[21] Raffoul, Y. N.: General theorems for stability and boundedness for nonlinear functional discrete systems. J. Math. Anal. Appl. 279 (2003), 639-650. DOI 10.1016/S0022-247X(03)00051-9 | MR 1974051 | Zbl 1022.39004
[22] Smart, D. R.: Fixed Point Theorems. Cambridge Tracts in Mathematics 66. Cambridge University Press, London (1974). MR 0467717 | Zbl 0297.47042
[23] Yankson, E.: Stability in discrete equations with variable delays Electronic J. Qual. Theory Differ. Equ. 2009, electronic only 2009, 7 p. MR 2480418
[24] Yankson, E.: Stability of Volterra difference delay equations. Electronic J. Qual. Theory Differ. Equ. 2006, electronic only (2006), 14 p. MR 2263079 | Zbl 1113.39013
[25] Zhang, B.: Fixed points and stability in differential equations with variable delays. Nonlinear Anal., Theory Methods Appl. 63 (2005), e233--e242. DOI 10.1016/j.na.2005.02.081 | Zbl 1159.34348
[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 1708451 | Zbl 0938.39016
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