October  2014, 19(8): 2691-2696. doi: 10.3934/dcdsb.2014.19.2691

Asymptotically zero solution of a class of higher nonlinear neutral difference equations with quasidifferences

1. 

University of Bialystok, ul. Akademicka 2, 15-267 Białystok

2. 

Lodz Unviersity of Technology, Wólczańska 215, 90-924 Łódź, Poland

Received  November 2013 Revised  May 2014 Published  August 2014

A class of higher order nonlinear neutral difference equations with quasidifferences is studied. Sufficient conditions under which considered equation has a solution which converges to zero are presented.
Citation: Ewa Schmeidel, Robert Jankowski. Asymptotically zero solution of a class of higher nonlinear neutral difference equations with quasidifferences. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2691-2696. doi: 10.3934/dcdsb.2014.19.2691
References:
[1]

R. P. Agarwal, M. Bohner, S. R. Grace and D. O'Regan, Discrete Oscillation Theory,, Contemporary Mathematics and Its Applications, (2005).  doi: 10.1155/9789775945198.  Google Scholar

[2]

R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations,, Kluwer, (1997).  doi: 10.1007/978-94-015-8899-7.  Google Scholar

[3]

J. Banaş and B. Rzepka, An application of measure of noncompactness in study of asymptotic stability,, Appl. Math. Lett., 16 (2003), 1.  doi: 10.1016/S0893-9659(02)00136-2.  Google Scholar

[4]

O. Došlý, J. Graef and J. Jaroš, Forced oscillation of second order linear and half-linear difference equations,, Proc. Amer. Math. Soc., 131 (2003), 2859.  doi: 10.1090/S0002-9939-02-06811-9.  Google Scholar

[5]

M. Galewski and E. Schmeidel, On the well posed solutions for nonlinear second order neutral difference equations,, to appear in Mathematica Slovaca., ().   Google Scholar

[6]

S. R Grace, R. P. Agarwal, M. Bohner and S. Pinelas, Oscillation of some fourth-order difference equations,, Int. J. Difference Equ., 6 (2011), 105.   Google Scholar

[7]

R. Jankowski and E. Schmeidel, Almost oscillatory solutions of second order difference equations of neutral type,, Recent Advances in Delay Differential and Difference Equations, (2014).   Google Scholar

[8]

R. Jankowski and E. Schmeidel, Almost oscillation criteria for second order neutral difference equation with quasidifferences,, Int. J. Difference Equ., 9 (2014), 77.   Google Scholar

[9]

W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications,, Academic Press, (2001).   Google Scholar

[10]

V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications,, Mathematics and its Applications, (1993).  doi: 10.1007/978-94-017-1703-8.  Google Scholar

[11]

G. Ladas, C. Qian and J. Yan, Oscillations of higher order neutral differential equations,, Portugal. Math., 48 (1991), 291.   Google Scholar

[12]

J. Migda and M. Migda, On the asymptotic behavior of solutions of higher order nonlinear difference equations,, Nonlinear Anal., 47 (2001), 4687.  doi: 10.1016/S0362-546X(01)00581-8.  Google Scholar

[13]

J. Migda and M. Migda, On unstable neutral difference equations of higher order,, Indian J. Pure Appl. Math., 36 (2005), 557.   Google Scholar

[14]

J. Migda and M. Migda, Oscillatory and asymptotic properties of solutions of even order neutral difference equations,, J.Difference Equ. Appl., 15 (2009), 1077.  doi: 10.1080/10236190903032708.  Google Scholar

[15]

M. Migda, A. Musielak and E. Schmeidel, On a class of fourth order nonlinear difference equations,, Advances in Difference Equations, 1 (2004), 23.  doi: 10.1155/S1687183904308083.  Google Scholar

[16]

E. Schmeidel, Asymptotic trichotomy of solutions of a class of even order difference equations with quasidifferences,, Difference Equations, (2007), 600.  doi: 10.1142/9789812770752_0052.  Google Scholar

[17]

E. Schmeidel, An application of measures of noncompactness in investigation of boundedness of solutions of second order neutral difference equations,, Adv. Difference Equ., 2013 (2013), 1.  doi: 10.1186/1687-1847-2013-91.  Google Scholar

[18]

E. Schmeidel and Z. Zbąszyniak, An application of Darbo's fixed point theorem in the investigation of periodicity of solutions of difference equations,, Comput. Math. Appl., 64 (2012), 2185.  doi: 10.1016/j.camwa.2011.12.025.  Google Scholar

[19]

E. Thandapani, N. Kavitha and S. Pinelas, Oscillation criteria for second-order nonlinear neutral difference equations of mixed type,, Adv. Difference Equ., 2012 (2012).  doi: 10.1186/1687-1847-2012-4.  Google Scholar

[20]

E. Thandapani, N. Kavitha and S. Pinelas, Comparison and oscillation theorem for second-order nonlinear neutral difference equations of mixed type,, Dynam. Systems Appl., 21 (2012), 83.   Google Scholar

show all references

References:
[1]

R. P. Agarwal, M. Bohner, S. R. Grace and D. O'Regan, Discrete Oscillation Theory,, Contemporary Mathematics and Its Applications, (2005).  doi: 10.1155/9789775945198.  Google Scholar

[2]

R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations,, Kluwer, (1997).  doi: 10.1007/978-94-015-8899-7.  Google Scholar

[3]

J. Banaş and B. Rzepka, An application of measure of noncompactness in study of asymptotic stability,, Appl. Math. Lett., 16 (2003), 1.  doi: 10.1016/S0893-9659(02)00136-2.  Google Scholar

[4]

O. Došlý, J. Graef and J. Jaroš, Forced oscillation of second order linear and half-linear difference equations,, Proc. Amer. Math. Soc., 131 (2003), 2859.  doi: 10.1090/S0002-9939-02-06811-9.  Google Scholar

[5]

M. Galewski and E. Schmeidel, On the well posed solutions for nonlinear second order neutral difference equations,, to appear in Mathematica Slovaca., ().   Google Scholar

[6]

S. R Grace, R. P. Agarwal, M. Bohner and S. Pinelas, Oscillation of some fourth-order difference equations,, Int. J. Difference Equ., 6 (2011), 105.   Google Scholar

[7]

R. Jankowski and E. Schmeidel, Almost oscillatory solutions of second order difference equations of neutral type,, Recent Advances in Delay Differential and Difference Equations, (2014).   Google Scholar

[8]

R. Jankowski and E. Schmeidel, Almost oscillation criteria for second order neutral difference equation with quasidifferences,, Int. J. Difference Equ., 9 (2014), 77.   Google Scholar

[9]

W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications,, Academic Press, (2001).   Google Scholar

[10]

V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications,, Mathematics and its Applications, (1993).  doi: 10.1007/978-94-017-1703-8.  Google Scholar

[11]

G. Ladas, C. Qian and J. Yan, Oscillations of higher order neutral differential equations,, Portugal. Math., 48 (1991), 291.   Google Scholar

[12]

J. Migda and M. Migda, On the asymptotic behavior of solutions of higher order nonlinear difference equations,, Nonlinear Anal., 47 (2001), 4687.  doi: 10.1016/S0362-546X(01)00581-8.  Google Scholar

[13]

J. Migda and M. Migda, On unstable neutral difference equations of higher order,, Indian J. Pure Appl. Math., 36 (2005), 557.   Google Scholar

[14]

J. Migda and M. Migda, Oscillatory and asymptotic properties of solutions of even order neutral difference equations,, J.Difference Equ. Appl., 15 (2009), 1077.  doi: 10.1080/10236190903032708.  Google Scholar

[15]

M. Migda, A. Musielak and E. Schmeidel, On a class of fourth order nonlinear difference equations,, Advances in Difference Equations, 1 (2004), 23.  doi: 10.1155/S1687183904308083.  Google Scholar

[16]

E. Schmeidel, Asymptotic trichotomy of solutions of a class of even order difference equations with quasidifferences,, Difference Equations, (2007), 600.  doi: 10.1142/9789812770752_0052.  Google Scholar

[17]

E. Schmeidel, An application of measures of noncompactness in investigation of boundedness of solutions of second order neutral difference equations,, Adv. Difference Equ., 2013 (2013), 1.  doi: 10.1186/1687-1847-2013-91.  Google Scholar

[18]

E. Schmeidel and Z. Zbąszyniak, An application of Darbo's fixed point theorem in the investigation of periodicity of solutions of difference equations,, Comput. Math. Appl., 64 (2012), 2185.  doi: 10.1016/j.camwa.2011.12.025.  Google Scholar

[19]

E. Thandapani, N. Kavitha and S. Pinelas, Oscillation criteria for second-order nonlinear neutral difference equations of mixed type,, Adv. Difference Equ., 2012 (2012).  doi: 10.1186/1687-1847-2012-4.  Google Scholar

[20]

E. Thandapani, N. Kavitha and S. Pinelas, Comparison and oscillation theorem for second-order nonlinear neutral difference equations of mixed type,, Dynam. Systems Appl., 21 (2012), 83.   Google Scholar

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