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, Hindawi Publishing Corporation, New York, 2005. doi: 10.1155/9789775945198.  Google Scholar

[2]

R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, Kluwer, Dordrecht, 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), 1-6. 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), 2859-2867. 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), 105-112.  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, Edited by Ferenc Hartung, Mihaly Pituk 2014, to appear in PROMST. 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), 77-86. Google Scholar

[9]

W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, San Diego, 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, 256, Kluwer Academic Publishers Group, Dordrecht, 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), 291-307.  Google Scholar

[12]

J. Migda and M. Migda, On the asymptotic behavior of solutions of higher order nonlinear difference equations, Nonlinear Anal., 47 (2001), 4687-4695. 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), 557-567.  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), 1077-1084. 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), 23-36. 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, Special Functions and Orthogonal Polynomials, (2007), 600-609. 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), 1-9. 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), 2185-2191. 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), 10 pp. 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), 83-92.  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, Hindawi Publishing Corporation, New York, 2005. doi: 10.1155/9789775945198.  Google Scholar

[2]

R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, Kluwer, Dordrecht, 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), 1-6. 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), 2859-2867. 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), 105-112.  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, Edited by Ferenc Hartung, Mihaly Pituk 2014, to appear in PROMST. 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), 77-86. Google Scholar

[9]

W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, San Diego, 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, 256, Kluwer Academic Publishers Group, Dordrecht, 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), 291-307.  Google Scholar

[12]

J. Migda and M. Migda, On the asymptotic behavior of solutions of higher order nonlinear difference equations, Nonlinear Anal., 47 (2001), 4687-4695. 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), 557-567.  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), 1077-1084. 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), 23-36. 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, Special Functions and Orthogonal Polynomials, (2007), 600-609. 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), 1-9. 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), 2185-2191. 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), 10 pp. 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), 83-92.  Google Scholar

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