American Institute of Mathematical Sciences

Advanced
January  2018, 23(1): 239-252. doi: 10.3934/dcdsb.2018016

Qualitative properties of solutions of higher order difference equations with deviating arguments

Alina Gleska , and  Małgorzata Migda

Institute of Mathematics, Poznań University of Technology, Piotrowo 3A, 60-965 Poznań, Poland

* Corresponding author: Alina Gleska

Received  December 2016 Published  January 2018

In the paper the general higher order difference equation
$(-1)^z Δ^m x(n)=f(n, x(σ_1(n)),x(σ_2(n)),..., x(σ_k(n)))$
with several deviating arguments is considered. According to the kind of the deviations
$σ_i$
sufficient conditions for the equation to have property A and B are established.
Keywords: Oscillatory solutions, sufficient conditions, difference equations.
Mathematics Subject Classification: Primary: 39A10.
Citation: Alina Gleska, Małgorzata Migda. Qualitative properties of solutions of higher order difference equations with deviating arguments. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 239-252. doi: 10.3934/dcdsb.2018016
References:
[1]

R. P. Agarwal, M. Bohner, S. R. Grace and D. O'Regan, Discrete Oscillation Theory Hindawi Publishing Corporation, New York, 2005. doi: 10.1155/9789775945198. Google Scholar

[2]

R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications Second Edition, Revised and Expanded, Marcel Dekker, New York, 2000. Google Scholar

[3]

J. Baštinec, L. Berezansky, J. Diblík and Z. šmarda, A final result on the oscillation of solutions of the linear discrete delayed equation $Δ x(n)=-p(n)x(n-k)$ with a positive coefficient Abstract and Applied Analysis 2011 (2011), Art. ID 586328, 28 pp. doi: 10.1155/2011/586328. Google Scholar

[4]

G. E. Chatzarakis, R. Koplatadze and I. P. Stavroulakis, Oscillation criteria of first order linear difference equations with delay argument, Nonlinear Anal., 68 (2008), 994-1005. doi: 10.1016/j.na.2006.11.055. Google Scholar

[5]

G. E. Chatzarakis, M. Lafci and I. P. Stavroulakis, Oscillation results for difference equations with several oscillating coefficients, Applied Mathematics and Computation, 251 (2015), 81-91. doi: 10.1016/j.amc.2014.11.045. Google Scholar

[6]

G. E. Chatzarakis, Ch. G. Philos and I. P. Stavroulakis, On the oscillation of the solutions to linear difference equations with variable delay, Electron. J. Differ. Equ., 50 (2008), 1-15. Google Scholar

[7]

G. E. Chatzarakis and Ö. Öcalan, Oscillation of difference equations with non-monotone retarded arguments, Applied Mathematics and Computation, 258 (2015), 60-66. doi: 10.1016/j.amc.2015.01.110. Google Scholar

[8]

G. E. Chatzarakis, H. Péics, S. Pinelas and I. P. Stavroulakis, Oscillation results for difference equations with oscillating coefficients Advances in Difference Equations 2015 (2015), 17pp. doi: 10.1186/s13662-015-0391-0. Google Scholar

[9]

L. H. Erbe and B. G. Zhang, Oscillation of discrete analogues of delay equations, Differential Integral Equations, 2 (1989), 300-309. Google Scholar

[10]

G. Grzegorczyk and J. Werbowski, On oscillatory solutions of certain difference equations, Opuscula Mathematica, 26 (2006), 317-326. Google Scholar

[11]

G. Grzegorczyk and J. Werbowski, Oscillation of higher-order linear difference equations, Comput. Math. Appl., 42 (2001), 711-717. doi: 10.1016/S0898-1221(01)00190-0. Google Scholar

[12]

I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications Clarendon Press, Oxford, New York, 1991. Google Scholar

[13]

W. G. Kelley and A. C. Peterson, Difference Equations. An Introduction with Applications Second edition, Harcourt/Academic Press, San Diego, CA, 2001. Google Scholar

[14]

G. Ladas, Ch. G. Philos and Y. G. Sficas, Sharp conditions for the oscillation of delay difference equations, J. Appl. Math. Simulation, 2 (1989), 101-111. doi: 10.1155/S1048953389000080. Google Scholar

[15]

J. Migda, Approximative solutions of difference equations, Electron. J. Qual. Theory Differ. Equ. 13 (2014), 26 pp. Google Scholar

[16]

M. Migda, Existence of nonoscillatory solutions of some higher order difference equations, Appl. Math. E-Notes, 4 (2004), 33-39. Google Scholar

[17]

M. Migda and J. Migda, On the asymptotic behavior of solutions of higher order nonlinear difference equations, Nonlinear Analysis, 47 (2001), 4687-4695. doi: 10.1016/S0362-546X(01)00581-8. Google Scholar

[18]

M. Migda, Asymptotic properties of nonoscillatory solutions of higher order neutral difference equations, Opuscula Mathematica, 26 (2006), 507-514. Google Scholar

[19]

W. Nowakowska and J. Werbowski, On connections between oscillatory solutions of functional, difference and differential equations, Fasc. Math., 44 (2010), 95-106. Google Scholar

[20]

Ch. G. Philos, I. K. Purnaras and I. P. Stavroulakis, Sufficient conditions for the oscillation of delay difference equations, J. Difference Equ. Appl., 10 (2004), 419-435. doi: 10.1080/10236190410001648239. Google Scholar

[21]

J. Werbowski, Bounded oscillations of differential equations generated by deviating arguments, Utilitas Math., 31 (1987), 191-198. Google Scholar

[22]

A. Wyrwińska, Oscillation criteria of a higher order linear difference equation, Bull. Inst. Math. Acad. Sinica, 22 (1994), 259-266. Google Scholar

show all references

References:
[1]

R. P. Agarwal, M. Bohner, S. R. Grace and D. O'Regan, Discrete Oscillation Theory Hindawi Publishing Corporation, New York, 2005. doi: 10.1155/9789775945198. Google Scholar

[2]

R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications Second Edition, Revised and Expanded, Marcel Dekker, New York, 2000. Google Scholar

[3]

J. Baštinec, L. Berezansky, J. Diblík and Z. šmarda, A final result on the oscillation of solutions of the linear discrete delayed equation $Δ x(n)=-p(n)x(n-k)$ with a positive coefficient Abstract and Applied Analysis 2011 (2011), Art. ID 586328, 28 pp. doi: 10.1155/2011/586328. Google Scholar

[4]

G. E. Chatzarakis, R. Koplatadze and I. P. Stavroulakis, Oscillation criteria of first order linear difference equations with delay argument, Nonlinear Anal., 68 (2008), 994-1005. doi: 10.1016/j.na.2006.11.055. Google Scholar

[5]

G. E. Chatzarakis, M. Lafci and I. P. Stavroulakis, Oscillation results for difference equations with several oscillating coefficients, Applied Mathematics and Computation, 251 (2015), 81-91. doi: 10.1016/j.amc.2014.11.045. Google Scholar

[6]

G. E. Chatzarakis, Ch. G. Philos and I. P. Stavroulakis, On the oscillation of the solutions to linear difference equations with variable delay, Electron. J. Differ. Equ., 50 (2008), 1-15. Google Scholar

[7]

G. E. Chatzarakis and Ö. Öcalan, Oscillation of difference equations with non-monotone retarded arguments, Applied Mathematics and Computation, 258 (2015), 60-66. doi: 10.1016/j.amc.2015.01.110. Google Scholar

[8]

G. E. Chatzarakis, H. Péics, S. Pinelas and I. P. Stavroulakis, Oscillation results for difference equations with oscillating coefficients Advances in Difference Equations 2015 (2015), 17pp. doi: 10.1186/s13662-015-0391-0. Google Scholar

[9]

L. H. Erbe and B. G. Zhang, Oscillation of discrete analogues of delay equations, Differential Integral Equations, 2 (1989), 300-309. Google Scholar

[10]

G. Grzegorczyk and J. Werbowski, On oscillatory solutions of certain difference equations, Opuscula Mathematica, 26 (2006), 317-326. Google Scholar

[11]

G. Grzegorczyk and J. Werbowski, Oscillation of higher-order linear difference equations, Comput. Math. Appl., 42 (2001), 711-717. doi: 10.1016/S0898-1221(01)00190-0. Google Scholar

[12]

I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications Clarendon Press, Oxford, New York, 1991. Google Scholar

[13]

W. G. Kelley and A. C. Peterson, Difference Equations. An Introduction with Applications Second edition, Harcourt/Academic Press, San Diego, CA, 2001. Google Scholar

[14]

G. Ladas, Ch. G. Philos and Y. G. Sficas, Sharp conditions for the oscillation of delay difference equations, J. Appl. Math. Simulation, 2 (1989), 101-111. doi: 10.1155/S1048953389000080. Google Scholar

[15]

J. Migda, Approximative solutions of difference equations, Electron. J. Qual. Theory Differ. Equ. 13 (2014), 26 pp. Google Scholar

[16]

M. Migda, Existence of nonoscillatory solutions of some higher order difference equations, Appl. Math. E-Notes, 4 (2004), 33-39. Google Scholar

[17]

M. Migda and J. Migda, On the asymptotic behavior of solutions of higher order nonlinear difference equations, Nonlinear Analysis, 47 (2001), 4687-4695. doi: 10.1016/S0362-546X(01)00581-8. Google Scholar

[18]

M. Migda, Asymptotic properties of nonoscillatory solutions of higher order neutral difference equations, Opuscula Mathematica, 26 (2006), 507-514. Google Scholar

[19]

W. Nowakowska and J. Werbowski, On connections between oscillatory solutions of functional, difference and differential equations, Fasc. Math., 44 (2010), 95-106. Google Scholar

[20]

Ch. G. Philos, I. K. Purnaras and I. P. Stavroulakis, Sufficient conditions for the oscillation of delay difference equations, J. Difference Equ. Appl., 10 (2004), 419-435. doi: 10.1080/10236190410001648239. Google Scholar

[21]

J. Werbowski, Bounded oscillations of differential equations generated by deviating arguments, Utilitas Math., 31 (1987), 191-198. Google Scholar

[22]

A. Wyrwińska, Oscillation criteria of a higher order linear difference equation, Bull. Inst. Math. Acad. Sinica, 22 (1994), 259-266. Google Scholar

[1]

Cristopher Hermosilla. Stratified discontinuous differential equations and sufficient conditions for robustness. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4415-4437. doi: 10.3934/dcds.2015.35.4415
[2]

Ana Cristina Barroso, José Matias. Necessary and sufficient conditions for existence of solutions of a variational problem involving the curl. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 97-114. doi: 10.3934/dcds.2005.12.97
[3]

Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141
[4]

Samuel Bernard, Jacques Bélair, Michael C Mackey. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 233-256. doi: 10.3934/dcdsb.2001.1.233
[5]

Baruch Cahlon. Sufficient conditions for oscillations of higher order neutral delay differential equations. Conference Publications, 1998, 1998 (Special) : 124-137. doi: 10.3934/proc.1998.1998.124
[6]

Kyril Tintarev. Positive solutions of elliptic equations with a critical oscillatory nonlinearity. Conference Publications, 2007, 2007 (Special) : 974-981. doi: 10.3934/proc.2007.2007.974
[7]

Alexandra Rodkina, Henri Schurz. On positivity and boundedness of solutions of nonlinear stochastic difference equations. Conference Publications, 2009, 2009 (Special) : 640-649. doi: 10.3934/proc.2009.2009.640
[8]

M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070
[9]

Angelo Favini, Rabah Labbas, Stéphane Maingot, Hiroki Tanabe, Atsushi Yagi. Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 973-987. doi: 10.3934/dcds.2008.22.973
[10]

Samir Hodžić, Enes Pasalic. Generalized bent functions -sufficient conditions and related constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 549-566. doi: 10.3934/amc.2017043
[11]

Hans Weinberger. On sufficient conditions for a linearly determinate spreading speed. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2267-2280. doi: 10.3934/dcdsb.2012.17.2267
[12]

Małgorzata Migda, Ewa Schmeidel, Małgorzata Zdanowicz. Periodic solutions of a $2$-dimensional system of neutral difference equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 359-367. doi: 10.3934/dcdsb.2018024
[13]

Ewa Schmeidel, Karol Gajda, Tomasz Gronek. On the existence of weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2681-2690. doi: 10.3934/dcdsb.2014.19.2681
[14]

John R. Graef, Lingju Kong, Min Wang. Existence of homoclinic solutions for second order difference equations with $p$-laplacian. Conference Publications, 2015, 2015 (special) : 533-539. doi: 10.3934/proc.2015.0533
[15]

Yoshihiro Hamaya. Stability properties and existence of almost periodic solutions of volterra difference equations. Conference Publications, 2009, 2009 (Special) : 315-321. doi: 10.3934/proc.2009.2009.315
[16]

Jehad O. Alzabut. A necessary and sufficient condition for the existence of periodic solutions of linear impulsive differential equations with distributed delay. Conference Publications, 2007, 2007 (Special) : 35-43. doi: 10.3934/proc.2007.2007.35
[17]

Zuji Guo, Zhaoli Liu. Perturbed elliptic equations with oscillatory nonlinearities. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3567-3585. doi: 10.3934/dcds.2012.32.3567
[18]

Gennaro Infante. Positive solutions of differential equations with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 432-438. doi: 10.3934/proc.2003.2003.432
[19]

John A. D. Appleby, Xuerong Mao, Alexandra Rodkina. On stochastic stabilization of difference equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 843-857. doi: 10.3934/dcds.2006.15.843
[20]

Bernard Dacorogna. Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 257-263. doi: 10.3934/dcdsb.2001.1.257

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (29)
  • HTML views (70)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences