Opuscula Math. 39, no. 1 (2019), 61-75
https://doi.org/10.7494/OpMath.2019.39.1.61
Opuscula Mathematica
On the convergence of solutions to second-order neutral difference equations
Małgorzata Migda
Janusz Migda
Małgorzata Zdanowicz
Abstract. A second-order nonlinear neutral difference equation with a quasi-difference is studied. Sufficient conditions are established under which for every real constant there exists a solution of the considered equation convergent to this constant.
Keywords: second-order difference equation, asymptotic behavior, quasi-differences, Krasnoselskii's fixed point theorem.
Mathematics Subject Classification: 39A10, 39A22, 39A30.
- R.P. Agarwal, S.R. Grace, D. O'Regan, Nonoscillatory solutions for discrete equations, Comput. Math. Appl. 45 (2003), 1297-1302.
- A. Bezubik, M. Migda, M. Nockowska-Rosiak, E. Schmeidel, Trichotomy of nonoscillatory solutions to second-order neutral difference equation with quasi-difference, Adv. Difference Equ. 192 (2015), 14 pp.
- A. Drozdowicz, J. Popenda, Asymptotic behavior of solutions of the second order difference equation, Proc. Amer. Math. Soc. 99 (1987) 1, 135-140.
- M. Galewski, E. Schmeidel, On the well posed solutions for nonlinear second order neutral difference equations, Math. Slovac. 66 (2016) 4, 933-944.
- M. Galewski, R. Jankowski, M. Nockowska-Rosiak, E. Schmeidel, On the existence of bounded solutions for nonlinear second order neutral difference equations, Electron. J. Qual. Theory Differ. Equ. 72 (2014), 1-12.
- R. Jankowski, E. Schmeidel, Almost oscillation criteria for second order neutral difference equation with quasidifferences, Int. J. Difference Equ. 9 (2014), 77-86.
- B.S. Lalli, B.G. Zhang, On existence of positive solutions and bounded oscillations of neutral difference equations, J. Math. Anal. Appl. 166 (1992), 272-287.
- Z. Liu, Y. Xu, S.M. Kang, Global solvability for a second order nonlinear neutral delay difference equation, Comput. Math. Appl. 57 (2009) 4, 587-595.
- Z. Liu, M. Jia, S.M. Kang, Y.C. Kwun, Bounded positive solutions for a third order discrete equation, Abst. Appl. Anal. 2012 (2012), Article ID 237036, 12 pp.
- J. Migda, Approximative solutions to difference equations of neutral type, Appl. Math. Comput. 268 (2015), 763-774.
- M. Migda, Asymptotic behaviour of solutions of nonlinear difference equations, Fasc. Math. 31 (2001), 57-63.
- M. Migda, J. Migda, On a class of first order nonlinear difference equations of neutral type, Math. Comput. Modelling 40 (2004), 297-306.
- M. Migda, J. Migda, Asymptotic properties of solutions of second-order neutral difference equations, Nonlinear Anal. 63 (2005), e789-e799.
- M. Migda, E. Schmeidel, M. Zdanowicz, Bounded solutions of \(k\)-dimensional system of nonlinear difference equations of neutral type, Electron. J. Qual. Theory Differ. Equ. 80 (2015), 17 pp.
- E. Schmeidel, An application of measures of noncompactness in investigation of boundedness of solutions of second order neutral difference equations, Adv. Difference Equ. 91 (2013), 9 pp.
- E. Schmeidel, 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.
- E. Thandapani, M.M.S. Manuel, J.R. Graef, P.W. Spikes, Monotone properties of certain classes of solutions of second-order difference equations, Comput. Math. Appl. 36 (1998), 291-297.
- Małgorzata Migda
https://orcid.org/0000-0003-3188-1173- Poznań University of Technology, Piotrowo 3A, 60-965 Poznań, Poland
- Janusz Migda
- https://orcid.org/0000-0002-0855-6180
- A. Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
- Małgorzata Zdanowicz
- https://orcid.org/0000-0001-6951-5721
- University of Bialystok, Ciołkowskiego 1M, 15-245 Białystok, Poland
- Communicated by Marek Galewski.
- Received: 2017-12-13.
- Revised: 2018-04-15.
- Accepted: 2018-06-07.
- Published online: 2018-08-07.
