# American Institute of Mathematical Sciences

January  2018, 23(1): 369-375. doi: 10.3934/dcdsb.2018025

## Existence of uncountably many asymptotically constant solutions to discrete nonlinear three-dimensional system with $p$-Laplacian

 1 Institute of Mathematics, Lodz University of Technology, Wolczanska 215, 90-924 Lodz, Poland 2 Institute of Logistics and Warehousing, Estkowskiego 6, 61-755 Poznan, Poland 3 Institute of Mathematics, University of Białystok, Ciolkowskiego 1M, 15-245 Bialystok, Poland

* Corresponding author: Piotr Hachuła

Received  July 2016 Revised  December 2016 Published  January 2018

This work is devoted to the study of the existence of uncountably many asymptotically constant solutions to discrete nonlinear three-dimensional system with p-Laplacian.

Citation: Magdalena Nockowska-Rosiak, Piotr Hachuła, Ewa Schmeidel. Existence of uncountably many asymptotically constant solutions to discrete nonlinear three-dimensional system with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 369-375. doi: 10.3934/dcdsb.2018025
##### References:
 [1] R. Agarwal, K. Perera and D. O'Regan, Multiple positive solutions of singular discrete p-Laplacian problems via variational methods, Adv. Difference Equ., 2005 (2005), 93-99.   Google Scholar [2] G. Bisci and D. Repovš, Existence of solutions for $p$-{L}aplacian discrete equations, Appl. Math. Comput., 242 (2014), 454-461.  doi: 10.1016/j.amc.2014.05.118.  Google Scholar [3] A. Burton, Stability by Fixed Point Theory for Functional Differential Equations Dover Publications, Inc., Mineola, NY, 2006.  Google Scholar [4] A. Cabada, Extremal solutions for the difference ϕ-laplacian problem with nonlinear functional boundary conditions, Comput. Math. Appl., 42 (2001), 593-601.  doi: 10.1016/S0898-1221(01)00179-1.  Google Scholar [5] A. Cabada and V. Otero-Espinar, Existence and comparison results for difference ϕ-Laplacian boundary value problems with lower and upper solutions in reverse order, J. Math. Anal. Appl., 267 (2002), 501-521.  doi: 10.1006/jmaa.2001.7783.  Google Scholar [6] M. Cecchi, Z. Došlá and M. Marini, Intermediate solutions for nonlinear difference equations with p-Laplacian, Adv. Stud. Pure Math., 53 (2009), 33-40.   Google Scholar [7] C. Costara and D. Popa, Exercises in Functional Analysis Kluwer Academic Publisher Group, Dordrecht, 2003. doi: 10.1007/978-94-017-0223-2.  Google Scholar [8] Z. He, On the existence of positive solutions of $p$-{L}aplacian difference equations, J. Comput. Appl. Math., 161 (2003), 193-201.  doi: 10.1016/j.cam.2003.08.004.  Google Scholar [9] E. Lee and Y. Lee, A result on three solutions theorem and its application to p-Laplacian systems with singular weights, Boundary Value Problems, 2012 (2012), 20pp.  doi: 10.1186/1687-2770-2012-63.  Google Scholar [10] M. Migda, E. Schmeidel and M. Zdanowicz, Existence of nonoscillatory solutions for system of neutral difference equations, Appl. Anal. Discrete Math., 9 (2015), 271-284.  doi: 10.2298/AADM150811016M.  Google Scholar [11] E. Schmeidel, Boundedness of solutions of nonlinear three-dimensional difference systems with delays, Fasc. Math., 44 (2010), 107-113.   Google Scholar [12] E. Schmeidel, Oscillation of nonlinear three-dimensional difference systems with delays, Math. Bohem., 135 (2010), 163-170.   Google Scholar [13] E. Schmeidel, Properties of solutions of system of difference equations with neutral term, Funct. Differ. Equ., 18 (2011), 293-302.   Google Scholar [14] E. Thandapani and B. Ponnammal, Oscillatory properties of solutions of three dimensional difference systems, Math. Comput. Modelling, 42 (2005), 641-650.  doi: 10.1016/j.mcm.2004.04.010.  Google Scholar

show all references

##### References:
 [1] R. Agarwal, K. Perera and D. O'Regan, Multiple positive solutions of singular discrete p-Laplacian problems via variational methods, Adv. Difference Equ., 2005 (2005), 93-99.   Google Scholar [2] G. Bisci and D. Repovš, Existence of solutions for $p$-{L}aplacian discrete equations, Appl. Math. Comput., 242 (2014), 454-461.  doi: 10.1016/j.amc.2014.05.118.  Google Scholar [3] A. Burton, Stability by Fixed Point Theory for Functional Differential Equations Dover Publications, Inc., Mineola, NY, 2006.  Google Scholar [4] A. Cabada, Extremal solutions for the difference ϕ-laplacian problem with nonlinear functional boundary conditions, Comput. Math. Appl., 42 (2001), 593-601.  doi: 10.1016/S0898-1221(01)00179-1.  Google Scholar [5] A. Cabada and V. Otero-Espinar, Existence and comparison results for difference ϕ-Laplacian boundary value problems with lower and upper solutions in reverse order, J. Math. Anal. Appl., 267 (2002), 501-521.  doi: 10.1006/jmaa.2001.7783.  Google Scholar [6] M. Cecchi, Z. Došlá and M. Marini, Intermediate solutions for nonlinear difference equations with p-Laplacian, Adv. Stud. Pure Math., 53 (2009), 33-40.   Google Scholar [7] C. Costara and D. Popa, Exercises in Functional Analysis Kluwer Academic Publisher Group, Dordrecht, 2003. doi: 10.1007/978-94-017-0223-2.  Google Scholar [8] Z. He, On the existence of positive solutions of $p$-{L}aplacian difference equations, J. Comput. Appl. Math., 161 (2003), 193-201.  doi: 10.1016/j.cam.2003.08.004.  Google Scholar [9] E. Lee and Y. Lee, A result on three solutions theorem and its application to p-Laplacian systems with singular weights, Boundary Value Problems, 2012 (2012), 20pp.  doi: 10.1186/1687-2770-2012-63.  Google Scholar [10] M. Migda, E. Schmeidel and M. Zdanowicz, Existence of nonoscillatory solutions for system of neutral difference equations, Appl. Anal. Discrete Math., 9 (2015), 271-284.  doi: 10.2298/AADM150811016M.  Google Scholar [11] E. Schmeidel, Boundedness of solutions of nonlinear three-dimensional difference systems with delays, Fasc. Math., 44 (2010), 107-113.   Google Scholar [12] E. Schmeidel, Oscillation of nonlinear three-dimensional difference systems with delays, Math. Bohem., 135 (2010), 163-170.   Google Scholar [13] E. Schmeidel, Properties of solutions of system of difference equations with neutral term, Funct. Differ. Equ., 18 (2011), 293-302.   Google Scholar [14] E. Thandapani and B. Ponnammal, Oscillatory properties of solutions of three dimensional difference systems, Math. Comput. Modelling, 42 (2005), 641-650.  doi: 10.1016/j.mcm.2004.04.010.  Google Scholar
 [1] Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033 [2] Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371 [3] Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922 [4] Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593 [5] Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Dead cores and bursts for p-Laplacian elliptic equations with weights. Conference Publications, 2007, 2007 (Special) : 191-200. doi: 10.3934/proc.2007.2007.191 [6] Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069 [7] Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040 [8] Yuming Qin, Yang Wang, Xing Su, Jianlin Zhang. Global existence of solutions for the three-dimensional Boussinesq system with anisotropic data. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1563-1581. doi: 10.3934/dcds.2016.36.1563 [9] 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 [10] Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239 [11] Igor Kukavica, Vlad C. Vicol. The domain of analyticity of solutions to the three-dimensional Euler equations in a half space. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 285-303. doi: 10.3934/dcds.2011.29.285 [12] Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060 [13] Madalina Petcu, Roger Temam, Djoko Wirosoetisno. Averaging method applied to the three-dimensional primitive equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5681-5707. doi: 10.3934/dcds.2016049 [14] Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012 [15] Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020 [16] Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130 [17] Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure & Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475 [18] Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254 [19] Wanli Yang, Jie Sun, Su Zhang. Analysis of optimal boundary control for a three-dimensional reaction-diffusion system. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 325-344. doi: 10.3934/naco.2017021 [20] Leo Howden, Donald Giddings, Henry Power, Michael Vloeberghs. Three-dimensional cerebrospinal fluid flow within the human central nervous system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 957-969. doi: 10.3934/dcdsb.2011.15.957

2018 Impact Factor: 1.008