References
- R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker Inc. 2000.
- R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Second edition, Marcel Dekker, Inc, 2000.
- R. P. Agarwal, M. Bohner and D. O'Regan, Time scale boundary value problems on infinite intervals, J. Comput. Appl. Math. 141 (2002), 27-34. https://doi.org/10.1016/S0377-0427(01)00433-2
- R. P. Agarwal and D. O'Regan, Cone compression and expansion and fixed point theorems in Frchet spaces with application, J. Differ. Equ. 171 (2001), 412-422. https://doi.org/10.1006/jdeq.2000.3831
- R. P. Agarwal and D. O'Regan, Nonlinear Urysohn discrete equations on the infinite interval: a fixed-point approach, Comput. Math. Appl. 42 (2001), 273-281. https://doi.org/10.1016/S0898-1221(01)00152-3
- R. P. Agarwal and D. O'Regan, Boundary value problems for general discrete systems on infinite intervals, Comput. Math. Appl. 33 (1997), 85-99. https://doi.org/10.1016/S0898-1221(97)00044-8
- R. P. Agarwal and D. O'Regan, Discrete systems on infinite intervals, Comput. Math. Appl. 35 (1998) 97-105. https://doi.org/10.1016/S0898-1221(98)00061-3
- R. P. Agarwal, K. Perera and D. O'Regan, Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal. 58 (2004), 69-73. https://doi.org/10.1016/j.na.2003.11.012
- R. I. Avery, A generalization of Leggett-Williams fixed point theorem, Math. Sci. Res. Hot Line 3 (1993), 9-14.
- R. I. Avery and A. C. Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl. 42 (2001), 313-322. https://doi.org/10.1016/S0898-1221(01)00156-0
- P. Chen, Existence of homoclinic orbits in discrete Hamiltonian systems without Palais-Smale condition, J. Differ. Equ. Appl. 19(11) (2013), 1781-1794. https://doi.org/10.1080/10236198.2013.777716
- A. Cabada and J. Cid, Solvability of some p-Laplacian singular difference equations defined on the integers, ASJE-Mathematics. 34 (2009), 75-81.
- A. Cabada and S. Tersian, Existence of heteroclinic solutions for discrete p-Laplacian problems with a parameter, Nonlinear Anal. RWA. 12 (2011), 2429-2434. https://doi.org/10.1016/j.nonrwa.2011.02.022
- A. Cabada, A. Iannizzotto and S. Tersian, Multiple solutions for discrete boundary value problems, J. Math. Anal. Appl. 356 (2009), 418-428. https://doi.org/10.1016/j.jmaa.2009.02.038
- A. Cabada, L. Li and S. Tersian, On Homoclinic solutions of a semilinear p-Laplacian difference equation with periodic coefficients, Adv. Differ. Equ. 2010 (2010), Article ID 195376, 17 pages.
- X. Cai, Z. Guo and J. Yu, Periodic solutions of a class of nonlinear diffrence equations via critical point method, Comput. Math. Appl. 52 (2006), 1639-1647. https://doi.org/10.1016/j.camwa.2006.09.003
- P. Candito and N. Giovannelli, Multiple solutions for a discrete boundary value problem involving the p-Laplacian, Comput. Math. Appl. 56 (2008), 959-964. https://doi.org/10.1016/j.camwa.2008.01.025
- W. Cheung, J. Ren, P. J. Y. Wong and D. Zhao, Multiple positive solutions for discrete nonlocal boundary value problems, J. Math. Anal. Appl. 330 (2007), 900-915. https://doi.org/10.1016/j.jmaa.2006.08.034
- P. Chen and X. Tang, Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations, Adv. Differ. Equ. 2010 (2010), Article ID 470375, 19 pages.
- E. M. Elsayed, Solutions of rational difference system of order two, Math. Comput. Modelling, 55 (2012), 378-384. https://doi.org/10.1016/j.mcm.2011.08.012
- E. M. Elsayed, Behavior and expression of the solutions of some rational difference equations, J. Comput. Anal. Appl. 15 (1) (2013), 73-81.
- E. M. Elsayed, Solution for systems of difference equations of rational form of order two, Comput. Appl. Math. 33(3) (2014), 751-765. https://doi.org/10.1007/s40314-013-0092-9
- F. Faraci and A. Iannizzotto, Multiplicity theorems for discrete boundary value problems, Aequationes Math. 74 (2007), 111-118. https://doi.org/10.1007/s00010-006-2855-5
- J. R. Graef, L. Kong and B. Yang, Positive solutions for third order multi-point singular boundary value problems, Czechoslovak Math. J. 60 (2010), 173-182. https://doi.org/10.1007/s10587-010-0007-5
- Z. Guo and J. Yu, Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems, Nonlinear Anal. 55 (2003), 969-983. https://doi.org/10.1016/j.na.2003.07.019
- Z. Guo and J. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. Lond. Math. Soc. 68 (2003), 419-430. https://doi.org/10.1112/S0024610703004563
- X. He and P. Hen, Homoclinic solutions for second order discrete p-Laplacian systems, Adv. Differ. Equ. 57 (2011), 20 pages.
- J. Henderson and R. Luca, Existence of positive solutions for a system of second-order multi-point discrete boundary value problems, J. Differ. Equ. Appl. 19 (11) (2013), 1889-1906. https://doi.org/10.1080/10236198.2013.788646
- L. Jodar and R. J. Villanueva, Explicit solutions of implicit second-order difference systems in unbounded bilateral domains, Comput. Math. Appl. 32 (9) (1996), 19-28. https://doi.org/10.1016/0898-1221(96)00173-3
- L. Jiang and Z. Zhou, Three solutions to Dirichlet boundary value problems for p-Laplacian difference equations, Adv. Differ. Equ. 2008 (2008), Article ID 345916, 10 pages.
- L. Kong, Homoclinic solutions for a second order difference equation with p-Laplacian, Appl. Math. Comput. 247 (15) (2014), 1113-1121. https://doi.org/10.1016/j.amc.2014.09.069
- A. R. Kanth and Y. Reddy, A numerical method for solving two point boundary value problems over infinite intervals, Appl. Math. Comput. 144 (2003), 483-494. https://doi.org/10.1016/S0096-3003(02)00422-8
- W. G. Kelley and A. Peterson, Difference equations, Harcourt/Academic Press. 2001.
- V. Lakshmikantham and D. Trigiante, Theory of difference equations: numerical methods and applications, Marcel Dekker Inc. 2002.
- Y. Liu, Positive Solutions of BVPs for finite Difference Equations with One-Dimensional p-Laplacian, Commun. Math. Anal. 4 (2008), 58-77.
- Y. Long, Homoclinic solutions of some second-order nonperiodic discrete systems, Adv. Differ. Equ. 64 (2011), 1-12.
- Y. Liu and S. Chen, Multiple Heteroclinic solutions of bilateral difference systems with Laplacian operators, Math. Sci. 126 (8) (2014), 13 pages.
- Y. Liu and W. Ge, Twin positive solutions of boundary value problems for finite difference equations with p-Laplacian operator, J. Math. Anal. Appl. 278 (2003), 551-561. https://doi.org/10.1016/S0022-247X(03)00018-0
- Y. Li and L. Lu, Existence of positive solutions of p-Laplacian difference equations, Appl. Math. Letters 19 (2006), 1019-1023. https://doi.org/10.1016/j.aml.2005.10.020
- Y. Long and H. Shi, Multiple slutions for the discrete-Laplacian boundary value problems, Disc. Dyn. Nature Soc. 2014 (2014), Article ID 213702, 6 pages.
- Y. Li and L. Zhu, Existence of periodic solutions discrete Lotka-Volterra systems with delays, Bull. of Inst. of Math. Academia Sinica 33 (4) (2005), 369-380.
- X. Liu, Y. Zhang and H. Shi, Periodic solutions for fourth-order nonlinear functional difference equations, Math. Meth. Appl. Sci. 38 (1) (2015), 1-10. https://doi.org/10.1002/mma.3045
- X. Liu, Y. Zhang and H. Shi, Homoclinic orbits of second order nonlinear functional difference equations with Jacobi operators, Indagationes Math. 26 (1) (2015), 75-87. https://doi.org/10.1016/j.indag.2014.06.002
- X. Liu, Y. Zhang and H. Shi, Nonexistence and existence results for a class of fourth-order difference Neumann boundary value problems, Indagationes Math. 26 (1) (2015), 293-305. https://doi.org/10.1016/j.indag.2014.05.001
- X. Liu, Y. Zhang and H. Shi, Periodic and subharmonic solutions for fourth-order nonlinear difference equations, Appl. Math. Comput. 236 (2014), 613-620. https://doi.org/10.1016/j.amc.2014.03.086
- X. Liu, Y. Zhang and H. Shi, Nonexistence and existence results for a class of fourth-order difference Dirichlet boundary value problems, Math. Meth. Appl. Sci. 38 (4) (2015), 691-700. https://doi.org/10.1002/mma.3100
- X. Liu, Y. Zhang and H. Shi, Existence of Periodic Solutions for a 2nth-Order Difference Equation Involving p-Laplacian, Bull. Malaysian Math. Sci. Soc. 38 (3) (2015), 1107-1125. https://doi.org/10.1007/s40840-014-0066-0
- X. Liu, Y. Zhang and H. Shi, Existence and nonexistence results for a fourth-order discrete neumann boundary value problem, Studia Sci. Math. Hungarica, 51 (2) (2014), 186-200. https://doi.org/10.1556/SScMath.51.2014.2.1275
- X. Liu, Y. Zhang and H. Shi, Existence of periodic solutions for a class of nonlinear difference equations, Qual. Theory Dyn. Syst. 14 (1) (2015), 51-69. https://doi.org/10.1007/s12346-014-0125-9
- X. Liu, Y. Zhang and H. Shi, Nonexistence and existence of solutions for a fourth-order discrete mixed boundary value problem, Proceedings-Math. Sci. 124 (2) (2014), 179-191. https://doi.org/10.1007/s12044-014-0176-5
- X. Liu, Y. Zhang and H. Shi, Nonexistence and existence results for a 2nth-order discrete mixed boundary value problem, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 109 (2) (2015), 303-314. https://doi.org/10.1007/s13398-014-0182-6
- R. Ma and I. Raffoul, Positive solutions of three-point nonlinear discrete second order boundary value problem, J. Differ. Eqns. Appl. 10 (2004), 129-138. https://doi.org/10.1080/1023619031000114323
- M. Mihuailescu, V. Radulescu and S. Tersian, Homoclinic solutions of difference equations with variable exponents, Topological Meth. Nonl. Anal. Journal of the Juliusz Schauder University Centre, 38 (2011), 277-289.
- M. Mihailescu, V. Radulescu and S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems, J. Differ. Equ. Appl. 15 (2009), 557-567. https://doi.org/10.1080/10236190802214977
- H. Pang, H. Feng and W. Ge, Multiple positive solutions of quasi-linear boundary value problems for finite difference equations, Appl. Math. Comput. 197 (2008), 451-456. https://doi.org/10.1016/j.amc.2007.06.027
- L. Rachunek and I. Rachunkoa, Homoclinic solutions of non-autonomous dif- ference equations arising in hydrodynamics, Nonlinear Anal. RWA. 12 (2011), 14-23. https://doi.org/10.1016/j.nonrwa.2010.05.031
-
B. Ricceri, A multiplicity theorem in
$R^n$ , J. Convex Anal. 16 (2009), 987-992. - H. Shi, Periodic and subharmonic solutions for second-order nonlinear difference equations, J. Appl. Math. Comput. 48 (1-2) (2014), 1-15. https://doi.org/10.1007/s12190-014-0788-z
- H. Shi, X. Liu and Y. Zhang, Nonexistence and existence results for a 2nth-order discrete Dirichlet boundary value problem, Kodai Math. J. 37 (2) (2014), 492-505. https://doi.org/10.2996/kmj/1404393901
- H. Shi, X. Liu and Y. Zhang, Homoclinic orbits for second order p-Laplacian difference equations containing both advance and retardation, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, DOI 10.1007/s13398-015-0221-y, 2015: 1-14.
- H. Shi, X. Liu, Y. Zhang and X. Deng, Existence of periodic solutions of fourth-order nonlinear difference equations, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 108 (2) (2014), 811-825. https://doi.org/10.1007/s13398-013-0143-5
- Y. Tian and W. Ge, Multiple positive solutions of boundary value problems for second-order discrete equations on the half-line, J. Differ. Eqns. Appl. 12 (2006),191-208. https://doi.org/10.1080/10236190500539329
- P. J. Y. Wong and L. Xie, Three symmetric solutions of lidstone boundary value problems for difference and partial difference equations, Comput. Math. Appl. 45 (2003), 1445-1460. https://doi.org/10.1016/S0898-1221(03)00102-0
- J. Yu and Z. Guo, On generalized discrete boundary value problems of Emden-Fowler equation, Sci. China Math. 36 (2006), 721-732.
- Q. Zhang, Existence of homoclinic solutions for a class of difference systems involving p-Laplacian, Adv. Differ. Equ. 291 (2014), 1-14.
- Z. Zhou, J. Yu and Y. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Sci. China Math. 54 (2011), 83-93. https://doi.org/10.1007/s11425-010-4101-9