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On positive solutions and Mann iterative schemes of a third order difference equation. (English) Zbl 1470.39014

Summary: The existence of uncountably many positive solutions and convergence of the Mann iterative schemes for a third order nonlinear neutral delay difference equation are proved. Six examples are given to illustrate the results presented in this paper.

MSC:

39A12 Discrete version of topics in analysis

References:

[1] Abu-Risha, M. H., Oscillation of second-order linear difference equations, Applied Mathematics Letters, 13, 1, 129-135 (2000) · Zbl 0947.39005 · doi:10.1016/S0893-9659(99)00156-1
[2] Agarwal, R. P., Difference Equations and Inequalities (2000), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0952.39001
[3] Agarwal, R. P.; Henderson, J., Positive solutions and nonlinear eigenvalue problems for third-order difference equations, Computers & Mathematics with Applications, 36, 10-12, 347-355 (1998) · Zbl 0933.39003 · doi:10.1016/S0898-1221(98)80035-7
[4] Andruch-Sobiło, A.; Migda, M., On the oscillation of solutions of third order linear difference equations of neutral type, Mathematica Bohemica, 130, 1, 19-33 (2005) · Zbl 1110.39002
[5] Došlá, Z.; Kobza, A., Global asymptotic properties of third-order difference equations, Computers & Mathematics with Applications, 48, 1-2, 191-200 (2004) · Zbl 1068.39006 · doi:10.1016/j.camwa.2003.05.008
[6] Grace, S. R.; Hamedani, G. G., On the oscillation of certain neutral difference equations, Mathematica Bohemica, 125, 3, 307-321 (2000) · Zbl 0969.39006
[7] Cheng, J., Existence of a nonoscillatory solution of a second-order linear neutral difference equation, Applied Mathematics Letters, 20, 8, 892-899 (2007) · Zbl 1144.39004 · doi:10.1016/j.aml.2006.06.021
[8] Kong, L.; Kong, Q.; Zhang, B., Positive solutions of boundary value problems for third-order functional difference equations, Computers & Mathematics with Applications, 44, 3-4, 481-489 (2002) · Zbl 1057.39014 · doi:10.1016/S0898-1221(02)00170-0
[9] Karaca, I. Y., Discrete third-order three-point boundary value problem, Journal of Computational and Applied Mathematics, 205, 1, 458-468 (2007) · Zbl 1127.39028 · doi:10.1016/j.cam.2006.05.030
[10] Li, W.-T.; Sun, J. P., Existence of positive solutions of BVPs for third-order discrete nonlinear difference systems, Applied Mathematics and Computation, 157, 1, 53-64 (2004) · Zbl 1073.39005 · doi:10.1016/j.amc.2003.06.017
[11] Li, W.-T.; Sun, J.-P., Multiple positive solutions of BVPs for third-order discrete difference systems, Applied Mathematics and Computation, 149, 2, 389-398 (2004) · Zbl 1042.39003 · doi:10.1016/S0096-3003(03)00147-4
[12] Liu, Z.; Jia, M.; Kang, S. M.; Kwun, Y. C., Bounded positive solutions for a third order discrete equation, Abstract and Applied Analysis, 2012 (2012) · Zbl 1250.39003 · doi:10.1155/2012/237036
[13] Liu, Z.; Kang, S. M.; Ume, J. S., Existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations, Applied Mathematics and Computation, 213, 2, 554-576 (2009) · Zbl 1182.39002 · doi:10.1016/j.amc.2009.03.050
[14] Liu, Z.; Xu, Y.; Kang, S. M., Bounded oscillation criteria for certain third order nonlinear difference equations with several delays and advances, Computers & Mathematics with Applications, 61, 4, 1145-1161 (2011) · Zbl 1217.39018 · doi:10.1016/j.camwa.2010.12.064
[15] Migda, M.; Migda, J., Asymptotic properties of solutions of second-order neutral difference equations, Nonlinear Analysis: Theory, Methods and Applications, 63, 5-7, e789-e799 (2005) · Zbl 1160.39306 · doi:10.1016/j.na.2005.02.005
[16] Parhi, N., Non-oscillation of solutions of difference equations of third order, Computers & Mathematics with Applications, 62, 10, 3812-3820 (2011) · Zbl 1236.39017 · doi:10.1016/j.camwa.2011.09.029
[17] Parhi, N.; Panda, A., Nonoscillation and oscillation of solutions of a class of third order difference equations, Journal of Mathematical Analysis and Applications, 336, 1, 213-223 (2007) · Zbl 1125.39010 · doi:10.1016/j.jmaa.2007.02.054
[18] Saker, S. H., New oscillation criteria for second-order nonlinear neutral delay difference equations, Applied Mathematics and Computation, 142, 1, 99-111 (2003) · Zbl 1028.39003 · doi:10.1016/S0096-3003(02)00286-2
[19] Saker, S. H., Oscillation of third-order difference equations, Portugaliae Mathematica, 61, 3, 249-257 (2004) · Zbl 1069.39015
[20] Stević, S., On a third-order system of difference equations, Applied Mathematics and Computation, 218, 14, 7649-7654 (2012) · Zbl 1243.39011 · doi:10.1016/j.amc.2012.01.034
[21] Tang, X. H., Bounded oscillation of second-order delay difference equations of unstable type, Computers & Mathematics with Applications, 44, 8-9, 1147-1156 (2002) · Zbl 1035.39009 · doi:10.1016/S0898-1221(02)00222-5
[22] Yan, J.; Liu, B., Asymptotic behavior of a nonlinear delay difference equation, Applied Mathematics Letters, 8, 6, 1-5 (1995) · Zbl 0840.39007
[23] Zhang, Z. G.; Li, Q. L., Oscillation theorems for second-order advanced functional difference equations, Computers & Mathematics with Applications, 36, 6, 11-18 (1998) · Zbl 0935.39005 · doi:10.1016/S0898-1221(98)00157-6
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