This paper studies the nonlinear two-dimensional difference system with multiple delays $\mathrm{\Delta}({x}_{n}+{p}_{1n}{x}_{n-{\tau}_{1}})$ $+$ ${f}_{1}(n,{x}_{{a}_{1n}},\dots ,{x}_{{a}_{hn}},{y}_{{b}_{1n}},\dots ,{y}_{{b}_{kn}})={q}_{1n},\mathrm{\Delta}({y}_{n}+{p}_{2n}{y}_{n-{\tau}_{2}})+{f}_{2}(n,{x}_{{c}_{1n}},\dots ,{x}_{{c}_{hn}},{y}_{{d}_{1n}},\dots ,{y}_{{d}_{kn}})={q}_{2n},n\ge {n}_{0}$. Using the Banach fixed point theorem and a few new analysis techniques, we show the existence of uncountably many bounded positive solutions for the system, suggest Mann iterative algorithms with errors, and discussthe error estimates between the positive solutions and iterative sequences generated by the Mann iterativealgorithms. Examples to illustrates the results are included.

## References

*Difference Equations and Inequalities*, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. 1160.34313 MR1740241 R. P. Agarwal,

*Difference Equations and Inequalities*, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. 1160.34313 MR1740241

*Applied Mathematics and Computation*, vol. 140, no. 2-3, pp. 307–316, 2003. 1030.39004 MR1953902 10.1016/S0096-3003(02)00228-X R. P. Agarwal, W.-T. Li, and P. Y. H. Pang, “Asymptotic behavior of nonlinear difference systems,”

*Applied Mathematics and Computation*, vol. 140, no. 2-3, pp. 307–316, 2003. 1030.39004 MR1953902 10.1016/S0096-3003(02)00228-X

*Abstract and Applied Analysis*, Article ID 417869, 20 pages, 2010. 1209.34080 MR2720031 J. Baštinec, L. Berezansky, J. Diblík, and Z. Šmarda, “On the critical case in oscillation for differential equations with a single delay and with several delays,”

*Abstract and Applied Analysis*, Article ID 417869, 20 pages, 2010. 1209.34080 MR2720031

*Abstract and Applied Analysis*, Article ID 586328, 28 pages, 2011. 1223.39008 MR2824906 J. Baštinec, L. Berezansky, J. Diblík, and Z. Šmarda, “A final result on the oscillation of solutions of the linear discrete delayed equation $\Delta x(n)=-p(n)x(n-k)$ with a positive coefficient,”

*Abstract and Applied Analysis*, Article ID 586328, 28 pages, 2011. 1223.39008 MR2824906

*Abstract and Applied Analysis*, no. 6, pp. 461–470, 2004. 1101.39003 MR2063054 J. Baštinec and J. Diblík, “Subdominant positive solutions of the discrete equation $\Delta u(k+n)=-p(k)u(k)$,”

*Abstract and Applied Analysis*, no. 6, pp. 461–470, 2004. 1101.39003 MR2063054

*Nonlinear Analysis, Theory, Methods and Applications*, vol. 63, no. 5–7, pp. e2145–e2151, 2005. J. Baštinec and J. Diblík, “Remark on positive solutions of discrete equation $\Delta u(k+n)=-p(k)u(k)$,”

*Nonlinear Analysis, Theory, Methods and Applications*, vol. 63, no. 5–7, pp. e2145–e2151, 2005.

*Journal of Difference Equations and Applications*, vol. 16, no. 9, pp. 1047–1056, 2010. 1207.39014 MR2722821 10.1080/10236190902718026 J. Baštinec, J. Diblík, and Z. Šmarda, “Existence of positive solutions of discrete linear equations with a single delay,”

*Journal of Difference Equations and Applications*, vol. 16, no. 9, pp. 1047–1056, 2010. 1207.39014 MR2722821 10.1080/10236190902718026

*$\dot{x}$*$(t)=-c(t)x(t-\tau (t))$,”

*Abstract and Applied Analysis*, vol. 2011, Article ID 561902, 2011. J. Baštinec, J. Diblík, and Z. Šmarda, “An Explicit criterion for the existence of positive solutions of the linear delayed equation

*$\dot{x}$*$(t)=-c(t)x(t-\tau (t))$,”

*Abstract and Applied Analysis*, vol. 2011, Article ID 561902, 2011.

*Computers and Mathematics with Applications*, vol. 60, no. 5, pp. 1332–1342, 2010. 1201.34095 MR2672932 L. Berezansky, J. Diblík, and Z. Šmarda, “Positive solutions of second-order delay differential equations with a damping term,”

*Computers and Mathematics with Applications*, vol. 60, no. 5, pp. 1332–1342, 2010. 1201.34095 MR2672932

*Abstract and Applied Analysis*, vol. 2012, Article ID 764351, 12 pages, 2012. MR2922943 1242.39001 J. Diblík and I. Hlavickova, “Asymptotic upper and lower estimates of a class of positive solutions of a discrete linear equation with a single delay,”

*Abstract and Applied Analysis*, vol. 2012, Article ID 764351, 12 pages, 2012. MR2922943 1242.39001

*Nonlinear Analysis. Theory, Methods and Applications A*, vol. 70, no. 10, pp. 3702–3714, 2009. 1169.34050 MR2504457 J. Diblík and M. Kúdelčíková, “Two classes of asymptotically different positive solutions of the equation $\dot{y}(t)=-f(t,y(t))$,”

*Nonlinear Analysis. Theory, Methods and Applications A*, vol. 70, no. 10, pp. 3702–3714, 2009. 1169.34050 MR2504457

*Abstract and Applied Analysis*, Article ID 754701, 16 pages, 2011. 1210.34106 MR2739690 J. Diblík and M. Kúdelčíková, “Existence and asymptotic behavior of positive solutions of functional differential equations of delayed type,”

*Abstract and Applied Analysis*, Article ID 754701, 16 pages, 2011. 1210.34106 MR2739690

*Nonlinear Analysis. Theory, Methods and Applications A*, vol. 64, no. 8, pp. 1831–1848, 2006. MR2197363 J. Diblík and Z. Svoboda, “Positive solutions of p-type retarded functional differential equations,”

*Nonlinear Analysis. Theory, Methods and Applications A*, vol. 64, no. 8, pp. 1831–1848, 2006. MR2197363

*Computers and Mathematics with Applications*, vol. 38, no. 7-8, pp. 157–165, 1999. 0964.39012 MR1713170 J. R. Graef and E. Thandapani, “Oscillation of two-dimensional difference systems,”

*Computers and Mathematics with Applications*, vol. 38, no. 7-8, pp. 157–165, 1999. 0964.39012 MR1713170

*Journal of Mathematical Analysis and Applications*, vol. 256, no. 2, pp. 478–485, 2001. 0976.39003 MR1821751 10.1006/jmaa.2000.7296 H.-F. Huo and W.-T. Li, “Oscillation of the Emden-Fowler difference systems,”

*Journal of Mathematical Analysis and Applications*, vol. 256, no. 2, pp. 478–485, 2001. 0976.39003 MR1821751 10.1006/jmaa.2000.7296

*Computers and Mathematics with Applications*, vol. 45, no. 6–9, pp. 1221–1226, 2003. 1056.39009 MR2000592 H.-F. Huo and W.-T. Li, “Oscillation of certain two-dimensional nonlinear difference systems,”

*Computers and Mathematics with Applications*, vol. 45, no. 6–9, pp. 1221–1226, 2003. 1056.39009 MR2000592

*Journal of Computational and Applied Mathematics*, vol. 188, no. 1, pp. 77–88, 2006. 1117.39004 MR2192628 10.1016/j.cam.2005.03.054 J. Jiang and X. Li, “Oscillation and nonoscillation of two-dimensional difference systems,”

*Journal of Computational and Applied Mathematics*, vol. 188, no. 1, pp. 77–88, 2006. 1117.39004 MR2192628 10.1016/j.cam.2005.03.054

*Computers and Mathematics with Applications*, vol. 54, no. 6, pp. 808–818, 2007. 1175.39004 MR2348265 J. Jiang and X. Tang, “Oscillation criteria for two-dimensional difference systems of first order linear difference equations,”

*Computers and Mathematics with Applications*, vol. 54, no. 6, pp. 808–818, 2007. 1175.39004 MR2348265

*Computers and Mathematics with Applications*, vol. 54, no. 9-10, pp. 1240–1249, 2007. 1148.39005 MR2397675 J. Jiang and X. Tang, “Oscillation and asymptotic behavior of two-dimensional difference systems,”

*Computers and Mathematics with Applications*, vol. 54, no. 9-10, pp. 1240–1249, 2007. 1148.39005 MR2397675

*Applied Mathematics Letters*, vol. 20, no. 8, pp. 892–899, 2007. 1144.39004 MR2323128 10.1016/j.aml.2006.06.021 J. Cheng, “Existence of a nonoscillatory solution of a second-order linear neutral difference equation,”

*Applied Mathematics Letters*, vol. 20, no. 8, pp. 892–899, 2007. 1144.39004 MR2323128 10.1016/j.aml.2006.06.021

*Computers and Mathematics with Applications*, vol. 42, no. 3–5, pp. 341–355, 2001. 1006.39013 MR1837996 W.-T. Li, “Classification schemes for nonoscillatory solutions of two-dimensional nonlinear difference systems,”

*Computers and Mathematics with Applications*, vol. 42, no. 3–5, pp. 341–355, 2001. 1006.39013 MR1837996

*Applied Mathematics Letters*, vol. 16, no. 2, pp. 161–163, 2003. 1052.39009 MR1962310 10.1016/S0893-9659(03)80026-5 W.-T. Li and S.-S. Cheng, “Remarks on two recent oscillation theorems for second-order linear difference equations,”

*Applied Mathematics Letters*, vol. 16, no. 2, pp. 161–163, 2003. 1052.39009 MR1962310 10.1016/S0893-9659(03)80026-5

*Applied Mathematics and Computation*, vol. 146, no. 2-3, pp. 543–551, 2003. 1052.39010 MR2008571 10.1016/S0096-3003(02)00604-5 W.-T. Li and S. H. Saker, “Oscillation of second-order sublinear neutral delay difference equations,”

*Applied Mathematics and Computation*, vol. 146, no. 2-3, pp. 543–551, 2003. 1052.39010 MR2008571 10.1016/S0096-3003(02)00604-5

*Applied Mathematics Letters*, vol. 16, no. 5, pp. 627–633, 2003. 1049.39007 MR1986025 10.1016/S0893-9659(03)00057-0 X. Li and D. Zhu, “New results for the asymptotic behavior of a nonlinear second-order difference equation,”

*Applied Mathematics Letters*, vol. 16, no. 5, pp. 627–633, 2003. 1049.39007 MR1986025 10.1016/S0893-9659(03)00057-0

*Journal of Mathematical Analysis and Applications*, vol. 326, no. 1, pp. 511–520, 2007. 1112.39008 MR2277799 10.1016/j.jmaa.2006.03.017 H. Liang and P. Weng, “Existence and multiple solutions for a second-order difference boundary value problem via critical point theory,”

*Journal of Mathematical Analysis and Applications*, vol. 326, no. 1, pp. 511–520, 2007. 1112.39008 MR2277799 10.1016/j.jmaa.2006.03.017

*Applied Mathematics and Computation*, vol. 213, no. 2, pp. 554–576, 2009. 1182.39002 MR2536681 10.1016/j.amc.2009.03.050 Z. Liu, S. M. Kang, and J. S. Ume, “Existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations,”

*Applied Mathematics and Computation*, vol. 213, no. 2, pp. 554–576, 2009. 1182.39002 MR2536681 10.1016/j.amc.2009.03.050

*Computers and Mathematics with Applications*, vol. 57, no. 4, pp. 587–595, 2009. 1165.39307 MR2489080 Z. Liu, Y. Xu, and S. M. Kang, “Global solvability for a second order nonlinear neutral delay difference equation,”

*Computers and Mathematics with Applications*, vol. 57, no. 4, pp. 587–595, 2009. 1165.39307 MR2489080

*Applied Mathematics and Computation*, vol. 184, no. 2, pp. 685–694, 2007. 1123.39005 MR2294933 10.1016/j.amc.2006.06.069 M. Ma, H. Tang, and W. Luo, “Periodic solutions for nonlinear second-order difference equations,”

*Applied Mathematics and Computation*, vol. 184, no. 2, pp. 685–694, 2007. 1123.39005 MR2294933 10.1016/j.amc.2006.06.069

*Nonlinear Analysis, Theory, Methods and Applications*, vol. 63, no. 5–7, pp. e789–799, 2005. M. Migda and J. Migda, “Asymptotic properties of solutions of second-order neutral difference equations,”

*Nonlinear Analysis, Theory, Methods and Applications*, vol. 63, no. 5–7, pp. e789–799, 2005.

*Computers and Mathematics with Applications*, vol. 44, no. 8-9, pp. 1147–1156, 2002. 1035.39009 MR1937574 X. H. Tang, “Bounded oscillation of second-order delay difference equations of unstable type,”

*Computers and Mathematics with Applications*, vol. 44, no. 8-9, pp. 1147–1156, 2002. 1035.39009 MR1937574

*Computers and Mathematics with Applications*, vol. 54, no. 4, pp. 556–566, 2007. 1130.39010 MR2340846 E. Thandapani and P. M. Kumar, “Oscillation of difference systems of the neutral type,”

*Computers and Mathematics with Applications*, vol. 54, no. 4, pp. 556–566, 2007. 1130.39010 MR2340846

*Journal of Difference Equations and Applications*, vol. 12, no. 2, pp. 191–208, 2006. 1103.39016 MR2205088 10.1080/10236190500539329 Y. Tian and W. Ge, “Multiple positive solutions of boundary value problems for second-order discrete equations on the half-line,”

*Journal of Difference Equations and Applications*, vol. 12, no. 2, pp. 191–208, 2006. 1103.39016 MR2205088 10.1080/10236190500539329

*Journal of Computational and Applied Mathematics*, vol. 206, no. 2, pp. 713–725, 2007. 1120.39010 MR2333708 10.1016/j.cam.2006.08.023 J. Wu and Y. Liu, “Two periodic solutions of neutral difference systems depending on two parameters,”

*Journal of Computational and Applied Mathematics*, vol. 206, no. 2, pp. 713–725, 2007. 1120.39010 MR2333708 10.1016/j.cam.2006.08.023

*Computers and Mathematics with Applications*, vol. 42, no. 3–5, pp. 283–290, 2001. 0998.39004 MR1837990 Z. Zhou, J. Yu, and L. Huang, “Asymptotic behavior of delay difference systems,”

*Computers and Mathematics with Applications*, vol. 42, no. 3–5, pp. 283–290, 2001. 0998.39004 MR1837990

*Journal of Mathematical Analysis and Applications*, vol. 194, no. 1, pp. 114–125, 1995. 0872.47031 MR1353071 10.1006/jmaa.1995.1289 L. S. Liu, “Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces,”

*Journal of Mathematical Analysis and Applications*, vol. 194, no. 1, pp. 114–125, 1995. 0872.47031 MR1353071 10.1006/jmaa.1995.1289

*Science in China A*, vol. 8, pp. 785–790, 1992 (Chinese). Z. G. Zhang and J. S. Yu, “On existence of positive solutions for neutral differential equations,”

*Science in China A*, vol. 8, pp. 785–790, 1992 (Chinese).