we show the existence and multiplicity of positive solutions of the nonlineardiscrete fourth-order boundary value problem ${\Delta}^{4}u\left(t-2\right)=\lambda h\left(t\right)f\left(u\left(t\right)\right)$, $t\in {\mathbb{T}}_{2}$, $u\left(1\right)=u\left(T+1\right)={\Delta}^{2}u\left(0\right)={\Delta}^{2}u\left(T\right)=0$, where $\lambda >0$, $h:{\mathbb{T}}_{2}\to (0,\infty )$ is continuous, and $f:\mathbb{R}\to [0,\infty )$ is continuous, $T>4$, ${\mathbb{T}}_{2}=\left\{\mathrm{2,3},\dots ,T\right\}$. The main tool is the Dancer's global bifurcation theorem.

## References

R. P. Agarwal, “On fourth order boundary value problems arising in beam analysis,”

*Differential and Integral Equations*, vol. 2, no. 1, pp. 91–110, 1989. 0715.34032 MR960017 R. P. Agarwal, “On fourth order boundary value problems arising in beam analysis,”*Differential and Integral Equations*, vol. 2, no. 1, pp. 91–110, 1989. 0715.34032 MR960017 A. Cabada, “The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems,”

*Journal of Mathematical Analysis and Applications*, vol. 185, no. 2, pp. 302–320, 1994. 0807.34023 MR1283059 10.1006/jmaa.1994.1250 A. Cabada, “The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems,”*Journal of Mathematical Analysis and Applications*, vol. 185, no. 2, pp. 302–320, 1994. 0807.34023 MR1283059 10.1006/jmaa.1994.1250 C. De Coster and L. Sanchez, “Upper and lower solutions, Ambrosetti-Prodi problem and positive solutions for fourth order O.D.E,”

*Rivista di Matematica Pura ed Applicata*, no. 14, pp. 57–82, 1994. 0979.34015 MR1275354 C. De Coster and L. Sanchez, “Upper and lower solutions, Ambrosetti-Prodi problem and positive solutions for fourth order O.D.E,”*Rivista di Matematica Pura ed Applicata*, no. 14, pp. 57–82, 1994. 0979.34015 MR1275354 Z. Liu and F. Li, “Multiple positive solutions of nonlinear two-point boundary value problems,”

*Journal of Mathematical Analysis and Applications*, vol. 203, no. 3, pp. 610–625, 1996. 0878.34016 MR1417118 10.1006/jmaa.1996.0400 Z. Liu and F. Li, “Multiple positive solutions of nonlinear two-point boundary value problems,”*Journal of Mathematical Analysis and Applications*, vol. 203, no. 3, pp. 610–625, 1996. 0878.34016 MR1417118 10.1006/jmaa.1996.0400 Z.-C. Hao and L. Debnath, “On eigenvalue intervals and eigenfunctions of fourth-order singular boundary value problems,”

*Applied Mathematics Letters*, vol. 18, no. 5, pp. 543–553, 2005. 1074.34079 MR2127818 10.1016/j.aml.2004.03.018 Z.-C. Hao and L. Debnath, “On eigenvalue intervals and eigenfunctions of fourth-order singular boundary value problems,”*Applied Mathematics Letters*, vol. 18, no. 5, pp. 543–553, 2005. 1074.34079 MR2127818 10.1016/j.aml.2004.03.018 Y. Guo and Y. Gao, “The method of upper and lower solutions for a Lidstone boundary value problem,”

*Czechoslovak Mathematical Journal*, vol. 55(130), no. 3, pp. 639–652, 2005. 1081.34019 MR2153088 10.1007/s10587-005-0051-8 Y. Guo and Y. Gao, “The method of upper and lower solutions for a Lidstone boundary value problem,”*Czechoslovak Mathematical Journal*, vol. 55(130), no. 3, pp. 639–652, 2005. 1081.34019 MR2153088 10.1007/s10587-005-0051-8 J. Chu and D. O'Regan, “Positive solutions for regular and singular fourth-order boundary value problems,”

*Communications in Applied Analysis*, vol. 10, no. 2-3, pp. 185–199, 2006. 1123.34015 MR2286503 J. Chu and D. O'Regan, “Positive solutions for regular and singular fourth-order boundary value problems,”*Communications in Applied Analysis*, vol. 10, no. 2-3, pp. 185–199, 2006. 1123.34015 MR2286503 G. Han and Z. Xu, “Multiple solutions of some nonlinear fourth-order beam equations,”

*Nonlinear Analysis*, vol. 68, no. 12, pp. 3646–3656, 2008. 1145.34008 MR2416072 G. Han and Z. Xu, “Multiple solutions of some nonlinear fourth-order beam equations,”*Nonlinear Analysis*, vol. 68, no. 12, pp. 3646–3656, 2008. 1145.34008 MR2416072 J. R. L. Webb, G. Infante, and D. Franco, “Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions,”

*Proceedings of the Royal Society of Edinburgh A*, vol. 138, no. 2, pp. 427–446, 2008. 1167.34004 MR2406699 10.1017/S0308210506001041 J. R. L. Webb, G. Infante, and D. Franco, “Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions,”*Proceedings of the Royal Society of Edinburgh A*, vol. 138, no. 2, pp. 427–446, 2008. 1167.34004 MR2406699 10.1017/S0308210506001041 R. Ma and H. Wang, “On the existence of positive solutions of fourth-order ordinary differential equations,”

*Applicable Analysis*, vol. 59, no. 1–4, pp. 225–231, 1995. 0841.34019 MR1378037 10.1080/00036819508840401 R. Ma and H. Wang, “On the existence of positive solutions of fourth-order ordinary differential equations,”*Applicable Analysis*, vol. 59, no. 1–4, pp. 225–231, 1995. 0841.34019 MR1378037 10.1080/00036819508840401 Z. Bai and H. Wang, “On positive solutions of some nonlinear fourth-order beam equations,”

*Journal of Mathematical Analysis and Applications*, vol. 270, no. 2, pp. 357–368, 2002. 1006.34023 MR1915704 10.1016/S0022-247X(02)00071-9 Z. Bai and H. Wang, “On positive solutions of some nonlinear fourth-order beam equations,”*Journal of Mathematical Analysis and Applications*, vol. 270, no. 2, pp. 357–368, 2002. 1006.34023 MR1915704 10.1016/S0022-247X(02)00071-9 J. A. Cid, D. Franco, and F. Minhós, “Positive fixed points and fourth-order equations,”

*Bulletin of the London Mathematical Society*, vol. 41, no. 1, pp. 72–78, 2009. 1179.34020 MR2481991 10.1112/blms/bdn105 J. A. Cid, D. Franco, and F. Minhós, “Positive fixed points and fourth-order equations,”*Bulletin of the London Mathematical Society*, vol. 41, no. 1, pp. 72–78, 2009. 1179.34020 MR2481991 10.1112/blms/bdn105 B. Zhang, L. Kong, Y. Sun, and X. Deng, “Existence of positive solutions for BVPs of fourth-order difference equations,”

*Applied Mathematics and Computation*, vol. 131, no. 2-3, pp. 583–591, 2002. 1025.39006 MR1920247 10.1016/S0096-3003(01)00171-0 B. Zhang, L. Kong, Y. Sun, and X. Deng, “Existence of positive solutions for BVPs of fourth-order difference equations,”*Applied Mathematics and Computation*, vol. 131, no. 2-3, pp. 583–591, 2002. 1025.39006 MR1920247 10.1016/S0096-3003(01)00171-0 Z. He and J. Yu, “On the existence of positive solutions of fourth-order difference equations,”

*Applied Mathematics and Computation*, vol. 161, no. 1, pp. 139–148, 2005. 1068.39008 MR2111336 10.1016/j.amc.2003.12.016 Z. He and J. Yu, “On the existence of positive solutions of fourth-order difference equations,”*Applied Mathematics and Computation*, vol. 161, no. 1, pp. 139–148, 2005. 1068.39008 MR2111336 10.1016/j.amc.2003.12.016 R. Ma and Y. Xu, “Existence of positive solution for nonlinear fourth-order difference equations,”

*Computers & Mathematics with Applications*, vol. 59, no. 12, pp. 3770–3777, 2010. 1220.39008 MR2651852 R. Ma and Y. Xu, “Existence of positive solution for nonlinear fourth-order difference equations,”*Computers & Mathematics with Applications*, vol. 59, no. 12, pp. 3770–3777, 2010. 1220.39008 MR2651852 E. Zeidler,

*Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems*, Springer, New York, NY, USA, 1986, Translated from the German by Peter R. Wadsac. MR816732 0583.47050 E. Zeidler,*Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems*, Springer, New York, NY, USA, 1986, Translated from the German by Peter R. Wadsac. MR816732 0583.47050 B. P. Rynne, “Infinitely many solutions of superlinear fourth order boundary value problems,”

*Topological Methods in Nonlinear Analysis*, vol. 19, no. 2, pp. 303–312, 2002. 1017.34015 MR1921051 B. P. Rynne, “Infinitely many solutions of superlinear fourth order boundary value problems,”*Topological Methods in Nonlinear Analysis*, vol. 19, no. 2, pp. 303–312, 2002. 1017.34015 MR1921051 R. Ma, “Existence of positive solutions of a fourth-order boundary value problem,”

*Applied Mathematics and Computation*, vol. 168, no. 2, pp. 1219–1231, 2005. 1082.34023 MR2171774 10.1016/j.amc.2004.10.014 R. Ma, “Existence of positive solutions of a fourth-order boundary value problem,”*Applied Mathematics and Computation*, vol. 168, no. 2, pp. 1219–1231, 2005. 1082.34023 MR2171774 10.1016/j.amc.2004.10.014 R. Ma and J. Xu, “Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem,”

*Nonlinear Analysis*, vol. 72, no. 1, pp. 113–122, 2010. 1200.34023 MR2574922 R. Ma and J. Xu, “Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem,”*Nonlinear Analysis*, vol. 72, no. 1, pp. 113–122, 2010. 1200.34023 MR2574922 R. Ma, “Nodal solutions of boundary value problems of fourth-order ordinary differential equations,”

*Journal of Mathematical Analysis and Applications*, vol. 319, no. 2, pp. 424–434, 2006. 1098.34012 MR2227914 10.1016/j.jmaa.2005.06.045 R. Ma, “Nodal solutions of boundary value problems of fourth-order ordinary differential equations,”*Journal of Mathematical Analysis and Applications*, vol. 319, no. 2, pp. 424–434, 2006. 1098.34012 MR2227914 10.1016/j.jmaa.2005.06.045 R. Ma, “Nodal solutions for a fourth-order two-point boundary value problem,”

*Journal of Mathematical Analysis and Applications*, vol. 314, no. 1, pp. 254–265, 2006. 1085.34015 MR2183550 10.1016/j.jmaa.2005.03.078 R. Ma, “Nodal solutions for a fourth-order two-point boundary value problem,”*Journal of Mathematical Analysis and Applications*, vol. 314, no. 1, pp. 254–265, 2006. 1085.34015 MR2183550 10.1016/j.jmaa.2005.03.078 Y. Xu, C. Gao, and R. Ma, “Solvability of a nonlinear fourth-order discrete problem at resonance,”

*Applied Mathematics and Computation*, vol. 216, no. 2, pp. 662–670, 2010. 1193.39002 MR2601534 10.1016/j.amc.2010.01.112 Y. Xu, C. Gao, and R. Ma, “Solvability of a nonlinear fourth-order discrete problem at resonance,”*Applied Mathematics and Computation*, vol. 216, no. 2, pp. 662–670, 2010. 1193.39002 MR2601534 10.1016/j.amc.2010.01.112