## Positive solutions for discrete boundary value problems to one-dimensional $$p$$-Laplacian with delay.(English)Zbl 1271.39006

Summary: We study the existence of positive solutions for discrete boundary value problems to one-dimensional $$p$$-Laplacian with delay. The proof is based on the Guo-Krasnoselskii fixed-point theorem in cones. Two numerical examples are also provided to illustrate the theoretical results.

### MSC:

 39A14 Partial difference equations 35B09 Positive solutions to PDEs 35J92 Quasilinear elliptic equations with $$p$$-Laplacian 47H10 Fixed-point theorems
Full Text:

### References:

 [1] D. Q. Jiang, J. F. Chu, D. O’Regan, and R. P. Agarwal, “Positive solutions for continuous and discrete boundary value problems to the one-dimension p-Laplacian,” Mathematical Inequalities & Applications, vol. 7, no. 4, pp. 523-534, 2004. · Zbl 1072.34021 [2] W.-T. Li and H.-F. Huo, “Positive periodic solutions of delay difference equations and applications in population dynamics,” Journal of Computational and Applied Mathematics, vol. 176, no. 2, pp. 357-369, 2005. · Zbl 1068.39013 [3] Y. J. Liu, “A study on periodic solutions of higher order nonlinear functional difference equations with p-Laplacian,” Journal of Difference Equations and Applications, vol. 13, no. 12, pp. 1105-1114, 2007. · Zbl 1146.39013 [4] H. Y. Feng, W. G. Ge, and M. Jiang, “Multiple positive solutions for m-point boundary-value problems with a one-dimensional p-Laplacian,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 8, pp. 2269-2279, 2008. · Zbl 1138.34005 [5] C.-G. Kim, “Existence of positive solutions for singular boundary value problems involving the one-dimensional p-Laplacian,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 12, pp. 4259-4267, 2009. · Zbl 1162.34315 [6] R. Glowinski and J. Rappaz, “Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology,” Mathematical Modelling and Numerical Analysis, vol. 37, no. 1, pp. 175-186, 2003. · Zbl 1046.76002 [7] C. H. Jin, J. X. Yin, and Z. J. Wang, “Positive radial solutions of p-Laplacian equation with sign changing nonlinear sources,” Mathematical Methods in the Applied Sciences, vol. 30, no. 1, pp. 1-14, 2007. · Zbl 1146.34019 [8] D. Y. Bai and Y. T. Xu, “Existence of positive solutions for boundary-value problems of second-order delay differential equations,” Applied Mathematics Letters, vol. 18, no. 6, pp. 621-630, 2005. · Zbl 1080.34048 [9] C. H. Jin and J. X. Yin, “Positive solutions for the boundary value problems of one-dimensional p-Laplacian with delay,” Journal of Mathematical Analysis and Applications, vol. 330, no. 2, pp. 1238-1248, 2007. · Zbl 1124.34040 [10] L. Wei and J. Zhu, “The existence and blow-up rate of large solutions of one-dimensional p-Laplacian equations,” Nonlinear Analysis. Real World Applications, vol. 13, no. 2, pp. 665-676, 2012. · Zbl 1253.34035 [11] Z. L. Yang and D. O’Regan, “Positive solutions of a focal problem for one-dimensional p-Laplacian equations,” Mathematical and Computer Modelling, vol. 55, no. 7-8, pp. 1942-1950, 2012. · Zbl 1260.34052 [12] Z. M. He, “On the existence of positive solutions of p-Laplacian difference equations,” Journal of Computational and Applied Mathematics, vol. 161, no. 1, pp. 193-201, 2003. · Zbl 1041.39002 [13] R. I. Avery and J. Henderson, “Existence of three positive pseudo-symmetric solutions for a one-dimensional p-Laplacian,” Journal of Mathematical Analysis and Applications, vol. 277, no. 2, pp. 395-404, 2003. · Zbl 1028.34022 [14] D.-B. Wang and W. Guan, “Three positive solutions of boundary value problems for p-Laplacian difference equations,” Computers & Mathematics with Applications, vol. 55, no. 9, pp. 1943-1949, 2008. · Zbl 1147.39008 [15] P. Candito and N. Giovannelli, “Multiple solutions for a discrete boundary value problem involving the p-Laplacian,” Computers & Mathematics with Applications, vol. 56, no. 4, pp. 959-964, 2008. · Zbl 1155.39301 [16] A. Iannizzotto and S. A. Tersian, “Multiple homoclinic solutions for the discrete p-Laplacian via critical point theory,” Journal of Mathematical Analysis and Applications, vol. 403, no. 1, pp. 173-182, 2013. · Zbl 1282.39003 [17] Y. J. Liu and X. Y. Liu, “The existence of periodic solutions of higher order nonlinear periodic difference equations,” Mathematical Methods in the Applied Sciences, vol. 36, pp. 1459-1470, 2013. · Zbl 1276.39007 [18] H. H. Liang and P. X. 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. · Zbl 1112.39008 [19] D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5, Academic Press, Boston, Mass, USA, 1988. · Zbl 0661.47045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.