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Three positive solutions of nonhomogeneous multi-point BVPs for second order \(p\)-Laplacian functional difference equations. (English) Zbl 1181.39004

There is a large number of papers on the existence of multiple positive solutions of boundary value problems (BVPs) for difference equations. In this paper sufficient conditions to guarantee the existence of at least three positive solutions of a multi-point boundary value problem of second order difference equations with one-dimensional \(p\)-Laplacian are given. The author wants to show that the approach to positive solutions of BVPs by using multi-fixed-point theorems can be extended to nonhomogeneous BVPs.

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
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[1] Pang, H., Feng, H., Ge, W.: Multiple positive solutions of quasi-linear boundary value problems for finite difference equations. Appl. Math. Comput. (2007). doi: 10.1016/j.amc.2007.06.027 · Zbl 1143.39006
[2] Cheung, W., Ren, J., Wong, P.J.Y., Zhao, D.: Multiple positive solutions for discrete nonlocal boundary value problems. J. Math. Anal. Appl. 330, 900–915 (2007) · Zbl 1120.39016 · doi:10.1016/j.jmaa.2006.08.034
[3] Li, Y., Lu, L.: Existence of positive solutions of p-Laplacian difference equations. Appl. Math. Lett. 19, 1019–1023 (2006) · Zbl 1125.39007 · doi:10.1016/j.aml.2005.10.020
[4] Cai, X., Yu, J.: Existence theorems for second-order discrete boundary value problems. J. Math. Anal. Appl. 320, 649–661 (2006) · Zbl 1113.39019 · doi:10.1016/j.jmaa.2005.07.029
[5] Zhang, G., Medina, R.: Three-point boundary value problems for difference equations. Comput. Math. Appl. 48, 1791–1799 (2004) · Zbl 1075.39015 · doi:10.1016/j.camwa.2004.09.002
[6] Aykut, N.: Existence of positive solutions for boundary value problems of second-order functional difference equations. Comput. Math. Appl. 48, 517–527 (2004) · Zbl 1066.39015 · doi:10.1016/j.camwa.2003.10.007
[7] He, Z.: On the existence of positive solutions of p-Laplacian difference equations. J. Comput. Appl. Math. 161, 193–201 (2003) · Zbl 1041.39002 · doi:10.1016/j.cam.2003.08.004
[8] Wong, P.J.Y., Agarwal, R.P.: Existence theorems for a system of difference equations with (n,p)-type conditions. Appl. Math. Comput. 123, 389–407 (2001) · Zbl 1025.39002 · doi:10.1016/S0096-3003(00)00078-3
[9] Graef, J.R., Henderson, J.: Double solutions of boundary value problems for 2nd-order differential equations and difference equations. Comput. Math. Appl. 45, 873–885 (2003) · Zbl 1070.34036 · doi:10.1016/S0898-1221(03)00063-4
[10] Avery, R.I., Chyan, C., Henderson, J.: Twin solutions of boundary value problems for ordinary differential equations and finite difference equations. Comput. Math. Appl. 42, 695–704 (2001) · Zbl 1006.34022 · doi:10.1016/S0898-1221(01)00188-2
[11] Liu, Y., Ge, W.: Twin positive solutions of boundary value problems for finite difference equations with p-Laplacian operator. J. Math. Anal. Appl. 278, 551–561 (2003) · Zbl 1019.39002 · doi:10.1016/S0022-247X(03)00018-0
[12] Graef, J.R., Henderson, J.: Double solutions of boundary value problems for 2nd-order differential equations and difference equations. Comput. Math. Appl. 45, 873–885 (2003) · Zbl 1070.34036 · doi:10.1016/S0898-1221(03)00063-4
[13] Leggett, R., Williams, L.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28, 673–688 (1979) · Zbl 0421.47033 · doi:10.1512/iumj.1979.28.28046
[14] Avery, R.: A generalization of the Leggett-Williams fixed point theorem. Math. Sci. Res. Hot-line 2, 9–14 (1998) · Zbl 0965.47038
[15] Avery, R., Peterson, A.: Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 42, 313–322 (2001) · Zbl 1005.47051 · doi:10.1016/S0898-1221(01)00156-0
[16] Chyan, C.J., Henderson, J., Lo, H.: Existence of triple solutions of discrete (n,p) boundary value problems. Appl. Math. Lett. 14, 347–352 (2001) · Zbl 0982.39002 · doi:10.1016/S0893-9659(00)00160-9
[17] Avery, R.I., Peterson, A.C.: Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 42, 313–322 (2001) · Zbl 1005.47051 · doi:10.1016/S0898-1221(01)00156-0
[18] Wong, P.J.Y., Xie, L.: Three symmetric solutions of lidstone boundary value problems for difference and partial difference equations. Comput. Math. Appl. 45, 1445–1460 (2003) · Zbl 1057.39020 · doi:10.1016/S0898-1221(03)00102-0
[19] Yang, C., Weng, P.: Green functions and positive solutions for boundary value problems of third-order difference equations. Comput. Math. Appl. 54, 567–578 (2007) · Zbl 1130.39012 · doi:10.1016/j.camwa.2007.01.032
[20] Karaca, I.Y.: Discrete third-order three-point boundary value problem. J. Comput. Appl. Math. 205, 458–468 (2007) · Zbl 1127.39028 · doi:10.1016/j.cam.2006.05.030
[21] Anderson, D.R.: Discrete third-order three-point right-focal boundary value problems. Comput. Math. Appl. 45, 861–871 (2003) · Zbl 1054.39010 · doi:10.1016/S0898-1221(03)80157-8
[22] Anderson, D., Avery, R.I.: Multiple positive solutions to a third-order discrete focal boundary value problem. Comput. Math. Appl. 42, 333–340 (2001) · Zbl 1001.39022 · doi:10.1016/S0898-1221(01)00158-4
[23] Cai, X., Yu, J.: Existence of periodic solutions for a 2nd-order nonlinear difference equation. J. Math. Anal. Appl. 329, 870–878 (2007) · Zbl 1153.39302 · doi:10.1016/j.jmaa.2006.07.022
[24] Cai, X., Yu, J., Guo, Z.: Periodic solutions of a class of nonlinear difference equations via critical point method. Comput. Math. Appl. 52, 1639–1647 (2006) · Zbl 1134.39003 · doi:10.1016/j.camwa.2006.09.003
[25] Yu, J., Guo, Z.: On boundary value problems for a discrete generalized Emden-Fowler equation. J. Differ. Equ. 231, 18–31 (2006) · Zbl 1112.39011 · doi:10.1016/j.jde.2006.08.011
[26] Bai, D., Xu, Y.: Positive solutions and eigenvalue intervals of nonlocal boundary value problems with delays. J. Math. Anal. Appl. 334, 1152–1166 (2007) · Zbl 1123.34047 · doi:10.1016/j.jmaa.2006.12.062
[27] Bai, D., Xu, Y.: Nontrivial solutions of boundary value problems of second-order difference equations. J. Math. Anal. Appl. 326, 297–302 (2007) · Zbl 1113.39018 · doi:10.1016/j.jmaa.2006.02.091
[28] Wang, D., Guan, W.: Three positive solutions of boundary value problems for p-Laplacian difference equations. Comput. Math. Appl. (2007). doi: 10.1016/j.camwa.2007.08.033
[29] Avery, R., Henderson, J.: Existence of three positive pseudo-symmetric solutions for a one dimensional p-Laplacian. J. Math. Anal. Appl. 277, 395–404 (2003) · Zbl 1028.34022 · doi:10.1016/S0022-247X(02)00308-6
[30] Su, Y., Li, W., Sun, H.: Triple positive pseudo-symmetric solutions of three-point BVPs for p-Laplacian dynamic equations on time scales. Nonlinear Anal. 68, 1442–1452 (2008) · Zbl 1139.34019
[31] Su, Y., Li, W.: Triple positive solutions of m-point BVPs for p-Laplacian dynamic equations on time scales. Nonlinear Anal. (2007). doi: 10.1016/j.na.2007.10.018
[32] He, Z., Long, Z.: Three positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales. Nonlinear Anal. (2007). doi: 10.1016/j.na.2007.06.001 · Zbl 1157.39005
[33] He, Z., Li, L.: Multiple positive solutions for the one-dimensional p-Laplacian dynamic equations on time scales. Math. Comput. Model. 45, 68–79 (2007) · Zbl 1143.34012 · doi:10.1016/j.mcm.2006.03.021
[34] Bai, Z., Ge, W.: Existence of three positive solutions for some second-order boundary value problems. Comput. Math. Appl. 48, 699–707 (2004) · Zbl 1066.34019 · doi:10.1016/j.camwa.2004.03.002
[35] Chu, J.: Eigenvalues and discrete boundary value problems for the one-dimensional p-Laplacian. J. Math. Anal. Appl. 305, 452–465 (2005) · Zbl 1074.39022 · doi:10.1016/j.jmaa.2004.10.055
[36] He, Z.: Double positive solutions of three-point boundary value problem for p-Laplacian difference equations. Z. Anal. Anwend. 24, 302–315 (2005) · Zbl 1093.39003
[37] Kong, L., Kong, Q.: Second-order boundary value problems with nonhomogeneous boundary conditions (II). J. Math. Anal. Appl. 330, 1393–1441 (2007) · Zbl 1119.34009 · doi:10.1016/j.jmaa.2006.08.064
[38] Kong, L., Kong, Q.: Multi-point boundary value problems of second-order differential equations (I). Nonlinear Anal. 58, 909–931 (2004) · Zbl 1066.34012 · doi:10.1016/j.na.2004.03.033
[39] Chen, S., Zhang, Q., Wang, C.: Existence of positive solutions to n-point nonhomogeneous boundary value problem. J. Math. Anal. Appl. 330, 612–621 (2007) · Zbl 1113.35100 · doi:10.1016/j.jmaa.2006.08.014
[40] Chen, H.: Positive solutions for the nonhomogeneous three-point boundary value problem of second-order differential equations. Math. Comput. Model. 45, 844–852 (2007) · Zbl 1137.34319 · doi:10.1016/j.mcm.2006.08.004
[41] Chen, S., Zhang, Q., Chen, L.: Positive solutions for an n-point nonhomogeneous boundary value problem. Math. Comput. Model. 40, 1405–1412 (2004) · Zbl 1084.34022 · doi:10.1016/j.mcm.2005.01.001
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