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Positive solutions of singular \(p\)-Laplacian dynamic equations with sign changing nonlinearity. (English) Zbl 1146.39033

Let \(\mathbb{T}\) be a time scale including \(0\) and \(T>0\). Using the Schauder’s fixed point theorem and the upper and lower solution method, the authors obtained some results of existence of the positive solution for an \(m\)-point singular \(p\)-Laplacian dynamic equation with a boundary condition.

MSC:

39A12 Discrete version of topics in analysis
39A10 Additive difference equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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[1] Agarwal, R.P.; Bohner, M.; Rehak, P., Half-linear dynamic equations, nonlinear analysis and applications: to V. lakshmikantham on his 80th birthday, vol. 1, (2003), Kluwer Academic Publishers Dordrecht, pp. 1-57 · Zbl 1056.34049
[2] Agarwal, R.P.; Lü, H.; O’Regan, D., Existence theorem for the one-dimensional singular p-Laplacian equation with sign changing nonlinearities, Appl. math. comput., 143, 15-38, (2003) · Zbl 1031.34023
[3] Agarwal, R.P.; Bohner, M.; Li, W.T., Nonoscillation and oscillation theory for functional differential equations, Pure and applied mathematics ser., vol. 267, (2004), Marcel Dekker
[4] Atici, F.M.; Guseinov, G.Sh., On green’s functions and positive solutions for boundary value problems on time scales, J. comput. appl. math., 141, 75-99, (2002) · Zbl 1007.34025
[5] Anderson, D.R.; Avery, R.; Henderson, J., Existence of solutions for a one-dimensional p-Laplacian on time scales, J. diff. equ. appl., 10, 889-896, (2004) · Zbl 1058.39010
[6] Aulbach, B.; Neidhart, L., Integration on measure chains, (), 239-252 · Zbl 1083.26005
[7] Bohner, M.; Peterson, A., Dynamic equation on time scales, An introduction with applications, (2001), Birkhäuser Boston · Zbl 0978.39001
[8] Bohner, M.; Peterson, A., Advances in dynamic equation on time scales, (2003), Birkhäuser Boston
[9] He, Z., Double positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales, J. comput. appl. math., 182, 304-315, (2005) · Zbl 1075.39011
[10] Hilger, S., Analysis on measure chains – a unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001
[11] Spedding, V., Taming nature’s numbers, New sci., July, 28-32, (2003)
[12] Jones, M.A.; Song, B.; Thomas, D.M., Controlling wound healing through debridement, Math. comput. model., 40, 1057-1064, (2004) · Zbl 1061.92036
[13] Jiang, D.Q.; O’Regan, D.; Agarwal, R.P., A generalized upper and lower solution method for singular boundary value problems for the one-dimensional p-Laplacian, Appl. anal., 11, 35-47, (2005) · Zbl 1086.39022
[14] Lakshmikantham, V.; Sivasundaram, S.; Kaymakcalan, B., Dynamic systems on measure chains, (1996), Kluwer Academic Publishers Boston · Zbl 0869.34039
[15] Li, W.T.; Sun, H.R., Multiple positive solutions for nonlinear dynamic systems on a measure chain, J. comput. appl. math., 162, 421-430, (2004) · Zbl 1045.39007
[16] Li, W.T.; Liu, X.L., Eigenvalue problems for second-order nonlinear dynamic equations on time scales, J. math. anal. appl., 318, 578-592, (2006) · Zbl 1099.34026
[17] Li, W.T.; Sun, H.R., Positive solutions for second-order m-point boundary value problems on time scales, Acta math. sin. English series, 22, 1797-1804, (2006) · Zbl 1119.34021
[18] Lü, H.; O’Regan, D.; Agarwal, R.P., Upper and lower solutions for the singular p-Laplacian with sign changing nonlinearities and nonlinear boundary data, J. comput. appl. math., 181, 442-466, (2005) · Zbl 1082.34022
[19] Lü, H.; O’Regan, D.; Agarwal, R.P., Existence theorem for the one-dimensional singular p-Laplacian equation with a nonlinear boundary condition, J. comput. appl. math., 182, 188-210, (2005) · Zbl 1071.34019
[20] Lü, H.; O’Regan, D.; Agarwal, R.P., Positive solutions for singular p-Laplacian equations with sign changing nonlinearities using inequality theory, Appl. math. comput., 165, 587-597, (2005) · Zbl 1071.34018
[21] H.R. Sun, Boundary value problems for dynamic equations on measure chains, Ph.D thesis, Lanzhou University, 2004.
[22] Sun, H.R., Existence of positive solutions to second-order time scale systems, Comput. math. appl., 49, 131-145, (2005) · Zbl 1075.34019
[23] Sun, H.R.; Li, W.T., Multiple positive solutions for p-Laplacian m-point boundary value problems on time scales, Appl. math. comput., 182, 478-491, (2006) · Zbl 1126.34019
[24] Sun, H.R.; Li, W.T., Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales, J. diff. equ., 240, 217-248, (2007) · Zbl 1139.34047
[25] H.R. Sun, W.T. Li, Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales, Taiwanese J. Math., in press. · Zbl 1139.34047
[26] Thomas, D.M.; Vandemuelebroeke, L.; Yamaguchi, K., A mathematical evolution model for phytoremediation of metals, Discrete contin. dyn. syst. ser. B, 5, 411-422, (2005) · Zbl 1085.34530
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