×
Sun, Hong-Rui; Li, Wan-Tong

Existence theory for positive solutions to one-dimensional \(p\)-Laplacian boundary value problems on time scales. (English) Zbl 1139.34047

J. Differ. Equations 240, No. 2, 217-248 (2007).
This paper studies the boundary value problem
\[ \left(\phi_p \left(u^{\triangle}(t)\right)\right)^{\triangle} + h(t) f(u(\sigma(t)) = 0, \quad t \in [a,b], \]
\[ u(a) - B_0\left(u^{\triangle}(a)\right) = 0, \quad u^{\triangle}\left(\sigma(b)\right) = 0, \]
where \(\phi_p(u)\) is the \(p\)-Laplacian operator. The problem is considered on a time scale. Using fixed point theorems of Krasnosel’skii, Avery and Henderson, and Leggett and Williams, the authors obtain several existence and multiplicity results for positive solutions.
Reviewer: Nickolai Kosmatov (Little Rock)

Cited in 63 Documents

MSC:

34K10 Boundary value problems for functional-differential equations
39A10 Additive difference equations

Keywords:

time scales; positive solution; cone; fixed point
PDF BibTeX XML Cite
\textit{H.-R. Sun} and \textit{W.-T. Li}, J. Differ. Equations 240, No. 2, 217--248 (2007; Zbl 1139.34047)
Full Text: DOI

References:

[1] Agarwal, R. P.; Bohner, M., Basic calculus on time scales and some of its applications, Results Math., 35, 3-22 (1999) · Zbl 0927.39003
[2] Agarwal, R. P.; Bohner, M.; Li, W. T., Nonoscillation and Oscillation Theory for Functional Differential Equations, Pure Appl. Math., vol. 267 (2004), Marcel Dekker
[3] 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 Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 1-57 · Zbl 1056.34049
[4] Agarwal, R. P.; O’Regan, D., Triple solutions to boundary value problems on time scales, Appl. Math. Lett., 13, 7-11 (2000) · Zbl 0958.34021
[5] Agarwal, R. P.; O’Regan, D.; Wong, P. J.Y., Positive Solutions of Differential, Difference and Integral Equations (1999), Kluwer: Kluwer Dordrecht · Zbl 0923.39002
[6] Agarwal, R. P.; Wong, P. J.Y., Advanced Topics in Difference Equations (1997), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0914.39005
[7] Akin, E., Boundary value problems for a differential equation on a measure chain, Panamer. Math. J., 10, 17-30 (2000) · Zbl 0973.39010
[8] Anderson, D. R., Solutions to second-order three-point problems on time scales, J. Difference Equ. Appl., 8, 673-688 (2002) · Zbl 1021.34011
[9] 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
[10] Aulbach, B.; Neidhart, L., Integration on measure chains, (Proceedings of the Sixth International Conference on Difference Equations (2004), CRC: CRC Boca Raton, FL), 239-252 · Zbl 1083.26005
[11] Avery, R. I.; Anderson, D. R., Existence of three positive solutions to a second-order boundary value problem on a measure chain, J. Comput. Appl. Math., 141, 65-73 (2002) · Zbl 1032.39009
[12] Avery, R. I.; Chyan, C. J.; 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
[13] Avery, R. I.; Henderson, J., Two positive fixed points of nonlinear operator on ordered Banach spaces, Comm. Appl. Nonlinear Anal., 8, 27-36 (2001) · Zbl 1014.47025
[14] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (2001), Birkhäuser: Birkhäuser Boston, MA · Zbl 0978.39001
[15] Bohner, M.; Peterson, A., Advances in Dynamic Equations on Time Scales (2003), Birkhäuser: Birkhäuser Boston · Zbl 1025.34001
[16] Chyan, C. J.; Henderson, J., Twin solutions of boundary value problems for differential equations on measure chains, J. Comput. Appl. Math., 141, 123-131 (2002) · Zbl 1134.39301
[17] Chyan, C. J.; Henderson, J., Positive solutions in an annulus for nonlinear differential equations on a measure chain, Tamkang J. Math., 30, 231-240 (1999) · Zbl 0995.34017
[18] Deimling, K., Nonlinear Functional Analysis (1985), Springer: Springer New York · Zbl 0559.47040
[19] Erbe, L.; Peterson, A., Eigenvalue conditions and positive solutions, J. Difference Equ. Appl., 6, 165-191 (2000) · Zbl 0949.34015
[20] Erbe, L.; Peterson, A., Green’s functions and comparison theorems for differential equations on measure chains, Dyn. Contin. Discrete Impuls. Syst., 6, 121-137 (1999) · Zbl 0938.34027
[21] Erbe, L.; Peterson, A., Positive solutions for a nonlinear differential equation on a measure chain, Math. Comput. Modelling, 32, 571-585 (2000) · Zbl 0963.34020
[22] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press San Diego, CA · Zbl 0661.47045
[23] He, Z., On the existence of positive solutions of \(p\)-Laplacian difference equation, J. Comput. Appl. Math., 161, 193-201 (2003) · Zbl 1041.39002
[24] Hilger, S., Analysis on measure chains—A unified approach to continuous and discrete calculus, Results Math., 18, 18-56 (1990) · Zbl 0722.39001
[25] Spedding, V., Taming nature’s numbers, New Scientist: The Global Science and Technology Weekly, 2404, 28-31 (2003)
[26] Jones, M. A.; Song, B.; Thomas, D. M., Controlling wound healing through debridement, Math. Comput. Modelling, 40, 1057-1064 (2004) · Zbl 1061.92036
[27] Kaufmann, E. R., Positive solutions of a three-point boundary value problem on a time scale, Electron. J. Differential Equations, 2003, 82, 1-11 (2003) · Zbl 1047.34015
[28] Krasnosel’skii, M., Positive Solutions of Operator Equations (1964), Noordhoff: Noordhoff Groningen · Zbl 0121.10604
[29] Lakshmikantham, V.; Sivasundaram, S.; Kaymakcalan, B., Dynamic Systems on Measure Chains (1996), Kluwer Academic Publ.: Kluwer Academic Publ. Boston · Zbl 0869.34039
[30] 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
[31] Li, C.; Ge, W., Positive solutions of one-dimensional \(p\)-Laplacian singular Sturm-Liouville boundary value problems, Math. Appl., 15, 13-17 (2002), (in Chinese) · Zbl 1022.34020
[32] 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
[33] 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
[34] Lü, H.; O’Regan, D.; Zhong, C., Multiple positive solutions for the one-dimensional singular \(p\)-Laplacian, Appl. Math. Comput., 133, 407-422 (2002) · Zbl 1048.34047
[35] Sun, H. R., Existence of positive solutions to second-order time scale systems, Comput. Math. Appl., 49, 131-145 (2005) · Zbl 1075.34019
[36] Sun, H. R.; Li, W. T., Existence of positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl., 299, 508-524 (2004) · Zbl 1070.34029
[37] Sun, W.; Ge, W., The existence of positive solutions for a class of nonlinear boundary value problems, Acta Math. Sinica, 44, 577-580 (2001), (in Chinese) · Zbl 1024.34016
[38] 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
[39] Wang, H., Positive periodic solutions of functional differential equations, J. Differential Equations, 202, 354-366 (2004) · Zbl 1064.34052
[40] Wang, J., The existence of positive solutions for the one-dimensional \(p\)-Laplacian, Proc. Amer. Math. Soc., 125, 2275-2283 (1997) · Zbl 0884.34032
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.