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

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.

### MSC:

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

### Keywords:

time scales; positive solution; cone; fixed point
Full Text:

### 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
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