Abstract
In this work, we consider the following second-order m-point boundary value problem on time scales
We establish new criteria for the existence of at least three unbounded positive solutions. Our results are new even for the corresponding differential \(({\mathbb{T}}={\mathbb{R}})\) , difference equation \(({\mathbb{T}}={\mathbb{Z}})\) and for the general time-scale setting. An example is given to illustrate our results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Hilger, S.: Analysis on measure chains a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)
Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Application. Birkhäuser, Boston (2001)
Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)
Zhang, Z., Dong, W., Li, Q., Liang, H.: Existence of nonoscillatory solutions for higher order neutral dynamic equations on time scales. J. Appl. Math. Comput. 28, 29–38 (2008)
Rchid, S.A.M., Torres, D.F.M.: Existence of positive solutions for non local p-Laplacian thermistor problems on time scales. J. Inequal. Pure Appl. Math. 8, 1–10 (2007)
Rchid, S.A.M., Torres, D.F.M.: Existence of infinitely many solutions for a quasilinear elliptic problem on time scales. Int. J. Pure Appl. Math. 39, 239–248 (2007)
Feng, M., Zhang, X., Ge, W.: Positive solutions for a class of boundary value problems on time scales. Comput. Math. Appl. 54, 467–475 (2007)
Sun, H., Wang, Y.: Existence of positive solutions for p-Laplacian three-point boundary-value problems on time scales. Electron. J. Differ. Equ. 92, 1–14 (2008)
Karaca, I.Y.: Fourth-order four-point boundary value problem on time scales. Appl. Math. Lett. 21, 1057–1063 (2008)
Liang, S., Zhang, J., Wang, Z.: Existence of countably many positive solutions for nth-order m-point boundary-value problems on time scales. Electron. J. Differ. Equ. 123, 1–13 (2008)
Baxley, J.V.: Existence and uniqueness of nonlinear boundary value problems on infinite intervals. J. Math. Anal. Appl. 147, 127–133 (1990)
Kawano, N., Yanagida, E., Yotsutani, S.: Structure theorems for positive radial solutions to Δu+K(|x|)u p=0 in R n. Funkcial. Ekvac. 36, 557–579 (1993)
Aronson, D., Crandall, M.G., Peletier, L.A.: Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal. 6, 1001–1022 (1982)
Iffland, G.: Positive solutions of a problem Emden-Fowler type with a type free boundary. SIAM J. Math. Anal. 18, 283–292 (1987)
Brighi, B., Hoernel, J.: Asymptotic behavior of the unbounded solutions of some boundary layer equations. Arch. Math. 85, 161–166 (2005)
Yan, B., O’Regan, D., Agarwal, R.P.: Unbounded solutions for singular boundary value problems on the semi-infinite interval: Upper and lower solutions and multiplicity. J. Comput. Appl. Math. 197, 365–386 (2006)
Agarwal, R.P., Bohner, M., O’Regan, D.: Time scale boundary value problems on infinite intervals. J. Comput. Appl. Math. 141, 27–34 (2002)
Hao, Z.C., Liang, J., Xiao, T.J.: Singular boundary value problem on infinite time scale. Discrete Dyn. Nat. Soc. (2006). doi:10.1155/DDNS/2006/71580
Author information
Authors and Affiliations
Corresponding author
Additional information
X. Zhao supported by National Natural Science Foundation of China (10671012) and the Doctoral Program Foundation of Education Ministry of China (20050007011).
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Zhao, X., Ge, W. Unbounded positive solutions for m-point time-scale boundary value problems on infinite intervals. J. Appl. Math. Comput. 33, 103–123 (2010). https://doi.org/10.1007/s12190-009-0276-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-009-0276-z