Abstract
By using fixed-point index theory and a new fixed-point theorem in cones, some sufficient conditions for the existence of countably many positive solutions for singular third-order boundary value problem on the half-line are established, which are the complement of previously known results. The interesting point lies in the fact that the nonlinear term is allowed to depend on the first order derivative u′.
Similar content being viewed by others
References
Agarwal, R.P., O’Regan, D.: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht (2001)
Aroson, D., Crandall, M.G., Peletier, L.A.: Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal. 6, 1001–1022 (1982)
Baxley, J.V.: Existence and uniqueness of nonlinear boundary value problems on infinite intervals. J. Math. Anal. Appl. 147, 122–133 (1990)
Callegari, A., Nachman, A.: A nonlinear singular boundary value problem in the theory of pseudoplastic fluids. SIAM J. Appl. Math. 38, 275–282 (1980)
Fermi, E.: Un methodo statistico par la determinazione di alcune proprietá dell’ atoma. Rend. Accad. Naz. Lincei. Cl. Sci. Fis., Mat. Natur. 6, 602–607 (1927)
Gatica, J., Hernandez, G., Waltman, P.: Radially symmetric solutions of a class of singular elliptic equations. Proc. Edinb. Math. Soc. 33, 169–180 (1990)
Greguš, M.: On a special boundary value problem. Acta Math. Univ. Comen. 40, 161–168 (1982)
Iffland, G.: Positive solutions of a problem Emden-Flower type with a type free boundary. SIAM J. Math. Anal. 18, 283–292 (1987)
Kaufmann, E.R., Kosmatov, N.: A multiplicity result for a boundary value problem with infinitely many singularities. J. Math. Anal. Appl. 269, 444–453 (2002)
Liu, B.F., Zhang, J.H.: The existence of positive solutions for some nonlinear boundary value problems with linear mixed boundary conditions. J. Math. Anal. Appl. 309, 505–516 (2005)
Liang, S.H., Zhang, J.H.: The existence of countably many positive solutions for nonlinear singular m-point boundary value problems. J. Comput. Appl. Math. 214, 78–89 (2008)
Liang, S.H., Zhang, J.H.: The existence of countably many positive solutions for nonlinear singular m-point boundary value problems on the half-line. J. Comput. Appl. Math. 222, 229–243 (2008)
Liu, B.: Positive solutions three-points boundary value problems for one-dimensional p-Laplacian with infinitely many singularities. Appl. Math. Lett. 17, 655–661 (2004)
Lian, H., Ge, W., Pang, H.: Triple positive solutions for boundary value problems on infinite intervals. Nonlinear Anal. 67, 2199–2207 (2007)
Liu, Y.S.: Existence and unboundedness of positive solutions for singular boundary value problems on half-line. Appl. Math. Comput. 1404, 543–556 (2003)
Na, T.Y.: Computational Methods in Engineering Boundary Value Problems. Academic Press, San Diego (1979)
Ren, J.L., Ge, W.G., Ren, B.X.: Existence of positive solutions for quasi-linear boundary value problems. Acta, Math. Appl. Sinica 21(3), 353–358 (2005) (in Chinese)
Thomas, L.H.: The calculation of atomic fields. Proc. Camb. Philos. Soc. 23, 542–548 (1927)
Wang, Y.Y., Hou, C.: Existence of multiple positive solutions for one-dimensional p-Laplacian. J. Math. Anal. Appl. 315, 144–153 (2006)
Yan, B.Q.: Multiple unbounded solutions of boundary value problems for second-order differential equations on the half-line. Nonlinear Anal. 51, 1031–1044 (2002)
Zima, M.: On positive solution of boundary value problems on the half-line. J. Math. Anal. Appl. 259, 127–136 (2001)
Deimling, K.: Nonlinear Functional Analysis. Springer, New York (1985)
Liu, Y.Sh.: Boundary value problem for second order differential equations on unbounded domain. Acta Anal. Funct. Appl. 4(3), 211–216 (2002) (in Chinese)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liang, S., Zhang, J. Positive Solutions for Singular Third-Order Boundary Value Problem with Dependence on the First Order Derivative on the Half-Line. Acta Appl Math 111, 27–43 (2010). https://doi.org/10.1007/s10440-009-9528-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-009-9528-z