Abstract
This paper is concerned with the existence of positive solutions of a multipoint boundary value problem for higher-order differential equation with one-dimensional p-Laplacian. Examples are presented to illustrate the main results. The result in this paper generalizes those in existing papers.
Similar content being viewed by others
References
R.P. Agarwal, D. O’Regan, P. J.Y. Wong: Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht, 1999.
C. Bai, J. Fang: Existence of multiple positive solutions for nonlinear m-point boundary value problems. Appl. Math. Comput. 140 (2003), 297–305.
C. Bai, J. Fang: Existence of multiple positive solutions for nonlinear multi-point boundary value problems. J. Math. Anal. Appl. 281 (2003), 76–85.
M.A. delPino, M. Elgueta, R. F. Manásevich: A homotopic deformation along p of a Leray-Schauder degree result and existence for (|u′|p−2u′)′ + f(t, u) = 0, u(0) = u(T) = 0, p > 1. J. Differ. Equ. 80 (1989), 1–13.
M.A. delPino, R.F. Manásevich: Multiple solutions for the p-Laplacian under global nonresonance. Proc. Am. Math. Soc. 112 (1991), 131–138.
R.E. Gaines, J. L. Mawhin: Coincidence Degree and Nonlinear Differential Equations. Lecture Notes in Math., Vol. 568. Springer, Berlin, 1977.
Y. Guo, W. Ge: Three positive solutions for the one dimension p-Laplacian. J. Math. Anal. Appl. 286 (2003), 491–508.
D. Ji, M. Feng, W. Ge: Multiple positive solutions for multipoint boundary value problems with sign changing nonlinearity. Appl. Math. Comput. 196 (2008), 511–520.
G. L. Karakostas: Triple positive solutions for the Φ-Laplacian when Φ is a supmultiplicative-like function. Electron. Diff. Equ. 69 (2004), 1–13.
G. L. Karakostas: Positive solutions for the Φ-Laplacian when Φ is a sup-multiplicative-like function. Electron. Diff. Equ. 68 (2004), 1–12.
N. Kosmatov: Symmetric solutions of a multi-point boundary value problem. J. Math. Anal. Appl. 309 (2005), 25–36.
K.D. Lan: Multiple positive solutions of semilinear differential equations with singularities. J. Lond. Math. Soc., Ser. II 63 (2001), 690–704.
W.-C. Lian, F. Wong: Existence of positive solutions for higher-order generalized p-Laplacian BVPs. Appl. Math. Lett. 13 (2000), 35–43.
R. Liang, J. Peng, J. Shen: Positive solutions to a generalized second order three-point boundary value problem. Appl. Math. Comput. 196 (2008), 931–940.
B. Liu: Positive solutions of a nonlinear four-point boundary value problems. Appl. Math. Comput. 155 (2004), 179–203.
Y. Liu, W. Ge: Multiple positive solutions to a three-point boundary value problem with p-Laplacian. J. Math. Anal. Appl. 277 (2003), 293–302.
H. Lü, D. O’Regan, C. Zhong: Multiple positive solutions for the one dimensional singular p-Laplacian. Appl. Math. Comput. 133 (2002), 407–422.
R. Ma: Existence of positive solutions for superlinear semipositone m-point boundary-value problems. Proc. Edinb. Math. Soc., II. Ser. 46 (2003), 279–292.
R. Ma: Multiple nonnegative solutions of second-order systems of boundary value problems. Nonlinear Anal., Theory Methods Appl. 42 (2000), 1003–1010.
R. Ma: Nodal solutions for a second-order m-point boundary value problem. Czech. Math. J. 56 (2006), 1243–1263.
R. Ma, N. Castaneda: Existence of solutions of nonlinear m-point boundary value problems. J. Math. Anal. Appl. 256 (2001), 556–567.
R. Ma, B. Thompson: Positive solutions for nonlinear m-point eigenvalue problems. J. Math. Anal. Appl. 297 (2004), 24–37.
Y. Wang, W. Zhao, W. Ge: Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional p-Laplacian. J. Math. Anal. Appl. 326 (2007), 641–654.
J.R. L. Webb: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear Anal., Theory Methods Appl. 47 (2001), 4319–4332.
G. Zhang, J. Sun: Positive solutions of m-point boundary value problems. J. Math. Anal. Appl. 291 (2004), 406–418.
Z. Zhang, J. Wang: On existence and multiplicity of positive solutions to singular multi-point boundary value problems. J. Math. Anal. Appl. 295 (2004), 502–512.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author is supported by the Science Foundation of Hunan Province (06JJ5008) and the Natural Sciences Foundation of Guangdong province (No. 7004569).
Rights and permissions
About this article
Cite this article
Liu, Y. A note on the existence of positive solutions of one-dimensional p-Laplacian boundary value problems. Appl Math 55, 241–264 (2010). https://doi.org/10.1007/s10492-010-0010-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10492-010-0010-z
Keywords
- one-dimension p-Laplacian differential equation
- nonlocal boundary value problem
- positive solution
- fixed-point theorem