- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 305279, 14 pages
A Generalization of Mahadevan's Version of the Krein-Rutman Theorem and Applications to p-Laplacian Boundary Value Problems
1Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266590, China
2Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China
Received 12 January 2012; Revised 24 March 2012; Accepted 13 July 2012
Academic Editor: Lishan Liu
Copyright © 2012 Yujun Cui and Jingxian Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- M. G. Kreĭn and M. A. Rutman, “Linear operators leaving invariant a cone in a Banach space,” Uspekhi Matematicheskikh Nauk, vol. 23, no. 1, pp. 3–95, 1948 (Russian), English translation: American Mathematical Society Translations, vol. 26, 1950.
- M. G. Kreĭn and M. A. Rutman, “Linear operators leaving invariant a cone in a Banach space,” American Mathematical Society Translations, vol. 10, pp. 199–325, 1962.
- M. A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, Translated from the Russian by R. E. Flaherty, edited by L. F. Boron, P. Noordhoff Ltd., Groningen, The Netherlands, 1964.
- M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, The Macmillan, New York, NY, USA, 1964.
- R. D. Nussbaum, “Eigenvectors of nonlinear positive operators and the linear Kreĭn-Rutman theorem,” in Fixed Point Theory, vol. 886 of Lecture Notes in Mathematics, pp. 309–330, Springer, Berlin, Germany, 1981.
- R. D. Nussbaum, “Eigenvectors of order-preserving linear operators,” Journal of the London Mathematical Society Second Series, vol. 58, no. 2, pp. 480–496, 1998.
- J. R. L. Webb, “Remarks on -positive operators,” Journal of Fixed Point Theory and Applications, vol. 5, no. 1, pp. 37–45, 2009.
- J. Mallet-Paret and R. D. Nussbaum, “Eigenvalues for a class of homogeneous cone maps arising from max-plus operators,” Discrete and Continuous Dynamical Systems Series A, vol. 8, no. 3, pp. 519–562, 2002.
- J. Mallet-Paret and R. D. Nussbaum, “Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index,” Journal of Fixed Point Theory and Applications, vol. 7, no. 1, pp. 103–143, 2010.
- R. Mahadevan, “A note on a non-linear Krein-Rutman theorem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 11, pp. 3084–3090, 2007.
- K. C. Chang, “A nonlinear Krein Rutman theorem,” Journal of Systems Science & Complexity, vol. 22, no. 4, pp. 542–554, 2009.
- K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
- D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.
- H. Amann, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,” SIAM Review, vol. 18, no. 4, pp. 620–709, 1976.
- S. Jingxian and L. Xiaoying, “Computation for topological degree and its applications,” Journal of Mathematical Analysis and Applications, vol. 202, no. 3, pp. 785–796, 1996.
- F.-H. Wong, “Existence of positive solutions for -Laplacian boundary value problems,” Applied Mathematics Letters, vol. 12, no. 3, pp. 11–17, 1999.
- B. Liu, “Positive solutions of singular three-point boundary value problems for the one-dimensional -Laplacian,” Computers & Mathematics with Applications, vol. 48, no. 5-6, pp. 913–925, 2004.