## Abstract and Applied Analysis

### A Generalization of Mahadevan's Version of the Krein-Rutman Theorem and Applications to p-Laplacian Boundary Value Problems

#### Abstract

We will present a generalization of Mahadevan’s version of theKrein-Rutman theorem for a compact, positively 1-homogeneous operator on aBanach space having the properties of being increasing with respect to a cone $P$ and such that there is a nonzero $u\in P\setminus \{\theta \}-P$ for which $M{T}^{p}u\ge u$ for somepositive constant $M$ and some positive integer p. Moreover, we give some newresults on the uniqueness of positive eigenvalue with positive eigenfunction andcomputation of the fixed point index. As applications, the existence of positivesolutions for p-Laplacian boundary-value problems is considered under someconditions concerning the positive eigenvalues corresponding to the relevantpositively 1-homogeneous operators.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 305279, 14 pages.

Dates
First available in Project Euclid: 4 April 2013

https://projecteuclid.org/euclid.aaa/1365099926

Digital Object Identifier
doi:10.1155/2012/305279

Mathematical Reviews number (MathSciNet)
MR2965454

Zentralblatt MATH identifier
1252.47045

#### Citation

Cui, Yujun; Sun, Jingxian. A Generalization of Mahadevan's Version of the Krein-Rutman Theorem and Applications to p -Laplacian Boundary Value Problems. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 305279, 14 pages. doi:10.1155/2012/305279. https://projecteuclid.org/euclid.aaa/1365099926

#### References

• 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 ${u}_{0}$-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 $m$-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 $p$-Laplacian,” Computers & Mathematics with Applications, vol. 48, no. 5-6, pp. 913–925, 2004.