Original Paper
Positive periodic solutions and eigenvalue intervals for systems of second order differential equations
Jifeng Chu,
Hao Chen,
Donal O'Regan,
Jifeng Chu
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China
Search for more papers by this authorHao Chen
The Second Department, Army Command College, Nanjing 210045, P. R. China
Search for more papers by this authorDonal O'Regan
Department of Mathematics, National University of Ireland, Galway, Ireland
Search for more papers by this authorJifeng Chu,
Hao Chen,
Donal O'Regan,
Jifeng Chu
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China
Search for more papers by this authorHao Chen
The Second Department, Army Command College, Nanjing 210045, P. R. China
Search for more papers by this authorDonal O'Regan
Department of Mathematics, National University of Ireland, Galway, Ireland
Search for more papers by this authorAbstract
In this paper, we employ a well-known fixed point theorem for cones to study the existence of positive periodic solutions to the n -dimensional system x ″ + A (t)x = H (t)G (x). Moreover, the eigenvalue intervals for x ″ + A (t)x = λH (t)G (x) are easily characterized. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
References
- [1] R. P. Agarwal, H. Lü, and D. O'Regan, Eigenvalues and the one-dimensional p -Laplacian, J. Math. Anal. Appl. 266, 383–400 (2002).
- [2] R. P. Agarwal, M. Bohner, and P. J. Y. Wong, Positive solutions and eigenvalues of conjugate boundary-value problems, Proc. Edinb. Math. Soc. (2) 42, 349–374 (1999).
- [3] R. P. Agarwal, D. O'Regan, and P. J. Y. Wong, Constant-sign solutions of a system of Fredholm integral equations, Acta Appl. Math. 80, 57–94 (2004).
- [4] J. Chu, and D. Jiang, Eigenvalues and discrete boundary value problems for the one-dimensional p -Laplacian, J. Math. Anal. Appl. 305, 452–465 (2005).
- [5] J. Chu, D. O'Regan, and M. Zhang, Positive solutions and eigenvalue intervals for nonlinear systems, Proc. Indian Acad. Sci. Math. Sci. 117, 85–95 (2007).
- [6] L. Erbe, Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems, Math. Comput. Modelling 32, 529–539 (2000).
- [7] D. Franco, E. Liz, and P. J. Torres, Existence of periodic solutions for population models with periodic delay, Indian J. Pure Appl. Math. 38, 143–152 (2007).
- [8] D. Franco, G. Infante, and D. O'Regan, Nontrivial solutions in abstract cones for Hammerstein integral systems, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 14, 837–850 (2007).
- [9] J. Henderson, and H. Wang, Positive solutions for nonlinear eigenvalue problems, J. Math. Anal. Appl. 208, 252–259 (1997).
- [10] G. Infante, and J. R. L. Webb, Nonzero solutions of Hammerstein integral equations with discontinuous kernels, J. Math. Anal. Appl. 272, 30–42 (2002).
- [11] D. Jiang, J. Wei, and B. Zhang, Positive periodic solutions of functional differential equations and population models, Electron. J. Differential Equations 71, 1–13 (2002).
- [12] D. Jiang, J. Chu, D. O'Regan, and R. P. Agarwal, Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces, J. Math. Anal. Appl. 286, 563–576 (2003).
- [13] D. Jiang, J. Chu, and M. Zhang, Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. Differential Equations 211, 282–302 (2005).
- [14] M. A. Krasnosel'skii, Positive Solutions of Operator Equations (Noordhoff, Groningen, 1964).
- [15] K. Q. Lan, Multiple positive solutions of semilinear differential equations with singularities, J. London Math. Soc. (2) 63, 690–704 (2001).
- [16] K. Q. Lan, and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations 148, 407–421 (1998).
- [17] D. O'Regan, and H. Wang, Positive periodic solutions of systems of second ordinary differential equations, Positivity 10, 285–298 (2006).
- [18] P. J. Torres, and M. Zhang, A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. Nachr. 251, 101–107 (2003).
- [19] P. J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations 190, 643–662 (2003).
- [20] H. Wang, On the number of positive solutions of nonlinear systems, J. Math. Anal. Appl. 281, 287–306 (2003).
- [21] H. Wang, Positive periodic solutions of functional differential equations, J. Differential Equations 202, 354–366 (2004).
- [22] H. Wang, Y. Kuang, and M. Fan, Periodic solutions of systems of delay differential equations, preprint.
