Geodesic
Eigenvalue Approach to Even Order System Periodic Boundary Value Problems
Canadian mathematical bulletin, Tome 56 (2013) no. 1, pp. 102-115

Voir la notice de l'article provenant de la source Cambridge University Press

We study an even order system boundary value problem with periodic boundary conditions. By establishing the existence of a positive eigenvalue of an associated linear system Sturm-Liouville problem, we obtain new conditions for the boundary value problem to have a positive solution. Our major tools are the Krein-Rutman theorem for linear spectra and the fixed point index theory for compact operators.
DOI : 10.4153/CMB-2011-138-3
Mots-clés : 34B18, 34B24, Green's function, high order system boundary value problems, positive solutions, Sturm-Liouville problem. 
Kong, Qingkai; Wang, Min. Eigenvalue Approach to Even Order System Periodic Boundary Value Problems. Canadian mathematical bulletin, Tome 56 (2013) no. 1, pp. 102-115. doi: 10.4153/CMB-2011-138-3
@article{10_4153_CMB_2011_138_3,
     author = {Kong, Qingkai and Wang, Min},
     title = {Eigenvalue {Approach} to {Even} {Order} {System} {Periodic} {Boundary} {Value} {Problems}},
     journal = {Canadian mathematical bulletin},
     pages = {102--115},
     year = {2013},
     volume = {56},
     number = {1},
     doi = {10.4153/CMB-2011-138-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-138-3/}
}
TY  - JOUR
AU  - Kong, Qingkai
AU  - Wang, Min
TI  - Eigenvalue Approach to Even Order System Periodic Boundary Value Problems
JO  - Canadian mathematical bulletin
PY  - 2013
SP  - 102
EP  - 115
VL  - 56
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-138-3/
DO  - 10.4153/CMB-2011-138-3
ID  - 10_4153_CMB_2011_138_3
ER  - 
%0 Journal Article
%A Kong, Qingkai
%A Wang, Min
%T Eigenvalue Approach to Even Order System Periodic Boundary Value Problems
%J Canadian mathematical bulletin
%D 2013
%P 102-115
%V 56
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-138-3/
%R 10.4153/CMB-2011-138-3
%F 10_4153_CMB_2011_138_3

[1] [1] Atici, F. M. and Guseinov, , On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions,. J. Comput. Applied Math. 132 (2001), no. 2, 341–356. Google Scholar | DOI

[2] [2] Deimling, K., Nonlinear Functional Analysis. Springer-Verlag, Berlin, 1985. Google Scholar

[3] [3] Clark, S. and Henderson, J., Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations. Proc. Amer. Math. Soc. 134 (2006), 3363–3372. Google Scholar | DOI

[4] [4] Erbe, L., Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems. Boundary value problems and related topics. Math. Comput. Modelling 32 (2000), no. 5-6, 529–539. Google Scholar | DOI

[5] [5] Ge, and Xue, C., Some fixed point theorems and existence of positive solutions of two-point boundary-value problems. Nonlinear Anal. 70 (2009), no. 1, 16–31. Google Scholar | DOI

[6] [6] Graef, J. R. and Kong, L., Existence results for nonlinear periodic boundary-value problems. Proc. Edinb. Math. Soc. 52 (2009), no. 1, 79–95. Google Scholar | DOI

[7] [7] Graef, J. R., Kong, L., and H.Wang, Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. J. Differential Equations 245 (2008), no. 5, 1185–1197. Google Scholar | DOI

[8] [8] Graef, J. R. and Yang, B., Positive solutions to a multi-point higher order boundary value problem. J. Math. Anal. Appl. 316 (2006), no. 2, 409–421. Google Scholar | DOI

[9] [9] Guo, D. and Lakshmikantham, V., Nonlinear Problems in Abstract Cones. Notes and Reports in Mathematics in Science and Engineering 5. Academic Press, Boston, MA, 1988. Google Scholar

[10] [10] Henderson, J., Karna, B., and Tisdell, C. C., Existence of solutions for three-point boundary value problems for second order equations. Proc. Amer. Math. Soc. 133 (2005), no. 5, 1365–1369. Google Scholar | DOI

[11] [11] Jiang, D., Chua, J., O’Regan, D., and Agarwal, R., Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces. J. Math. Anal. Appl. 286 (2003), no. 2, 563–576. Google Scholar | DOI

[12] [12] Kong, L. and Kong, Q., Positive solutions of higher-order boundary-value problems. Proc. Edinb. Math. Soc. 48 (2005), no. 2, 445–464. Google Scholar | DOI

[13] [13] Kong, L. and Kong, Q., Second-order boundary value problems with nonhomogeneous boundary conditions. Math. Nachr. 278 (2005), no. 1-2, 173–193. Google Scholar | DOI

[14] [14] Kong, L. and Kong, Q., Multi-point boundary value problems of second-order differential equations. I. Nonlinear Anal. 58 (2004), no. 7-8, 909–931. Google Scholar | DOI

[15] [15] Kong, Q., Existence and nonexistence of solutions of second-order nonlinear boundary value problems. Nonlinear Anal. 66 (2007), no. 11, 2635–2651. Google Scholar | DOI

[16] [16] Kong, Q. and Wang, M., Positive solutions of even order periodic boundary value problems. Rocky Mountain J. Math., to appear. Google Scholar

[17] [17] Kong, Q. and Wang, , Positive solutions of even order system periodic boundary value problems. Nonlinear Anal. 72 (2010), no. 3-4, 1778–1791. Google Scholar | DOI

[18] [18] Kwong, M. K., The topological nature of Krasnoselskii's cone fixed point theorem. Nonlinear Anal. 69 (2008), no. 3, 891–897. Google Scholar | DOI

[19] [19] Lan, K. Q., Multiple positive solutions of Hammerstein integral equations and applications to periodic boundary value problems. Applied Math. Comp. 154 (2004), no. 2, 531–542. Google Scholar | DOI

[20] [20] Li, Y., Positive solutions of fourth-order periodic boundary value problems. Nonlinear Anal. 54 (2003), no. 6, 1069–1078. Google Scholar | DOI

[21] [21] Li, Y., Existence and uniqueness for higher order periodic boundary value problems under spectral separation conditions. J. Math. Anal. Appl. 322 (2006), no. 2, 530–539. Google Scholar | DOI

[22] [22] Li, F., Li, Y., and Liang, Z., Existence and multiplicity of solutions to2mth-order ordinary differential equations. J. Math. Anal. Appl. 331 (2007), no. 2, 958–977. Google Scholar | DOI

[23] [23] Mawhin, J. and Willem, , Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, 74. Springer, New York, 1989. Google Scholar

[24] [24] O’Regan, D. and Wang, H., Positive periodic solutions of systems of second order ordinary differential equations. Positivity 10 (2006), no. 2, 285—298. Google Scholar | DOI

[25] [25] Rach°unková, I., Tvrdý, , and Vrkoč, , Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J. Differential Equations 176 (2001), no. 2, 445–469. Google Scholar | DOI

[26] [26] Torres, P. J., Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differential Equations 190 (2003), no. 2, 643–662. Google Scholar | DOI

[27] [27] Wang, H., On the number of positive solutions of nonlinear systems. J. Math. Anal. Appl. 281 (2003), no. 1, 287–306. Google Scholar

[28] [28] Yao, Q.. Positive solutions of nonlinear second-order periodic boundary value problems. Applied Math Lett. 20 (2007), no. 5, 583–590. Google Scholar | DOI

[29] [29] Zeidler, E., Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems. Springer-Verlag, New York, 1986. Google Scholar

[30] [30] Zhang, Z. and Wang, J., On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations. J. Math. Anal. Appl. 281 (2003), no. 1, 99–107. Google Scholar

[31] [31] Zhao, F. and Wu, X., Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity. Nonlinear Anal. 60 (2005), no. 2, 325–335. Google Scholar

Cité par Sources :