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Eigenvalue Approach to Even Order System Periodic Boundary Value Problems

Published online by Cambridge University Press:  20 November 2018

Qingkai Kong  and
Min Wang
Qingkai Kong
Affiliation:
Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USA e-mail: kong@math.niu.edu mwang@math.niu.edu
Min Wang
Affiliation:
Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USA e-mail: kong@math.niu.edu mwang@math.niu.edu
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Abstract

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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.

Keywords

34B1834B24Green's functionhigh order system boundary value problemspositive solutionsSturm-Liouville problem.

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 1 , 01 March 2013 , pp. 102 - 115
Copyright
Copyright © Canadian Mathematical Society 2013

References

[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. http://dx.doi.org/10.1016/S0377-0427(00)00438-6 Google Scholar
[2] Deimling, K., Nonlinear Functional Analysis. Springer-Verlag, Berlin, 1985.Google Scholar
[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. http://dx.doi.org/10.1090/S0002-9939-06-08368-7 Google Scholar
[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. http://dx.doi.org/10.1016/S0895-7177(00)00150-3 Google Scholar
[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. http://dx.doi.org/10.1016/j.na.2007.11.040 Google Scholar
[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. http://dx.doi.org/10.1017/S0013091507000788 Google Scholar
[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. http://dx.doi.org/10.1016/j.jde.2008.06.012 Google Scholar
[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. http://dx.doi.org/10.1016/j.jmaa.2005.04.049 Google Scholar
[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] 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. http://dx.doi.org/10.1090/S0002-9939-04-07647-6 Google Scholar
[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. http://dx.doi.org/10.1016/S0022-247X(03)00493-1 Google Scholar
[12] Kong, L. and Kong, Q., Positive solutions of higher-order boundary-value problems. Proc. Edinb. Math. Soc. 48 (2005), no. 2, 445–464. http://dx.doi.org/10.1017/S0013091504000860 Google Scholar
[13] Kong, L. and Kong, Q., Second-order boundary value problems with nonhomogeneous boundary conditions. Math. Nachr. 278 (2005), no. 1-2, 173–193. http://dx.doi.org/10.1002/mana.200410234 Google Scholar
[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. http://dx.doi.org/10.1016/j.na.2004.03.033 Google Scholar
[15] Kong, Q., Existence and nonexistence of solutions of second-order nonlinear boundary value problems. Nonlinear Anal. 66 (2007), no. 11, 2635–2651. http://dx.doi.org/10.1016/j.na.2006.03.045 Google Scholar
[16] Kong, Q. and Wang, M., Positive solutions of even order periodic boundary value problems. Rocky Mountain J. Math., to appear.Google Scholar
[17] Kong, Q. and Wang, , Positive solutions of even order system periodic boundary value problems. Nonlinear Anal. 72 (2010), no. 3-4, 1778–1791. http://dx.doi.org/10.1016/j.na.2009.09.019 Google Scholar
[18] Kwong, M. K., The topological nature of Krasnoselskii's cone fixed point theorem. Nonlinear Anal. 69 (2008), no. 3, 891–897. http://dx.doi.org/10.1016/j.na.2008.02.060 Google Scholar
[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. http://dx.doi.org/10.1016/S0096-3003(03)00733-1 Google Scholar
[20] Li, Y., Positive solutions of fourth-order periodic boundary value problems. Nonlinear Anal. 54 (2003), no. 6, 1069–1078. http://dx.doi.org/10.1016/S0362-546X(03)00127-5 Google Scholar
[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. http://dx.doi.org/10.1016/j.jmaa.2005.08.054 Google Scholar
[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. http://dx.doi.org/10.1016/j.jmaa.2006.09.025 Google Scholar
[23] Mawhin, J. and Willem, , Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, 74. Springer, New York, 1989.Google Scholar
[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. http://dx.doi.org/10.1007/s11117-005-0021-2 Google Scholar
[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. http://dx.doi.org/10.1006/jdeq.2000.3995 Google Scholar
[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. http://dx.doi.org/10.1016/S0022-0396(02)00152-3 Google Scholar
[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] Yao, Q.. Positive solutions of nonlinear second-order periodic boundary value problems. Applied Math Lett. 20 (2007), no. 5, 583–590. http://dx.doi.org/10.1016/j.aml.2006.08.003 Google Scholar
[29] Zeidler, E., Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems. Springer-Verlag, New York, 1986.Google Scholar
[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] 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