General Periodic Boundary Value Problem for Systems

The paper deals with the existence of nonzero periodic solution of systems, where k∈(0, π/T), α, β are n×n real nonsingular matrices, μ=(μ1…μn), f(t, u)=(f1(t, u),…,fn(t, u))∈C([0, T]×□n+,□+) is periodic of period T in the t variable are continuous and nonnegative functions. We determine the Green’s function and prove that the existence of nonzero periodic positive solutions if one of . In addition, if all i=（1…n）where λ1 is the principle eigenvalues of the corresponding linear systems. The proof based on the fixed point index theorem in cones. Application of our result is given to such systems with specific nonlinearities.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

M. Elnagi, "General Periodic Boundary Value Problem for Systems," Applied Mathematics, Vol. 3 No. 8, 2012, pp. 882-887. doi: 10.4236/am.2012.38130.

 [1] L. H. Erbe and P. K. Palamides, “Boundary Value Problems for Second-Order Differential Systems,” Journal of Mathematical Analysis and Applications, Vol. 127, No. 1, 1987, pp. 80-92. doi:10.1016/0022-247X(87)90141-7 [2] J. W. Bebernes and K. Schmitt, “Periodic Boundary Value Problems for Systems of Second Order Differential Equations,” Journal of Differential Equations, Vol. 13, No. 1, 1973, pp. 32-47. doi:10.1016/0022-0396(73)90030-2 [3] L. H. Erbe and K. Schmitt, “Boundary Value Problems for Second-Order Differential Systems,” In: V. Lakshmikantham, Ed., Nonlinear Analysis and Applications, New York and Basel, 1987, pp. 179-183. [4] H. Wang, “Periodic Solutions to Non-Autonomous Second-Order Systems,” Nonlinear Analysis, Vol. 71, No. 3-4, 2009, pp. 1271-1275. doi:10.1016/j.na.2008.11.079 [5] D. Franco and J. R. L. Webb, “Collisionless Orbits of Singular and Nonsingular Dynamical Systems,” Discrete and Continuous Dynamical Systems, Vol. 15, No. 3, 2006, pp. 747-757. doi:10.3934/dcds.2006.15.747 [6] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, “Constant-Sign Solutions of a System of Fredholm Integral Equations,” Acta Applicandae Mathematicae, Vol. 80, No. 1, 2004, pp. 57-94. doi:10.1023/B:ACAP.0000013257.42126.ca [7] D. ORegan and H. Wang, “Positive Periodic Solutions of Systems of Second Order Ordinary Differential Equations,” Positivity, Vol. 10, No. 2, 2006, pp. 285-298. doi:10.1007/s11117-005-0021-2 [8] X. Lin, D. Jiang, D. ORegan and R. Agarwal, “Twin Positive Periodic Solutions of Second Order Singular Differential Systems,” Topological Methods in Nonlinear Analysis, Vol. 25, 2005, pp. 263-273. [9] L. H. Erbe and K. Schmitt, “On Solvability of Boundary Value Problems for Systems of Differential Equations,” Journal of Applied Mathematics and Physics, Vol. 38, 1987. [10] H. Wang, “Positive Periodic Solutions of Singular Systems with a Parameter,” Journal of Differential Equations, Vol. 249, No. 12, 2010, pp. 2986-3002. doi:10.1016/j.jde.2010.08.027 [11] X. Li and Z. Zhang, “On the Existence of Positive Periodic Solutions of Systems of Second Order Differential Equations,” Mathematische Nachrichten, Vol. 284, No. 11-12, 2011, pp. 1472-1482. doi:10.1002/mana.200710145 [12] Z. Cao and D. Jiang, “Periodic Solutions of Second Order Singular Coupled Systems,” Nonlinear Analysis, Vol. 71, No. 9, 2009, pp. 3661-3667. doi:10.1016/j.na.2009.02.053 [13] J. R. Graef, L. Kong and H. Wang, “A Periodic Boundary Value Problem with Vanishing Greens Function,” Applied Mathematics Letters, Vol. 21, No. 2, 2008, pp. 176-180. doi:10.1016/j.aml.2007.02.019 [14] I. Rachunkova, M. Tvrdy and I. Vrkoc, “Existence of Nonnegative and Nonpositive Solutions for Second Order Periodic Boundary Value Problems,” Journal of Differential Equations, Vol. 176, No. 2, 2001, pp. 445-469. doi:10.1006/jdeq.2000.3995 [15] P. Torres, “Existence of One-Signed Periodic Solutions of Some Second-Order Differential Equations via a Krasnoselskii’s Fixed Point Theorem,” Journal of Differential Equations, Vol. 190, No. 2, 2003, pp. 643-662. doi:10.1016/S0022-0396(02)00152-3 [16] F. Li and Z. Liang, “Existence of Positive Periodic Solutions to Nonlinear Second Order Differential Equations,” Applied Mathematics Letters, Vol. 18, No. 11, 2005, pp. 1256-1264. doi:10.1016/j.aml.2005.02.014 [17] D. Jiang, J. Chu, D. ORegan, R. Agarwal, “Multiple Positive Solutions to Superlinear Periodic Boundary Value Prob lem, with Repulsive Singular Forces,” Journal of Mathematical Analysis and Applications, Vol. 286, No. 2, 2003, pp. 563-576. doi:10.1016/S0022-247X(03)00493-1 [18] X. Li and Z. Zhang, “Periodic Solutions for Second-Order Differential Equations with a Singular Nonlinearity,” Nonlinear Analysis, Vol. 69, No. 11, 2008, pp. 3866-3876. doi:10.1016/j.na.2007.10.023 [19] D. Jiang, J. Chu and M. Zhang, “Multiplicity of Positive Periodic Solutions to Superlinear Repulsive Singular Equations,” Journal of Differential Equations, Vol. 211, No. 2, 2005, pp. 282-302. doi:10.1016/j.jde.2004.10.031 [20] P. J. Torres and M. Zhang, “A Monotone Iterative Scheme for a Nonlinear Second Order Equation Based on a Geeralized Anti-Maximum Principle,” Mathematische Nachrichten, Vol. 251, No. 1, 2003, pp. 101-107. doi:10.1002/mana.200310033 [21] R. Ma, “Nonlinear Periodic Boundary Value Problems with Sign-Changing Green’s Function,” Nonlinear Analysis, Vol. 74, No. 5, 2011, pp. 1714-1720. doi:10.1016/j.na.2010.10.043 [22] B. Liu, L. Liu and Y. Wu, “Existence of Nontrivial Periodic Solutions for a Nonlinear Second Order Periodic Boundary Value Problem,” Nonlinear Analysis, Vol. 72, No. 7-8, 2010, pp. 3337-3345. doi:10.1016/j.na.2009.12.014 [23] H. Amann, “Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces,” SIAM Review, Vol. 18, No. 4, 1976, pp. 620-709. [24] K. Q. Lan, “Nonzero Positive Solutions of Systems of Elliptic Boundary Value Problems,” AMS, Vol. 139, No. 12, 2011, pp. 4343-4349. doi:10.1090/S0002-9939-2011-10840-2 [25] D. D. Hai and H. Wang, “Nontrivial Solutions for p-Laplacian Systems,” Journal of Mathematical Analysis and Applications, Vol. 330, No. 1, 2007, pp. 186-194. doi:10.1016/j.jmaa.2006.07.072 [26] K. Q. Lan and W. Lin, “Multiple Positive Solutions of Systems of Hammerstein Integral Equations with Applications to Fractional Differential Equations,” Journal London Mathematical Society, Vol. 83 No. 2, 2011, pp. 449-469. doi:10.1112/jlms/jdq090 [27] R. D. Nussbaum, “Eigenvectors of Nonlinear Positive Operators and the Linear Krein-Rutman Theorem, in Fixed Point Theory,” In: E. Fadell and G. Fournier, Eds., Lecture Notes in Math, Vol. 886, Springer, 1981, pp. 309-330.