Rocky Mountain Journal of Mathematics

Positive, negative and mixed-type solutions for periodic vector differential equations

Xiao Han and Jinchao Ji

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This paper is devoted to the study of periodic solutions of the first-order vector differential equations $x'(t)+f(t,x(t))=0$. We first introduce the concepts of positive, negative and mixed-type solutions. Then, by using a fixed point theorem in cones, we obtain some existence and multiplicity results of such solutions. Furthermore, we also present some examples to illustrate our main results.

Article information

Rocky Mountain J. Math., Volume 44, Number 4 (2014), 1183-1201.

First available in Project Euclid: 31 October 2014

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Zentralblatt MATH identifier

Primary: 34C25: Periodic solutions

Existence multiplicity positive solutions negative solutions mixed-type solutions fixed point theorem in cones


Han, Xiao; Ji, Jinchao. Positive, negative and mixed-type solutions for periodic vector differential equations. Rocky Mountain J. Math. 44 (2014), no. 4, 1183--1201. doi:10.1216/RMJ-2014-44-4-1183.

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  • R.I. Avery, Existence of multiple positive solutions to a conjugate boundary value problem, Math. Sci. Res. 2 (1998), 1–6.
  • Y. Ding and C. Lee, Periodic solutions of an infinite dimensional Hamiltonian system, Rocky Mountain J. Math. 35 (2005), 1881–1908.
  • L.H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994), 743–748.
  • Z. Feng, X. Xu and S. Ji, Finding the periodic solution of differential equation via solving optimization problem, J. Optim. Theor. Appl. 143 (2009), 75–86.
  • G. Gentile and M. Procesi, Periodic solutions for a class of nonlinear partial differential equations in higher dimension, Comm. Math. Phys. 289 (2009), 863–906.
  • D. Guo and V. Lakshmikantham, Nonlinear problems in abstract cones, Academic Press, New York, 1988.
  • Y.P. Guo and J.W. Tian, Two positive solutions for second-order quasilinear differential equation boundary value problems with sign-changing nonlinearities, J. Comp. Appl. Math. 169 (2004), 345–357.
  • J.R. Haddock and M.N. Nkashama, Periodic boundary value problems and monotone iterative methods for functional differential equations, Nonlin. Anal. 22 (1994), 267–276.
  • X. Han, S. Ji and Z. Ma, On the existence and multiplicity of positive periodic solutions for first-order vector differential equation, J. Math. Anal. Appl. 329 (2007), 977–986.
  • S. Ji, Time periodic solutions to a nonlinear wave equation with $x$-dependent coefficients, Calc. Var. Part. Diff. Eq. 32 (2008), 137–153.
  • ––––, Periodic solutions on the Sturm-Liouville boundary value problem for two-dimensional wave equation, J. Math. Phys. 50 (2009), 16 pages.
  • ––––, Time-periodic solutions to a nonlinear wave equation with periodic or anti-periodic boundary conditions, Proc. Roy. Soc. Lond. 465 (2009), 895–913.
  • S. Ji and Y. Li, Periodic solutions to one-dimensional wave equation with $x$-dependent coefficients, J. Diff. Eq. 229 (2006), 466–493.
  • ––––, Time-periodic solutions to the one-dimensional wave equation with periodic or anti-periodic boundary conditions, Proc. Roy. Soc. Edinb. 137 (2007), 349–371.
  • ––––, Time periodic solutions to the one-dimensional nonlinear wave equation, Arch. Rat. Mech. Anal. 199 (2011), 435–451.
  • S. Ji and S. Shi, Periodic solutions for a class of second-order ordinary differential equations, J. Optim. Theor. Appl. 130 (2006), 125–137.
  • V. Lakshmikantham, Periodic boundary value problems of first and second order differential equations, J. Appl. Math. Simul. 2 (1989), 131–138.
  • V. Lakshmikantham and S. Leela, Existence and monotone method for periodic solutions of first order differential equations, J. Math. Anal. Appl. 91 (1983), 237–243.
  • ––––, Remarks on first and second order periodic boundary value problems, Nonlin. Anal. 8 (1984), 281–287.
  • S. Leela and M.N. Oğuztöreli, Periodic boundary value problem for differential equations with delay and monotone iterative method, J. Math. Anal. Appl. 122 (1987), 301–307.
  • E. Liz and J.J. Nieto, Periodic boundary value problems for a class of functional differential equations, J. Math. Anal. Appl. 200 (1996), 680–686.
  • S.G. Peng, Positive solutions for first order periodic boundary value problem, Appl. Math. Comp. 158 (2004), 345–351.