The boundary value problem of higher order differential equations with delay

Zhimin He 1  and Jianhua Shen 2
  • 1 College of Science, Zhejiang Forestry University, Hangzhou, Zhejiang 311300, P.R. China
  • 2 Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, P.R. China

Abstract

In the paper, Guo–Krasnoselskii’s fixed point theorem is adapted to study the existence of positive solutions to a class of boundary value problems for higher order differential equations with delay. The sufficient conditions, which assure that the equation has one positive solution or two positive solutions, are derived. These conclusions generalize some existing ones.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] J. W. Lee, D. O’Regan, Existence results for differential delay equations-I, J. Differential Equations 102 (1993), 342–359.

  • [2] A. Carvalho, L. A. Ladeira, M. Martelli, Forbidden periods in delay differential equation, Portugal. Math. 57(3) (2000), 259–271.

  • [3] J. K. Hale, W. Huang, Global geometry of stable regions for two delay differential equations, J. Math. Anal. Appl. 178 (1993), 344–362.

  • [4] Y. Li, Y. Kuang, Periodic solutions in periodic state-dependent delay differential equations and population models, J. Math. Anal. Appl. 255 (2001), 265–280.

  • [5] D. Q. Jiang, Multiple positive solutions for boundary value problems of second-order delay differential equations, Appl. Math. Lett. 15 (2002), 575–583.

  • [6] D. Bai, Y. Xu, Existence of positive solutions for boundary-value problems of second-order delay differential equations, Appl. Math. Lett. 18 (2005), 621–630.

  • [7] W. B. Wang, J. H. Shen, Positive solutions to a multi-point boundary value problem with delay, Appl. Math. Comput. 188 (2007), 96–102.

  • [8] T. Jankowski, Solvability of three point boundary value problems for second order ordinary differential equations with deviating arguments, J. Math. Anal. Appl. 312 (2005), 620–636.

  • [9] B. Du, X. P. Hu, W. G. Ge, Positive solutions to a type of multi-point boundary value problem with delay and one-dimensional p-Laplacian, Appl. Math. Comput. 208 (2009), 501–510.

  • [10] J. R. Graef, B. Yang, Positive solutions to a multi-point higher order boundary-value problem, J. Math. Anal. Appl. 316 (2006), 409–421.

  • [11] J. H. Shen, J. Dong, Existence of positive solutions to BVPS of higher delay differential equations, Demonstratio Math. 42 (2009), 53–64.

  • [12] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988.

  • [13] J. Henderson, Boundary Value Problems for Functional Equations, World Scientific, 1995.

OPEN ACCESS

Journal + Issues

Demonstratio Mathematica, founded in 1969, is a fully peer-reviewed, open access journal that publishes original and significant research works and review articles devoted to functional analysis, approximation theory, and related topics. The journal provides the readers with free, instant, and permanent access to all content worldwide (all 53 volumes are available online!)

Search