Skip to main content
  • Research Article
  • Open access
  • Published:

Nonlocal Four-Point Boundary Value Problem for the Singularly Perturbed Semilinear Differential Equations

Abstract

This paper deals with the existence and asymptotic behavior of the solutions to the singularly perturbed second-order nonlinear differential equations. For example, feedback control problems, such as the steady states of the thermostats, where the controllers add or remove heat, depending upon the temperature detected by the sensors in other places, can be interpreted with a second-order ordinary differential equation subject to a nonlocal four-point boundary condition. Singular perturbation problems arise in the heat transfer problems with large Peclet numbers. We show that the solutions of mathematical model, in general, start with fast transient which is the so-called boundary layer phenomenon, and after decay of this transient they remain close to the solution of reduced problem with an arising new fast transient at the end of considered interval. Our analysis relies on the method of lower and upper solutions.

Publisher note

To access the full article, please see PDF.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Robert Vrabel.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Vrabel, R. Nonlocal Four-Point Boundary Value Problem for the Singularly Perturbed Semilinear Differential Equations. Bound Value Probl 2011, 570493 (2011). https://doi.org/10.1186/1687-2770-2011-570493

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2011-570493

Keywords