Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Nonlocal Four-Point Boundary Value Problem for the Singularly Perturbed Semilinear | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 570493 9 pages doi 2011 570493 Research Article Nonlocal Four-Point Boundary Value Problem for the Singularly Perturbed Semilinear Differential Equations Robert Vrabel Institute of Applied Informatics Automation and Mathematics Faculty of Materials Science and Technology Hajdoczyho 1 917 01 Trnava Slovakia Correspondence should be addressed to Robert Vrabel Received 21 April 2010 Revised 9 September 2010 Accepted 13 September 2010 Academic Editor Daniel Franco Copyright 2011 Robert Vrabel. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 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. 1. Motivation and Introduction We will consider the nonlocal four-point boundary value problem ey ky f t y t e a b k 0 0 e 1 yC - y a 0 y b - y d 0 a c d b. We focus our attention on the existence and asymptotic behavior of the solutions ye t for .
