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

Boundary Value Problems20102011:570493

DOI: 10.1186/1687-2770-2011-570493

Received: 21 April 2010

Accepted: 13 September 2010

Published: 19 September 2010

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.

1. Motivation and Introduction

We will consider the nonlocal four-point boundary value problem
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ1_HTML.gif
(1.1)
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ2_HTML.gif
(1.2)

We focus our attention on the existence and asymptotic behavior of the solutions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq1_HTML.gif for singularly perturbed boundary value problem (1.1), (1.2) and on an estimate of the difference between http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq2_HTML.gif and a solution http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq3_HTML.gif of the reduced equation http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq4_HTML.gif when a small parameter http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq5_HTML.gif tends to zero.

Singularly perturbed systems (SPS) normally occur due to the presence of small "parasitic" parameters, armature inductance in a common model for most DC motors, small time constants, and so forth. The literature on control of nonlinear SPS is extensive, at least starting with the pioneering work of Kokotović et al. nearly 30 years ago [1] and continuing to the present including authors such as Artstein [2, 3], Gaitsgory et al. [46].

Such boundary value problems can also arise in the study of the steady-states of a heated bar with the thermostats, where the controllers at http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq6_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq7_HTML.gif maintain a temperature according to the temperature registered by the sensors at http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq9_HTML.gif respectively. In this case, we consider a uniform bar of length http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq10_HTML.gif with nonuniform temperature lying on the http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq11_HTML.gif -axis from http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq12_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq13_HTML.gif The parameter http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq14_HTML.gif represents the thermal diffusivity. Thus, the singular perturbation problems are of common occurrence in modeling the heat-transport problems with large Peclet number [7].

We show that the solutions of (1.1), (1.2), in general, start with fast transient ( http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq15_HTML.gif ) of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq16_HTML.gif from http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq17_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq18_HTML.gif which is the so-called boundary layer phenomenon, and after decay of this transient they remain close to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq19_HTML.gif with an arising new fast transient of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq20_HTML.gif from http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq21_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq22_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq23_HTML.gif ). Boundary thermal layers are formed due to the nonuniform convergence of the exact solution http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq24_HTML.gif to the solution http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq25_HTML.gif of a reduced problem in the neighborhood of the ends http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq26_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq27_HTML.gif of the bar.

The differential equations of the form (1.1) have also been discussed in [8] but with the boundary conditions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq28_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq29_HTML.gif that is, with free end http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq30_HTML.gif Moreover, we show that the convergence rate of solutions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq31_HTML.gif toward the solution http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq32_HTML.gif of a reduced problem is at least http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq33_HTML.gif on every compact subset of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq34_HTML.gif (in [8], the rate of convergence is only of the order http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq35_HTML.gif ). We will write http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq36_HTML.gif when http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq37_HTML.gif

The situation in the case of nonlocal boundary value problem is complicated by the fact that there are the inner points in the boundary conditions, in contrast to the "standard" boundary conditions as the Dirichlet problem, Neumann problem, Robin problem, periodic boundary value problem [912], for example. In the problem considered; there is not positive solution http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq38_HTML.gif of differential equation http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq39_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq40_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq41_HTML.gif (i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq42_HTML.gif is convex) such that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq44_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq45_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq46_HTML.gif which could be used to solve this problem by the method of lower and upper solutions. The application of convex functions is essential for composing the appropriate barrier functions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq47_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq48_HTML.gif for two-endpoint boundary conditions, (see, e.g., [10]). We will define the correction function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq49_HTML.gif which will allow us to apply the method.

In the past few years the multipoint boundary value problem has received a wide attention (see, e.g., [13, 14]) and the references therein. For example, Khan [14] have studied a four-point boundary value problem of type http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq50_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq51_HTML.gif where the constants http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq52_HTML.gif http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq53_HTML.gif are not simultaneously equal to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq55_HTML.gif

As was said before, we apply the method of lower and upper solutions to prove the existence of a solution for problem (1.1), (1.2) which converges uniformly to the solution http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq56_HTML.gif of the reduced problem (i.e., if we let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq57_HTML.gif in (1.1)) on every compact subset of interval http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq58_HTML.gif As usual, we say that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq59_HTML.gif is a lower solution for problem (1.1), (1.2) if http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq62_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq63_HTML.gif An upper solution http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq64_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq66_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq67_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq68_HTML.gif

Lemma 1.1 (see [15]).

If http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq69_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq70_HTML.gif are respectively lower and upper solutions for (1.1), (1.2) such that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq71_HTML.gif then there exists solution http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq72_HTML.gif of (1.1), (1.2) with http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq73_HTML.gif

Proof of uniqueness of solution for (1.1), (1.2) will be based on the following lemmas.

Lemma 1.2 (cf. [16, Theorem http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq74_HTML.gif (Peano's phenomenon)]).

Assume that

(i)the function
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ3_HTML.gif
(1.3)

is nondecreasing with respect to the variable http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq75_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq76_HTML.gif

(ii) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq77_HTML.gif

If http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq78_HTML.gif http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq79_HTML.gif are two solutions of (1.1), (1.2), then

(a) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq80_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq81_HTML.gif

(b)if http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq82_HTML.gif , then for each http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq83_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq84_HTML.gif the function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq85_HTML.gif is a solution of the problem (1.1), (1.2).

Lemma 1.3.

If http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq86_HTML.gif satisfies the strengthened condition (i)

( http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq202_HTML.gif )the function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq88_HTML.gif is increasing with respect to the variable http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq89_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq90_HTML.gif

then there exists at most one solution of (1.1), (1.2).

Proof.

Assume to the contrary that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq91_HTML.gif are two solutions of the problem (1.1), (1.2). Lemma 1.2 implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq92_HTML.gif on http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq93_HTML.gif for some constant http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq94_HTML.gif Thus
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ4_HTML.gif
(1.4)

This is a contradiction.

The following assumptions will be made throughout the paper.

(A1)For a reduced problem http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq95_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq96_HTML.gif function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq97_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq98_HTML.gif on http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq99_HTML.gif

Denote http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq100_HTML.gif where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq101_HTML.gif is the positive continuous function on http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq102_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ5_HTML.gif
(1.5)

http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq103_HTML.gif is a small positive constant.

(A2)The function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq104_HTML.gif satisfies the condition

http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ6_HTML.gif
(1.6)

2. Main Result

Theorem 2.1.

Under the assumptions (A1) and (A2) there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq105_HTML.gif such that for every http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq106_HTML.gif the problem (1.1), (1.2) has in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq107_HTML.gif a unique solution, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq108_HTML.gif satisfying the inequality
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ7_HTML.gif
(2.1)
on http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq109_HTML.gif where
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ8_HTML.gif
(2.2)
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq110_HTML.gif and the positive function
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ9_HTML.gif
(2.3)

http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq111_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq113_HTML.gif

Remark 2.2.

The function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq114_HTML.gif satisfies the following:

(1) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq115_HTML.gif ;

(2) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq116_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq117_HTML.gif ;

(3) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq118_HTML.gif is decreasing (increasing) for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq119_HTML.gif and increasing (decreasing) for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq120_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq121_HTML.gif http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq122_HTML.gif ;

(4) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq123_HTML.gif converges uniformly to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq124_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq125_HTML.gif on every compact subset of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq126_HTML.gif ;

(5) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq127_HTML.gif where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq128_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq129_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq130_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq131_HTML.gif

The function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq132_HTML.gif satisfies the following:

(1) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq133_HTML.gif ;

(2) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq134_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq135_HTML.gif ;

(3) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq136_HTML.gif is decreasing for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq137_HTML.gif and increasing for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq138_HTML.gif ;

(4) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq139_HTML.gif converges uniformly to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq140_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq141_HTML.gif on every compact subset of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq142_HTML.gif ;

(5) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq143_HTML.gif where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq144_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq145_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq146_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq147_HTML.gif

The correction function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq148_HTML.gif will be determined precisely in the next section.

3. The Correction Function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq149_HTML.gif

Consider the linear problem
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ10_HTML.gif
(3.1)

with the boundary conditions (1.2).

We apply the method of lower and upper solutions. We define
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ11_HTML.gif
(3.2)
Obviously, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq150_HTML.gif and the constant functions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq151_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq152_HTML.gif satisfy the differential and boundary inequalities required on the lower and upper solutions for (3.1) and the boundary conditions (1.2). Thus on the basis of Lemma 1.1 for every http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq153_HTML.gif the unique solution http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq154_HTML.gif of linear problem (3.1), (1.2) satisfies
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ12_HTML.gif
(3.3)
on http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq155_HTML.gif The solution we denote by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq156_HTML.gif , that is, the function
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ13_HTML.gif
(3.4)
and we compute http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq157_HTML.gif exactly as following:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ14_HTML.gif
(3.5)
where
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ15_HTML.gif
(3.6)
Hence
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ16_HTML.gif
(3.7)
Thus, we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ17_HTML.gif
(3.8)

Hence, taking into consideration (3.8) and the fact that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq158_HTML.gif the correction function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq159_HTML.gif converges uniformly to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq160_HTML.gif on http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq161_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq162_HTML.gif

4. Proof of Theorem 2.1

First we will consider the case http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq163_HTML.gif . We define the lower solutions by
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ18_HTML.gif
(4.1)
and the upper solutions by
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ19_HTML.gif
(4.2)

Here http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq164_HTML.gif where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq165_HTML.gif is the constant which shall be defined below, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq166_HTML.gif on http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq167_HTML.gif and satisfy the boundary conditions prescribed for the lower and upper solutions of (1.1), (1.2).

Now we show that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq168_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq169_HTML.gif

Denote http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq170_HTML.gif By the Taylor theorem, we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ20_HTML.gif
(4.3)
where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq171_HTML.gif is a point between http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq172_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq173_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq174_HTML.gif for sufficiently small http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq175_HTML.gif Hence, from the inequalities http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq176_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq177_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ21_HTML.gif
(4.4)
Because http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq178_HTML.gif we have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq179_HTML.gif ; as follows from differential equation (3.1), we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ22_HTML.gif
(4.5)
For http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq180_HTML.gif we have the inequality
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ23_HTML.gif
(4.6)

where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq181_HTML.gif is a point between http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq182_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq183_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq184_HTML.gif for sufficiently small http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq185_HTML.gif

The Case: http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq186_HTML.gif

The lower solution
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ24_HTML.gif
(4.7)
and the upper solution
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ25_HTML.gif
(4.8)
satisfy
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_Equ26_HTML.gif
(4.9)

Now, if we choose a constant http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq187_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq188_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq189_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq190_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq191_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq192_HTML.gif

The existence of a solution for (1.1), (1.2) satisfying the above inequality follows from Lemma 1.1 and the uniqueness of solution in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq193_HTML.gif follows from Lemma 1.3.

Remark 4.1.

Theorem 2.1 implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq194_HTML.gif on every compact subset of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq195_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq196_HTML.gif The boundary layer effect occurs at the point http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq197_HTML.gif or/and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq198_HTML.gif in the case when http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq199_HTML.gif or/and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-570493/MediaObjects/13661_2010_47_Article_IEq200_HTML.gif

Declarations

Acknowledgments

This research was supported by Slovak Grant Agency, Ministry of Education of Slovak Republic under Grant no. 1/0068/08. The author would like to thank the reviewers for helpful comments on an earlier draft of this article.

Authors’ Affiliations

(1)
Faculty of Materials Science and Technology, Institute of Applied Informatics, Automation and Mathematics

References

  1. Kokotović PV, Khalil HK, O'Reilly J: Singular Perturbation Methods in Control: Analysis and Design. Academic Press, London, UK; 1986:xii+371.MATH
  2. Artstein Z: Stability in the presence of singular perturbations.Nonlinear Analysis: Theory, Methods & Applications 1998,34(6):817–827.View ArticleMathSciNetMATH
  3. Artstein Z: Singularly perturbed ordinary differential equations with nonautonomous fast dynamics.Journal of Dynamics and Differential Equations 1999,11(2):297–318.View ArticleMathSciNetMATH
  4. Artstein Z, Gaitsgory V: The value function of singularly perturbed control systems.Applied Mathematics and Optimization 2000,41(3):425–445.View ArticleMathSciNetMATH
  5. Gaitsgory V: On a representation of the limit occupational measures set of a control system with applications to singularly perturbed control systems.SIAM Journal on Control and Optimization 2004,43(1):325–340.View ArticleMathSciNetMATH
  6. Gaitsgory V, Nguyen M-T: Multiscale singularly perturbed control systems: limit occupational measures sets and averaging.SIAM Journal on Control and Optimization 2002,41(3):954–974.View ArticleMathSciNetMATH
  7. Khan A, Khan I, Aziz T, Stojanovic M: A variable-mesh approximation method for singularly perturbed boundary-value problems using cubic spline in tension.International Journal of Computer Mathematics 2004,81(12):1513–1518.View ArticleMathSciNet
  8. Vrábel R: Three point boundary value problem for singularly perturbed semilinear differential equations.Electronic Journal of Qualitative Theory of Differential Equations 2009, 70:1–4.View Article
  9. De Coster C, Habets P: Two-Point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering. Volume 205. 1st edition. Elsevier, Amsterdam, The Netherlands; 2006:xii+489.
  10. Chang KW, Howes FA: Nonlinear Singular Perturbation Phenomena: Theory and Applications, Applied Mathematical Sciences. Volume 56. Springer, New York, NY, USA; 1984:viii+180.
  11. Vrábel R: Asymptotic behavior of -periodic solutions of singularly perturbed second-order differential equation.Mathematica Bohemica 1996,121(1):73–76.MathSciNetMATH
  12. Vrábel R: Semilinear singular perturbation.Nonlinear Analysis: Theory, Methods & Applications 1995,25(1):17–26.View ArticleMathSciNetMATH
  13. Guo Y, Ge W: Positive solutions for three-point boundary value problems with dependence on the first order derivative.Journal of Mathematical Analysis and Applications 2004,290(1):291–301.View ArticleMathSciNetMATH
  14. Khan RA: Positive solutions of four-point singular boundary value problems.Applied Mathematics and Computation 2008,201(1–2):762–773.View ArticleMathSciNetMATH
  15. Mawhin J: Points Fixes, Points Critiques et Problèmes aux Limites, Séminaire de Mathématiques Supérieures, no. 92. Presses de l'Université de Montréal, Montreal, Canada; 1985:162.
  16. Šeda V: On some nonlinear boundary value problems for ordinary differential equations.Archivum Mathematicum 1989,25(4):207–222.MathSciNetMATH

Copyright

© Robert Vrabel 2011

This article is published under license to BioMed Central Ltd. 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.