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


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

We focus our attention on the existence and asymptotic behavior of the solutions for singularly perturbed boundary value problem (1.1), (1.2) and on an estimate of the difference between and a solution of the reduced equation when a small parameter 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 and maintain a temperature according to the temperature registered by the sensors at and respectively. In this case, we consider a uniform bar of length with nonuniform temperature lying on the -axis from to The parameter 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 ( ) of from to which is the so-called boundary layer phenomenon, and after decay of this transient they remain close to with an arising new fast transient of from to ( ). Boundary thermal layers are formed due to the nonuniform convergence of the exact solution to the solution of a reduced problem in the neighborhood of the ends and of the bar.

The differential equations of the form (1.1) have also been discussed in [8] but with the boundary conditions , that is, with free end Moreover, we show that the convergence rate of solutions toward the solution of a reduced problem is at least on every compact subset of (in [8], the rate of convergence is only of the order ). We will write when

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 of differential equation , , (i.e., is convex) such that and for and 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 , for two-endpoint boundary conditions, (see, e.g., [10]). We will define the correction function 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 , where the constants are not simultaneously equal to and

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 of the reduced problem (i.e., if we let in (1.1)) on every compact subset of interval As usual, we say that is a lower solution for problem (1.1), (1.2) if and , for every An upper solution satisfies and , for every

Lemma 1.1 (see [15]).

If , are respectively lower and upper solutions for (1.1), (1.2) such that then there exists solution of (1.1), (1.2) with

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

Lemma 1.2 (cf. [16, Theorem (Peano's phenomenon)]).

Assume that

(i)the function

is nondecreasing with respect to the variable for each


If are two solutions of (1.1), (1.2), then

(a) in

(b)if , then for each , the function is a solution of the problem (1.1), (1.2).

Lemma 1.3.

If satisfies the strengthened condition (i)

( )the function is increasing with respect to the variable for each

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


Assume to the contrary that are two solutions of the problem (1.1), (1.2). Lemma 1.2 implies that on for some constant Thus

This is a contradiction.

The following assumptions will be made throughout the paper.

(A1)For a reduced problem , there exists function such that on

Denote where is the positive continuous function on such that
(1.5) is a small positive constant.

(A2)The function satisfies the condition

2. Main Result

Theorem 2.1.

Under the assumptions (A1) and (A2) there exists such that for every the problem (1.1), (1.2) has in a unique solution, satisfying the inequality
on where
(2.2) and the positive function
(2.3) for and

Remark 2.2.

The function satisfies the following:

(1) ;

(2) , ;

(3) is decreasing (increasing) for and increasing (decreasing) for if ;

(4) converges uniformly to for on every compact subset of ;

(5) where for and for

The function satisfies the following:

(1) ;

(2) , ;

(3) is decreasing for and increasing for ;

(4) converges uniformly to for on every compact subset of ;

(5) where for and for

The correction function will be determined precisely in the next section.

3. The Correction Function

Consider the linear problem

with the boundary conditions (1.2).

We apply the method of lower and upper solutions. We define
Obviously, and the constant functions , 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 the unique solution of linear problem (3.1), (1.2) satisfies
on The solution we denote by , that is, the function
and we compute exactly as following:
Thus, we obtain

Hence, taking into consideration (3.8) and the fact that the correction function converges uniformly to on for

4. Proof of Theorem 2.1

First we will consider the case . We define the lower solutions by
and the upper solutions by

Here where is the constant which shall be defined below, on and satisfy the boundary conditions prescribed for the lower and upper solutions of (1.1), (1.2).

Now we show that and

Denote By the Taylor theorem, we obtain
where is a point between and and for sufficiently small Hence, from the inequalities in we have
Because we have ; as follows from differential equation (3.1), we get
For we have the inequality

where is a point between and and for sufficiently small

The Case:

The lower solution
and the upper solution

Now, if we choose a constant such that , , then and in

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 follows from Lemma 1.3.

Remark 4.1.

Theorem 2.1 implies that on every compact subset of and The boundary layer effect occurs at the point or/and in the case when or/and



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

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


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© Robert Vrabel 2011

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