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. [4–6].

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 [9–12], 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

is nondecreasing with respect to the variable
for each

(ii)

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).

Proof.

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

is a small positive constant.

(A2)The function
satisfies the condition