Received 10 September 2015; accepted 17 January 2016; published 20 January 2016
1. Introduction
In this paper, we are going to obtain the asymptotic bounds for the following parameterized singularly perturbed boundary value problem (BVP):
(1.1)
(1.2)
where 
is the perturbation parameter, 
are given constants and 
is a sufficiently smooth function in
. Further, the function 
is assumed to be sufficiently continuously differentiable for our purpose function in 
and
(1.3)
By a solution of (1.1), (1.2), we mean pair 
for which problem (1.1), (1.2) is satisfied.
An overview of some existence and uniqueness results and applications of parameterized equations may be obtained, for example, in [1] - [10] . In [11] - [14] have also been considered some approxi-mating aspects of this kind of problems. The qualitative analysis of singular perturbation situations has always been far from trivial because of the boundary layer behavior of the solution. In singular perturbation cases, problems depend on a small parameter 
in such a way that the solution exhibits a multiscale character, i.e., there are thin transition layers where the solution varies rapidly while away from layers it behaves regularly and varies slowly [15] -[18] . In this note, we establish the boundary layer behaviour for 
of the solution of (1.1)-(1.2) and its first and second derivatives. Example that agrees with the analytical results is given.
2. The Continuous Problem
Lemma 2.1. Let 
and 
be the continuous functions on
. Then, the solution of the boundary-value problem
(2.1)
(2.2)
satisfies the inequality
(2.3)
where
Proof. Under the above conditions, the operatör 
admits the folloving maximum principle:
Suppose 
be any function satisfiying 
, 
and
. Then, 
for all
.
Now, for the barrier fonction
taking also into consideration that, 
is a solution of the problem
it follows that,
therefore
, which immediayely leads to (2.3).
Remark 1. The inequality (2.3) yields.
(2.4)
Theorem 2.1. For 
and under conditions (1.3), the solution 
of the problem (1.1), (1.2), satisfies,
(2.5)
(2.6)
where
and
(2.7)
provided 
and 
for 
and
.
Proof. We rewrite Equation (1.1) in form
(2.8)
where, 
intermediate values.
From (2.8) for the first derivate, we have
(2.9)
from which, after using the initial condition
, it follows that,
(2.10)
Applying the mean value theorem for integrals, we deduce that,
(2.11)
and
(2.12)
Also, for first and second terms in right side of (2.10) for 
values, we have
(2.13)
It then follows from (2.11)-(2.13),
(2.14)
Further from (2.4) by taking 
we get
(2.15)
The inequlities (2.14), (2.15) immediately leads to (2.5), (2.6). After taking into consideration the uniformly boundnees in 
of 
and
, it then follows from (2.9) that,
which proves (2.7) for
. To obtain (2.7) for
, first from (1.1) we have
from which after taking into consideration here 
and (2.5) we obtain
(2.16)
Next, differentiation (1.1) gives
(2.17)
(2.18)
with
and due to our assumptions clearly,
Consequently, from (2.17), (2.18) we have
which proves (2.7) for
. □
Example. Consider the following parameterized singular perturbation problem:
with
and 
selected so that the solution is
where,
First and second derivatives have the form
Therefore, we observe here the accordance in our theoretical results described above.
[1] Feckan, M. (1994) Parametrized Singularly Perturbed Boundary Value Problems. Journal of Mathematical Analysis and Applications, 188, 426-435.
http://dx.doi.org/10.1006/jmaa.1994.1436
[2] Goma, I.A. (1977) Method of Successive Approximations in a Two-Point Boundary Problem with Parameter. Ukrainian Mathematical Journal, 29, 594-599.
http://dx.doi.org/10.1007/BF01085968
[3] Jankowski, T. and Lakshmikantham, V. (1997) Monotone Iterations for Differential Equations with a Parameter. Journal of Applied Mathematics and Stochastic Analysis, 10, 273-278.
http://dx.doi.org/10.1155/S1048953397000348
[4] Jankowski, T. (1999) Generalization of the Method of Quasilinearization for Differential Problems with a Parameter. Dynamic Systems and Applications, 8, 53-72.
[5] Jankowski, T. (2001) Monotone Iterations for Differential Problems. Mathematical Notes (Miskolc), 2, 31-38.
[6] Pomentale, T. (1976) A Constructive Theorem of Existence and Uniqueness for the Problem, , Zeitschrift für Angewandte Mathematik und Mechanik, 56, 387-388.
http://dx.doi.org/10.1002/zamm.19760560806
[7] Rontó, M. and Csikos-Marinets, T. (2000) On the Investigation of Some Non-Linear Boundary Value Problems with Parameters. Mathematical Notes (Miskolc), 1, 157-166.
[8] Rontó, M. (2000) On Non-Linear Boundary Value Problems Containing Parameters. Archivum Mathematicum, 36, 585-593.
[9] Staněk, S. (1997) Nonlinear Boundary Value Problem for Second Order Differential Equations Depending on a Parameter. Mathematica Slovaca, 47, 439-449.
[10] Zhang, P. (2011) Existence of Positive Solutions for Nonlocal Second-Order Boundary Value Problem with Variable Parameter in Banach Spaces. Fixed Point Theory and Applications, 43, 1687-1812.
http://dx.doi.org/10.1186/1687-1812-2011-43
[11] Amiraliyev, G.M., Kudu, M. and Duru, H. (2004) Finite-difference Method for Parameterized Singularly Perturbed Problem. Journal of Applied Mathematics, 3, 191-199.
[12] Amiraliyev, G.M., Kudu, M. and Duru, H. (2006) Uniform Difference Method for A Parameterized Singular Perturbation Problem. Applied Mathematics and Computation, 175, 89-100.
http://dx.doi.org/10.1016/j.amc.2005.07.068
[13] Amiraliyeva, I.G. and Amiraliyev, G.M. (2009) Uniform Difference Method for Parameterized Singularly Perturbed Delay Differential Equations. Numerical Algorithms, 52, 509-552.
http://dx.doi.org/10.1007/s11075-009-9295-y
[14] Turkyilmazoglu, M. (2011) Analytic Approximate Solutions of Parameterized Unperturbed and Singularly Perturbed Boundary Value Problems. Applied Mathematical Modelling, 35, 3879-3886.
http://dx.doi.org/10.1016/j.apm.2011.02.011
[15] Farrel, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E. and Shishkin, G.I. (2000) Robust Computational Techniques for Boundary Layers. Chapman Hall/CRC, New York.
[16] Nayfeh, A.H. (1981) Introduction to Perturbation Techniques. John Wiley & Sons, New York.
[17] O’Malley Jr., R.E. (1991) Singular Perturbation Methods for Ordinary Differential Equations. Springer-Verlag, New York.
[18] Roos, H.G., Stynes, M. and Tobiska, L. (2008) Robust Numerical Methods for Singularly Perturbed Differential Equations, Convection Diffusion and Flow Problems. Springer-Verlag, Berlin, Heidelberg.