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^{2}

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In this paper, we consider a parameterized singularly perturbed second order quasilinear boundary value problem. Asymptotic estimates for the solution and its first and second derivatives have been established. The theoretical estimates have been justified by concrete example.

In this paper, we are going to obtain the asymptotic bounds for the following parameterized singularly perturbed boundary value problem (BVP):

where

By a solution of (1.1), (1.2), we mean pair

An overview of some existence and uniqueness results and applications of parameterized equations may be obtained, for example, in [

Lemma 2.1. Let

satisfies the inequality

where

Proof. Under the above conditions, the operatör

Suppose

Now, for the barrier fonction

taking also into consideration that,

it follows that,

therefore

Remark 1. The inequality (2.3) yields.

Theorem 2.1. For

where

and

provided

Proof. We rewrite Equation (1.1) in form

where,

From (2.8) for the first derivate, we have

from which, after using the initial condition

Applying the mean value theorem for integrals, we deduce that,

and

Also, for first and second terms in right side of (2.10) for

It then follows from (2.11)-(2.13),

Further from (2.4) by taking

The inequlities (2.14), (2.15) immediately leads to (2.5), (2.6). After taking into consideration the uniformly boundnees in

which proves (2.7) for

from which after taking into consideration here

Next, differentiation (1.1) gives

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

where,

First and second derivatives have the form

Therefore, we observe here the accordance in our theoretical results described above.

MustafaKudu,IlhameAmirali, (2016) A Priori Estimates of Solution of Parametrized Singularly Perturbed Problem. Journal of Applied Mathematics and Physics,04,73-78. doi: 10.4236/jamp.2016.41011