1. Introduction

The history of spectral theory is the history of a beautiful and important area of mathematics with close links to physics and with a strong influence on the development of functional analysis. Its roots lies in three areas :1) discrete systems described by matrices (or quadratics forms) and continuous systems described by 2) differential equations or 3) integral equations.

The spectral theory of linear operators has as its basic origins, on the one hand, linear algebra-more precise theorems on the reduction of quadratic forms to sums of squares-and on the other hand, problems in the theory oscillations(vibration strings, membranes, etc).

Many authors studied the asymptotic formulas for eigenvalues and corresponding eigen functions to different types of spectral problems ^{[1, 2, 3]} and ^{[5, 6, 7]}.

In this paper, we study the behavior of the solutions and asymptotic behaviors of eigenvalues of the fourth order boundary value problem of the form:

(1)

Where and are real-valued functions and and is a spectral parameter in which .

Here we assume that , , and . We have introduced the sectors and their conjugates (relative to the -axis). Let be located in some fixed sector or , and let for be different roots of unity of degree 4, and ordered so that for all satisfied the inequality:

(2)

Numbering depends on the selected sector. Entire complex plane of is divided into 8 sector and ( in the plane which is determined by the inequalities and we assume that and

2. The Behavior of the Solution of Fourth Order Boundary Value Problem

The aim of this section is to estimate the behavior of the solutions to the given fourth order boundary value problem and finding their coefficients from the following theorem:

Theorem 1:

All coefficients in the linear independent solutions

k=0,1,2,3 of equation(1)

with sufficient large are :

Proof:

From ^[4], proved that the solutions of equation (1) for sufficient large which can be written in the form

(3)

By differentiating (3) up to fourth order with respect to x, the following relations are obtained:

(4)

(5)

Now, we substitute the resulting equations (3),(4) and (5)

With equation in the equation (1), we obtain the results.

3. Asymptotic Behaviors of Eigenvalues to the Problem (1)

The aim of this section is to study the asymptotic behaviors of eigenvalues in the following cases:

a.

b.

c.

d.

to the given spectral problem by the theorem :

Theorem 2:

Asymptotic behavior of eigenvalues for sufficiently large of the spectral boundary value problem in the irregular case and in the sector has the form:

Where

and

Where

is natural number,

Proof: Since from the given boundary conditions, we can write all as follows

Thus,

Where

Proceed to finding the zeros of determinant for in this irregular case .

Let .

Therefore, by ^[5], we have

Where , where the in front of the imaginary unit is taken as , for sector

Let

Solving this equation by the same way in ^[1], we obtain:

Taking the initial approximation , the method of successive approximations we obtain:

and

Where

References

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