ASYMPTOTIC BEHAVIOUR OF THE WRONSKIAN OF BOUNDARY CONDITION FUNCTIONS FOR A FOURTH ORDER BOUNDARY VALUE ROBLEM (A SPECIAL CASE)

In this paper, we prove that the Wronskian W (�) of the boundary condition functions for the following boundary value problem �: � : L� ≡ � (4) (x) +P2 (x)� (2) (x) +P3 (x)� (1) (x) +P4 (x)�(x) = ��(x) �(a) = � / (a) = �(b) = � / (b) = 0 is asymptotically equivalent for large values of |�|, to the Wronskian of the boundary condition functions of the corresponding Fourier problemF given

url: www.acadpubl.eu§ Correspondence author cians for some years now.The use of boundary condition functions for boundary value problems was first considered by Kodaira in [1].Since then, quite a number authors including [12] and [13] have worked on Boundary value problems.
In [5], D. N. Offei proved that the boundary condition functions, the Wronskian of the boundary conditions and the Green's function for the boundaryvalue problem: φ (a) = φ (b) = φ (1) (b) = 0, are asymptotically equivalent, for suitably large values of |λ|, to the corresponding functions, associated with the corresponding Fourier problem.
In [14], E. K. Essel et.al proved that the boundary condition functions of the Fourth order boundary value problem are asymptotically equivalent to the boundary condition functions of the corresponding Fourth order Fourier problem.

Notations
In this section we give some properties of the linear differential expression L and some notations used in subsequent sections of this paper.
1. (a) For a suitable set of functions, the symbol Φ (x) denotes the 4 x 4 Wronskian matrix φ . A similar notation is used if φ is replaced by another symbol; the respective capital always representing the Wronskian matrix.
(d) The Lagrange adjoint of L + is L and for suitable pair of functions g and where The A jk are dependent on the coefficients of the differential expression L + and A This implies that [φ (x, λ) ψ (x, λ)] (x) and {ψ (x, λ) φ (x, λ)} (x) may be denoted by [φψ] and {ψφ}, respectively.
(a) If there is a constant K such that |f x| ≤ Kφ for x ≥ x 0 we write x → ∞ where l = 0 we write f ∼ lφ.

Preliminaries
In this paper we consider the boundary value problem which is a special case of the boundary value problem The Fourier problem corresponding to (3) -( 4) is given by In this special case where, φ Substituting P 0 = 1 from (3) into (2) (i.e., Notation 3(b)) we see that We now state some Lemmas that will enable us to prove our main result.

Main Result
Let for suitably large values of |λ| .We prove our main result via two theorems.
where B (a) and B (b) are as in (8) .Substituting ( 7) and ( 8) into (9) we have F 2 (a) ψ From Lemma 1, with the matrices M and N as in (7) we have Substituting ( 12) into (13) we see that Comparing corresponding elements on the right and left hand sides of (11) we see that for x = b, (14) reduces to Evaluating we have Using Lemma 2 (ii) we find that where B (a) and B (b) are as in (8) .Substituting ( 7) and ( 8) into (19) we obtain ψ 4 (a) ψ Using similar deductions as in (12) we have and by substituting ( 22) into (23) it reduces to Applying Lemma 2a (ii) on the 2nd and 3rd terms on the right hand side of (27) we get = ψ If follows from the results of Theorem 1 (i.e., (18)) and Theorem 2 (i.e., (32)) that W (λ) ∼ W F (λ)

Conclusion
We have succesfully proved through Theorem 1 and Theorem 2, that the Wronskian of the boundary condition functions of the fourth order boundary value problem is asymptotically equivalent to the corresponding Wronskian of the fourth order Fourier problem.
Boundary condition functions have been studied widely by many mathemati-Received: August 23, 2014 c 2015 Academic Publications, Ltd.