Asymptotic behaviour of boundary condition functions for a fourth-order boundary value problem

Abstract

In this paper, we prove that the Boundary Condition Constants for the Boundary Value Problem π : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 (1) 3 (2) 2 (4) Lφ ≡ φ x + P x φ x + P x φ x + P x φ x = λφ x { ( ) ( )} 0 4 1 ( 1) ( 1) ≡ ∑ + = = − − s s rs s Urφ mrsφ a n φ b (1≤ r,s ≤ 4) can be replaced by Boundary Condition Functions and that the Boundary Condition Functions are asymptotically equivalent for large values of λ , to the Boundary Condition Functions for the corresponding Fourier problem �, given by π F : ( ) ( ) (4) x = λφ x { ( ) ( )} 0 4 ( 1) ( 1) ≡ ∑ + = − − s rs s Urφ mrsφ a n φ b (1≤ r,s ≤ 4)