Eigenfunction expansion for a regular fourth order eigenvalue problem with eigenvalue parameter in the boundry conditions

Regular eigenvalue problem with eigenparameter contained in the equation and the boundary conditions

“…Regular right-definite eigenvalue problems for ordinary differential equations with eigenvalue parameter in the boundary conditions have been studied by Fulton [I], Hinton [2], Ibrahim [3], Schneider [4], Walter [5], Zayed and Ibrahim [6], Zayed [7] and many others, while in the present paper we shall study regular right-definite eigenvalue problems for partial differential equations with eigenvalue parameter in Robin boundary conditions. The object of this paper is to prove the expansion theorem for the follow- where we assume throughout that n (i) A E is the Laplace operator in R n (n > 2) n i=l x. n 7. u (x) (x) denotes differentiation of u(x) along the outward unit (ii) u i=l x.…”