On a fourth-order problem with spectral and physical parameters in the boundary condition

We consider the following fourth-order boundary-value problem: [(py'')'-qy'] = lambdary, y(0) = y'(0) = y''(1) = [(py'')' - qy'] (1) + lambdamy(1) = 0 with spectral parameter lambda is an element of C and physical parameter m is an element of R. We assign to this problem a linear pencil of bounded operators T-m = T-m (lambda) depending on the physical parameter m and acting from H-2 = {y \ y is an element of W-2(2)[0, 1], y(0) = y'(0) = 0} to the dual space H-2. We study the spectral properties of Tmz and use the results of this study to describe properties of the eigenvalues of the problem for various values of m. In particular, we establish asymptotics of these eigenvalues as m NE arrow 0.