Fourth-Order Finite Spectrum Problems on Time Scales with Finitely Many Interior Nodes

In this paper, we investigate fourth-order boundary value problems with finite spectrum on hybrid time scales containing finitely many isolated interior nodes. Under fourth-order Atkinson-type conditions, we construct the global transfer matrix and show that the corresponding characteristic function is a polynomial in the spectral parameter. Consequently, the problem has only finitely many eigenvalues. An explicit upper bound for their number is also obtained in terms of the partition parameters of the continuous components and the number of isolated interior nodes. The results extend finite-spectrum theory to higher-order dynamic equations on hybrid time scales and clarify the spectral contribution of the interior nodes.