where we have defined ϕ$_{M}$ ≡ ¯ φ $^{′}$. Thus ϕ$_{M}$ is a massless solution ( m 2 n = 0) of the perturbation equation (11), although it satisfies mixed boundary conditions rather than the Dirichlet boundary conditions imposed on ϕ$_{y}$ . At this point we are already able to state a new result of this work, which is an extension of the previous result related to the impossibility of having stable kink solutions with nodes inside the interval. Suppose that ¯ φ ( y ) happens to have δ nodes inside the interval. We have just shown that ¯ φ ′ ≡ ϕ$_{M}$ will solve the equation for a massless excitation, but with mixed boundary conditions. Since ¯ φ has δ nodes, ¯ φ ′ = ϕ$_{M}$ must have δ + 1 nodes inside the interval. The following inequalities relating the eigenvalues λ D n for the Dirichlet case and the eigenvalues λ M n for a general mixed boundary condition case [44] hold from Sturm-Liouville theory Since we have λ M δ +1 = 0 (i.e. the eigenvalue of the solution with δ + 1 nodes), we can immediately deduce that the mass-squared of the lowest excitation of the Dirichlet problem must be negative since λ D δ − 1 ≤ λ M δ +1 = 0 with δ ≥ 1. Proposition 3 In a warped background on a slice of AdS$_{5}$ , any static solution to equa- tion (8), with δ > 0 nodes inside the interval is always unstable. However, for nodeless static solutions (when δ = 0) the results for the flat case obtained in [40, 41] cannot be extended here. Lacking a general stability condition, we will instead propose a weaker sufficient stability condition for these and other more generic solutions in the next subsection. In a general warped background with metric ansatz (7) the equation for a static scalar background configuration is In this situation, although we have been unable to extend the stability theorems found earlier, we are still able to find a general sufficient condition for perturbative stability of the background configurations. B. Kink Scalar on a General Background ¯ φ ′′ − 4 σ ′ ¯ φ ′ − ∂V ∂φ ∣ ¯ φ = 0 . (14) 9 λ D n ≤ λ M n +2 ≤ λ D n +2 . (13)