Strong Continuity of the Lidstone Eigenvalues of the Beam Equation in Potentials

In this paper we study the dependence of the Lidstone eigenvalues λm(q) , m ∈ N , of the fourth-order beam equation on potentials q ∈ Lp[0,1] , 1 p ∞ . The first result is that λm(q) have a strongly continuous dependence on potentials, i.e., as nonlinear functionals, λm(q) are continuous in q ∈ Lp[0,1] when the weak topology is considered. The second result is that λm(q) are continuously Fréchet differentiable in potentials q ∈ Lp[0,1] when the Lp norm is considered. These results will be used in studying the optimal estimations for these eigenvalues in later works. Mathematics subject classification (2010): 34L40, 34A30, 58C07.