Boundedness of Normalized Eigenfunctions of the Spectral Problem in the Case of Weight Function Satisfying the Lipschitz Condition.

In this paper we study the boundeness of the eigenfuctions of the spectral problem of the form: -y^'' (x)+p_1 (x) y^' (x)+q_1 (x)y(x)=λ^2 ρ(x)y(x),x?(0,a) (1) With the boundary conditions: y(0)=0 ,y^' (a)-iλy(a)=0, (2) and the normalized condition: (∫_0^a??ρ(x)/e^∫??p_1 (x) dx? |y(x) |^2 ? dx )^(1/2)=1, (3) whereλ is spectral parameter and ρ(x) is a weight function satisfy Lipschitz condition, thatis (ρ(x)∈Lip1)|ρ(x_2 )-ρ(x_1 ) |?k |x_2-x_1 | ,∀ x_1,x_2∈[0,a],k is Lipschitz constant, and p_1 (x)≠0,p_1 (x)∈C^1 [0,a],q_1 (x)∈C[0,a].