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

Karwan H.F. Jwamer, Aryan Ali. M

School of Science, University of Sulaimani  


In this paper we study the boundeness of the eigenfuctions of the spectral problem of the form:

-y” (x) + p1 (x) y’ (x) + q1 (x)y(x)= λ2 p(x)y(x), x∈(0,a)                                  (1)

With the boundary conditions:

y(0) = 0, y’ (a) - i λy(a)=0,                                                                                                      (2)

and the normalized condition:

where  λ is spectral parameter and p(x) is a wight function satisfy Lipschitz condition, that is 
( p(x) ∈ Lip1) /p(x2) – p(x1) | ≤ k | x2 – x1 | , ∀ x1, x2 ∈ [0, a], k is Lipschitz constant, and p1 (x) ≠ 0, p1 (x) ∈ C1 [0,a], q1 (x) ∈ C[0,a].
Keywords: Spectral problem, weight function, Lipschitz condition, Cauchy problem 2000 MR Subject Classification: 34B05, 34B15, 34L20

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