Inverse Nodal Problem for a Class of Nonlocal Sturm-Liouville Operator

. Inverse nodal problem consists in constructing operators from the given nodes (zeros) of their eigenfunctions. In this work, the Sturm-Liouville problem with one classical boundary condition and another nonlocal integral boundary condition is considered. We prove that a dense subset of nodal points uniquely determine the boundary condition parameter and the potential function of the Sturm-Liouville equation. We also provide a constructive procedure for the solution of the inverse nodal problem.


Introduction
Boundary problems with nonlocal conditions are a part of fast developing differential equations theory. Problems of this type arise in various fields of physics, biology, biotechnology and etc. Sturm-Liouville problems with integral conditions constitute a very interesting class of problems since they include as special cases two-, three-and multi-point boundary conditions. Nonlocal conditions come up when value of the function on the boundary is connected to values inside the domain. Theoretical investigation of problems with various types of nonlocal boundary conditions is actual problem and recently it is paid much attention for them in the literature. Originators of such problems were Samarskii and Bitsadze [2].
Quite a new area, related to problems of this type, deals with investigation of the spectrum of differential equations with nonlocal conditions. Eigenvalue problems with nonlocal conditions are closely linked to boundary problems for diffrential equations with nonlocal conditions [6,11,13,14]. In the papers [9,24,30,25,26] the similar problems are investigated for the operators with

Main Results
Let us consider the following boundary-value problem for the Sturm-Liouville operator on segment [0, π]: −y ′′ + q(x)y = λy, λ = s 2 (2.1) with one classical boundary condition and another nonlocal integral boundary condition where q(x) and µ(x) are real-valued, twice differentiable functions on segment [0, π]. In particular, if h = ∞, the classical boundary condition is The differential operation (2.1) together with boundary conditions (2.2) and (2.3) define a linear (non-self-adjoint) operator in the Hilbert space L 2 [0, π], and the problem (2.1)-(2.3) is an eigenvalue problem for this operator. From the results of [4] it follows that the spectrum of the operator is discrete and consists of single eigenvalues. Let λ 0 , λ 1 , · · · , λ n , · · · be the eigenvalues of the problem (2.1)-(2.3), ordered by increasing modulus.
Let φ(x, λ) be the solution of (2.1) satisfying the following initial conditions Then the spectra of the boundary-value problem (2.1)-(2.3) are the zerosequences {λ n } ∞ n=0 of the entire function Let φ(x, λ n ) be the eigenfunction corresponding to the eigenvalue λ n of the Sturm-Liouville operator (2.1)-(2.3). Using an analog of the Sturm oscillation theorem, for sufficiently large n we get φ(x, λ n ) has exactly n − 1 nodal points in (0, π) (h = ∞), and exactly n nodal points in (0, π) (h = ∞). Suppose x j n are the nodal points of the eigenfunction φ(x, λ n ). In other words, φ(x j n , λ n ) = 0. Let I j n = (x j n , x j+1 n ), and the nodal length l j n by l j n = x j+1 n − x j n . We also define the function j n (x) to be the largest index j such that 0 ≤ x j n ≤ x. Denote X := {x j n }. X is called the set of nodal points of the Sturm-Liouville operator (2.1)-(2.3). We consider the following inverse problem.
Problem: Given nodal points set X or its subset X 0 which is dense in (0, π), find the boundary condition parameter h and the potential q(x).
The main theorems are the following.
n } ⊂ X be chosen such that lim n→∞ x j n = x. Then the following finite limits exist and the corresponding equalities hold.
(a) In the case h = ∞.
In the case h = ∞ and n is chosen even number series.
Let us now formulate a uniqueness theorem and provide a constructive procedure for the solution of the inverse nodal problem.
Theorem 2. Let X 0 ⊂ X be a subset of nodal points which is dense in (0, π). Then, the specification of X 0 uniquely determines the potential and the coefficient h of the boundary conditions. The number h and the potential q(x) − 1 π π 0 q(x)dx can be constructed via the following algorithm: (1) for each x ∈ [0, π] choose a sequence {x j n } ⊂ X 0 such that x j n → x as n → ∞ (note in the case h = ∞ we choose n an even number series); (2) find the functions f (x) and g(x) via (2.4)-(2.7))and in turn, calculate (a) In the case h = ∞: Using only the nodal data, we can reconstruct the potential. Our reconstruction formulae are direct and automatically it implies the uniqueness of this inverse problem (2.1)-(2.3).

Proofs
We now proceed with the proof of theorems in the case h = ∞. The other case is treated similarly. First qualitative behaviour of eigenvalues for Sturm-Liouville operator (2.1)-(2.3) is given.
Proof. In the case h = ∞, the eigenfunction φ(x, λ n ) has the following asymptotic formula for n → ∞ uniformly in x [16]: Then, for the nodal point x j n of the eigenfunction φ(x, λ n ), from Using Taylor's expansions, we obtain the following asymptotic formula for nodal points as n → ∞ uniformly in j ∈ Z + :

C.F. Yang
which implies Moreover, using the asymptotic formulae thus, we have The equality (3.1) holds. It gives uniformly with respect to j. In the case h = ∞, for j = 0, 1, · · · , n + 1, the formula (3.1) gives Thus, according to the order of x j n , for large n, the eigenfunction φ(x, λ n ) of the Sturm-Liouville operator has exactly n nodes in the interval (0, π), i.e., x j n , j = 1, n. ⊓ ⊔ In the above results, the order estimate is independent of j. As a result, we get that l j n = π/n + o 1/n .  For the case h = ∞: • Step 1. Take x = π, it follows µ(π) = f (π) − π 4 . • Step 2. Take x = 0, then it yields h = f (0).
• Step 2. By taking derivatives we obtain Thus these formulae are constructed. Since each nodal data determines one set of reconstruction formulae which only depend on nodal data, the uniqueness holds obviously. ⊓ ⊔

Conclusions
In fact, using the methods in this paper we can obtain a extended conclusion. Consider the following boundary-value problem for the Sturm-Liouville operator on segment [0, π]: −y ′′ + q(x)y = λy where q(x) and µ(x) are real-valued, twice differentiable functions on segment [0, π]. In particular, if h = ∞, the classical boundary condition is y(0) = 0.
A uniqueness theorem of the inverse nodal problem (4.1), (4.2) and (4.3) is the following; its proof is similar to that of Theorem 2.

C.F. Yang
Theorem 3. Let X 0 be a subset of nodal points which is dense in (0, π) for the inverse nodal problem (4.1), (4.2) and (4.3). Then, the specification of X 0 uniquely determines the potential q(x) − 1 π π 0 q(x)dx in (0, π), and the coefficients h and H of the boundary conditions.