Asymptotic Distribution of Eigenvalues and Eigenfunctions of a Nonlocal Boundary Value Problem

. In this work, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the second order boundary-value problem with a Bitsadze–Samarskii type nonlocal boundary condition.

In some of problems of mathematical physics, biology and biotechnology subsidiary conditions are imposed locally. Asymptotic formulas for eigenvalues and eigenfunctions for these kinds of Boundary-Value Problems (BVPs) (the case γ = 0) are obtained in [1,2,9,13,14,16,19,26,27]. Asymptotic formulas for eigenvalues and eigenfunctions for BVPs which contains a spectral parameter in the local (classic) boundary conditions except from the differential equation obtained in [9,19].
There has been an increasing interest for spectral analysis of nonlocal boundary value problems (NBVPs) in the last decades. NBVPs are widely used for mathematical modelling of various processes of physics, ecology, chemistry and industry, when it is impossible to determine the boundary or initial values of the unknown function. For example, problems with feedback controls such as the steady-states of a thermostat, where a controller at one of its ends adds or removes heat, depending upon the temperature registered in another point, can be interpreted with a second-order ordinary differential equation subject to a nonlocal boundary conditions. The bibliography on the subject of NBVPs is very extensive and we refer to the list of the works in [6,22,23,24]. We should also note that an eigenvalue problem with the nonlocal boundary conditions is closely linked to boundary problems for differential equations with the nonlocal boundary conditions [3,4,5,11,12,18]. However, until this time, there was no work investigating asymptotic properties of eigenvalues and eigenfunctions of the second order nonlocal boundary value problems with potential function q(x) in differential equation.
The paper is organized as follows. In Section 2, notation and definitions used in the paper are stated. Also, we write the general solution of the (1.1) corresponding to the initial conditions and prove the simplicity of eigenvalues. In Section 3, we investigate the distribution of eigenvalues and obtain asymptotic formulas for eigenvalues and eigenfunctions of the boundary-value problem (1.1)-(1.3). Later on, in the same section, we obtain more exact formulas for eigenvalues and eigenfunctions under the condition q ∈ C 1 [0, 1]. Also, we calculate normalized eigenfunctions for the problem (1.1)-(1.3).

Fundamental solutions and simplicity of eigenvalues
In this section, first, we write the initial-value problem (1.1), (2.1) in terms of equivalent integral equation and then construct structure of the solutions of the initial-value problem (1.1), (2.1) and the space of these solutions. We see that any two solutions of the initial-value problem (1.1), (2.1) which are linearly independent on [0, 1] form a fundamental system of solutions.
Let ω s (t) be a solution of Equation ( is an analytic function of s.

Lemma 3.
Let s ∈ C s . Then there exists q 0 > 0 such that for |s| ≥ 2q 0 one has the estimate and more precisely These estimates hold uniformly for 0 ≤ t ≤ 1.

Theorem 1.
[see, Theorem II.2.2 in [17]] In order that the functions ω 1 s (t) and ω 2 s (t), solutions of the initial-value problem (1.1), (2.1) be linearly dependent on [0, 1] it is necessary and sufficient that The algebraic multiplicity of the eigenvalue is its multiplicity as a root of the characteristic polynomial.
We will say that geometric multiplicity of an eigenvalue of a boundary value problem is the maximum number of linearly independent eigenfunctions associated with the related eigenvalue. By the definition of eigenvalues and eigenfunctions, geometric multiplicity of an eigenvalue is equal or greater than one because each eigenvalue has at least one eigenfunction.

Spectral asymptotics for eigenvalues and eigenfunctions
In the case q(t) ≡ 0, the spectrum of the Sturm-Liouville problem (1.1)-(1.3) has countably many eigenvalues and all eigenvalues are positive for |γ| ≤ 1.
Let us denote D k = {s : |x| ≤ a k = (k + 1/2)π, |y| ≤ a k }, D s k = D k ∩ C s , k ∈ N, and a contour Γ s k = ∂D k ∩ C s . Then we have |s| ≥ 3π/2 on Γ s k , k ∈ N. The corresponding contour Γ λ k in the the plane C λ will be the boundary of the domain D λ k . Lemma 6. Let |γ| < 1. Then there exists q 1 > 0 such that all eigenvalues of the problem (1.1)-(1.3) in the domain {s ∈ C s : |s| > q 1 } are positive.
From formula (3.4) H(s) = h(s) + h 0 (s) where h 0 (s) = O(s −1 e |y| ). Hence, we have |h 0 (s)| ≤ c 1 |s| −1 e |y| < Ae |y| ≤ |h(s)| on the contours Γ k for sufficiently large k. Therefore, by Rouché theorem it follows that the number of zeros of H = h + h 0 and h are the same inside Γ k for sufficiently large k.
In the domain between contours Γ k−1 and Γ k there is exactly one positive root of the function h (see Remark 3). The function H has one root in this domain for sufficiently large k. But interval (k − 1/2)π, (k + 1/2)π belongs to this domain. So, the single root of H in this domain is positive (see Corollary 2).
We can enumerate the zeros of H as s k , k ∈ N. The first zeros can be complex numbers or not simple. From Lemma 6 we have that s k are positive for sufficiently large k. Now we will investigate the distribution of these positive eigenvalues of the problem (1. Since x k , s k ∈ (k − 1/2)π, (k + 1/2)π , we have Let us denote δ k = s k − x k . The functions H and h are analytic. So, from (3.12) and H(s) = 0 we have are valid for sufficiently large k.
Proof. Substituting s k = x k + δ k into (3.12) we obtain Since sin x k − γ sin(ξx k ) = 0 we rewrite this equality as Thus, by Corollary 3 we get δ k = O(k −1 ). Substituting s k = x k + δ k into equality (3.13), we find the asymptotic formula .
Under the condition that q ∈ C 1 [0, 1] the more exact asymptotic formulas may be obtained. In this case the following formulas are valid for t ∈ [0, 1] [14]. Let Q(t) = 1 2 t 0 q(τ )dτ . It is obvious that the function Q(t) is bounded for 0 ≤ t ≤ 1.
Substituting the expression (3.13) into the integrals in (2.2) we have Using the formula (3.14) formulas we derive are valid for sufficiently large k.
Proof. Substituting s k = x k + δ k into (3.16), we have Since sin x k − γ sin(ξx k ) = 0 we rewrite this equality as Now, we are ready to obtain a sharper asymptotic formula for the eigenfunctions. Substituting s k = x k + δ k into (3.15), we have Remark 6. To obtain this asymptotic expansion for the normalized eigenfunctions v k (t) = −α −1 k u k (t) let us consider the integral Thus, the normalizing coefficients .
So, normalized eigenfunctions are v k (t) = √ 2 sin(x k t) Remark 7. Assuming that q ∈ C 2 [0, 1], one can prove a more precise asymptotic formula . Here Q 2 is a bounded function. for eigenvalues and eigenfunctions, respectively [14].

Conclusions
In this paper, the spectrum and asymptotic formulas of eigenfunctions for Sturm-Liouville problem with one Bitsadze-Samarskii type nonlocal boundary condition was investigated. The results obtained in this work can be extended to differential equations with retarded argument [20]. Furthermore, asymptotics of eigenvalues and eigenfunctions of the same differential equation but with different boundary conditions such as integral boundary conditions can be also investigated.