Eigenvalues of the Sturm-Liouville problem with a frozen argument on time scales

In this study, we consider a boundary value problem generated by the Sturm-Liouville problem with a frozen argument and with non-separated boundary conditions on a time scale. Firstly, we present some solutions and characteristic function of the problem on an arbitrary bounded time scale. Secondly, we prove some properties of eigenvalues and obtain a formulation for the eigenvalues-number on a finite time scale. Finally, we give an asymptotic formula for eigenvalues of the problem on another special time scale.


Introduction
A Sturm-Liouville equation with a frozen argument has the form −y ′′ (t) + q(t)y(a) = λy(t), where q(t) is the potential function, a is the frozen argument and λ is the complex spectral parameter. The spectral analysis of boundary value problems generated with this equation is studied in several publications [3], [11], [12], [20], [27] and references therein. This kind problems are related strongly to non-local boundary value problems and appear in various applications [4], [8], [25] and [32].
In the present paper, we consider a boundary value problem which is generated by equation (1) and the following boundary conditions U (y) : = a 11 y (α) + a 12 y ∆ (α) + a 21 y (β) + a 22 y ∆ (β) where α = inf T, β = ρ(sup T), α = β and a ij , b ij ∈ R for i, j = 1, 2. We aim to give some properties of some solutions and eigenvalues of (1)-(3) for two different cases of T For the basic notation and terminology of time scales theory, we recommend to see [7], [9], [10] and [26].
We obtain by taking into account uniqueness of the solution of an initial value problem that y(t, λ) = ϕ(t, λ).

Eigenvalues of (1)-(3) on a finite time scale
Let T be a finite time scale such that there are m (or r) many elements which are larger (or smaller) than a in T. Assume m ≥ 1, r ≥ 0 and r + m ≥ 2. It is clear that the number of elements of T is n = m + r + 1. We can write T as follows T = ρ r (a) , ρ r−1 (a) , ..., ρ 2 (a) , ρ (a) , a, σ(a), σ 2 (a), ..., σ m−1 (a), σ m (a) , Lemma 2. i) If r ≥ 3 and m ≥ 2, the following equalities hold for all λ where O(λ l ) denotes polynomials whose degrees are l.
On the other hand, since S(t, λ) and C(t, λ) satisfy (1) then the following equalities hold for each t ∈ T κ and for all λ.
Lemma 3. The following equlaties hold for all λ ∈ C.
Proof. Consider the function It is clear that and it is the solution of initial value problem Therefore, we can obtain the following relations ϕ ρ (t, λ) = ϕ (t, λ) + µ ρ (t) q (ρ (t)) S (t, λ) .
where 0 ≤ a ≤ n − 2 and h, H ∈ R. According to Theorem 2, if h = 1, the eigenvalues-number of this problem is exactly n − 2, otherwise less than n − 2. Now, we want to give a theorem which includes some informations about the eigenvalues of (16)- (18). and The problem (16)- (18) and the matrix Q have the same eigenvalues.
Remark 1. The result in the Theorem 3 can be generalized easily to (1)-(3) on the general discrete time scale.

Remark 2.
As is known, all eigenvalues of the classical Sturm-Liouville problem with separated boundary conditions on time scales are real and algebraicly simple [2]. However, the Sturm-Liouville problem with the frozen argument may have non-real or non-simple eigenvalues even if it is equipped with separated boundary conditions.
We end this section with two examples. The problem in the first example has non-real or non-simple eigenvalues, unlike, all eigenvalues of the latter problem are real and simple.
Lemma 4. The following equlaties hold for all λ ∈ C and t ∈ T.
Thus, we establish the proof of (ii). On the other hand by using Rouche's theorem to (27) on G n , we can show clearly that (26) holds for sufficiently large n.
Remark 3. Since µ (α) = 0 in the considered time scale, the term a 22 b 12 − a 12 b 22 is not another than detA in section 3.