ON THE EIGENVALUES OF SECOND-ORDER BOUNDARY-VALUE PROBLEMS

In this paper we investigate the properties of eigenvalues of some boundary-value problems generated by second-order Sturm-Liouville equation with distributional potentials and suitable boundary conditions. Moreover, we share a necessary condition for the problem to have an infinitely many eigenvalues. Finally, we introduce some ordinary and Frechet derivatives of the eigenvalues with respect to some elements of the data.


Introduction
The problem of existence of eigenvalues of a boundary-value problem has been attacted by the authors extensively for years. The special attempt has been applied for the following problem where p, p ′ , q, w are real-valued and integrable functions, p > 0, w > 0 and α, β are some real numbers. One of the tools is the Prüfer's transformation. Indeed, with the following new variables y(x) = r(x) sin θ(x), p(x)y ′ (x) = r(x) cos θ(x), the differential equation in (1.1) is transformed into the equations [1,3,7,8,18] r ′ = 1 p − g r sin θ cos θ, and θ ′ = 1 p cos 2 θ + g sin 2 θ, (1.2) where This method is meaningful provided that y and py ′ do not vanish simultaneously. Note that Eq. (1.2) has a unique solution θ satisfying the initial condition θ(a, λ) = α.
Some certain properties of eigenvalues of the propblem (1.1) can be investigated with the help of the monotone increasing proporty of θ. In fact, one may examine the properties of θ using the following equation with p 1 > p 2 and g 2 > g 1 .
Another method belongs to Atkinson [1]. Indeed, for the problem He passed to the differential equations generated by y and z instead of the second-order equation such that where z = y ′ /r and investigated the properties of eigenvalues of the problem.
In [5], Eckhardt et al have investigated some properties of the eigenvalues and solutions of the following problem where α, β are some real numbers, p, p ′ , q, s, w are real-valued and integrable functions with w > 0 on the given interval. The differential equation appearing in (1.3) is called as differential equation with distributional potentials. This equation and the corresponding problems have also been studied in [2] and [16]. Clearly, this differential equation contains the differential equation in (1.1). However, this one provides a detailed analysis. It is better to note that a weaker version of the differential equation in (1.3) has also been introduced by Savchuk and Shkalikov [13] as Clearly for s ≡ 0, these equations turn out to be the ordinary Sturm-Liouville equations.
In this paper, we will investigate the properties of eigenvalues of a similar problem with (1.3) with the aid of Atkinson's approach. Moreover, at the end of the paper we will compare this method with Prüfer's transformation. Finally, we will investigate the dependence of the eigenvalues of the problem on some elements of data. This is also a remarkable area for the investigation of the eigenvalues of some Sturm-Liouville boundary value problems and the readers may see the papers [4, 6, 9-12, 14, 15, 17, 19] including recent works on the dependence of the eigenvalues of some ordinary Sturm-Liouville boundary value problems.

Boundary-value problem
Let us consider the following system of equations (2.1) Here the basic assumptions are as follows (i) r, s, q, w are real-valued and integrable functions on [a, b], We should note that the Eq. (2.1) can also be considered as the following second-order differential equation with distributional potentials However, we will continue with the system of Eq.s (2.1).
Proof. We obtain from the second equation in (2.1) that Then the inequality implies the following Therefore on an arbitrarily small interval (c 1 , Since y is continuous we may consider that |y( Then (2.2) and (2.4) give a contradiction.
or equivalently where k is a constant. Therefore we get for k ̸ = 0 that Since the right hand side of (2.5) is positive from (iii), y(b) and y(a) can not be zero at the same time. This implies by the continuity of y that there exists an interval (a, a + ϵ) or (b − ϵ, b) such that (2.2) is not satisfied. This completes the proof for k ̸ = 0.
For k = 0 one gets Consequently, as before, this gives a contradiction with (2.2) and this completes the proof for k = 0. The boundary conditions for the solutions of (2.1) are considered as follows where 0 ≤ γ < π and 0 < φ ≤ π.
The first property of the problem (2.1), (2.6) is the following.
Theorem 2.1. The eigenvalues of (2.1), (2.6) are all real and discrete with the possible limit point at infinity.

Proof. Consider the equation
Integration of both sides of (2.7) and the conditions (2.6) give Since y is a nontrivial solution this implies ℑλ = 0. The second assertion follows from the conditions (2.6) and the entire property of y(x, λ) and z(x, λ).
Therefore the proof is completed. A direct consequence of (2.1) is the following.

Theorem 2.2. Following equations are satisfied
Proof. Using Eq. (2.7) and (2.6) we get and the results follow from the last equation.
Corollary 2.1. The solutions of (2.1) have the following properties Proof. We get from (2.8) that The proof is completed by (2.9) and (2.10) because the right hand sides of (2.9) and (2.10) remain finite on the given intervals.

On the eigenvalues of the problem
In this section we investigate the properties of the problem (2.1), (2.6). For this purpose we shall construct the following function Clearly the roots of ψ coincide with the roots of y. Moreover ψ can be considered on (−π/2 + nπ, π/2 + nπ) for the fixed integer n.
One may obtain from (3.1) the following Therefore we may infer that (3.2) has a unique solution satisfying ψ(a, λ) = γ.

Theorem 3.3. The roots of the equation
where k is the nonnegative integer, are the eigenvalues of (2.1), (2.6).
Proof. Using (3.2) we obtain that is uniformly bounded for all real λ and for all negative λ. Therefore on (c 1 , c 2 ) we have for λ < 0 that For large negative λ we see that sin ψ is sufficiently small. Moreover using we obtain that ψ is sufficiently small for large negative λ and so one may infer that Now suppose that the inequality and so On the other side since ψ(d 1 , −∞) < π 2 for large negative λ we have (3.9) Replacing c 1 and c 2 by d 1 and d 2 , respectively in (3.5) we see that where b * < b and ϵ * > 0. Therefore we may infer for sufficiently large negative λ from previous calculations on So 0 < sin 2 (ϵ * ) < sin 2 ψ.

Therefore we again infer that
which contradicts with (iii). Consequently we should have This completes the proof.
It follows for λ > 0 that ψ ′ 1 ≥ −λ −1/2 |q| − |s| , and so for λ ≥ 1 1 (a, λ) is bounded from below, uniformly for λ ≥ 1. Now suppose that for λ ≥ 1 there exists a constant l 1 such that So ψ 1 is of bounded variation uniformly for λ ≥ 1. Using (3.12) we obtain Then at any rate x ∈ [c 2k , c 2k+1 ] we obtain Similarly we obtain For large values of λ one may find x ∈ [c 0 , c 1 ] such that ψ 1 is arbitrarily close to a multiple of π, and x ∈ [c 1 , c 2 ] such that ψ 1 is arbitrarily close to an odd multiple of π/2. Therefore taking λ large, the variation of ψ 1 (x, λ) over (a, b) can be made as large as we please and we have a contradiction. Therefore ψ 1 (b, λ)−ψ 1 (a, λ) can be made arbitrarily large and ψ 1 (x, λ) increases through an arbitrarily large number of multiples of π as x goes from a to b. This completes the proof.

Banach space
In this section we investigate the differentiable property of the eigenvalues of (2.1), (2.6) with respect to some elements of data. For this purpose we shall construct a suitable Banach space as follows with the norm where l 1 , ..., l 4 are real numbers and p 1 , ..., p 4 are integrable functions on (a ′ , b ′ ). Now we shall consider the following subspace B 1 of B consisting of all elements υ 1 such that υ 1 = (a, b, γ, φ, r, s, q, w) , where the corresponding function k is defined as follows If we define B as the set consisting of all elements υ such that υ = (a, b, γ, φ, r, s, q, w) , then B is not a subset of B. Therefore we identify B with B 1 to inherit the norm from B and the convergence in B.
Lemma 4.1. Let y and z be the solutions of (2.1) satisfying Then y = y(., x * , ξ 1 , ξ 2 , r, s, q, w) is continuous of all its variables.
Proof. The proof can be introduced as Theorem 2.7 in [10]. Proof. The roots of the function can be considered as the eigenvalues of the problem (2.1), (2.6) provided that y and z satisfy the certain conditions at a. Since Φ is entire in λ, for the point υ 0 ∈ B, there exists an η > 0 such that Φ(λ) ̸ = 0 for µ with |λ − µ| = η. Therefore by the theorem on continuity of roots of an equation as a function of parameters [4] the proof is completed.
Proof. For the proof we refer to [10] together with Lemma 4.1 and Lemma 4.2.
Using Theorem 4.1 we can introduce the following.

Theorem 4.2.
Let λ be an eigenvalue of (2.1), (2.6). Then the derivatives of λ with respect to some certain elements of data can be introduced as follows a, b).
Proof. The ordinary derivatives of λ with respect to γ and φ and the Frechet derivatives of λ with respect to q and w can be obtained using a similar method in the literature (for example, see [4, 6, 9-12, 14, 15, 17, 19]) . However, the proofs of the derivatives of λ with respect to r and s should be given as they are new in the literature. We shall consider the Eq. (2.7). Let y u be a normalized eigenfunction of λ and y u = y u (x, λ(1/r)) and 1 (a, b). Fixing all the other variables we get Therefore the result follows from the last equation. Now consider that y u be a normalized eigenfunction of λ and y u = y u (x, λ(s)) and y v = y u (x, λ(s+h)), where h ∈ L 2 (a, b). Fixing all the other variables we obtain Therefore the proof is completed.

Conclusion and remarks
In this paper we investigate some properties of the real eigenvalues of the problem (2.1), (2.6) and we have followed Atkinson's method. We should note that for the case s ≡ 0 on [a, b] the results are well known but in this work there is no need to consider s as identically zero. Therefore the results are new. Furthermore, as can be seen in Theorem 4.2, for the Frechet derivative of the spectral parameter with respect to s we need to consider h as an element of square integrable function space. This construction and result are also new in the literature as well as the results for the derivative of the eigenvalues with respect to the function r.
On the other side, as we have discussed in the introduction, there exists another way following the Prüfer's transformation. It is possible for the problem (2.1), (2.6) to pass to the new variables using Prüfer's transformation and obtain some results for this variables. Indeed, one may consider the following transformations y(x) = τ (x) sin ψ(x), z(x) = τ (x) cos ψ(x).