Remarks on the Periodic Conformable Sturm–Liouville Problems

Te conformable Sturm–Liouville problem (CSLP), − x 1 − α ( p ( x ) x 1 − α y ′ ( x )) ′ � ( λρ ( x ) − q ( x )) y ( x ) , for 0 < α ≤ 1, is studied under some certain conditions on the coefcients p , ρ , and q . According to an interesting idea proposed by P. Binding and H. Volkmer [Binding et al., 2012, Binding et al., 2013], we will derive how to reduce the periodic or antiperiodic (CSLP) to an analysis of the Pr¨ufer angle. Te eigenvalue interlacing property related to (CSLP) will be given.


Introduction
In 2014, Khalil et al. [1] introduced a new local derivative, called the conformable fractional derivative D α x .Tis concept was quickly adopted by Abdeljawad in [2] where he claimed to have developed some tools of fractional calculus.Recently, Abdelhakim and Machado [3,4] showed that a function f has a conformable fractional derivative of order α at x if and only if it is diferentiable at x and holds.From the above, the "conformability" of D α x comes precisely from the integer-order derivative, the factor f ′ , in (1).Also, Zhao and Luo [5] gave physical and geometrical interpretations of the conformable derivative.Tey generalized the defnition of the conformable derivative to general conformable derivative by means of linear extended Gâteaux derivative and used this defnition to demonstrate that the physical interpretation of the conformable derivative is a modifcation of classical derivative in direction and magnitude.Indeed, D α x is a weighted derivative but not a fractional one.Hence, we shall call D α x the conformable derivative in this paper.
Te purpose of this work is to investigate some eigenvalue properties related to the conformable Sturm-Liouville (CSL) equation

−D α
x p(x)D α x y(x)  + q(x)y(x) � λρ(x)y(x), (2) where D α x is the conformable derivative of order α ∈ (0, 1].From the above discussion, (2) is equivalent to Such an equation has been studied in a variety of contexts subject to separated boundary conditions of the form with β, c ∈ [0, π).Te basic eigenvalue existence and eigenfunction oscillation theory can be found in many results (see e.q., [6][7][8][9][10][11]).For the case of coupled (or nonseparated) boundary conditions, there is few work done so far on the existence of eigenvalues and some related properties.Here, we shall consider the periodic/antiperiodic conditions Following Hill's studies of planetary motion in the later part of the 19th century, Sturm-Liouville equations (as α � 1) with periodic (or antiperiodic) conditions became of interest, and one can remark that such boundary conditions also appear in the study of wave motion, separation of variables in classical boundary value problems, etc.A fascinating and interesting idea proposed by Binding and Volkmer [12,13] demonstrated a method to reduce the periodic or antiperiodic Sturm-Liouville problems to an analysis of the Prüfer angle.Tis provides a simple and fexible alternative to the usual approaches via operator theory or the Hill discriminant.Te Prüfer treatment is a simple and efcient method.It depends on elementary analysis of initial value problems, builds on standard ideas from the case of separated boundary conditions, and is less intricate (and shorter) than the Floquet/Hill theory.Motivated by the above, we intend to employ the method of Prüfer transformations to deduce the existence of periodic/ antiperiodic eigenvalues of (2) and give the inequalities which interlace the eigenvalues corresponding to separated and coupled boundary conditions.Te above mentioned results are well known for α � 1.How we try to extend them to the general case 0 < α ≤ 1.Here, for (4), denoted by λ k (β, c) the eigenvalue with "oscillation count" k for k ≥ 0 (except, that k ≥ 1 when c � 0).In this paper, we defne this as the number of zeros in (0, π] of any eigenfunction corresponding to λ � λ k (β, c).For later purposes, we also write λ k (β) � λ k (β, β).Now, we have the following result related to eigenvalue interlacing inequalities.
From the above, we shall conclude by connecting our approach with a version of conformable Floquet/Hill theory (see Teorems 3 and 4).Also, we can characterize the socalled stability and instability intervals in terms of the Prüfer angle.
Our plan of this paper is as follows.In Section 2, we recall some basic defnitions and properties of conformable calculus and the Prüfer substitution.Ten, we will show the existence of periodic/antiperiodic eigenvalues and connect our approach with a version of conformable Floquet/Hill theory in Section 3.

Some Preliminaries and Prüfer Substitution
Te conformable calculus [1-4, 6, 14-17] is defned and wellstudied from 2014.In this section, we frst recall the elementary defnitions and properties of conformable calculus for the reader's convenience.
(i) Te conformable derivative of f of order αatx > 0 is defned by and the conformable derivative at 0 is defned as (ii) Te conformable integral of f of order α is defned by Note that the space Ten, for all x > 0, we have (ii) Let f: (0, b) ⟶ R be diferentiable.Ten, for x > 0, we have with g ≠ 0. ( ). Ten, h(x) is α-diferentiable, and for all x with x ≠ 0 and g(x) ≠ 0, we have 2 Complexity Next, we apply the Prüfer substitution to deduce the existence of various eigenvalues for (2).First consider (2) coupled with the separated boundary conditions here employ the substitution for a nonzero solution y of (2) taking the form y � r sin θ, Similar manipulation as in the classical case α � 1 gives with the initial conditions (the latter can be arranged by scaling y).We use these initial value problems to defne θ and r as functions of (x, β, λ).
Next, we quote a result.
Te above sufces to give existence of a unique λ k � λ k (β, c) with oscillation count k for each k ≥ 0 (except, that k ≥ 1 when c � 0).Now for (15) and the given β, the eigenvalue condition gives In this paper, we defne k as the number of zeros in (0, π] for the eigenfunction corresponding to λ k .From Lemma 1, for fxed β and c, the unique existence of λ k (β, c) with oscillation count k for k ≥ 0 is valid.

Periodic/Antiperiodic Eigenvalues and Connections to Other Approaches
In this section, we will prove the existence of periodic/ antiperiodic eigenvalues and connect the approach with a version of conformable Floquet/Hill theory.Now, we prepare some groundwork for this issue.For any fxed λ, θ(x, β, λ) is C 1 in β, and θ β � zθ/zβ satisfes by ( 17) and (19).Applying ( 18) and ( 22), one can obtain Hence, holds whenever the solutions of ( 17) and ( 18) exist.By applying ( 16) and ( 19) (where β is as yet undetermined), ( 5) can be written as where k is even (resp.odd) for a periodic (resp.antiperiodic) condition.Ten, ( 24) and (25) yield so the eigenvalues can be found by studying the Prüfer angle θ without the radius r. (Tis means that (25) holds if the Prüfer angle satisfes the right endpoint condition (26).)Indeed, it is useful to defne a function δ(•, λ): β ⟶ δ(β, λ) by Ten, the periodic (or antiperiodic) eigenvalue condition implies that Tat is, a real number λ is a periodic (or antiperiodic) eigenvalue if and only if kπ is a critical value of the continuously diferentiable function δ(, λ).Besides, it is obvious that Ten, we can derive the following.
Lemma .For the functions m and M defned as in ( 31) and (32), the following is valid.

□
Now, it sufces to deduce the existence of periodic (or antiperiodic) eigenvalues.By Lemma 2, m (resp.M) can attain each kπ for k ≥ 0 (resp.k ≥ 1) so one can defne intervals Also, the end points of I k are denoted by (39) Now, the end points λ ± k represent the following properties.
Next, we will conclude by connecting our approach with a version of the conformable Floquet/Hill theory.Here, we seek nontrivial solutions of (2) that satisfy which evidently generalizes (5), for some complex "Floquet multipliers" ω.Now assume that y 1 (x, λ) and y 2 (x, λ) are solutions of (2) satisfying To determine the multipliers, the solution C 1 y 1 + C 2 y 2 is considered to satisfy (48), which yields For a nontrivial solution (C 1 , C 2 ) to exist, the determinant of the coefcients must vanish.Tat is, From constancy of W α (y 1 , y 2 ) with p(0) � p(π), one can obtain the quadratic equation where d(λ) ≔ y 2 (π, λ) + D α x y 1 (π, λ) is Hill's discriminant for (2).Te roots ω 1 and ω 2 of (52) are distinct complex conjugates of magnitude 1 if Now recall D α x (e c/αx α ) � ce c/αx α .In this case, two independent solutions exist, y 1 � u 1 (x)e a 1 /αx α and y 2 � u 2 (x)e a 2 /αx α , where u i (i � 1, 2) are periodic of period π and e a 2 /απ α � ω i .Tus, if (53) holds, all solutions of (2) are uniformly bounded on (−∞, ∞).Terefore, the value of λ for which (53) holds will be called the stability interval.Conversely, those λ for |d(λ)| > 2 can be called the instability interval.Next, from (52), we intend to distinguish the possibilities of ω via the Prüfer angle.By (38), defne intervals CI k complementary to I k by Te following results are related to the Floquet multipliers connecting with I k and CI k .Theorem 3. Te Floquet multipliers ω 1 and ω 2 corresponding λ are real (resp.nonreal) if and only if there exists an integer k ≥ 0 such that λ ∈ I k (resp.λ ∈ CI k ).

Conclusion
In this paper, we consider the periodic/antiperiodic conformable Sturm-Liouville problem (CSLP).We employ the Prüfer transformation to reduce the periodic or antiperiodic (CSLP) to an analysis of the Prüfer angle.By the efciency of this method, we give the inequalities which interlace the eigenvalues corresponding to separated and coupled boundary conditions.We also conclude by connecting our approach with a version of conformable Floquet/Hill theory and characterize the so-called stability and instability intervals in terms of the Prüfer angle [18][19][20].

Data Availability
All the data used to support the fndings of this study are included within the article.