Remarks on the Periodic Conformable Sturm–Liouville Problems

Abstract

The conformable Sturm–Liouville problem (CSLP), , for , is studied under some certain conditions on the coefficients , , and . 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 Prüfer angle. The eigenvalue interlacing property related to (CSLP) will be given.

1. Introduction

In 2014, Khalil et al. [1] introduced a new local derivative, called the conformable fractional derivative . This 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 has a conformable fractional derivative of order at if and only if it is differentiable at andholds. From the above, the “conformability” of comes precisely from the integer-order derivative, the factor , in (1). Also, Zhao and Luo [5] gave physical and geometrical interpretations of the conformable derivative. They generalized the definition of the conformable derivative to general conformable derivative by means of linear extended Gâteaux derivative and used this definition to demonstrate that the physical interpretation of the conformable derivative is a modification of classical derivative in direction and magnitude. Indeed, is a weighted derivative but not a fractional one. Hence, we shall call the conformable derivative in this paper.

The purpose of this work is to investigate some eigenvalue properties related to the conformable Sturm–Liouville (CSL) equationwhere is the conformable derivative of order (]. 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 formwith [). The basic eigenvalue existence and eigenfunction oscillation theory can be found in many results (see e.q., [6–11]). 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 ) 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. This provides a simple and flexible alternative to the usual approaches via operator theory or the Hill discriminant. The Prüfer treatment is a simple and efficient 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. The above mentioned results are well known for . How we try to extend them to the general case . Here, for (4), denoted by the eigenvalue with “oscillation count” for (except, that when ). In this paper, we define this as the number of zeros in of any eigenfunction corresponding to . For later purposes, we also write . Now, we have the following result related to eigenvalue interlacing inequalities.

Theorem 1. Assume that and .

(i)For each even (resp. odd) , the existence of two periodic (resp. antiperiodic) eigenvalues with oscillation count is valid.(ii)For all [), one has the inequalities

This also indicates the standard eigenvalue “interlacing inequalities” as in the classical Sturm–Liouville problemwhere and correspond to Dirichlet and Neumann eigenvalues, respectively.

From the above, we shall conclude by connecting our approach with a version of conformable Floquet/Hill theory (see Theorems 3 and 4). Also, we can characterize the so-called 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 definitions and properties of conformable calculus and the Prüfer substitution. Then, we will show the existence of periodic/antiperiodic eigenvalues and connect our approach with a version of conformable Floquet/Hill theory in Section 3.

2. Some Preliminaries and Prüfer Substitution

The conformable calculus [1–4, 6, 14–17] is defined and well-studied from 2014. In this section, we first recall the elementary definitions and properties of conformable calculus for the reader’s convenience.

Definition 1 (cf. [1, 2]). Let and f:[) .(i)The conformable derivative of of order at is defined by and the conformable derivative at 0 is defined as . If exists, one can say that is -differentiable at .(ii)The conformable integral of of order is defined byNote that the space consists of all function satisfying .

Proposition 1 (cf. [1, 2, 10]). (i)Let f:[) be any continuous function. Then, for all , we have(ii)Let be differentiable. Then, for , we have(iii)For all ,.(iv)Let be -differentiable. Then,(v)(-chain rule). Let be -differentiable and . Then, is -differentiable, and for all with and , we have(vi)(-integration by parts). Let be two functions such that is differentiable. Then,

Next, we apply the Prüfer substitution to deduce the existence of various eigenvalues for (2). First consider (2) coupled with the separated boundary conditionshere employ the substitution for a nonzero solution of (2) taking the form

Similar manipulation as in the classical case giveswith the initial conditions(the latter can be arranged by scaling ). We use these initial value problems to define and as functions of . Next, we quote a result.

Lemma 1 (cf. [9], Lemma 2.4]). For the phase function of (2), the following is valid.

(i)The function satisfying is continuous and strictly increasing in (ii)If , then for all (iii)For any (],

The above suffices to give existence of a unique with oscillation count for each (except, that when ). Now for (15) and the given , the eigenvalue condition gives

In this paper, we define as the number of zeros in (] for the eigenfunction corresponding to . From Lemma 1, for fixed and , the unique existence of with oscillation count for is valid.

3. 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 fixed , is in , and satisfiesby (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 is as yet undetermined), (5) can be written aswhere is even (resp. odd) for a periodic (resp. antiperiodic) condition. Then, (24) and (25) yieldso the eigenvalues can be found by studying the Prüfer angle without the radius . (This means that (25) holds if the Prüfer angle satisfies the right endpoint condition (26).) Indeed, it is useful to define a function by

Then, the periodic (or antiperiodic) eigenvalue condition implies that

That is, a real number is a periodic (or antiperiodic) eigenvalue if and only if is a critical value of the continuously differentiable function . Besides, it is obvious that

Now, one can extend the definition of from [) to , and then, is -periodic in . The above suggests the introduction of

Then, we can derive the following.

Lemma 2. For the functions and defined as in (31) and (32), the following is valid.

(i) and are continuous and strictly increasing in (ii)For each ,(iii) , , and .

Proof. (i)This follows from Lemma 1 (i) directly.(ii)For [), it is obvious that so . For a fixed , choose such that , and let (]. By (30) and , one gets This shows that .(iii)Note thatFor any [),so as by (20). From (36) and , the second limit follows. Also, (20) gives as so the third limit follows from .

Now, it suffices to deduce the existence of periodic (or antiperiodic) eigenvalues. By Lemma 2, (resp. ) can attain each for (resp. ) so one can define intervals

Also, the end points of are denoted by for each . Apart from , each is finite, and

Now, the end points represent the following properties.

Theorem 2. (i)Except for, each is an eigenvalue of (2) and (5), with oscillation count (ii) in are the only eigenvalues of (2) and (5) with oscillation count (iii)The interval are disjoint, so for each

Proof. (i)By (38) and (39), is an extreme value of , except for , so (29) holds. Hence, the results in are valid.(ii)Suppose to the contrary that there exists some such that has three distinct critical values corresponding to [), . Also assume that . Let be the solution of (2) with its Prüfer angle and Prüfer radius as in (16). For abbreviation, set and . From the Prüfer substitution (16), consider the conformable Wronskian Note that by using (2). Hence, is constant. Comparing the values at and and noting that by (19) and (24), one can obtain Now , so by (30) and . That is, From , one knows by (28). Then, (36) yields Similarly, and addition of the above three equations gives the contradiction . This proves .(iii)Suppose not, let for some . Then, by the monotonicity of and Lemma 2,This reaches a contradiction.

Now, recall that satisfies . This implies Then, Theorem 2 yields Theorem 1.

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 satisfywhich evidently generalizes (5), for some complex “Floquet multipliers” . Now assume that and are solutions of (2) satisfying

To determine the multipliers, the solution is considered to satisfy (48), which yields

For a nontrivial solution to exist, the determinant of the coefficients must vanish. That is,

From constancy of with , one can obtain the quadratic equationwhere is Hill’s discriminant for (2). The roots and of (52) are distinct complex conjugates of magnitude 1 if

Now recall . In this case, two independent solutions exist, and , where are periodic of period and . Thus, if (53) holds, all solutions of (2) are uniformly bounded on . Therefore, the value of for which (53) holds will be called the stability interval. Conversely, those for can be called the instability interval. Next, from (52), we intend to distinguish the possibilities of via the Prüfer angle. By (38), define intervals complementary to by

The following results are related to the Floquet multipliers connecting with and .

Theorem 3. The Floquet multipliers and corresponding are real (resp. nonreal) if and only if there exists an integer such that (resp. ).

Proof. For some , let be a solution of (2) satisfying (48) with . Without loss of generality, one can assume that and for some [). If (48) holds, then one gets for some integer . This means that . Conversely, if , then there exists such that . By the initial condition (19), one can see that (48) is satisfied with . This shows that . □

Theorem 4. The stability and instability intervals correspond to and the interiors of , respectively.

Proof. By the above discussion of , for (resp. ) lies in the stability interval (resp. instability interval). Besides, equals in the case . So (5) holds, and is one of the endpoints as in Theorem 2. Putting these cases together with Theorem 3, one can complete the proof.

4. 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 efficiency 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 [18–20].

Data Availability

All the data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.