On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative

: The fundamental objective of this article is to investigate about the boundary value problem with the uses of a generalized conformable fractional derivative introduced by Zarikaya et al. ( On generalized the conformable calculus , TWMS J. App. Eng. Math. 9 ( 2019 ) , no. 4, 792 – 799, http://jaem.isikun.edu.tr/web/images/ articles/vol.9.no.4/11.pdf ) . In the development of the this article, by using classical methods of fractional calculus, we ﬁ nd a de ﬁ nition of the generalized fractional Wronskian according to the fractional di ﬀ erential operator de ﬁ ned by Zarikaya, a fractional version of the Sturm - Picone theorem, and in addition, the stability criterion given by the Hyers - Ulam theorem is studied with the use of the aforementioned fractional derivatives.


Introduction
In 1965, L'Hopital gave the preliminary definition of the idea of fractional derivative.Since then, several related new definitions have been proposed.The most common ones are the Riemann-Liouville and Caputo definitions.For more information about the most known fractional definitions, we refer to [1][2][3].
The so-called fractional calculus has had a wide expansion, both from the theoretical and the applied point of view.In either case, the classical (global) fractional derivative has been used in differential equations, but in the case of local fractional derivatives, this type of research is very limited.
It is known that from 1960, certain differential operators have appeared which are called local fractional derivatives.It is not until 2014 that Khalil et  .Also in 2019, Abreu-Blaya et al. [7] introduced a generalized conformable fractional derivative and in 2020, Fleitas et al. [8] gave a note on this generalized conformable derivative.These definitions have properties suitable to that of the classical Riemman derivative with a better behavior than the classical fractional derivatives when used in different fields of application.To solve a given fractional problem, the question arises as to what type of fractional operator should be considered, since there are several different definitions of fractional derivative in the literature and the choice depends on the problem under consideration.
It can be seen from those articles that use the Riemann-Liouville or Caputo fractional derivative and the corresponding definitions of the conformable derivatives that there is a quantitative and qualitative difference between the two types of operators, local and global [9].Conformable fractional derivatives are new tools that have demonstrated their usefulness and potential in the modeling of different processes and phenomena.
As a result, several important elements of the mathematical analysis of functions of a real variable have been formulated, such as chain rule, fractional power series expansion and fractional integration by parts formulas, Rolle's theorem, and mean value theorem [10].The conformable partial derivative of the order α 0, 1 ( ] ∈ of the real-valued functions of several variables and conformable gradient vector are also defined.In addition, a conformable version of Clairaut's theorem for partial derivative is investigated in [11].In [12], the conformable version of Euler's theorem on homogeneous equations is introduced.Furthermore, in a short time, various research studies have been conducted on the theory and applications of fractional differential equations and fractional integral inequalities in the context of this newly introduced fractional derivative [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28].
With the motivation given by the aforementioned works, in this research article, we focus on the boundary value problems using a new definition of conformable fractional derivative.We have organized our present document in a subsection of preliminary knowledge, a section of main results where we define the Wronskian from the perspective of the conformable derivative defined in the preliminaries, some basic properties, and we proceed to deal with a conformable version of the conformable Sturm-Picone secondorder conformable identity, establish generalized conformable Sturm-Liouville comparison and separation theorems, construct the Green's function and study its properties, and then prove the generalized Hyers-Ulam stability of conformable nonhomogeneous linear differential equations with homogeneous boundary conditions.Also we include conclusions respect to the obtained results.

Preliminaries
In [35, Definition 2.1], a generalized conformable fractional derivative was defined, and some properties are given. )-conformable derivative of f of order α is defined by for all t a > , α 0, 1 ( ) ∈ . If this limit exists, then it will be said that the function f is α a , ( )-differentiable at the point t.
Remark 1.1.Note that if a 0 = , then this generalized conformable derivative coincides with that proposed in [4]  In the recently cited work, some important results for the calculation were also established for α a , ( )-conformable differentiable functions: the continuity of a function at a point from its conformable differentiability in it, Rolle's theorem, and the mean value and extended mean value theorems.
Also it was introduced a definition of α a , ( )-conformable fractional integral and some properties related.
exists and is finite.The set of all α a , ( )-conformable fractional integrable functions is denoted by L a b , α a , . The α a , ( )-conformable fractional integral operator is defined by where the integral is the usual Riemann improper integral.When the lower bound of the integral is any number c a > then we use the notation

Main results
Next, we give the following definition of an α a , ( )-Wronskian and α a , ( )-conformable partial derivative.
Then we set the function: and is called the (α a , )-conformable fractional partial derivative of f at c.

Generalized fractional conformable Sturm-Picone's theorem
We will focus on the following second-order fractional differential equation given by where p and q are continuous functions, α 0, 1 ( ] ∈ . Let us remember that two functions φ 1 and φ 2 are linearly dependent if there exists c c + ≡ ; otherwise, they are linearly independent.

(
)is a differentiable function on I. Then the fractional α a , ( )-Wronskian of φ φ , Proof.Let φ 1 and φ 2 be two solutions of (3), and some t a b , ( ), we obtain By using (3), we have , then we have The following equivalent condition of linear independence can be obtained from Lemma 2.1 using the classical method.
Theorem 2.1.Two solutions φ 1 and φ 2 of the fractional differential equation (3) defined on an interval I are linearly independent if and only if W φ φ t , 0 To continue this study, we introduce the following self-adjoint fractional differential equation of Sturm-Liouville-type: On some conformable boundary value problems  5 where p p q q D x , , , , Proof.After a straightforward D a α -differentiation, it follows the desired result.□ Theorem 2.3.Let a and b with a b 0 ≤ < be two consecutive zeroes of a nontrivial solution φ t ( ) of (4).Suppose that i q t p t a n d i i q t p t 0 . Then, every solution χ t ( ) of ( 5) has at least one zero in a b Proof.If φ t ( ) and χ t ( ) are solutions of (4) and (5), respectively, and χ t 0 ( ) ≠ for all.Then by substitution of these solutions and an applications of the algebraic properties of D a α , we have the Picone's identity Then, by taking the α a , ( )-integrating over a b , [ ], we have ], then the right-hand side side of 7 equals to zero.Also, since q t 0 1 ( ) > , then the third term in the integral is nonnegative, so we must have either In case (ii), we have contradiction because q t p t ).In other words, the zeros of any two linearly independent solutions of (4) are interlaced.
Proof.Suppose that χ t 0 ( ) ≠ for all t a b , .
Since φ and χ are linearly independent, we have that χ a 0 ( ) ≠ , and otherwise, we would have and therefore, φ and χ would be linearly dependent, contrary to our supposition.For the same reason, χ d 0 ( ) ≠ .If q t p t Since a and b are zeroes of φ, χ a 0 ( ) ≠ , and χ b 0 ( ) ≠ , then the right-hand side of (8) evaluates to zero.Also we have that p t 0 1 ( ) > and the kernel t a t a 0 α ( ) ( ) − / − > , then it must be that , from which we obtain that Hence, φ and χ are be linearly dependent on a b , ( ) contrary to the supposition.□

Green's function study
In this section, we consider the conformable Sturm-Liouville system with β β γ γ 0, 0 where in the tε-plane.A function G t ε , α ( ) defined in Q is called a conformable Green's function of the Sturm-Liouville system given by (9), if it has the following properties: )-conformable partial derivatives of first and second order with respect to the variable x, if t ε ≠ , and it satisfies .
Let φ 1 and φ 2 be two solutions of (9) that verify the second condition.Then, φ 1 and φ 2 are linearly dependent.
Proof.Similar to the proof of Lemma 2.2.□ Theorem 2.5.The system given by (9) has no Green's function if λ is an eigenvalue.
Proof.Let φ 1 an eigenfunction of the system given by (9).Let φ 2 be a solution of the fractional differential equation linearly independent of φ .
1 From Lemmas 2.2 and 2.3 we have that φ 2 does not satisfy the initial conditions in the system.
We know that G t ε , α ( ) satisfy the fractional differential equation in (9) over the intervals a ε , [ )and ε b , ( ], and so, it has the form and also the function G t ε , α ( ) fulfills the condition 4 in Definition 2.3, so Since φ 1 fulfill the initial conditions, then On the contrary, if ] From here, we can write which contradicts condition 2 in Definition 2.3.□ Theorem 2.6.System given by (9) has one and only one Green's Function if λ is not an eigenvalue.
Proof.Let φ 1 and φ 2 be two solutions of the considered system such that , and φ t 2 ( ) are no null, they also satisfy the initial conditions, respectively.
These solutions are linearly independent, since otherwise it would be φ t δφ t δ , for some 0. 1 Therefore, we have which would imply that φ 1 fulfills the initial conditions, but this is not possible because φ 1 is not an eigenfunction.
Reasoning as in the proof of Theorem 2.5, we have that and knowing that G t ε , α ( ) fulfill the condition 4 in Definition 2.3, it follows that and it can be reduced to and since φ 1 and φ 2 are not eigenfunctions, we have that ).By conditions 1 and 2 in Definition 2.3, we have which allows us to calculate the following: On some conformable boundary value problems  9 Note that the expression φ ε D φ ε is the α-Wronskian of two linearly independent solu- tions of (9), so it is not zero.Now, given the following by multiplying the first equation by φ 2 , the second by φ 1 , and subtracting, we have Note that is a constant K that does not depend on ε.Then we can define This conformable Green's function satisfies the conditions 1-4 in Definition 2.3.The uniqueness of this function is easily deduced from the method that we have followed to determine G x y , α ( ). □

The applicability of conformable Green's function
In this section, we consider the system obtained from (9) for λ 0 = .We now propose to solve the nonhomogeneous system: where h t ( ) is a real continuous function in the interval a b , [ ] for some a b 0 .≤ < Theorem 2.7.If the given homogeneous system (11) has the identically null function as its only solution, then the system given by (12) has only one solution, which is given by where G t ε , α ( ) is the conformable Green's function of (11).
Proof.Since the homogeneous system (11) has the identically null function as its only solution, then λ 0 = is not an eigenvalue of (9); therefore, there exists the conformable Green's function of (11).
Let φ 1 and φ 2 be two linearly independent solutions of (11) that verify the initial conditions, respectively.Let us apply the generalized conformable version of the method of variation of the parameters to solve the fractional differential equation in (11) that is, to say is a constant, and it is equal to K .Also, by using the initial conditions, we have Thus, we obtain that where the Green's function is Finally, we investigate the generalized Hyers-Ulam stability of the conformable linear nonhomogeneous differential equation of second order (12)   where K is a nonzero constant and φ t 1 ( ) and φ t 2 ( ) are two linearly independent solutions of (11) (Theorem 2. where K is a nonzero constant and φ t 1 ( ) and φ t 2 ( ) are two linearly independent solutions of (11) .We now define a function f a b R : , 0 [ ] → by for all t a b , [ ] ∈ . According to Theorem 2.7, it is obvious that f 0 is a solution of the system (12).Moreover, it follows from (15)-( 17) that .□

Conclusion
In the development of the present article, fractional versions of the Sturm-Picone Theorem and the study of the problem of boundary value determined with the use of a generalization of fractional derivatives introduced by Zarikaya et al. in [35] were established.In addition, the stability criterion given by the Hyers-Ulam Theorem was studied with the use of the aforementioned fractional derivatives.

Lemma 2 . 1 .
Let φ 1 and φ 2 be two solutions of the fractional differential equation (3) in the interval I and W φ φ , al. introduced in [4] a local derivative (conformable) In 2018, Nápoles Valdés et al. [6] introduced a definition of a nonconformable fractional derivative, denoted by N F α , with very good properties, and defined by . Then, with (11)me that the conformable homogeneous differential equation in(11)has the only null solution. 7)