1.Introduction

The idea of fractional derivative was first raised by L’Hospital in 1695. Since then, several related new definitions have been proposed. The most common ones are Riemann-Liouville and Caputo definitions. For more information about the most known fractional definitions, we refer to ^1,2. A new definition of fractional derivative and fractional integral has been recently proposed by Khalil et al. in ³. 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 ^3-5,40. 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 ⁶. In ⁷, conformable Jacobian matrix is defined, and chain rule for multivariable conformable derivative is proposed. In ⁸, the conformable version of Euler’s Theorem on homogeneous is introduced. Furthermore, in a short time, various research studies have been conducted on the theory and applications of fractional differential equations in the context of this newly introduced fractional derivative ^9-18,23,24.

In addition, another new definition of local fractional derivative is introduced by P. M. Guzmán et al. ²⁵, and it is called the non-conformable fractional derivative which is considered as a natural extension of the usual derivative of a function a point. The difference between conformable fractional derivative and non-conformable fractional derivative is that the tangent line angle is conserved in the conformable one, while it is not conserved in the sense of nonconformable one ²⁵ (see also ^26,34,36,37 for more new related results about this newly proposed definition of non-conformable fractional derivative). The definition of non-conformable fractional derivative has been investigated and applied in various research studies and applications of physics and natural sciences such as the stability analysis, oscillatory character, and boundedness of fractional Liénard-type systems ^27,28,33, analysis of the local fractional Drude model ²⁹, Hermite-Hadamard inequalities ³⁰, fractional Laplace transform ³¹, fractional logistic growth models ³², oscillatory character of fractional Emden-Fowler equation ³⁵, asymptotic behavior of fractional nonlinear equations ³⁸, and qualitative behavior of nonlinear differential equations ³⁹.

This paper is organized as follows: In Sec. 2, the main concepts of conformable fractional calculus are presented. Section 3, we proved a conformable version of the conformable second-order Sturm-Picone identity. From this result, we establish the conformable Sturm-Liouville comparison and separation theorems. Section 4, for a conformable Sturm-Liouville problem, the Green function is constructed, and its properties are studied. At the end, we prove the generalized Hyers-Ulam stability of conformable inhomogeneous linear differential equations with homogeneous boundary conditions.

2.Basic definitions and tools

Definition 1. Given a functionf:]0∞)→R. Then, the conformable fractional derivative of order α, ³, is defined by

for all t > 0, 0<α≤1. If f is α-differentiable in some (0,α), α > 0, andlimt→0+(Tαf)(t)exists, then it is defined as

Theorem 1. ³. If a functionf:]0,∞)→R is α-differentiable att0>0, 0<α≤1, then f is continuous at t ₀ .

Theorem 2. ³. Let0<α≤1, and let f,g be α-differentiable at a point t > 0. Then

The conformable fractional derivative of certain functions for the above definition is given as:

Definition 2. The (left) conformable derivative starting from a of a given function f:]a,∞)→R of order 0<α≤1, ⁴, is defined by

When α = 0, it is written as (Tαf)(t). If f is α-differentiable in some (a,b), then the following can be defined as:

Theorem 3 (Chain Rule). ⁴. Assume f,g:(a,∞)→R be (left) α-differentiable functions, where 0<α≤1. By letting h(t)=f(g(t)), h(t) is α-differentiable for all t≠a and g(t)≠0, therefore, we have the following:

if t = α, then

Theorem 4 (Rolle’s Theorem) . ³. Let α > 0, α∈(0,1] and f:[a,∞)→ be a given function that satisfies the following:

Then, there exists c∈(a,b), such that (Tαf)(c)=0.

Corollary 1. Let I⊂[0,∞), α∈(0,1] and f:I→R be a given function that satisfies

Then, there exists c∈(a,b), such that (Tαf)(c)=0.

Theorem 5. (Mean Value Theorem). ³. Let α > 0, α∈(0,1] and f:[a,∞)→R be a given function that satisfies

Then, exists c∈(a,b) such that

Theorem 6. ⁵. Let α > 0, α∈(0,1] and f:[a,∞)→R be a given function that satisfies

If (Tαf)(c)=0 for all t∈(a,b), then f is a constant on (a,b).

Corollary 7. [5]. Let α > 0, α∈(0,1] and F,G:[a,∞)→R be functions such that (TαF)(t)=(TαG)(t) for all t∈(a,b). Then, there exists a constant C such that

The following definition is the α-fractional integral of a function f starting from a≥0:

Definition 3.

Iαa(f)(t)=∫atf(x)x1-α⋅dx,

where the integral is the usual Riemann improper integral, and α∈(01] ².

Theorem 8. TαaIαa(f)(t)=f(t), for t≥a, where f is any continuous function in the domain of Iα.

Lemma 9. Let f:(a,b)→R be differentiable and α∈(0,1]. Then, for all α > 0,, we have ³,

Finally, we give the definition of non-conformable a-Wronskian, which is necessary in the next section.

Definition 4. Let x and 𝑦 be given conformable a-differentiable functions on [a,b] with a≥0 and α∈(0,1]. We set the following:

3.Sturm’s theorems

In this section, we consider the scalar fractional differential equation of order α + α as follows:

with continuous functions p and q, and α∈(0,1]. Traditionally, from ¹⁹, two functions x and y that are continuous on [a,b] for some 0≤a<b, will be called linearly dependent if there exist c₁, c2∈R such that |c1|+|c1|>0 and c1x(t)+c2y(t)≡0 for all t∈[a,b]. In the other case, they are linearly independent.

Remark 1. We can write

for two solutions x and y of ⁵ and some t0∈(a,b). In fact, we apply the operator Tα on Wα(x,y)(t) to obtain

Tα(Wα(x,y)(t))=Tα(x(t)Tαy(t)-y(t)Tαx(t)=Tαx(t)Tαy(t)+x(t)TαTαy(t)-Tαy(t)Tαx(t)-y(t)TαTαx(t).

However, x and y satisfies (11). Hence, we have:

TαTαx(t)=-p(t)Tαx(t)-q(t)x(t),

and

TαTαy(t)=-p(t)Tαy(t)-q(t)y(t).

Therefore, we get

Tα(Wα(x,y)(t))=-(x(t)Tαy(t)-y(t)Tαx(t))p(t)=-(Wα(x,y)(t))p(t).

Thus

Tα(Wα(x,y)(t))Wα(x,y)(t)=-p(t).

Consequently, we have

Wα(x,y)(t)=e-∫t0t(p(x)/1-α)dxWα(x,y)(t0).

This completes the proof.

Similar to the classical case, by using the above formula, we can immediately obtain the following equivalent condition of linear independence:

Theorem 10. Two solutions x and y of (11) defined on [a,b] for some 0≤a<b are linearly independent if and only if Wα(x,y)(t)≠0 for all t∈[a,b].

Now, we propose a conformable version of three classical results, the Sturm-Picone identity, Sturm’s comparison, and separation theorems of order a + a ²⁰.

Let us now introduce the conformable self-adjoint Sturm-Liouville equation as follows:

where p₀, p₁, q₀, q₁, Tα, p₁Tαq1 are continuous on some closed interval I⊂&#091;0,+∞), p₁ > 0, q₁ > 0 on I and α∈(0,1].

Theorem 11 (Conformable Picone Identity). If x(t), y(t) and p1(t)Tαx(t), q1(t)Tαy(t) are α-differentiable for t∈I and y(t)≠0 in I, then we obtain

Proof. This arises from the straightforward α-differentiation.

Theorem 12 (Conformable Sturm’s Comparison Theorem). Let 0? a < b be two consecutive zeros of a nontrivial solution x(t) of (3.3). Suppose that

and

for all t∈[a,b]. Then, every solution y(t) of (14) has at least one zero in the closed interval [a,b].

Proof. If x(t) and y(t) are solutions of (13) and (14), respectively, and y(t)≠0 for all t∈[a,b], then the conformable Picone identity (15) yields on substitution of (13) and (14) as follows:

Tαx(t)y(t)p1(t)y(t)Tαx(t)-q1(t)x(t)Tαy(t)=(p0(t)-q0(t))(x(t))\dos+(p1(t)-q1(t))(Tαx(t))2+q1(t)Tαx(t)-x(t)y(t)Tαy(t)2.

Integrating over [a,b]; therefore, we have (see Lemma 9),

The right-hand side of (16) evaluates to zero by assuming x(a)=x(b)=0, and y(a)≠0, y(b)≠0. Since q1(t)>0 in [a,b], the third term of the integrand is nonnegative over [a,b]. Hence, we must have either

or

∫ab([p0t-q0t]xt2+[p1t-q1t](Tαx(t))2)×(1/t1-α)dt<0.

However, case (ii) gives an immediate contradiction since p0(t)-q0(t)≥0 and p1(t)-q1(t)≥0 by assumption. In Case (i), we are also led to a contradiction since (i) implies

y(t)Tαx(t)-x(t)Tαy(t)(y(t))2=Tαx(t)y(t)≡0,

or (t)≡ky(t) for all t∈[a,b], for some k≠0, but y(a)=y(b)=0 which is a contrary to our assumption.

Theorem 13 (Conformable Sturm’s Separation Theorem) . Let 0≤a<b be two consecutive zeros of a nontrivial solution x(t) of (13). Let y(t) be any other solution of (13) which is linearly independent of x(t). Then, y(t) has exactly one zero of the intervall (a,b). In other words, the zeros of any two linearly independent solutions of (13) are interlaced.

Proof. On the contrary, suppose that y(t)≠0 for all t∈(a,b). Since x(t) and y(t) are linearly independent, it follows that y(a)≠0; otherwise, we would have

Wα(x,y)(a)=*20cx(a)y(a)Tαx(t)Tαy(t)=0,

which implies that the conformable Wronskian, Wα(x,y)(t), is zero for all t and that x(t) and y(t) are linearly dependent. For the same reason, we know that y(b)≠0, but when q1(t)≡p1(t) and q0(t)≡p0(t), (16) becomes

∫abp1(t)Tαx(t)-x(t)y(t)Tαy(t)21t1-αdt=x(t)y(t)p1(t)(y(t)Tαx(t)-x(t)Tαy(t))t=at=b.

Since y(a)≠0 and y(b)≠0, the right-hand side evaluates to zero. Since p1(t)>0 in [a,b], it follows that Tαx(t)-(x(t)/y(t))Tαy(t)≡0, or

Wα(x,y)(t)=y(t)Tαx(t)-x(t)Tαy(t)≡0,

for all t∈(a,b). Hence, x(t) and y(t) are linearly dependent on (a,b) which is a contrary to our assumption.

Remark 2.

Remark 3. An important application of Sturm’s Comparison Theorem is to provide a good understanding of the zero set on non-trivial solutions of Conformable Bessel’s Equation. The Conformable Bessel’s Equation is given by

where α∈(0,1] and p≥0. Clearly, if a = 0, the above equation is just the classical Bessel Equation, ¹⁹. For more information about the conformable Bessel’s function in the solution of wave equation, we refer to ²¹.

For t > 0, making a change variable y=v/tα/2, the (17) transforms into

(To obtain the above equation, we start differentiating the equation tα/2y=v).

Case 1: p>1/2. In this case, compare (18) with

TαTαy(t)+α2y(t)=0,

which has a solution sin(tα) with zeros at t=(nπ)1/α, n∈N. Therefore, a solution of (18) has at least one zero on each of the open interval ([{n-1}π]1α,(nπ)1α], n∈N.

Case 1: 0<p<1/2. In this case, compare (18) with

TαTαy(t)+α2y(t)=0,

and conclude that between any two consecutives zeros, a and b of v(t), there exists one zero of sin(tα). Thus, we have a<(nπ)1/α<b for some n∈N.

4.The study of conformable Green’s Functions