Lyapunov-type inequalities for differential equation with Caputo–Hadamard fractional derivative under multipoint boundary conditions

In this work, we establish Lyapunov-type inequalities for the fractional boundary value problems with Caputo–Hadamard fractional derivative subject to multipoint and integral boundary conditions. As far as we know, there is no literature that has studied these problems.

The study of Lyapunov-type inequalities for fractional differential equations has begun recently. The first result in this direction is due to Ferreira [14]. He obtained a Lyapunov inequality for Riemann-Liouville fractional differential equations; his main result is as follows. where D α a + is the Riemann-Liouville fractional derivative of order α.

Theorem 1.1 If the fractional boundary value problem
One year later, the same author Ferreira [15] obtained a Lyapunov-type inequality for the Caputo fractional boundary value problem.

Theorem 1.2 If the fractional boundary value problem
where q is a real continuous function, has a nontrivial continuous solution, where C D α a + is the Caputo fractional derivative of order α.
After the publication of [14,15], the research on Lyapunov inequalities for fractional differential equations has become a hot topic. The results in the literature can be divided into two categories. The first one is using other fractional derivatives instead of the Caputo fractional derivatives or Riemann-Liouville fractional derivatives in equation (1.3) or (1.6). Secondly, the boundary conditions (1.4) or (1.7) are replaced by multipoint boundary conditions or integral boundary conditions. For instance, in [16][17][18], Lyapunov inequalities for Hadamard fractional differential equations are given. Lyapunov-type inequalities regarding sequential fractional differential equations are obtained in [19][20][21]. The first paper considering integral boundary conditions is also duo to Ferreira [22]. For the results of multipoint boundary conditions, see [23,24].
Motivated by the above works, in this paper, we establish Lyapunov-type inequalities for the fractional boundary value problems with Caputo-Hadamard fractional derivative under multipoint boundary condition where C H D α a + denotes the Caputo-Hadamard fractional derivative of order α.

Preliminaries
In this section, we recall the concepts of the Riemann-Liouville fractional integral, the Riemann-Liouville fractional derivative, the Caputo fractional derivative of order α ≥ 0, and the definition of the Caputo-Hadamard fractional derivative.

Definition 2.2 ([25]) The Riemann-Liouville fractional derivative of order
for α > 0, where m is the smallest integer greater than or equal to α.

Definition 2.3 ([25])
The Caputo fractional derivative of order α ≥ 0 is defined by for α > 0, where m is the smallest integer greater than or equal to α.

Definition 2.4 ([25])
The Hadamard fractional integral of order α ∈ R + for a continuous function f : [a, ∞) → R is defined by

Main results
We begin by writing problem (1.9)-(1.10) in an equivalent integral form.
is a solution to the boundary value problem (1.9)-(1.10) if and only if where c 0 and c 1 are real constants. Since u(a) = 0, we immediately get that c 0 = 0, and thus Hence which concludes the proof.
By Lemma 3.2, r(s) ≥ 0, and we easily obtain Proof A proof of this lemma can be found in [15]. Here we give a new proof. Let 0 < a ≤ s ≤ b. It is easy to check that

By Lemmas 3.2 and 3.3 we obtain
Thus the proof is completed.

Lemma 3.6
The function G defined in Lemma 3.1 satisfies the following property: Proof The Green's function G(t, s) can be rewritten as the following form: Define two functions Obviously, g 2 (t, s) is an increasing function in t, and 0 ≤ g 2 (t, s) ≤ g 2 (s, s). By Lemma 3.3 we obtain Now we turn our attention to the function g 1 (t, s). We start by fixing an arbitrary s ∈ [a, b).
Differentiating g 1 (t, s) with respect to t, we get It follows that ∂g 1 (t * ,s) ∂t = 0 if and only if t * = se , and therefore ∂g 1 (t,s) ∂t < 0, g 1 (t, s) is strictly decreasing with respect to t, and thus we have 0 = g 1 (b, s) ≤ g 1 (t, s) ≤ g 1 (s, s) = g 2 (s, s).
From this we conclude that It remains to verify the result for s ≤ b(a/b) α-1 , that is, for t * ≤ b. It is easy to check that ∂g 1 (t,s) ∂t < 0 for t < t * and ∂g 1 (t,s) ∂t ≥ 0 for t ≥ t * . This, together with the fact that g 1 (b, s) = 0, implies that g 1 (t * , s) ≤ 0, and we only have to show that Indeed, by Lemmas 3.4 and 3.5 we obtain The proof is completed.
Now we are ready to prove our Lyapunov-type inequality.

Theorem 3.7 If a nontrivial continuous solution of the Caputo-Hadamard fractional boundary value problem
] be the Banach space endowed with norm u = sup t∈ [a,b] |u(t)|. It follows from Lemma 3.1 that a solution u to the boundary value problem satisfies the integral equation Now an application of Lemma 3.6 yields which implies that (3.1) holds.

Remarks
Applying the Green's approach, we can also obtain Lyapunov-type inequalities for Caputo-Hadamard fractional differential equations under integral boundary conditions,

Lemma 4.1 A function u ∈ C[a, b] is a solution to the boundary value problem (4.1)-(4.2)
if and only if it satisfies the integral equation where c 0 and c 1 are real constants. Since u(a) = 0, we immediately get that c 0 = 0, and thus