Lyapunov-type Inequalities for a Higher Order Fractional Differential Equation with Fractional Integral Boundary Conditions

New Lyapunov-type inequalities are derived for the fractional boundary value problem D α a u(t) + q(t)u(t) = 0, a < t < b, u(a) = u (a) = · · · = u (n−2) (a) = 0, u(b) = I α a (hu)(b), a denotes the Riemann–Liouville fractional derivative of order α, I α a denotes the Riemann–Liouville fractional integral of order α, and q, h ∈ C([a, b]; R). As an application, we obtain numerical approximations of lower bound for the eigenvalues of corresponding equations.


Introduction
In this paper, we obtain new Lyapunov-type inequalities for the fractional boundary value problem D α a u(t) + q(t)u(t) = 0, a < t < b, (1.1) where n ∈ N, n ≥ 2, n − 1 < α < n, D α a denotes the Riemann-Liouville fractional derivative of order α, I α a denotes the Riemann-Liouville fractional integral of order α, and q, h ∈ C([a, b]; R).We use a Green's function approach that consists in transforming the fractional boundary value problem (1.1)-(1.2) into an equivalent integral form and then find the maximum of the modulus of its Green's function.In the case n = 2, we obtain a generalization of the Lyapunovtype inequality established by Ferreira in [12].In the case n ≥ 3, the study of the Green's function is more complex.The obtained Lyapunov-type inequalities in such case involve the solution of a certain nonlinear equation that belongs to the interval 0, 2α−4 2α−3 α−2 α−1 .Some numerical results are presented in order to estimate such solution for different values of α.As an application of our obtained Lyapunov-type inequalities, we present some numerical approximations of lower bound for the eigenvalues of corresponding equations.
Let us start by describing some historical backgrounds about Lyapunov inequality and some related works.In the late 19th century, the mathematician A. M. Lyapunov established the following result (see [27]).Inequality (1.4) is known as "Lyapunov inequality".It proved to be very useful in various problems in connection with differential equations, including oscillation theory, asymptotic theory, eigenvalue problems, disconjugacy, etc.For more details, we refer the reader to [3-5, 8, 14, 16, 17, 30, 33, 35, 37] and references therein.
In [17] where Using the fact that Lyapunov inequality (1.4) follows immediately from inequality (1.5).Many other generalizations and extensions of inequality (1.4) exist in the literature, see for instance [6, 9-11, 15, 16, 19, 26, 29, 31, 32, 36, 38, 39] and references therein.Due to the positive impact of fractional calculus on several applied sciences (see for instance [25]), several authors investigated Lyapunov type inequalities for various classes of fractional boundary value problems.The first work in this direction is due to Ferreira [12], where he considered the fractional boundary value problem → R is a continuous function, and D α a is the Riemann-Liouville fractional derivative operator of order α.The main result obtained in [12] is the following fractional version of Theorem 1.1.where Γ is the Gamma function.
Observe that (1.4) can be deduced from Theorem 1.2 by passing to the limit as α → 2 in (1.7).
Before stating and proving the main results in this work, some preliminaries are needed.This is the aim of the next section.

Preliminaries
We start this section by briefly recalling some concepts on fractional calculus.
Let I be a certain interval in R. We denote by AC(I; R) the space of real valued and absolutely continuous functions on I.For n = 1, 2, . . ., we denote by AC n (I; R) the space of real valued functions f (x) which have continuous derivatives up to order n − 1 on I with f (n−1) ∈ AC(I; R), that is Cleraly, we have AC 1 (I; R) = AC(I; R).
Definition 2.1 (see [25]).Let f ∈ L 1 ((a, b); R), where (a, b) ∈ R 2 , a < b.The Riemann-Liouville fractional integral of order α > 0 of f is defined by Definition 2.2 (see [25]).Let α > 0 and n be the smallest integer greater or equal than α.The Riemann-Liouville fractional derivative of order α of a function f : provided that the right-hand side is defined almost everywhere on [a, b].
Let α > 0 and n be the smallest integer greater or equal than α.By AC α ([a, b]; R), where (a, b) ∈ R 2 , a < b, we denote the set of all functions f : [a, b] → R that have the representation: where c 0 , c 1 , . . ., The next lemma provides a necessary and sufficient condition for the existence of [18]).Let α > 0 and n be the smallest integer greater or equal than α.

1).
In such a case, we have with c 0 is the constant that appears in the representation (2.1).
Proof.By Lemma 2.3, we have On the other hand, observe that lim where (2.2) where d i are some constants.Next, we have The boundary condition v (a) = 0 yields d n−1 = 0. Continuing this process, we obtain Therefore, By the condition v(b) = 0, we get Then Hence, we have where G is the Green's function defined by (2.2).
that is, Observe that for all t ∈ (a, b), we have Therefore, using (1.1) we obtain On the other hand, using (2.3) we obtain is a solution of the fractional boundary value problem (2.4)-(2.5).Next, using Lemma 2.5 we have Therefore, by (2.3) we obtain which proves the desired result.

Estimates of the Green's function
In this section, we provide estimates of the Green's function G defined by (2.2) in both cases n = 2 and n ≥ 3.
Let us start with the case n = 2.We have the following result established by Ferreira [12].
Lemma 3.1.Let n = 2.The Green's function G defined by (2.2) satisfies the following conditions: The next lemma provides an estimate of the Green's function G in the case n ≥ 3. where Proof.It can be easily seen that Let s ∈ (a, b) be fixed.For s ≤ t ≤ b, we have Differentiating with respect to t, we obtain Observe that On the other hand, we have Therefore, for all s ∈ (a, b), we have s * ∈ (a, b).Moreover, for given s ∈ (a, b), we have G(t, s) arrives at maximum at s * , when s ≤ t.This together with the fact that G(t, s) is increasing on s > t, we obtain that (ii) holds.

Remark 3.3.
Observe that in the case n = 2, that is, 1 < α < 2, we have s * < a.Therefore, the estimates for G(t, s) for n ≥ 3 given in Lemma 3.2 cannot cover those for n = 2 given in Lemma 3.1.

Remark 3.4. A simple computation yields
where Differentiating with respect to z, we obtain where Clearly, for all 0 < z < 1, we have sign(µ (z)) = sign(ν(z)) = sign(P(z)), where Differentiating with respect to z, we obtain Further, we have .
Moreover, we have there exists a unique Hence, we obtain From the above analysis and Lemma 3.2, we deduce the following result.
where z α is the unique zero of the nonlinear equation Tables 3 Figure 3.1 shows the graph of functions y = µ(z) (normalized) and z = z α for α = 5  2 , where µ is defined by (3.4).Observe that the function µ attains its maximum at z = z α , which confirm the above theoretical analysis.

Lyapunov-type inequalities
We distinguish two cases.
We have the following Hartman-Wintner-type inequality for the fractional boundary value problem (4.1)-(4.2).
The following Lyapunov-type inequality for the fractional boundary value problem (4.1)-(4.2) holds.

The case n ≥ 3
We have the following Hartman-Wintner-type inequality for the fractional boundary value problem (1.1)-(1.2), in the case n ≥ 3.
Proof.Inequality (4.7) follows from Lemma (ii) 3.2 and by using similar argument as in the proof Theorem 4.1.
The following Lyapunov-type inequality for the fractional boundary value problem (1.1)-(1.2), in the case n ≥ 3, holds.
where z α is the unique zero of the nonlinear equation Proof.Inequality (4.8) follows immediately from inequality (4.7) and Lemma 3.5.
Taking h ≡ 0 in Theorem 4.7, we obtain the following Hartman-Wintner-type inequality for the fractional boundary value problem (5.1)-(5.2), in the case n ≥ 3.

Applications to eigenvalue problems
In this section, we present some applications of the obtained results to eigenvalue problems.More precisely, we provide lower bound for the eigenvalues of certain nonlocal boundary value problems.
We say that a scalar λ is an eigenvalue of the fractional boundary value problem fund FEDER, and XUNTA de Galicia under grant GRC2015-004.The third author extends his appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).

Table 3 .
(2,3)d 3.2 provide numerical values of z α for different values of α and n.The numerical results are obtained using the bisection method implemented in Matlab.1:Values of z α for α ∈(2,3)