Hartman-Type and Lyapunov-Type Inequalities for a Fractional Differential Equation with Fractional Boundary Conditions

We prove Hartman-type and Lyapunov-type inequalities for a class of Riemann–Liouville fractional boundary value problems with fractional boundary conditions. Some applications including a lower bound for the corresponding eigenvalue problem are obtained.


Introduction
In [1], Lyapunov established the following striking inequality: Theorem 1. Let q ∈ C([a, b], R). Assume that the problem has a solution ω ∈ C( [a, b], R) such that ω(x) ≠ 0 for x ∈ (a, b). en, and constant 4 is the best possible largest number.
In [11], the authors use the Hahn integral operator to prove a description of new generalization of Minkowski's inequality.
In [5], the authors improve inequality in (2) by proving the following Hartman-Winter inequality: where q + (z) � max(q(z), 0) is the nonnegative part of q(z).
Inequality (3) is also known as the best Lyapunov inequality.
In [17], Ferreira considered the following fractional differential problem: where q ∈ C([a, b], R) and D α a + denotes the Riemann-Liouville fractional derivative of order α (see Definition 2 in the following). e author established the following Lyapunov-type inequality for problem (4).
Theorem 2 (see [17]). Assume that problem (4) has a solution ω ∈ C( [a, b], R) such that ω(x) ≠ 0 for x ∈ (a, b). en, Remark 1. Note that if we let α � 2 in (5), one obtains Lyapunov's classical inequality (2). For the convenience of the reader, we recall the concept of fractional integral and derivative of order c ≥ 0.
Definition 1 (see [18,19]). e Riemann-Liouville fractional integral of order c ≥ 0 for a real-valued function ω is defined by where Γ(c) is the Euler gamma function.
Definition 2 (see [18,19]). e Riemann-Liouville fractional derivative of order c ≥ 0 for function ω is defined by where n � [c] + 1 with [c] the integer part of c. e new development in fractional calculus has attracted the attention of researchers of various disciplines. Different mathematical procedures have been considered by several authors through different research-oriented aspects of fractional differential equations (see, for instance, [20][21][22] and the references therein).
Our goal in this paper is to establish Hartman-type and Lyapunov-type inequalities for the following problem: where α ∈ (3, 4] and q ∈ C([a, b], R). Some applications are given to illustrate our result. e organization of the paper is as follows. In Section 2, we derive the explicit expression of the Green function corresponding to problem (8) and we establish some properties on it. is allows us to prove Hartman-type and Lyapunov-type inequalities for problem (8). In Section 3, we present some applications including a lower bound for the corresponding eigenvalue problem.

Green's Function.
First, we recall the following wellknown properties (see, for example, [18,19]). en, where ) be a solution of problem (8). en, where G α (x, y) is Green's function of problem (8) given by Proof. Let ω be such solution. By Lemma 1, we have Using 2 Discrete Dynamics in Nature and Society is ends the proof.

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To get a quick perspective, in Figure 1, we have the representation of Green's function G 7/2 (x, y) with the contours and some projections.
One can see from Figure 1 that Green's function G 7/2 (x, y) ≥ 0 and it is nondecreasing with respect to the first variable.
We say that if there exists c > 0 such that Remark 2. Let τ > 0 and x, z ∈ [0, 1]. en, Next, we establish some properties on Green's function G α (x, y) given by (11).
(iii) e function G α satisfies the following property: Proof (i) From Lemma 2, for x, y ∈ (a, b), we have where Hence, inequalities in (16) follow by observing that (ii) We have Similar to case (i), by using the fact that and applying Remark 2 with τ � α − 2 and z � (b − y)/(b − a) ∈ [0, 1], we obtain the required result. y) is nondecreasing on [a, b], we deduce that Discrete Dynamics in Nature and Society 3 is completes the proof.
Proof. From Lemma 2, we know that Without loss of generality, we may assume that ω(x) > 0 for x ∈ (a, b).

Lower Bound for the Eigenvalues.
Consider the following eigenvalue problem: Theorem 4. Assume that eigenvalue problem (42) has a solution ω ∈ C([a, b], R) such that ω(x) ≠ 0 for x ∈ (a, b). en, Proof. By Remark 3 (with a � 0 and b � 1), we have from which inequality (43) follows by observing that .
Proof. e assertion follows from Corollary 2.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.
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