Lyapunov-Type Inequalities for Riemann-Liouville Type Fractional Boundary Value Problems with Fractional Boundary Conditions

In this article, we establish Lyapunov-type inequalities for two-point Riemann-Liouville type fractional boundary value problems associated with well-posed fractional boundary conditions. To illustrate the applicability of established results, we estimate lower bounds for eigenvalues of the corresponding eigenvalue problems and deduce criteria for the nonexistence of real zeros of certain Mittag-Leffler functions.


Introduction
Lyapunov [10] established a necessary condition, known as the Lyapunov inequality, for the existence of a nontrivial solution of Hill's equation associated with Dirichlet boundary conditions. This inequality has several applications in various problems related to the theory of differential equations. Due to its importance, the Lyapunov inequality has been generalized in many forms. For a detailed discussion on Lyapunov-type inequalities and their applications, we refer [2,12,13,17,19,20] and the references therein.
Recently, many researchers have derived Lyapunov-type inequalities for various classes of fractional boundary value problems [8,15,16,18,21]. For the first time, Ferreira [5] obtained a Lyapunov-type inequality for a two-point Riemann-Liouville type fractional boundary value problem associated with Dirichlet boundary conditions as follows: Theorem 1.1. [5] If the fractional boundary value problem Recently, Ntouyas et al. [11] presented a survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions. This article shows a gap in the literature on Lyapunov-type inequalities for two-point Riemann-Liouville type fractional boundary value problems associated with fractional boundary conditions. In 2016, Dhar et al. [3] derived Lyapunov-type inequalities for two-point Riemann-Liouville type fractional boundary value problems associated with fractional integral boundary conditions. This article stresses the importance of choosing well-posed boundary conditions for Riemann-Liouville type fractional boundary value problems. In this line, the authors [7] have obtained Lyapunov-type inequalities for two-point Riemann-Liouville type fractional boundary value problems associated with well-posed mixed, Sturm-Liouville, Robin and general boundary conditions, recently.
Motivated by these developments, in this article, we establish Lyapunov-type inequalities for two-point Riemann-Liouville type fractional boundary value problems associated with well-posed fractional boundary conditions.

Preliminaries
Throughout, we shall use the following notations, definitions and known results of fractional calculus [9,14]. Denote the set of all real numbers and complex numbers by R and C, respectively. Definition 2.1. [9] The Euler gamma function is defined by Using the reduction formula Γ(z + 1) = zΓ(z), (z) > 0, the Euler gamma function can be extended to the half-plane (z) ≤ 0 except for z = 0, −1, −2, . . .

Definition 2.2.
[9] Let α > 0 and a ∈ R. The α th -order Riemann-Liouville fractional integral of a function y : [a, b] → R is defined by provided the right-hand side exists. For α = 0, define I α a to be the identity map. Moreover, let n denote a positive integer and assume n − 1 < α ≤ n. The α th -order Riemann-Liouville fractional derivative is defined as where D n denotes the classical n th -order derivative, if the right-hand side exists. |y(t)|.
[1] Let α > 0 and n be a positive integer such that n − 1 < α ≤ n. Then, the fractional differential equation D α a y(t) = 0, a < t < b, has a unique solution y ∈ C(a, b) ∩ L(a, b), and is given by for some C i ∈ R, i = 1, 2, · · · , n.

Main Results
In this section, we obtain two Lyapunov-type inequalities for the fractional boundary value problem using the properties of the corresponding Green's function.
has the unique solution where Proof. Applying I α a on both sides of (3) and using Lemma 2.8, we have for some C 1 , C 2 ∈ R. Applying D β a on both sides of (5), using Lemma 2.5 and Lemma 2.6, we get Using y(a) = 0 in (5), we get C 2 = 0. Using D β a y(b) = 0 in (6), we get Substituting C 1 and C 2 in (5), the unique solution of (3) is The proof is complete.
Then, the fractional boundary value problem (3) has the unique solution Proof. The proof is similar to the proof of Theorem 3.1.
The proof is complete.
Corollary 3.5. The Green's function G(t, s) for Corollary 3.2 satisfies implying that The proof is complete.
Proof. First, we show that for any fixed s ∈ [a, b], The proof is complete.
Proof. First, we show that for any fixed s ∈ [a, b], G(t, s) increases from G(a, s) to G(s, s), and then decreases to G(b, s). Let a ≤ t ≤ s ≤ b and consider To prove the second part, consider Differentiating G(s, s) with respect to s and equating it to 0, we obtain s = a+b 2 . Again, differentiating G (s, s) with respect to s, we observe that G (s, s) ≤ 0 at s = a+b 2 . So, G(s, s) attains its maximum at s = a+b 2 . The proof is complete.
Corollary 3.8. For the Green's functions G(t, s) defined in (4) and (7), Take Differentiating f (t) with respect to t and equating it to 0, we obtain Again, differentiating f (t) with respect to t, we observe that f (t) ≤ 0 at t = A. So, f (t) attains its maximum at t = A. The proof is complete.
We are now able to formulate Lyapunov-type inequalities for the fractional boundary value problem (3).
has a nontrivial solution, then Proof. Let B = C[a, b] be the Banach space of functions endowed with norm |y(t)|.
It follows from Theorem 3.1 that a solution to (8) satisfies the equation Hence, An application of Theorem 3.6 yields the result. Hence, An application of Corollary 3.7 yields the result.

Applications
In this section, we discuss two applications of Theorem 3.10 and Corollary 3.11. First, we estimate lower bounds for the eigenvalues of the Riemann-Liouville type fractional eigenvalue problems corresponding to (8).
The proof is complete.
Proof. The proof is similar to the proof of Theorem 4.3.