Lyapunov-type inequalities for an anti-periodic fractional boundary value problem involving ψ-Caputo fractional derivative

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Introduction
In this paper, we are concerned with the anti-periodic fractional boundary value problem is the ψ-Caputo fractional derivative of order α, and f : [a, b] × R → R is a given function.
A Lyapunov-type inequality is derived for problem (1.1). Next, as an application of the obtained inequality, an upper bound of possible eigenvalues of the corresponding problem is obtained.
Let us mention some motivations for studying problem (1.1). Suppose that u ∈ C 2 ([a, b]), (a, b) ∈ R 2 , a < b, is a nontrivial solution to the boundary value problem where w ∈ C( [a, b]) is a given function. Then (see [17] Inequality (1.3) is known in the literature as Lyapunov's inequality, which provides a necessary condition for the existence of a nontrivial solution to (1.2). Many generalizations and extensions of (1.3) were derived by many authors. In particular, Hartman and Wintner [9] proved that if It can be easily seen that (1.3) follows from (1.4). For other results related to Lyapunovtype inequalities, see, for example, [3,5,16,18,19,21] and the references therein. On the other hand, due to the importance of fractional calculus in applications, the study of Lyapunov-type inequalities was extended to fractional boundary value problems by many authors. The first contribution in this direction is due to Ferreira [6], where the fractional boundary value problem with w ∈ C([a, b]), 1 < α < 2 and D α a is the Riemann-Liouville fractional derivative of order α, was studied. The main result in [6] is the following: If u is a nontrivial solution to (1.6) Note that in the limit case α = 2, (1.5) reduces to (1.2). Moreover, taking α = 2 in (1.6), we obtain (1.3). For other works related to Lyapunov-type inequalities for fractional boundary value problems, see, for example, [4,7,8,[10][11][12]20] and the references therein. In particular, in [8], the anti-periodic fractional boundary value problem where w ∈ C([a, b]), 1 < α < 2 and C D α a is the Caputo fractional derivative of order α, was studied. Note that (1.7) is a special case of (1.1) with ψ(x) = x and f (x, z) = w(x)z.
Motivated by the above cited works, the problem (1.1) is investigated in this paper. The rest of the paper is organized as follows. In Sect. 2, we recall some basic concepts on fractional calculus and prove some preliminary results. In Sect. 3, a Lyapunov-type inequality is established for problem (1.1). Moreover, some particular cases are discussed. Next, an application to fractional eigenvalue problems is given. In Sect. 4, we end the paper with some open questions.

Methods and preliminaries
The main idea in this paper consists to reduce (1.1) to a fractional boundary value problem involving Caputo fractional derivative by using an adequate change of variable. Next, using an integral representation of the solution and an estimate of the corresponding Green's function, a Lyapunov-type inequality is derived for (1.1) under certain assumptions on the functions f and ψ. Before stating and proving the main results, we need some preliminaries on fractional calculus. The main references used in this part are [2,13]. For other references related to fractional calculus, see, for example, [1,14,15].
Let β > 0. The Riemann-Liouville fractional integral of order β of a function f ∈ C([a, b]) is given by (see [13]) where is the Gamma function.
The Caputo fractional derivative of order α of a function f ∈ C 2 ([a, b]) is given by (see [13]) The fractional integral of order β > 0 of a function f ∈ C([a, b]) with respect to ψ is given by (see [13]) The ψ-Caputo fractional derivative of order α of a function f ∈ C 2 ([a, b]) is given by (see [2]) The following lemma is crucial for the proof of our main result.
Let us consider the change of variable Using the chain rule, we have Hence, we obtain i.e., We refer the reader to Ferreira [8] for the proofs of the following results.
if and only if

A Lyapunov-type inequality for problem (1.1)
In this section, problem (1.1) is investigated under the following assumptions: where q ∈ C([a, b]). Observe that by (A3), we have f (x, 0) = 0, for all x ∈ ]a, b[. Therefore, 0 is a trivial solution to (1.1).
Our main result is given by the following theorem.

.3, for all
Since v ∞ > 0 (because v is nontrivial), we obtain Finally, using the change of variable inequality (3.1) follows.

Corollary 3.2 Let u ∈ C 2 ([a, b]) be a nontrivial solution to
Next, let us consider the fractional boundary value problem where 1 < α < 2 and w ∈ C ([a, b]). Problem (3.6) is a special case of (3.5) with Observe that the function f satisfies assumption (A3) with Therefore, by Corollary 3.2, we deduce the following result, which was derived in [8] (with strict inequality). (3.6).

Corollary 3.3 Let u ∈ C 2 ([a, b]) be a nontrivial solution to
. (3.7) Let us consider the fractional boundary value problem where 1 < α < 2 and w ∈ C([a, b]). Problem (3.6) is a special case of (3.5) with Observe that the function f satisfies assumption (A3) with Therefore, by Corollary 3.2, we deduce the following result. Let us consider the fractional boundary value problem where 1 < α < 2 and w ∈ C ([a, b]). Problem (3.9) is a special case of (3.5) with Note that the function f satisfies assumption (A3) with Therefore, by Corollary 3.2, we deduce the following result. Further, we consider the case 10) where N ≥ 1 is a natural number, c 1 > 0 and c 2 ∈ R. Observe that ψ ∈ C 2 ([-1, 1]). Moreover, we have Observe also that ψ (-1) = ψ (1) = c 1 + 1.

Corollary 3.7 Let u ∈ C 2 ([a, b]) be a nontrivial solution to
.

An application to eigenvalue problems
Let ψ ∈ C 2 ([a, b]) be a given function satisfying assumptions (A1) and (A2). We say that λ ∈ R is an eigenvalue of the fractional boundary value problem where 1 < α < 2, if and only if (3.14) admits a nontrivial solution u λ ∈ C 2 ([a, b]).
The following result provides an upper bound of possible eigenvalues of (3.14).

Theorem 3.8
If λ is an eigenvalue of (3.14), then Proof Let λ ∈ R be an eigenvalue of (3.14). Then (3.14) admits a nontrivial solution u λ ∈ C 2 ([a, b]). On the other hand, observe that (3.14) is a special case of (1.1) with Moreover, the function f satisfies assumption (A3) with Hence, by Theorem 3.1, we obtain Therefore, we proved (3.15). Taking in (3.14), we deduce the following result, which was obtained in [8].

Conclusion
In this paper, a Lyapunov-type inequality is established for the fractional boundary value problem (1.1) under assumptions (A1), (A2) and (A3). Next, the obtained inequality is used to obtain bounds on possible eigenvalues of the corresponding problem. We end the paper with the following open questions. First, it would be interesting to compute the Green's function for the fractional boundary value problem where μ > 0, h ∈ C([A, B]), and to obtain an estimate similar to that given by Lemma 2.3. Next, the obtained estimate can be used to derive a Lyapunov-type inequality for problem (1.1) by considering a more general class of functions ψ without assumption (A2). In fact, from the proof of Theorem 3.1, the function v given by