A Lyapunov-Type Inequality for Partial Differential Equation Involving the Mixed Caputo Derivative

: In this work, we derive a Lyapunov-type inequality for a partial differential equation on a rectangular domain with the mixed Caputo derivative subject to Dirichlet-type boundary conditions. The obtained inequality provides a necessary condition for the existence of nontrivial solutions to the considered problem and an example is given to illustrate it. Moreover, we present some applications to demonstrate the effectiveness of the new results.


Introduction
In this paper, we focus on the representation of the Lyapunov-type inequality for the following boundary value problem: where a, b > 0, r = (r 1 , r 2 ), 1 < r 1 , r 2 < 2, C D r 0 is the mixed Caputo derivative of order r and q : [0, a] × [0, b] → R is a given Lebesgue integrable function. To do that, we convert problem (1) into an integral equation. With the help of the properties of its Green function, we establish a new Lyapunov-type inequality, which provides a necessary condition for the existence of nontrivial solutions to problem (1). Furthermore, an example is given to illustrate it. In the end, we apply the obtained inequality to prove the uniqueness of solutions for the nonhomogenous boundary value problem and derive an estimation related to the eigenvalue of the corresponding equation.
The well-known Lyapunov inequality [1] states the fact that if the boundary value problem has a nontrival solution u ∈ C 2 [a, b], then the inequality b a |q(t)|dt holds, where q is a real-valued continuous function. The Lyapunov inequality (3) was regarded as a very important and useful tool in the study of differential equations, especially in the aspect of stability theory, oscillation theory, intervals of disconjugacy, and eigenvalue problems [2][3][4][5]. Subsequently, there were many improvements and extensions of the inequality (3) related to integer-order derivative, Definition 1 ([14]). For f (x, y) ∈ L(P), r 1 , r 2 > 0, the expression is called the left mixed Riemann-Liouville integral of order r of f (x, y).
In particular,
In particular,

Lemma 4.
Let 0 < r 1 , r 2 ≤ 1. If f (x, y) ∈ C(P) and I r 0 f (x, y) ∈ AC(P), then Proof. Since f (x, y) ∈ C(P), it exists a constant M > 0 such that Therefore, It follows from (4) and By virtue of Lemmas 2 and 3 and (5), we get The proof is completed. Now, we pass to mixed Caputo derivative of large order r = (r 1 , r 2 ) ∈ (1, 2] × (1, 2]. In the higher order case, we can generalize to the following. is called the mixed Caputo derivative of order r of f (x, y), where D 2 xy = D xy D xy .
(1) According to Definition 4 and Lemmas 1 and 4, we have (2) Using Definition 4 and Lemmas 1 and 5, we get The proof is completed.

A Lyapunov-Type Inequality for Problem (1)
In order to obtain the Lyapunov-type inequality for problem (1), we first give an expression for the Green's function of the the boundary value problem (1) and its properties. Then, a Lyapunov-type inequality for problem (1) is presented by making use of the properties of the obtained Green's function.

Lemma 7.
Assume that q ∈ L(P). A function u is a solution of problem (1), then it satisfies the integral equation where the Green function G(x, y, s, t) is given by where H(x, s) and K(y, t) are given by Proof. If u(x, y) is a solution of (1), applying the integral operator I r 0 to (1) and making use of Lemma 6, we have u(x, y) = γ(x, y) − I r 0 (q(x, y)u(x, y)). Let By virtue of boundary value conditions u(0, y) = u(x, 0) = u(a, y) = u(x, b) = 0, we get and Applying (11), (12), and u(x, 0) = u(0, y) = 0 to (10), we have Let x = a and y = b in (15) at the same time, we can calculate Furthermore, let x = a and y = b in (15) respectively, we can obtain and Applying (16) x a I r 0 (q(a, y)u(a, y)) − I r 0 (q(x, y)u(x, y)) where G(x, y, s, t) is given by (7)- (9). The proof is completed. (7) satisfies

Lemma 8. The Green function G given by
where v(r) = 1 Proof. It follows from (7) that In the case 0 ≤ s ≤ x ≤ a, for fixed x ∈ [0, a], we have Obviously,
Analogously, we can obtain the fact that In conclusion, (19) is obtained with the help of (21), (28), and (29). The proof is completed.
Our main aim is the following Lyapunov-type inequality.

Applications
In this section, some applications of the obtained Lyapunov-type inequality (30) in Section 3 are presented.
Proof. Assume that u 1 (x, y), u 2 (x, y) are both solutions to problem (34), then u(x, y) = u 1 (x, y) − u 2 (x, y) is a solution of the corresponding homogenous boundary value problem. By virtue of Remark 1, the corresponding homogeneous boundary value problem has only zero solution in C(P) ∩ C 2 (P). Therefore, problem (34) has a unique solution.
The other application is that we derive an estimation related to the eigenvalue of the corresponding equation by using our obtained Lyapunov-type inequality (30). For given λ ∈ R, we consider the following boundary value problem where C D r 0 is the mixed Caputo derivative of order r and a, b > 0, r = (r 1 , r 2 ), 1 < r 1 , r 2 < 2. If problem (36) admits a nontrivial solution u λ ∈ C(P) ∩ C 2 (P), we say that λ is an eigenvalue of problem (36).
The proof is completed.

Conclusions
In this article, we consider a partial differential equation on a rectangular domain with the mixed Caputo derivative subject to Dirichlet-type boundary conditions. A new Lyapunov-type inequality for the considered problem is derived. The obtained inequality provides a necessary condition for the existence of nontrivial solutions. Our approach is based on converting the boundary value problem into an integral equation and then finding the maximum value of its Green's function. We give two applications related to our obtained inequality. The new results generalize some existing results in the literature. We expect that the proposed approaches and the obtained results in this paper can be adapted to study other fractional boundary value problems.