Keywords:
1. Introduction
Lyapunov-type inequality is an important and useful tool for studying differential equations. The classical Lyapunov-type inequality for differential equations was studied in [1] :
(1.1)
if (1.1) has a nontrivial solution, then
(1.2)
Furthermore, the constant 4 in (1.2) is sharp.
More authors paid attention to study Lyapunov-type inequality for differential equations and got many results. In recent years, a series of achievements have been made in the Lyapunov-type inequalities of fractional differential equations. We refer to [2] - [12] . In [3] , Ferreira studied the following equations:
(1.3)
if (1.3) has a nontrivial solution, then
In [7] , Abdeljanad and Baleanu obtained a Lyapunov-type inequality for ABR fractional boundary value problem
(1.4)
if (1.4) has a nontrivial solution, then
where
In [10] , Abdeljawad studied a generalized Lyapunov-type inequalities for conformable BVP
(1.5)
if (1.5) has a nontrivial solution, then
Furthermore, Abdeljawad proved a Lyapunov-type inequalitiy for a sequential conformable BVP
(1.6)
if (1.6) has a nontrivial solution, then
In this paper, we establish a Lyapunov-type inequalities for conformable BVP
(1.7)
and
(1.8)
where
is conformable fractional derivative starting at a of order
, and
are real-valued continuous. The introduction and background of conformable fractional are given in [2] [10] . Then, we give the definition and lemma about conformable fractional derivative in the following.
Definition 1.1. [4] Let
. Then
is called the left conformable fractional derivative starting at c of order
.
Lemma 1.1. [4] Let
be
times differentiable for
,
. Then, we have the following result:
2. A Lyapunov-Type Inequality for Conformable Fractional Derivative of
Theorem 2.1.
is a solution of the BVP (1.7) if and only if y satisfies the integral equation
(2.1)
where
is the Green’s function defined as
(2.2)
Proof. Applying the integral
in the (1.7), we have
Then, using definition 1.1 and lemma 1.1, we obtain
(2.3)
Since
, we get immediately that
.
By the boundary condition
, we obtain
Hence, equation (2.3) becomes
(2.4)
Then, equation (2.4) can be written in the form of (2.1), where the Green’s function is defined in (2.2).
The proof is completed.
Corollary 2.1. The function G defined in Theorem 2.1 satisfied the following property:
(2.5)
Proof. We define the function
and
For
, differentiating
with respect to s, we get
(2.6)
While for
, differentiating
with respect to s, we get
(2.7)
Hence,
is a decreasing function,
is an increasing function in s. Consequently, G (t, s) gets the maximum at s = t, we obtain (2.5).
Corollary 2.2. If (1.7) has a nontrivial continuous solution, then
(2.8)
Proof. Let
be a nontrivial solution of the BVP (1.7), where the norm
Form (2.1), we have
(2.9)
Taking the norm leads to
Then,
This completes the proof.
Corollary 2.3. If the BVP (1.7) has a nontrivial continuous solution, then
(2.10)
Proof. In (2.8), let
Differentiating
on
, we have
hence,
is a increasing function, we have
Then,
Hence, we get the inequality (2.10). The proof is complete.
Example 2.1. If the BVP
has a nontrivial solution, then
(2.11)
Proof. Assume that
is an eigenvalue of (1.7). By using Corollary 2.3, we have
Hence, we get the desired result (2.11). The proof is complete.
3. A Lyapunov-Type Inequality for Conformable Fractional Derivative of
Theorem 3.1.
is a solution of the BVP (1.8) if and only if f satisfies the integral equation
(3.12)
where
is the Green’s function defined as
(3.13)
Proof. Applying the integral
in the (1.8), we have
Then, using definition 1.1 and lemma 1.1, we obtain
(3.14)
Since
, we get immediately that
.
By the boundary condition
, we obtain
Hence, equation (3.14) becomes
(3.15)
Then equation (3.15) can be written in the form of (3.12), where the Green’s function is defined in (3.13). The proof is completed.
Corollary 3.1. The function H defined in Theorem 3.1 satisfied the following property:
Proof. We define the function
and
For
, differentiating
with respect to t, we get
(3.16)
Hence,
is an increasing function in t.
While for
, differentiating
with respect to t, we get
Let
then, we have
Hence,
That we obtain
is an increasing function in t. Consequently,
gets the maximum at
. We have
and
Hence,
, we obtain
Furthermore, we have
Hence,
The proof is completed.
Corollary 3.2. If (1.8) has a nontrivial continuous solution, then
(3.17)
Proof. Let
be a nontrivial solution of the BVP (1.8), where the norm
Form (3.1), we have
(3.18)
Taking the norm leads to
Then,
Hence, we get the inequality in (3.17). This completes the proof.
Example 3.1. If the BVP
has a nontrivial continuous solution, then
(3.19)
Proof. Assume that
is an eigenvalue of (1.8). By using Corollary 3.2, we have
Then, we obtain
We get the desired result (3.19). The proof is complete.
4. Conclusion
On the base of [10] , by changing and increasing the edge value conditions, we establish some new Lyapunov-type inequalities for conformable BVP with the conformable derivative of order
and
. In Section 2 and Section 3, by Green’s function and its corresponding maximum value, we obtain new results about Lyapunov-type inequalities for conformable BVP.
Funding
This research is supported by National Science Foundation of China (11671227) and Academic dissertation research innovation funding fund.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.