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Lyapunov-type inequalities for sequential fractional boundary value problems using Hilfer’s fractional derivative
Journal of Inequalities and Applications volume 2019, Article number: 98 (2019)
Abstract
This paper is devoted to studying the Lyapunov-type inequality for sequential Hilfer fractional boundary value problems. We first provide some properties of Hilfer fractional derivative, and then establish Lyapunov-type inequalities for a sequential Hilfer fractional differential equation with two types of multi-point boundary conditions. Our results generalize and compliment the existing results in the literature.
1 Introduction
As is well known, Lyapunov inequality was first introduced by Lyapunov [1], who established a necessary condition for the existence of nontrivial solution of the boundary value problem (BVP for short):
as the form
where \(q \in C([a,b],\mathbb{R})\). Since then, Lyapunov inequality and Lyapunov-type inequality have been studied with great interest, and they have been proved to be an effective tool in the study of differential and difference equations, such as oscillation theory, disconjugacy, eigenvalue problems, etc. (see [2,3,4,5]).
In recent years, by the rise of theoretical research in fractional differential equations, there has been tremendous interest in the research of Lyapunov-type inequalities for fractional BVP, see [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] and the references cited therein.
In [11], Ferreira discussed the Lyapunov-type inequality for the following fractional BVP:
where \(D_{a+}^{\alpha }\) is the left Riemann–Liouville fractional derivative of order α, \(\alpha \in (1,2]\) and \(q \in C([a,b],\mathbb{R})\). The Lyapunov-type inequality for problem (1.3) was established as follows:
Furthermore, in 2016, Ferreira [12] considered the Lyapunov-type inequality for a sequential fractional BVP
where \({}_{a}{D^{\gamma }}\), \(\gamma = \alpha \), β stand for the left Riemann–Liouville fractional derivative or the left Caputo fractional derivative of order γ, \(\gamma \in (0,1]\) and \(1 < \alpha + \beta \leqslant 2\), \(q \in C([a,b],\mathbb{R})\). Two interesting results have been obtained as follows:
-
(i)
Take \({}_{a}{D^{\gamma }}\), \(\gamma = \alpha \), β, is the left Riemann–Liouville fractional derivative, then problem (1.5) has a nontrivial continuous solution provided that
$$ \int _{a}^{b} { \bigl\vert q(s) \bigr\vert \,ds > \varGamma (\alpha + \beta )} { \biggl( {\frac{4}{ {b - a}}} \biggr)^{\alpha + \beta - 1}}. $$(1.6) -
(ii)
Take \({}_{a}{D^{\gamma }}\), \(\gamma = \alpha \), β, is the left Caputo fractional derivative, then problem (1.5) has a nontrivial continuous solution provided that
$$ \int _{a}^{b} { \bigl\vert q(s) \bigr\vert } \,ds>\frac{{\varGamma (\alpha + \beta )}}{{{{(b-a)} ^{\alpha + \beta - 1}}}}\frac{{{{(\alpha + 2\beta - 1)}^{\alpha + 2 \beta - 1}}}}{{{{(\alpha + \beta - 1)}^{\alpha + \beta - 1}}{\beta ^{\beta }}}}. $$(1.7)
It is worth noting that the study of the Hilfer fractional differential equations has received a significant amount of attention in the last few years. Hilfer fractional derivative was proposed by Hilfer in 2000, which is a generalization of both Riemann–Liouville and Caputo fractional derivatives (see [31]). Meanwhile, the discussion of Lyapunov-type inequalities for fractional BVP with Hilfer fractional derivative can be found in papers [25, 29, 30].
In [25], Pathak investigated Lyapunov-type inequalities for the following Hilfer fractional differential equation:
with boundary conditions
or
where \(D_{a+}^{\alpha ,\beta }\) is the left Hilfer fractional derivative of order α and type β, \(\alpha \in (1,2]\), \(\beta \in [0,1]\), \(q \in C([a,b],\mathbb{R})\). The author got two Lyapunov-type inequalities: for BVP (1.8), (1.9) as
In [30], Wang obtained Lyapunov-type inequality for the fractional multi-point BVP involving Hilfer derivative:
where \(q \in C([a,b],\mathbb{R})\), \(D_{a+}^{\alpha ,\beta }\) is the left Hilfer fractional derivative of order α and type β with \(\alpha \in (1,2]\), \(\beta \in [0,1]\), \(a <{\xi _{1}} <{\xi _{2}} <\cdots <{\xi _{m-2}}< b\), \({\beta _{i}}\geqslant 0\) (\(i=1,2,\ldots ,m-2\)), \({(b-a)^{1-(2- \alpha )(1-\beta )}}> \sum_{i=1}^{m-2} {{\beta _{i}}} {( {\xi _{i}}-a)^{1-(2-\alpha )(1-\beta )}}\). The Lyapunov-type inequality for problem (1.13) is given as follows:
where
Motivated by these results, in this paper we study Lyapunov-type inequalities for a sequential Hilfer fractional differential equation
with multi-point boundary conditions
or
where \(q \in C([a,b],\mathbb{R})\), \(D_{a+}^{{\alpha _{i}},{\beta _{i}}}\), \(i=1\), 2, are two left Hilfer fractional derivatives of order \({\alpha _{i}}\) and types \({\beta _{i}}\) with \({\alpha _{i}} \in (0,1]\), \(1<{\alpha _{1}}+{\alpha _{2}} \leqslant 2\), \({\beta _{i}} \in [0,1]\). For the definition of Hilfer fractional derivative, see Sect. 2. A remarkable characteristic of this kind of fractional derivative is that the type \({\beta _{i}}\) allows \(D_{a+}^{{\alpha _{i}},{\beta _{i}}}\) to interpolate continuously from the Riemann–Liouville case \(D_{a+}^{{\alpha _{i}},0} \equiv D_{a+}^{\alpha _{i}}\) to the Caputo case \(D_{a+}^{{\alpha _{i}},1} \equiv {}^{C}D_{a+}^{\alpha _{i}}\) (see [32]). To state our main results, we assume that the following conditions hold:
- \((A_{1})\) :
-
\(a < {\xi _{1}} < {\xi _{2}} < \cdots < {\xi _{m - 2}} < b, {\sigma _{i}} \geqslant 0\) (\(i = 1,2, \ldots ,m - 2\)) and
$$ {\Delta _{1}} := {(b-a)^{{\alpha _{2}}-(1-{\alpha _{1}})(1-{\beta _{1}})}} -\sum _{i = 1}^{m - 2} {{\sigma _{i}}} {({\xi _{i}} - a)^{ {\alpha _{2}} - (1 - {\alpha _{1}})(1 - {\beta _{1}})}} > 0. $$ - \((A_{2})\) :
-
\(a < {\eta _{1}} < {\eta _{2}} < \cdots < {\eta _{m - 2}} < b, {\delta _{i}} \geqslant 0\) (\(i = 1,2, \ldots ,m - 2\)) and
$$\begin{aligned} {\Delta _{2}} :=& \bigl[{\alpha _{2}} - (1 - {\alpha _{1}}) (1 - {\beta _{1}})\bigr]{(b - a)^{{\alpha _{2}} - 1 - (1 - {\alpha _{1}})(1 - {\beta _{1}})}} \\ &{} - \sum_{i = 1}^{m - 2} {{\delta _{i}}} {({\eta _{i}} - a)^{{\alpha _{2}} - (1 - {\alpha _{1}})(1 - {\beta _{1}})}} > 0. \end{aligned}$$
In the present work, we are focused on establishing the Lyapunov-type inequalities for a sequential Hilfer fractional differential equation with two types of multi-point boundary conditions. As far as we know, the Lyapunov-type inequality for fractional BVP with Hilfer derivative has seldom been considered up to now. The new insights of this paper can be presented as follows. On the one hand, we provide some properties of Hilfer fractional derivative, which are not introduced in the previous paper (see Sect. 2, Lemmas 2.6, 2.7, and 2.8). On the other hand, we extend the previous results. Many previous results are a special case of our work, which is embodied in Sect. 3. The main difficulties in this article are as follows. First, we have to construct Green’s function for BVPs (1.14), (1.15) and (1.14), (1.16). Second, it is difficult to estimate the maximum of Green’s function because Green’s functions do not satisfy the non-negativity.
The rest of this paper is organized as follows. In Sect. 2, we recall some definitions, lemmas of fractional calculus. In Sect. 3, by constructing Green’s functions and finding its corresponding maximum value, we prove our main results. Finally, a conclusion is given in Sect. 4.
2 Preliminaries
In this section, we recall some definitions and lemmas which are used throughout this paper.
Definition 2.1
Let \(J = [a,b]\) (\(-\infty < a < b < \infty \)). The left Riemann–Liouville fractional integral of order \(\alpha > 0\) of a function \(x:(a,b) \to \mathbb{R}\) is defined by
provided that the right-hand side integral is pointwise defined on \([a,b]\).
Definition 2.2
The left-sided Riemann–Liouville fractional derivative of order \(\alpha > 0\) (\(n-1 < \alpha \leqslant n\), \(n \in {\mathbb{N}^{+}}\)) of a function \(x:(a,b) \to \mathbb{R}\) is defined by
provided the right-hand side integral is pointwise defined on \([a,b]\).
Definition 2.3
The left-sided Caputo fractional derivative of order \(\alpha > 0\) (\(n - 1 < \alpha < n\), \(n \in {\mathbb{N}^{+}}\)) of a function \(x:(a,b) \to \mathbb{R}\) is defined by
provided the right-hand side integral is pointwise defined on \([a,b]\).
Definition 2.4
([32])
The left-sided Hilfer fractional derivative of order \(\alpha >0\) (\(n-1 <\alpha \leqslant n\), \(n \in {\mathbb{N}^{+}}\)) and type \(0\leq \beta \leq 1\) of a function \(x:(a,b) \to \mathbb{R}\) is defined by
where \({D^{n}} = {{{d^{n}}} / {d{t^{n}}}}\).
Lemma 2.1
Let \(\alpha > 0\), \(n=[\alpha ] + 1\). If \(x \in {L^{1}}(a,b)\), \(I_{a+}^{n-\alpha }x \in A{C^{n}}[a,b]\), then
Lemma 2.2
If \(\alpha > 0\), \(\lambda > - 1\), then
Lemma 2.3
([36])
Let \(\alpha > 0\), \(n = [\alpha ]+1\), \(0 \leqslant \beta \leqslant 1\). If \(x \in {L^{1}}(a,b)\), \(I_{a+} ^{(n-\alpha )(1-\beta )}x \in A{C^{n}}[a,b]\), then
Lemma 2.4
([35])
Let \(\alpha >0\), \(n \in N\), and \(D={d / {dx}}\). If the fractional derivatives \((D_{a+}^{\alpha }x)(t)\) and \((D_{a+}^{\alpha +n}x)(t)\) exist, then
Lemma 2.5
([35])
Let \(\alpha >0\), \(n \in N\), and \(D={d / {dx}}\). If the fractional derivatives \(({D^{n}}x)(t)\) and \(({}^{C}D _{a+}^{\alpha +n}x)(t)\) exist, then
Lemma 2.6
Let \(\alpha > 0\), \(n=[\alpha ]+1\), \(0 \leqslant \beta \leqslant 1\), \(m \in \mathbb{N}\), and \(D={d / {dx}}\). If the fractional derivatives \(({D^{m}}x)(t)\) and \((D_{a+}^{\alpha + m, \beta }x)(t)\) exist, then
provided that
Proof
Since \({x^{(j)}}(a)=0\), \(j = 0,1,2,\ldots ,m - 1\), then we get
which yields the following equalities hold:
The proof is completed. □
Lemma 2.7
Let \(\alpha > 0\), \(n=[\alpha ]+1\), \(0 \leqslant \beta \leqslant 1\). If \(x \in C[a,b]\), \(I_{a+}^{1-\beta (n- \alpha )}x \in AC[a,b]\), then
Proof
On the one hand, if \(\beta (n-\alpha ) =0\), i.e., \(\beta =0\) or \(\alpha =n\), then
or
On the other hand, if \(\beta (n-\alpha )\neq 0\), by Definitions 2.2, 2.4 and Lemma 2.1, we have
Since \(x \in C[a,b]\), one has
Thus, (2.1) holds for \(\beta (n-\alpha ) \neq 0\). The proof is completed. □
Lemma 2.8
Let \(\alpha > 0\), \(n=[\alpha ]+1\), \(0 \leqslant \beta \leqslant 1\), \(\lambda > - 1\), then
In particular,
Proof
For \(\lambda > - 1\), by Definition 2.4 and Lemma 2.2, we have
In particular,
Thus the proof of Lemma 2.8 is completed. □
3 Main result
Take the Banach space \((X,\| \cdot \|{_{\infty }})\),
Lemma 3.1
Let \(0 < {\alpha _{1}}\), \({\alpha _{2}} \leqslant 1\), \(1 < {\alpha _{1}} + {\alpha _{2}} \leqslant 2\), \(0 \leqslant {\beta _{1}}\), \({\beta _{2}} \leqslant 1\). Assume that \((A_{1})\) holds. Then, for \(y\in X\), the function \(x\in X\) is a solution of the following BVP:
if and only if x satisfies the integral equation
where
and
Proof
By using Lemma 2.3 twice and combining Lemma 2.2, we get that x is a solution of (3.1) if and only if
where \({c_{1}},{c_{2}} \in \mathbb{R}\). Considering the boundary conditions \(x(a) = 0\), \(x(b)=\sum_{i=1}^{m-2} {{\sigma _{i}}x({\xi _{i}})}\), we get
Thus,
Therefore,
That is,
By substituting (3.5) into (3.4), we obtain
The proof is completed. □
Lemma 3.2
Let \(0 < {\alpha _{1}}, {\alpha _{2}} \leqslant 1\), \(1 < {\alpha _{1}} + {\alpha _{2}} \leqslant 2\), \(0 \leqslant {\beta _{1}}, {\beta _{2}} \leqslant 1\). Assume that \((A_{2})\) holds. Then, for \(y \in X\), the function \(x\in X\) is a solution of the following BVP:
if and only if x satisfies the integral equation
where
and
Proof
By a similar method used in Lemma 3.1, we obtain
where \({c_{1}} \in \mathbb{R}\). Then, taking derivative to the both sides of the above equality, we have
Using the boundary condition \(x'(b) = \sum_{i = 1}^{m - 2} {{\delta _{i}}x({\eta _{i}})}\), we get
Hence,
Therefore,
It follows that
If we plug (3.10) back into (3.9), we obtain
which completes the proof. □
Lemma 3.3
(See [14])
If \(1<\omega <2\), then
Lemma 3.4
The functions \(K(t,s)\) and \({H_{1}}(t,s)\) defined in (3.3) and (3.8) satisfy the following properties:
-
(i)
\(K(t,s)\) and \({H_{1}}(t,s)\) are two continuous functions for any \((t,s) \in [a,b] \times [a,b]\);
-
(ii)
\(|K(t,s)| \leqslant \frac{{{{[(b-a)({\alpha _{1}}+{\alpha _{2}}-1)]}^{{\alpha _{1}}+{\alpha _{2}}-1}}{{[{\alpha _{2}}-(1-{\alpha _{1}})(1-{\beta _{1}})]} ^{{\alpha _{2}}-(1-{\alpha _{1}})(1-{\beta _{1}})}}}}{{\varGamma ({\alpha _{1}}+{\alpha _{2}}){{[2{\alpha _{2}}-(1-{\alpha _{1}})(2-{\beta _{1}})]} ^{2{\alpha _{2}}-(1-{\alpha _{1}})(2-{\beta _{1}})}}}}\) for all \((t,s) \in [a,b] \times [a,b]\);
-
(iii)
\(|{H_{1}}(t,s)| \leqslant (b-a)\max \{ {\alpha _{1}}+ {\alpha _{2}}-1,{\beta _{1}}(1-{\alpha _{1}})\}\) for every \((t,s) \in [a,b] \times [a,b]\).
Proof
Obviously, (i) is true. To prove (ii), for \((t,s) \in [a,b] \times [a,b]\), it is straightforward to show that
Differentiating \({k_{1}}(t,s)\) with respect to s, we have
which shows \({k_{1}}(t,s)\) is increasing with respect to \(s \in [a,t]\). Thus,
Since
Thus,
We consider the functions
Differentiating \(f(t)\) on \((a,b)\), we have
and
By calculating, we get \(f'(t)=0\) has a unique zero in \((a,b)\) as follows:
Because
it is easy to verify that
Hence, we obtain
We now prove that \(\max_{t \in [a,b]} \tilde{f}(t) \leqslant \max_{t \in [a,b]} f(t)\). In fact, if \({\beta _{1}}(1-{\alpha _{1}})=0\), then \(\tilde{f}(t) \equiv 0\), and the conclusion is obvious. If \({\beta _{1}}(1-{\alpha _{1}}) \ne 0\), differentiating \(\tilde{f}(t)\) on \((a,b)\), we have
and
By calculating, we get \(\tilde{f}'(t) = 0\) has a unique zero in \((a,b)\) as follows:
Submitting (3.13) into (3.12), we have
Thus,
Take \(\omega = \frac{{2{\alpha _{2}}-(1-{\alpha _{1}})(2-{\beta _{1}})}}{{{\alpha _{2}}-(1- {\alpha _{1}})(1-{\beta _{1}})}}\), then \(1< \omega <2\). It follows from Lemma 3.3 that
From the above we get
Therefore,
To prove (iii), for \((t,s) \in [a,b] \times [a,b]\), obviously, we have that the following inequalities hold:
Differentiating \({h_{1}}(t,s)\) with respect to t, we have
Hence, \({h_{1}}(t,s)\) is a decreasing function of \(t \in [s,b]\), which implies that
It is trivial to show that \({h_{1}}(t,s)\) is an increasing function with respect to \(s \in [a,b]\). Thus,
Note that
We get
Combining (3.14) and (3.15), we obtain
Therefore,
Then we complete the proof of Lemma 3.4. □
Remark 3.1
From the proof of Lemma 3.4, we have the following conclusions:
-
(i)
\(K(t,s)\) has a unique maximum, given by
$$\begin{aligned}& \max_{(t,s) \in {{[a,b]}^{2}}} \bigl\vert K(t,s) \bigr\vert \\& \quad = K\bigl({t ^{*}},{t^{*}}\bigr) \\& \quad = \frac{{{{[(b-a)({\alpha _{1}}+{\alpha _{2}}-1)]} ^{{\alpha _{1}}+{\alpha _{2}}-1}}{{[{\alpha _{2}}-(1-{\alpha _{1}})(1- {\beta _{1}})]}^{{\alpha _{2}}-(1-{\alpha _{1}})(1-{\beta _{1}})}}}}{ {\varGamma ({\alpha _{1}}+{\alpha _{2}}){{[2{\alpha _{2}}-(1-{\alpha _{1}})(2- {\beta _{1}})]}^{2{\alpha _{2}}-(1-{\alpha _{1}})(2-{\beta _{1}})}}}}, \end{aligned}$$where \(t^{*}\) is defined by (3.11);
-
(ii)
\(\max_{(t,s) \in {{[a,b]}^{2}}} | {H_{1}}(t,s)| = (b-a)\max \{ {\alpha _{1}}+{\alpha _{2}}-1,{\beta _{1}}(1- {\alpha _{1}})\}\) and
$$ \bigl\vert {H_{1}}(t,s) \bigr\vert = \textstyle\begin{cases} (b-a)({\alpha _{1}}+{\alpha _{2}}-1),\quad \text{if and only if } t=s=b, \\ (b-a){\beta _{1}}(1-{\alpha _{1}}),\quad \text{if and only if } t=b, s=a. \end{cases} $$
Theorem 3.1
Assume that \((\mathrm{{A}_{1})}\) holds. If the fractional BVP (1.14), (1.15) has a nontrivial continuous solution for a real-valued continuous function q, then
where
Proof
Assume \(x(t)\) is a nontrivial solution of BVP (1.14), (1.15), then
and
Since x is a nontrivial solution, which will require \(q(s)\not \equiv 0\) on \([a,b]\). Moreover, from \(q(s) \in C[a,b]\), we obtain that there exists an interval \([{a_{1}},{b_{1}}] \subset [a,b]\) such that \(|q(s)| > 0\) on \([{a_{1}},{b_{1}}]\). Then, by Lemma 3.4 and Remark 3.1, we have
Thus, inequality (3.16) holds. This completes the proof of Theorem 3.1. □
Theorem 3.2
Assume that \((\mathrm{{A}_{2})}\) holds. If the fractional BVP (1.14), (1.16) has a nontrivial continuous solution for a real-valued continuous function q, then
where
Proof
Assume that \(x(t)\) is a nontrivial solution of BVP (1.14), (1.16), then
and
An argument similar to the one used in Theorem 3.1 shows that there exists an interval \([{a_{2}},{b_{2}}] \subset [a,b]\) such that \(|q(s)| > 0\) on \([{a_{2}},{b_{2}}]\). Now, applying Lemma 3.4 and Remark 3.1, we have
from which inequality in (3.17) follows. The proof is completed. □
Theorem 3.1 gives the following corollaries.
Corollary 3.1
The necessary condition for the existence of a nontrivial solution for BVP (1.1) is
Proof
Apply Theorem 3.1 for \({\alpha _{1}}= {\alpha _{2}}=1\), \({\sigma _{i}}=0\) (\(i = 1,2, \ldots ,m - 2\)), then (3.18) holds. Obviously, (3.18) coincides with the classical Lyapunov inequality, i.e., inequality (1.2). □
Corollary 3.2
The necessary condition for the existence of a nontrivial solution for BVP (1.5) of case (i) is
Proof
Apply Theorem 3.1 for \({\alpha _{1}}= \alpha \), \({\alpha _{2}}=\beta \), \({\beta _{1}}={\beta _{2}}=0\), \({\sigma _{i}}=0\) (\(i=1,2,\ldots ,m-2\)), then (3.19) holds. Obviously, (3.19) coincides with inequality (1.6). □
Corollary 3.3
The necessary condition for the existence of a nontrivial solution for BVP (1.3) is
Proof
Apply Theorem 3.1 for \({\alpha _{1}}=1\), \({\beta _{2}}=0\), \(\alpha =1+{\alpha _{2}}\), \({\sigma _{i}}=0\) (\(i=1,2, \ldots ,m-2\)), then (3.20) holds. Obviously, (3.20) coincides with inequality (1.4). □
Corollary 3.4
The necessary condition for the existence of a nontrivial solution for BVP (1.5) of case (ii) is
Proof
Apply Theorem 3.1 for \({\alpha _{1}} =\alpha , {\alpha _{2}}=\beta , {\beta _{1}}={\beta _{2}}=1, {\sigma _{i}}=0\) (\(i=1,2,\ldots ,m-2\)), then (3.21) holds, which coincides with inequality (1.7). □
Corollary 3.5
Consider the following fractional BVP:
where \(q\in C([a,b], \mathbb{R})\), \({}^{C}D_{a+}^{\alpha }\) is the left Caputo fractional derivative. If (3.22) has a nontrivial continuous solution in \([a,b]\), then
Proof
Apply Theorem 3.1 for \({\alpha _{2}}=1\), \({\beta _{1}}=1\), \(\alpha ={\alpha _{1}}+1\), \({\sigma _{i}}=0\) (\(i=1,2, \ldots ,m-2T\), then (3.23) holds. Corollary 3.5 coincides with [14, Theorem 1]. □
Corollary 3.6
The necessary condition for the existence of a nontrivial solution for BVP (1.8), (1.9) is
Proof
Apply Theorem 3.1 for \({\alpha _{2}} =1\), \(\alpha ={\alpha _{1}}+1\), \(\beta ={\beta _{1}}\), \({\sigma _{i}} =0\) (\(i=1,2,\ldots ,m-2\)), then (3.24) holds, which coincides with inequality (1.11). By Remark 3.1, we show that the non-strict inequality (1.11) can be replaced by strict inequality (3.24). □
Corollary 3.7
Assume that the following boundary value problem
where \(q\in C([a,b], \mathbb{R})\), \(D_{a+}^{\alpha ,\beta }\) is the Hilfer fractional derivative of order α and type \(\beta \in [0,1]\), has a nontrivial continuous solution in \([a,b]\), then
where
Proof
Apply Theorem 3.1 for \({\alpha _{2}} =1\), \(\alpha ={\alpha _{1}}+1\), \(\beta ={\beta _{1}}\), then (3.25) holds, which coincides with [30, Theorem 3.1]. □
Theorem 3.2 gives the following corollaries.
Corollary 3.8
If a nontrivial continuous solution of the following boundary value problem
exists, where \(q\in C([a,b], \mathbb{R})\), then
Proof
Apply Theorem 3.2 for \({\alpha _{1}} ={\alpha _{2}}=1\), \({\delta _{i}}=0\) (\(i=1,2,\ldots ,m-2\)), then (3.26) holds, which coincides with [16, Corollary 5]. □
Corollary 3.9
Suppose that the following boundary value problem
where \(q\in C([a,b], \mathbb{R})\), \(D_{a+}^{(\cdot )}\) is the left Riemann–Liouville fractional derivative, has a nontrivial continuous solution in \([a,b]\), then
Proof
Apply Theorem 3.2 for \({\beta _{1}}={\beta _{2}}=0\), \({\delta _{i}}=0\) (\(i=1,2,\ldots ,m-2\)), then (3.27) holds. □
Corollary 3.10
Consider the following fractional BVP:
where \(q\in C([a,b], \mathbb{R})\), \(D_{a+}^{\alpha }\) is the Riemann–Liouville fractional derivative of fractional order α. If (3.28) has a nontrivial continuous solution in \([a,b]\), then
Proof
Apply Theorem 3.2 for \({\alpha _{1}}=1\), \({\beta _{2}} =0\), \(\alpha =1+{\alpha _{2}}\), \({\delta _{i}}=0\) (\(i=1,2,\ldots ,m -2\)), then (3.29) holds. □
Corollary 3.11
Consider the following fractional BVP:
where \(q\in C([a,b], \mathbb{R})\), \({}^{C}D_{a+}^{(\cdot )}\) is the left Caputo fractional derivative. If (3.30) has a nontrivial continuous solution in \([a,b]\), then
Proof
Apply Theorem 3.2 for \({\beta _{1}}={\beta _{2}}=1, {\delta _{i}}=0\) (\(i = 1,2,\ldots ,m-2\)), then (3.31) holds. □
Corollary 3.12
Consider the following fractional BVP:
where \(q\in C([a,b], \mathbb{R})\), \({}^{C}D_{a+}^{\alpha }\) is the left Caputo fractional derivative of order α. If (3.32) has a nontrivial continuous solution in \([a,b]\), then
Proof
Apply Theorem 3.2 for \({\alpha _{2}} =1,{\beta _{1}}=1,\alpha ={\alpha _{1}}+1,{\delta _{i}}=0\) (\(i =1,2,\ldots ,m-2\)), then (3.33) holds, which coincides with [17] and [23, Theorem 3]. □
Corollary 3.13
The necessary condition for the existence of a nontrivial solution for BVP (1.8), (1.10) is
Proof
Apply Theorem 3.2 for \({\alpha _{2}} =1\), \(\alpha ={\alpha _{1}}+1\), \(\beta ={\beta _{1}}\), \({\delta _{i}} =0\) (\(i=1,2,\ldots ,m-2\)), then (3.34) holds, which coincides with inequality (1.12). By Remark 3.1, we show that the non-strict inequality (1.12) can be replaced by strict inequality (3.34). □
Corollary 3.14
If a nontrivial continuous solution of the following fractional boundary value problem
exists, \(q\in C([a,b], \mathbb{R})\), \(D_{a+}^{\alpha ,\beta }\) is the Hilfer fractional derivative of order α and type \(\beta \in [0,1]\), then
where
Proof
Apply Theorem 3.2 for \({\alpha _{2}}=1\), \(\alpha = {\alpha _{1}}+1\), \(\beta = {\beta _{1}}\), then (3.35) holds. □
4 Conclusion
In this paper, the Lyapunov-type inequalities of sequential Hilfer fractional BVPs were investigated for the first time. Since the Hilfer fractional derivative is a generalization of both Riemann–Liouville and Caputo types fractional derivatives, this requires that our results can be reduced to the corresponding classical results, and we do it. So our work is meaningful and the results we obtained are more general. There is some work to be done in the future such as: finding Lyapunov-type inequalities for higher order Hilfer fractional BVPs; studying Lyapunov-type inequalities for Hilfer fractional p-Laplacian equation, and so on.
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This research is supported by the National Natural Science Foundation of China (No. 11271364).
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Zhang, W., Liu, W. Lyapunov-type inequalities for sequential fractional boundary value problems using Hilfer’s fractional derivative. J Inequal Appl 2019, 98 (2019). https://doi.org/10.1186/s13660-019-2050-6
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DOI: https://doi.org/10.1186/s13660-019-2050-6
MSC
- 34A08
- 34B15
Keywords
- Lyapunov-type inequality
- Sequential fractional differential equation
- Hilfer fractional derivative
- Multi-point boundary condition