Advances in Fractional Lyapunov-Type Inequalities: A Comprehensive Review

Abstract

In this survey, we have included the recent results on Lyapunov-type inequalities for differential equations of fractional order associated with Dirichlet, nonlocal, multi-point, anti-periodic, and discrete boundary conditions. Our results involve a variety of fractional derivatives such as Riemann–Liouville, Caputo, Hilfer–Hadamard, $\psi$-Riemann–Liouville, Atangana–Baleanu, tempered, half-linear, and discrete fractional derivatives.

1. Introduction and Preliminaries

Lyapunov inequality, introduced by Lyapunov [1], plays an important role in studying the existence criteria, integral inequalities, stability, and bounds for solutions to boundary value problems of different types of differential equations. One can find applications of Lyapunov inequality in diverse areas of mathematics and mathematical physics like control theory, fractional calculus, mathematical biology, etc. Its modern interpretations often involve integral inequalities, comparison principles, and applications in stability analysis in dynamical systems.

The initial version of Lyapunov inequality was related to a second-order linear differential equation of the following form:

$$y^{″}+q\left(x\right)y=0,a_1\le x\le a_2,$$

(1)

where $q\left(x\right)$ is a continuous function. It was shown by Lyapunov that the existence of a nontrivial solution to the above equation supplemented with prescribed boundary conditions was subject to certain integral constraints. Specifically, if a nontrivial solution exists to the Equation (1), then

$${\int }_{a_1}^{a_2}\left|q\left(x\right)\right|dx>{\displaystyle \frac{4}{{(a_2-a_1)}^2}}.$$

This result elegantly links the properties of the coefficient function $q\left(x\right)$ with the geometry of the interval $[a_1,a_2]$ and provides an existence criterion for a nontrivial solution to (1).

For an up-to-date detailed account of the Lyapunov-type inequalities for fractional differential equations, see the survey [2] and references cited therein. In continuation with the survey [2], we gathered the latest findings on Lyapunov-type inequalities for differential equations of fractional order.

The content of this survey article is organized as follows: in Section 2, the Lyapunov-type inequalities for Caputo fractional differential equations are described, while the Lyapunov-type inequalities for higher-order Caputo fractional differential equations are presented in Section 3. For left Riemann–Liouville and right Caputo fractional differential equations, the results on Lyapunov-type inequalities are discussed in Section 4. Section 5 deals with the Lyapunov-type inequalities for Hilfer–Hadamard-type fractional differential equations. For a sequential $\psi$-Riemann–Liouville fractional differential equation with nonlocal conditions, the Lyapunov-type inequalities are discussed in Section 6. We examine Lyapunov-type inequalities related to a fractional hybrid boundary value problem in Section 7. The Lyapunov-type inequalities for Atangana–Baleanu fractional differential equations are presented in Section 8. We summarize Lyapunov-type inequalities for tempered fractional differential equations in Section 9. We include the results on a Lyapunov-type inequality for a half-linear local fractional differential equation with Dirichlet boundary conditions in Section 10, and the last section contains the results on a Lyapunov-type inequality for discrete fractional equations.

Our main objective in this survey is to highlight the progress on the Lyapunov-type inequalities for fractional boundary value problems in a comprehensive manner by including the most recent work on the topic. We omit the proofs of the results presented in this survey. However, we include the necessary details (lemmas and properties of Green’s function) for results on the Lyapunov-type inequalities offered in this survey and refer the reader to the related material for detailed proofs and explanations.

2. Lyapunov-Type Inequalities for Caputo Fractional Differential Equations

We begin this section with some definitions [3,4].

Definition 1.

For a real valued function $\phi \in L_1[a,b]$, the (left) Riemann–Liouville fractional integral of order $\delta \ge 0$ is given by $\left(I^0\phi \right)\left(x\right)=\phi \left(x\right)$ and

$$\left(I^{\delta }\phi \right)\left(x\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\delta \right)}}{\int }_a^x{(x-s)}^{\delta -1}\phi \left(s\right)ds,\delta >0,$$

where $\mathrm{\Gamma }\left(\delta \right)$ represents the Euler Gamma function: ${\displaystyle \mathrm{\Gamma }\left(\delta \right)={\int }_0^{\infty }{\tau }^{\delta -1}{e}^{-\tau }d\tau .}$

Definition 2.

For $\phi ,{\phi }^{\left(m\right)}\in L_1[a,b]$, the Riemann–Liouville fractional derivative of order $\delta \in (m-1,m],m\in \mathbb{N},$ for the function φ is given by

$$\left(D^{\delta }\phi \right)\left(x\right)=\left(D^m{I}^{m-\delta }\phi \right)\left(x\right),$$

while the Caputo fractional derivative of order $\delta \in (m-1,m]$ for an m-times continuously differentiable function φ is defined by

$${(}^C{D}^{\delta }\phi )\left(x\right)=\left(I^{m-\delta }{D}^m\phi \right)\left(x\right).$$

In [5], Xiao et al. considered the fractional boundary value problem (BVP for short):

$$\left\{\begin{array}{c}{(}^C{D}_0^{\zeta }\psi )\left(t\right)+r\left(t\right)\psi \left(t\right)=0,\zeta \in (1,2),t\in (0,1),\hfill \\ \psi \left(0\right)=\kappa \psi \left(1\right),{\psi }^{\prime }\left(0\right)=\vartheta z^{\prime }\left(1\right),\kappa ,\vartheta >0,\hfill \end{array}\right.$$

(2)

where $r\in L(0,1)$ is an identically non-zero function on any compact subinterval of $(0,1).$

Lemma 1.

A function ψ is a solution to the problem (2) if and only if

$$\psi \left(t\right)={\int }_0^{1}G(t,\upsilon )r\left(\upsilon \right)z\left(\upsilon \right)d\upsilon ,$$

where

$$G(t,\upsilon )={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\zeta \right)}}\left\{\begin{array}{c}{\displaystyle (1-\zeta )\frac{\kappa \vartheta (1-t)+\vartheta t}{(1-\vartheta )(1-\kappa )}{(1-\upsilon )}^{\zeta -2}-\frac{\kappa }{1-\kappa }{(1-\upsilon )}^{\zeta -1}}\hfill \\ -{(t-\upsilon )}^{\zeta -1},\hfill & 0\le \upsilon \le t\le 1,\hfill \\ {\displaystyle (1-\zeta )\frac{\kappa \vartheta (1-t)+\vartheta t}{(1-\vartheta )(1-\kappa )}{(1-v)}^{\zeta -2}-\frac{\kappa }{1-\kappa }{(1-\upsilon )}^{\zeta -1},}\hfill & 0\le t\le \upsilon \le 1.\hfill \end{array}\right.$$

Lemma 2.

For $\kappa ,\vartheta \in (0,1)$ with $A_1=\mathrm{\Gamma }\left(\zeta \right)(1-\kappa )(1-\vartheta )$, Green’s function $G(t,\upsilon )$ given in Lemma 1 satisfies the following properties:

(i) 

$G(t,\upsilon )\le 0,(t,\upsilon )\in [0,1]\times [0,1];$

(ii) 

${\displaystyle \underset{0\le t\le 1}{max}\left|G(t,\upsilon )\right|=-G(1,\upsilon )=\frac{1}{A_1}{(1-s)}^{\zeta -2}\left[\vartheta (\zeta -1)+(1-\vartheta )(1-\upsilon )\right],}$ for $\upsilon \in [0,1];$

(iii) 

${\displaystyle {\int }_0^1\left|G(t,\upsilon )\right|ds\le \frac{\vartheta (\zeta -1)+1}{\zeta A_1}.}$

Lemma 3.

For $\kappa \in (1,\infty )$ and $\vartheta \in (0,1)$ with $A_2=\mathrm{\Gamma }\left(\zeta \right)(\kappa -1)(1-\vartheta )$, the following properties hold for Green’s function $G(t,\upsilon )$:

(i) 

$G(t,\upsilon )\ge 0,(t,v)\in [0,1]\times [0,1];$

(ii) 

${\displaystyle \underset{0\le t\le 1}{max}\left|G(t,\upsilon )\right|=G(0,v)=\frac{\kappa {(1-\upsilon )}^{\zeta -2}}{A_2}\left[\vartheta (\zeta -1)+(1-\vartheta )(1-\upsilon )\right],}$ for $\upsilon \in [0,1];$

(iii) 

${\displaystyle {\int }_0^1\left|G(t,\upsilon )\right|ds\le \frac{\kappa (\vartheta \zeta +1-\vartheta )}{\zeta A_2}.}$

Lemma 4.

For $\kappa \in (0,1)$ and ${\displaystyle \vartheta \in \left(1,1+\frac{(\zeta -1)\kappa }{2-\zeta }\right)}$ with $A_3=\mathrm{\Gamma }\left(\zeta \right)(1-\kappa )(\vartheta -1)$, Green’s function $G(t,\upsilon )$ satisfies the following properties:

(i) 

$G(t,\upsilon )\ge 0,(t,v)\in [0,1]\times [0,1];$

(ii) 

${\displaystyle \underset{0\le t\le 1}{max}\left|G(t,\upsilon )\right|=G(1,v)=\frac{{(1-s)}^{\zeta -2}}{A_3}\left[\vartheta (\zeta -1)+(1-\vartheta )(1-\upsilon )\right],}$ for $s\in [0,1];$

(iii) 

${\displaystyle {\int }_0^1\left|G(t,\upsilon )\right|ds\le \frac{\vartheta (\zeta -1)+1)}{\zeta A_3}.}$

Lemma 5.

For $\kappa \in (1,\infty )$ and ${\displaystyle \vartheta \in \left(1,\frac{1}{2-\zeta }\right],}$ Green’s function $G(t,\upsilon )$ exhibits the following properties:

(i) 

$G(t,\upsilon )\le 0,(t,\upsilon )\in [0,1]\times [0,1];$

(ii) 

For $\upsilon \in [0,1]$, ${\displaystyle \underset{0\le t\le 1}{max}\left|G(t,\upsilon )\right|\le \frac{{(1-\upsilon )}^{\zeta -2}}{\mathrm{\Gamma }\left(\zeta \right)(\kappa -1)(\vartheta -1)}\times \left\{\kappa \vartheta (\zeta -1)-\left[\kappa (\vartheta -1)-\vartheta (2-\zeta )(\kappa -1){\left(\frac{\vartheta -1}{\vartheta }\right)}^{1/(2-\zeta )}\right](1-\upsilon )\right\};}$

(iii) 

${\displaystyle {\int }_0^1\left|G(t,v)\right|ds\le \frac{1}{\mathrm{\Gamma }(\zeta +1)(\kappa -1)(\vartheta -1)}\times \left\{\kappa [1+\vartheta (\zeta -1)]+(\kappa -1)\vartheta (2-\zeta ){\left(\frac{\vartheta -1}{\vartheta }\right)}^{1/(2-\zeta )}\right\}.}$

Lemma 6.

When $\vartheta >1,$ we obtain the following properties for Green’s function $G(t,\upsilon )$:

${\displaystyle \underset{0\le t\le 1}{max}\left|G(t,\upsilon )\right|=\frac{1}{\mathrm{\Gamma }\left(\zeta \right)|1-\kappa |}{(1-s)}^{\zeta -2}\underset{1\le i\le 4}{max}\left\{\right|f_i\left(\upsilon \right)\left|\right\},}$ where

$$\begin{array}{ccc}\hfill f_1\left(\upsilon \right)& =& \left[{\displaystyle \frac{\vartheta (1-\zeta )}{1-\vartheta }}+\upsilon -1\right]\kappa ,\hfill \\ \hfill f_2\left(\upsilon \right)& =& {\displaystyle \frac{(1-\zeta )\kappa \vartheta +(1-\zeta )\vartheta (1-\kappa )s}{1-\vartheta }}+\kappa (\upsilon -1),\hfill \\ \hfill f_3\left(\upsilon \right)& =& {\displaystyle \frac{\kappa \vartheta (\zeta -1)}{\vartheta -1}}+\left[(2-\zeta ){\left({\displaystyle \frac{\vartheta }{\vartheta -1}}\right)}^{{\displaystyle \frac{\zeta -1}{\zeta -2}}}(1-\kappa )+\kappa \right](\upsilon -1)+{\displaystyle \frac{\vartheta (\zeta -1)(1-\kappa )}{\vartheta -1}}\upsilon ,\hfill \\ \hfill f_4\left(\upsilon \right)& =& {\displaystyle \frac{1}{\upsilon }}{f}_1\left(\upsilon \right).\hfill \end{array}$$

In the next lemmas, we provide the properties and bounds of ${max}_{1\le i\le 4}\left\{\right|f_i\left(s\right)\left|\right\}.$

Lemma 7.

Let $1<\zeta <2,\vartheta >1$ and $f_i\left(\upsilon \right)$ be given in Lemma 6; then, the following results hold:

(1) 

If $\kappa >1,$ then $f_1\left(\upsilon \right)\ge f_2\left(\upsilon \right),f_2\left(\upsilon \right)\le f_3\left(\upsilon \right),f_3\left(\upsilon \right)\ge f_4\left(\upsilon \right),\forall \upsilon \in [0,1];$

(2) 

If $0<\kappa <1,$ then $f_1\left(\upsilon \right)\le f_2\left(\upsilon \right),f_2\left(\upsilon \right)\ge f_3\left(\upsilon \right),f_3\left(\upsilon \right)\le f_4\left(\upsilon \right),\forall \upsilon \in [0,1].$

Lemma 8.

Let $f_i\left(\upsilon \right)$ be given in Lemma 6 and

$${\kappa }^{*}={\displaystyle \frac{{\displaystyle (2-\zeta ){\left(\frac{\vartheta }{\vartheta -1}\right)}^{\frac{\zeta -1}{\zeta -2}}}}{{\displaystyle \frac{(2-\zeta )\vartheta -1}{1-\vartheta }+(2-\zeta ){\left(\frac{\vartheta }{\vartheta -1}\right)}^{\frac{\zeta -1}{\zeta -2}}}}}.$$

Then, we have:

(1) 

If $1<\zeta <2,\vartheta >1$ with $(2-\zeta )\vartheta >1,$ then ${\kappa }^{*}>1$ and

$$\underset{1\le i\le 4}{max}\left|f_i\left(0\right)\right|=\left\{\begin{array}{cc}-f_3\left(0\right),\hfill & 0<\kappa <1,\hfill \\ -f_2\left(0\right),\hfill & 1<\kappa <{\kappa }^{*},\hfill \end{array}\right.$$

$$\underset{1\le i\le 4}{max}\left|f_i\left(0\right)\right|\le f_3\left(0\right)-f_2\left(0\right),\kappa >{\kappa }^{*};$$

(2) 

If $1<\zeta <2,\vartheta >1$ with $(2-\zeta )\vartheta <1,$ then $0<{\kappa }^{*}<1$ and

$$\underset{1\le i\le 4}{max}\left|f_i\left(0\right)\right|=\left\{\begin{array}{cc}{f}_4\left(0\right),\hfill & {\kappa }^{*}<\kappa <1,\hfill \\ f_3\left(0\right),\hfill & \kappa >1,\hfill \end{array}\right.$$

$$\underset{1\le i\le 4}{max}\left|f_i\left(0\right)\right|\le f_4\left(0\right)-f_3\left(0\right),0<\kappa <{\kappa }^{*};$$

(3) 

If $1<\zeta <2,\vartheta >1,$ then

$$\underset{1\le i\le 4}{max}\left|f_i\left(1\right)\right|=\left\{\begin{array}{cc}{f}_2\left(1\right),\hfill & 0<\kappa <1,\hfill \\ f_1\left(1\right),\hfill & \kappa >1.\hfill \end{array}\right.$$

Now, we present the results on Lyapunov-type inequalities.

Theorem 1.

Assume that the fractional BVP (2) has a non-zero solution ψ and let $g_1\left(\upsilon \right)=\vartheta (\zeta -1)+(1-\vartheta )(1-\upsilon )$. Then, we obtain the following results:

(i) 

If $\kappa \in (0,1)$ and $\vartheta \in (0,1)$, then

$${\int }_0^1{(1-\upsilon )}^{\zeta -2}{g}_1\left(\upsilon \right)\left|r\left(\upsilon \right)\right|d\upsilon >\mathrm{\Gamma }\left(\zeta \right)(1-\kappa )(1-\vartheta ).$$

(ii) 

If $\kappa \in (1,\infty )$ and $\vartheta \in (0,1),$ then

$${\int }_0^1{(1-\upsilon )}^{\zeta -2}{g}_1\left(\upsilon \right)\left|r\left(\upsilon \right)\right|d\upsilon >{\displaystyle \frac{\mathrm{\Gamma }\left(\zeta \right)(\kappa -1)(1-\vartheta )}{\kappa }}.$$

(iii) 

If $\kappa \in (0,1)$ and ${\displaystyle \vartheta \in \left(1,1+\frac{(\zeta -1)\kappa }{2-\zeta }\right],}$ then

$${\int }_0^1{(1-\upsilon )}^{\zeta -2}{g}_1\left(\upsilon \right)\left|r\left(\upsilon \right)\right|d\upsilon >\mathrm{\Gamma }\left(\zeta \right)(1-\kappa )(\vartheta -1).$$

(iv) 

If $\kappa \in (1,\infty )$ and ${\displaystyle \vartheta \in \left(1,\frac{1}{2-\zeta }\right],}$ then

${\displaystyle {\int }_0^1{(1-\upsilon )}^{\zeta -2}\left\{\kappa \vartheta (\zeta -1)-\left[\kappa (\vartheta -1)-(2-\zeta )\vartheta (\kappa -1){\left(\frac{\vartheta -1}{\vartheta }\right)}^{1/(2-\zeta )}\right](1-\upsilon )\right\}\left|r\left(\upsilon \right)\right|d\upsilon >\mathrm{\Gamma }\left(\zeta \right)(\kappa -1)(\vartheta -1).}$

Theorem 2.

Let ψ be a nontrivial solution to the problem (2) and $\vartheta >1$ with $(2-\zeta )\vartheta >1.$ Let

$$M=M(\zeta ,\kappa ,\vartheta )={\displaystyle \frac{\kappa \left[\right(2-\zeta )\vartheta -1]}{1-\vartheta }}+(2-\zeta )(\kappa -1){\left({\displaystyle \frac{\vartheta }{\vartheta -1}}\right)}^{{\displaystyle \frac{\zeta -1}{\zeta -2}}}.$$

(3)

Then, for ${\kappa }^{*}>1,$ the following inequalities hold:

(i) 

If $0<\kappa <1,$ then

$$\mathrm{\Gamma }\left(\zeta \right)(1-\kappa )<{\displaystyle \frac{(1-\zeta )\vartheta }{1-\vartheta }}{\int }_0^1{(1-\upsilon )}^{\zeta -1}\left|r\left(\upsilon \right)\right|d\upsilon -M{\int }_0^1\upsilon {(1-\upsilon )}^{\zeta -2}\left|r\left(\upsilon \right)\right|d\upsilon .$$

(iii) 

If $1<\kappa <{\kappa }^{*},$ then

$$\begin{array}{ccc}\hfill \mathrm{\Gamma }\left(\zeta \right)(\kappa -1)& <& {\displaystyle \frac{\kappa (1-\zeta )\vartheta }{1-\vartheta }}{\int }_0^1{(1-\upsilon )}^{\zeta -1}\left|r\left(\upsilon \right)\right|d\upsilon \hfill \\ & & -{\displaystyle \frac{\kappa \left[\right(2-\zeta )\vartheta -1]}{1-\vartheta }}{\int }_0^1\upsilon {(1-\upsilon )}^{\zeta -2}\left|r\left(\upsilon \right)\right|d\upsilon .\hfill \end{array}$$

(ii) 

If $\kappa >{\kappa }^{*},$ then

$$\begin{array}{ccc}\hfill \mathrm{\Gamma }\left(\zeta \right)(\kappa -1)& <& {\displaystyle \frac{\kappa (1-\zeta )\vartheta }{1-\vartheta }}{\int }_0^1{(1-\upsilon )}^{\zeta -1}\left|r\left(\upsilon \right)\right|d\upsilon \hfill \\ & & +\left\{M-{\displaystyle \frac{\kappa \left[\right(2-\zeta )\vartheta -1]}{1-\vartheta }}\right\}{\int }_0^1\upsilon {(1-\upsilon )}^{\zeta -2}\left|r\left(\upsilon \right)\right|d\upsilon .\hfill \end{array}$$

Theorem 3.

Let ψ be a nontrivial solution to the problem (2) and $\vartheta >1$ with $(2-\zeta )\vartheta <1$, and M is defined as in (3). Then, we obtain $0<{\kappa }^{*}<1,$ and the following results hold:

(i) 

If $0<\kappa <{\kappa }^{*},$ then

$$\begin{array}{ccc}\hfill \mathrm{\Gamma }\left(\zeta \right)(1-\kappa )& <& {\displaystyle \frac{(1-\zeta )\vartheta }{1-\vartheta }}{\int }_0^1{(1-\upsilon )}^{\zeta -1}\left|r\left(\upsilon \right)\right|d\upsilon \hfill \\ & & +\left[{\displaystyle \frac{(2-\zeta )\vartheta -1}{1-\vartheta }}-M\right]{\int }_0^1\upsilon {(1-\upsilon )}^{\zeta -2}\left|r\left(\upsilon \right)\right|d\upsilon .\hfill \end{array}$$

(ii) 

If ${\kappa }^{*}<\kappa <1,$ then

$$\begin{array}{ccc}\hfill \mathrm{\Gamma }\left(\zeta \right)(1-\kappa )& <& {\displaystyle \frac{(1-\zeta )\vartheta }{1-\vartheta }}{\int }_0^1{(1-\upsilon )}^{\zeta -1}\left|r\left(\upsilon \right)\right|d\upsilon \hfill \\ & & +{\displaystyle \frac{(2-\zeta )\vartheta -1}{1-\vartheta }}{\int }_0^1\upsilon {(1-\upsilon )}^{\zeta -2}\left|r\left(\upsilon \right)\right|d\upsilon .\hfill \end{array}$$

(iii) 

If $\kappa >1,$ then

$$\mathrm{\Gamma }\left(\zeta \right)(\kappa -1)<{\displaystyle \frac{\kappa (1-\zeta )\vartheta }{1-\vartheta }}{\int }_0^1{(1-\upsilon )}^{\zeta -1}\left|r\left(\upsilon \right)\right|d\upsilon +M{\int }_0^1\upsilon {(1-\upsilon )}^{\zeta -2}\left|r\left(\upsilon \right)\right|d\upsilon .$$

3. Higher-Order Caputo Fractional Differential Equations and Lyapunov-Type Inequalities

In [6], Srivastava el al. presented three different Lyapunov-type inequalities for a Caputo-type higher-order fractional BVP given by

$$\left\{\begin{array}{c}{(}^C{D}_{y_1+}^{\epsilon }\psi )\left(t\right)+r\left(t\right)\psi \left(t\right)=0,y_1<t<y_2,n-1<\epsilon \le n,n\ge 3,\hfill \\ {\psi }^{\left(i\right)}\left(y_1\right)=0,{\psi }^{\left(k\right)}\left(y_2\right)=0,0\le i\le n-1,\mathrm{and}i\ne k,\hfill \end{array}\right.$$

(4)

where k is a natural number between 1 and $n-1.$

Lemma 9.

Assume that $\epsilon \in (n-1,n],1\le k\le n-1.$ Then, the unique solution to the problem (4) is given by

$$\psi \left(t\right)={\int }_{y_1}^{y_2}{G}_k(t,\upsilon )r\left(\upsilon \right)z\left(\upsilon \right)d\upsilon ,$$

where

$$G_k(t,\upsilon )={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\epsilon \right)}}\left\{\begin{array}{c}{\displaystyle \frac{1}{k!}(\epsilon -1)(\epsilon -2)\dots (\epsilon -k){(t-y_1)}^k{(y_2-\upsilon )}^{\epsilon -k-1}}\hfill \\ -{(t-\upsilon )}^{\epsilon -1},\hfill & y_1\le \upsilon \le t\le y_2,\hfill \\ {\displaystyle \frac{1}{k!}(\epsilon -1)(\epsilon -2)\dots (\epsilon -k){(t-y_1)}^k{(y_2-\upsilon )}^{\epsilon -k-1},}\hfill & y_1\le t\le \upsilon \le y_2.\hfill \end{array}\right.$$

The following lemma provides some properties for the function $G_k(t,\upsilon )$ defined in the above lemma.

Lemma 10.

Let $\epsilon \in (n-1,n]$ and $\epsilon >k+1.$ Then, we have

(i) 

$G_k(t,\upsilon )>0$ for all $t,\upsilon \in [y_1,y_2]$;

(ii) 

${\displaystyle \frac{\partial G_k(t,\upsilon )}{\partial t}>0}$ for all $t,\upsilon \in [y_1,y_2]$;

(iii) 

$G_k(t,\upsilon )\le G_k(y_2,\upsilon )\equiv {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\epsilon \right)}}\left[{\displaystyle \frac{1}{k!}}(\epsilon -1)(\epsilon -2)\dots (\epsilon -k){(y_2-y_1)}^k{(y_2-s)}^{\epsilon -k-1}-{(y_2-\upsilon )}^{\epsilon -1}\right].$

Theorem 4.

Assume that $\epsilon \in (n-1,n]$ and $\epsilon >k+1.$ If a nontrivial continuous solution to the problem (4) exists, then

$${\int }_{y_1}^{y_2}\left[{\displaystyle \frac{1}{k!}}(\epsilon -1)(\epsilon -2)\dots (\epsilon -k){(y_2-y_1)}^k{(y_2-\upsilon )}^{\epsilon -k-1}-{(y_2-\upsilon )}^{\epsilon -1}\right]\left|r\left(\upsilon \right)\right|d\upsilon \ge \mathrm{\Gamma }\left(\zeta \right).$$

Corollary 1.

Under the conditions of Theorem 4, if a nontrivial continuous solution to the problem (4) exists, then

$${\int }_{y_1}^{y_2}{(y_2-s)}^{\epsilon -k-1}\left|r\left(\upsilon \right)\right|d\upsilon \ge {\displaystyle \frac{k!\mathrm{\Gamma }(\epsilon -k)}{{(y_2-y_1)}^k}}.$$

Corollary 2.

Under the conditions of Theorem 4, if a nontrivial continuous solution to the problem (4) exists, then

$${\int }_{y_1}^{y_2}\left|r\left(\upsilon \right)\right|d\upsilon \ge {\displaystyle \frac{k!\mathrm{\Gamma }(\epsilon -k)}{{(y_2-y_1)}^{\zeta -1}}}.$$

For a general k with $1\le k\le n-1$, we consider a restriction for the following results:

$$k!>(\epsilon -1)(\epsilon -2)\dots (\epsilon -k-1).$$

(5)

Lemma 11.

Let $\epsilon >k+1$ and assume that (5) is satisfied. Then,

$$G_k(t,\upsilon )\le {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\epsilon \right)}}{\displaystyle \frac{k{(y_2-y_1)}^{\epsilon -1}}{\epsilon -k-1}}{\left({\displaystyle \frac{(\epsilon -2)(\epsilon -3)\dots (\epsilon -k)(\epsilon -k-1)}{k!}}\right)}^{{\displaystyle \frac{\epsilon -1}{k}}}.$$

Based on the above lemma, we obtain the Lyapunov-type inequality associated with the BVP (4).

Theorem 5.

If $\epsilon >k+1$ (5) is satisfied and a nontrivial continuous solution of (4) exists, then

$${\int }_{y_1}^{y_2}\left|r\left(\upsilon \right)\right|d\upsilon \ge {\displaystyle \frac{\mathrm{\Gamma }\left(\epsilon \right)(\epsilon -k-1)}{k{(y_2-y_1)}^{\epsilon -1}}}{\left({\displaystyle \frac{k!}{(\epsilon -2)(\epsilon -3)\dots (\epsilon -k)(\epsilon -k-1)}}\right)}^{{\displaystyle \frac{\epsilon -1}{k}}}.$$

Now, we avoid condition (5) and find a general Lyapunov-type inequality for (4), which is valid for the case $1\le k\le n-2.$

Lemma 12.

Let $\epsilon >k+1.$ Then,

$$\underset{t,\upsilon \in [y_1,y_2]}{max}{G}_k(t,\upsilon )\le {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\epsilon \right)}}M,$$

where

$$\begin{array}{ccc}\hfill M& =& max\{{\displaystyle \frac{1}{k!}}(\epsilon -1)(\epsilon -2)\dots (\epsilon -k){\displaystyle \frac{k^k{(y_2-y_1)}^{\epsilon -1}{(\epsilon -k-1)}^{\epsilon -k-1}}{{(\epsilon -1)}^{\epsilon -1}}},\hfill \\ & & {\displaystyle \frac{k{(y_2-y_1)}^{\epsilon -1}}{\epsilon -k-1}}{\left({\displaystyle \frac{(\epsilon -2)(\epsilon -3)\dots (\epsilon -k-1)}{k!}}\right)}^{{\displaystyle \frac{\epsilon -1}{k}}},\hfill \\ & & {\displaystyle \frac{{(y_2-y_1)}^{\epsilon -1}}{k!}}((\epsilon -1)(\epsilon -2)\dots (\epsilon -k)-k!)\}.\hfill \end{array}$$

Theorem 6.

Let $\epsilon >k+1.$ If ψ is a non-zero solution of (4), then

$${\int }_{y_1}^{y_2}\left|r\left(\upsilon \right)\right|d\upsilon >{\displaystyle \frac{\mathrm{\Gamma }\left(\epsilon \right)}{M}}.$$

4. Fractional Differential Equations with Left Riemann–Liouville and Right Caputo Fractional Derivatives and Lyapunov-Type Inequalities

In 2024, Kassymov and Torebek [7] considered a nonlinear Dirichlet BVP containing the left Riemann–Liouville fractional derivative operator $D_{a+}^{\beta }={\displaystyle \frac{d}{dt}}{I}_{a+}^{1-\beta }y\left(t\right)$ and right Caputo fractional derivative operator $D_{b-}^{\beta }y\left(t\right)=I_{b-}^{1-\beta }{y}^{\prime }\left(t\right)$ given by

$$\left\{\begin{array}{c}{\displaystyle D_{a+}^{\beta }{D}_{b-}^{\beta }y\left(t\right)-q\left(t\right)f\left(y\right)=0,a<t<b,}\hfill \\ y\left(a\right)=0,y\left(b\right)=0,\hfill \end{array}\right.$$

(6)

where $1/2<\beta \le 1,$ $q\in C[a,b],$ f is a continuous function and

$$I_{a+}^{\beta }y\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\beta \right)}}{\int }_a^t{(t-\upsilon )}^{\beta -1}y\left(\upsilon \right)d\upsilon ,I_{b-}^{\beta }y\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\beta \right)}}{\int }_t^b{(\upsilon -t)}^{\beta -1}y\left(\upsilon \right)d\upsilon ,t\in (a,b).$$

The next theorem deals with the Lyapunov-type inequality for problem (6).

Theorem 7.

Suppose that ${\displaystyle \beta \in \left(\frac{1}{2},1\right]}$, $q\in C[a,b]$, and f is a positive increasing function. If y is a nontrivial solution to (6), then

$${\int }_a^{b}q\left(\upsilon \right)d\upsilon >{\left({\displaystyle \frac{(2^{2-2\beta }-1+2^{2(1-2\beta )}}{{\mathrm{\Gamma }}^2\left(\beta \right)(2\beta -1)}}{(b-a)}^{2\beta -1}\right)}^{-1}{\displaystyle \frac{\parallel y\parallel }{f(\parallel y\parallel )}},$$

(7)

where $\parallel y\parallel ={max}_{t\in [a,b]}\left|y\left(t\right)\right|.$

Corollary 3.

Let ${\displaystyle \beta \in \left(\frac{1}{2},1\right]}$, $q\in C([a,b],R^{+})$, and f be a positive increasing function. If there exists $0<r_1<r_2$ such that

(F1) 

$f\left(y\right)\ge {\theta }^{*}{r}_1,y\in (0,r_1]$ and

(F2) 

$f\left(y\right)\le \theta r_2,y\in (0,r_2],$

then

$${\int }_a^{b}q\left(\upsilon \right)d\upsilon >{\left({\displaystyle \frac{(2^{2-2\beta }-1+2^{2(1-2\beta )}}{{\mathrm{\Gamma }}^2\left(\beta \right)(2\beta -1)}}{(b-a)}^{2\beta -1}\right)}^{-1}{\displaystyle \frac{r_1}{f\left(r_2\right)}}.$$

Remark 1.

Letting $\beta =1$ in (7), we obtain the Lyapunov-type inequality:

$${\int }_a^{b}q\left(\upsilon \right)d\upsilon >{\displaystyle \frac{4}{b-a}}\frac{\parallel y\parallel }{f(\parallel y\parallel )},$$

which for $f\left(y\right)=y$ takes the classical form of the Lyapunov inequality:

$${\int }_a^{b}q\left(\upsilon \right)d\upsilon >{\displaystyle \frac{4}{b-a}}.$$

In the next theorem, we describe the Hartman–Wintner-type inequality for problem (6).

Theorem 8.

Let ${\displaystyle \beta \in \left(\frac{1}{2},1\right],}$ $q\in C[a,b]$, and $f>0$ be an increasing function. If y is a nontrivial solution to (6), then

$$\begin{array}{ccc}\hfill & & {\int }_a^b\left[{\displaystyle \frac{{(b-a)}^{2\beta -1}\left({(b-\upsilon )}^{2\beta -1}+{(\upsilon -a)}^{2\beta -1}-{(b-a)}^{2\beta -1}\right)+{(b-\upsilon )}^{2\beta -1}{(\upsilon -a)}^{2\beta -1}}{{(b-a)}^{2\beta -1}}}\right]\times \hfill \\ & & \times q^{+}\left(\upsilon \right)d\upsilon \hfill \\ & >& {\mathrm{\Gamma }}^2\left(\beta \right)(2\beta -1){\displaystyle \frac{\parallel y\parallel }{f(\parallel y\parallel )}},\hfill \end{array}$$

(8)

where $q^{+}=max\{q,0\}.$

Remark 2.

If $\beta =1$, then the Hartman–Wintner-type inequality (8) takes the form of

$${\int }_a^b{\displaystyle \frac{(b-\upsilon )(\upsilon -a)}{b-a}}{q}^{+}\left(\upsilon \right)d\upsilon >{\displaystyle \frac{4}{b-a}}{\displaystyle \frac{\parallel y\parallel }{f(\parallel y\parallel )}},$$

which for $f\left(y\right)=y$ becomes the classical Hartman–Wintner inequality:

$${\int }_a^b{\displaystyle \frac{(b-\upsilon )(\upsilon -a)}{b-a}}{q}^{+}\left(\upsilon \right)d\upsilon >{\displaystyle \frac{4}{b-a}}.$$

In the following example, we demonstrate the application of Corollary 3.

Example 1.

Consider the fractional BVP:

$$\left\{\begin{array}{c}{\displaystyle D_{0+}^{\frac{3}{4}}{D}_{1-}^{\frac{3}{4}}y\left(t\right)-\sqrt{t}exp\left(-\frac{1}{y\left(t\right)+1}\right)=0,0<t<1,}\hfill \\ y\left(0\right)=0,y\left(1\right)=0.\hfill \end{array}\right.$$

(9)

Clearly, ${\displaystyle f\left(y\right)=exp\left(-\frac{1}{y\left(t\right)+1}\right)}$ is a positive increasing function and $q\left(t\right)=\sqrt{t}$ is a positive and Lebesgue integrable function with ${\parallel q\parallel }_{L^1(0,1)}={\displaystyle \frac{2}{3}}.$ Also, ${\theta }^{*}\approx 0.4328$ and $\theta \approx 0.5272$ ( ${\theta }^{*}$ and θ are, respectively, defined in $\left(F1\right)$ and $\left(F2\right)$). Letting $r_2={\displaystyle \frac{1}{5}}$ and $r_1={\displaystyle \frac{1}{20}},$ we obtain

$$f\left(y\right)=exp\left(-{\displaystyle \frac{1}{y+1}}\right)\ge {\theta }^{*}{r}_1,fory\in (0,r_1]$$

and

$$f\left(y\right)=exp\left(-{\displaystyle \frac{1}{y+1}}\right)\le \theta r_2,fory\in (0,r_2].$$

Then by Corollary 3, we obtain

$${\displaystyle \frac{2}{3}}={\int }_0^{1}q\left(s\right)ds>{\displaystyle \frac{\left(\sqrt{2}-1+\frac{1}{2}\right)\frac{1}{20}}{{\mathrm{\Gamma }}^2\left(\frac{3}{4}\right)\frac{1}{2}exp\left(-\frac{5}{6}\right)}}\approx 0.14.$$

5. Hilfer–Hadamard Fractional Differential Equations and Lyapunov-Type Inequalities

Definition 3.

For a function $z:[y_1,y_2]\to \mathbb{R}$, we define the ζ-th-order Hadamard fractional integral of z as

$${\displaystyle \left(I_{y_1}^{\zeta }z\right)\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\zeta \right)}{\int }_{y_1}^t{\left(log\frac{t}{\upsilon }\right)}^{\zeta -1}\frac{z\left(\upsilon \right)}{\upsilon }d\upsilon ,\zeta >0,t>y_1,y_1>0,}$$

if the right-hand side exists, and $I_{y_1}^{\zeta }$ defines an identity map when $\zeta =0.$

Definition 4.

Let $\zeta >0,0\le \rho \le 1$ and choose $n\in \mathbb{N}$ such that $n-1<\zeta \le n.$ The Hilfer–Hadamard fractional derivative of a function $z:[y_1,y_2]\to \mathbb{R}$ of order ζ and type ρ is defined by

$$\left(D^{\zeta ,\rho }z\right)\left(t\right)=\left(I_{y_1}^{\rho (n-\zeta )}{\mathcal{D}}^n{I}_{y_1}^{(n-\zeta )(1-\rho )}z\right)\left(t\right),{\mathcal{D}}^n=t^n{\displaystyle \frac{d^n}{d{t}^n}}t>y_1,$$

if the right-hand side exists.

In [8], the author obtained the Lyapunov-type inequalities for Hilfer–Hadamard-type fractional BVPs:

$$\left\{\begin{array}{c}\left(D_{y_1+}^{\zeta ,\rho }\psi \right)\left(t\right)+r\left(t\right)\psi \left(t\right)=0,1\le y_1<t<y_2,\hfill \\ l\left(I_{y_1+}^{(2-\zeta )(1-\rho )}\psi \right)\left(y_1\right)-m\left(\mathcal{D}{I}_{y_1+}^{(2-\zeta )(1-\rho )}\psi \right)\left(y_1\right)=0,\hfill \\ n\left(I_{y_1+}^{(2-\zeta )(1-\rho )}\psi \right)\left(y_2\right)+p\left(\mathcal{D}{I}_{y_1+}^{(2-\zeta )(1-\rho )}\psi \right)\left(y_2\right)=0,\hfill \end{array}\right.$$

(10)

and

$$\left\{\begin{array}{c}\left(D_{y_1+}^{\zeta ,\rho }\psi \right)\left(t\right)+r\left(t\right)\psi \left(t\right)=0,y_1<t<y_2,\hfill \\ \left(I_{y_1+}^{(2-\zeta )(1-\rho )}\psi \right)\left(y_1\right)+\left(I_{y_1+}^{(2-\zeta )(1-\rho )}\psi \right)\left(y_2\right)=0,\hfill \\ \left(\mathcal{D}{I}_{y_1+}^{(2-\zeta )(1-\rho )}\psi \right)\left(y_1\right)+\left(\mathcal{D}{I}_{y_1+}^{(2-\zeta )(1-\rho )}\psi \right)\left(y_2\right)=0,\hfill \end{array}\right.$$

(11)

where $D_{y_1+}^{\zeta ,\rho }$ denotes the Hilfer–Hadamard fractional derivative operator of order $\zeta \in (1,2]$ and type $\rho \in [0,1],$ $r:[y_1,y_2]\to \mathbb{R}$ is a continuous function. $l,m,n,p$ are real constants satisfying $l^2+m^2>0$ and $n^2+p^2>0,$ $\mathcal{D}=t{\displaystyle \frac{d}{dt}}$.

Lemma 13.

If ${\displaystyle mn+lp+ln\left(log\frac{y_2}{y_1}\right)\ne 0,}$ then problem (10) has a unique solution:

$$\psi \left(t\right)={\int }_{y_1}^{y_2}G(t,\upsilon )r\left(\upsilon \right)z\left(\upsilon \right)d\upsilon ,y_1<t<y_2,$$

where

$$G(t,\upsilon )={\displaystyle \frac{1}{\upsilon }}\left\{\begin{array}{cc}{G}_1(t,\upsilon ),\hfill & y_1<\upsilon \le t<y_2,\hfill \\ G_2(t,v),\hfill & y_1<t\le \upsilon <y_2,\hfill \end{array}\right.$$

with

$$G_1(t,\upsilon )=G_2(t,\upsilon )-{\displaystyle \frac{{\displaystyle {\left(log\frac{t}{s}\right)}^{\zeta -1}}}{\mathrm{\Gamma }\left(\zeta \right)}},$$

and

$$\begin{array}{ccc}\hfill G_2(t,\upsilon )& =& {\displaystyle {\left(log\frac{t}{y_1}\right)}^{-(2-\zeta )(1-\rho )}{\left(log\frac{y_2}{\upsilon }\right)}^{-\rho (2-\zeta )}}\hfill \\ & & \times {\displaystyle \frac{{\displaystyle \left[l\left(log\frac{t}{y_1}\right)+m(\zeta -1+\rho (2-\zeta ))\right][n\left(log\frac{y_2}{\upsilon }\right)+p(1-2\rho +\zeta \rho )]}}{{\displaystyle \left[mn+lp+ln\left(log\frac{y_2}{y_1}\right)\right]\mathrm{\Gamma }(2-2\rho +\zeta \rho )\mathrm{\Gamma }(2-(2-\zeta )(1-\rho ))}}}.\hfill \end{array}$$

Lemma 14.

The unique solution to problem (11) is given by

$$\psi \left(t\right)={\int }_{y_1}^{y_2}\widehat{G}(t,\upsilon )r\left(\upsilon \right)z\left(\upsilon \right)d\upsilon ,y_1<t<y_2,$$

where

$$\widehat{G}(t,\upsilon )={\displaystyle \frac{1}{s}}\left\{\begin{array}{cc}{\widehat{G}}_1(t,\upsilon ),\hfill & y_1<\upsilon \le t<y_2,\hfill \\ {\widehat{G}}_2(t,\upsilon ),\hfill & y_1<t\le \upsilon <y_2,\hfill \end{array}\right.$$

with

$${\widehat{G}}_1(t,\upsilon )={\widehat{G}}_2(t,\upsilon )-{\displaystyle \frac{{\displaystyle {\left(log\frac{t}{\upsilon }\right)}^{\zeta -1}}}{\mathrm{\Gamma }\left(\zeta \right)}},$$

and

$$\begin{array}{ccc}\hfill {\widehat{G}}_2(t,\upsilon )& =& {\displaystyle \frac{1}{2\mathrm{\Gamma }(1-2\rho +\zeta \rho )\mathrm{\Gamma }(1-(2-\zeta \left)\right(1-\rho \left)\right) } \{{\left(log\frac{t}{y_1}\right)}^{-(2-\zeta )(1-\rho )}}\hfill \\ & & {\displaystyle \times {\left(log\frac{y_2}{\upsilon }\right)}^{-\rho (2-\zeta )}\left[\frac{{\displaystyle \left(log\frac{t}{y_1}\right)}}{1-(2-\zeta )(1-\rho )}+\frac{{\displaystyle \left(log\frac{y_2}{\upsilon }\right)}}{1-\rho (2-\zeta )}-\frac{{\displaystyle \left(log\frac{y_2}{y_1}\right)}}{2}\right]\}.}\hfill \end{array}$$

Lemma 15.

Let $l,m,n,p\ge 0$ with ${\displaystyle mn+lp+ln\left(log\frac{y_2}{y_1}\right)>0}$ and

$$\begin{array}{ccc}\hfill H(t,\upsilon )& =& {\displaystyle {\left(log\frac{t}{y_1}\right)}^{(2-\zeta )(1-\beta )}{\left(log\frac{y_2}{\upsilon }\right)}^{\beta (2-\zeta )}G(t,\upsilon )}\hfill \\ & =& {\displaystyle \frac{1}{\upsilon }}\left\{\begin{array}{cc}{\displaystyle {\left(log\frac{t}{y_1}\right)}^{(2-\zeta )(1-\beta )}{\left(log\frac{y_2}{\upsilon }\right)}^{\beta (2-\zeta )}{G}_1(t,\upsilon ),}\hfill & y_1\le \upsilon \le t\le y_2,\hfill \\ {\displaystyle {\left(log\frac{t}{y_1}\right)}^{(2-\zeta )(1-\beta )}{\left(log\frac{y_2}{\upsilon }\right)}^{\beta (2-\zeta )}{G}_2(t,\upsilon ),}\hfill & y_1\le t\le \upsilon \le y_2,\hfill \end{array}\right.\hfill \end{array}$$

where $G(t,\upsilon )$, $G_1(t,\upsilon )$, and $G_2(t,\upsilon )$ are given in Lemma 13. Then,

$$\left|\upsilon H(t,\upsilon )\right|\le max\left\{\mathrm{\Omega },{\displaystyle \frac{{\displaystyle \left(log\frac{y_2}{y_1}\right)}}{\mathrm{\Gamma }\left(\zeta \right)}}\right\},(t,\upsilon )\in [y_1,y_2]\times [y_1,y_2],$$

where

$$\mathrm{\Omega }={\displaystyle \frac{{\displaystyle \left[l\left(log\frac{y_2}{y_1}\right)+m(\zeta -1+\beta (2-\zeta ))\right]\left[n\left(log\frac{y_2}{y_1}\right)+p(1-2\beta +\zeta \beta )\right]}}{{\displaystyle \left[mn+lp+ln\left(log\frac{y_2}{y_1}\right)\right]\mathrm{\Gamma }(2-2\beta +\zeta \beta )\mathrm{\Gamma }(2-(2-\zeta )(1-\beta ))}}}.$$

Lemma 16.

Let

$$\begin{array}{ccc}\hfill \overline{H}(t,\upsilon )& =& {\displaystyle {\left(log\frac{t}{y_1}\right)}^{(2-\zeta )(1-\rho )}{\left(log\frac{y_2}{\upsilon }\right)}^{\rho (2-\zeta )}\widehat{G}(t,\upsilon )}\hfill \\ & =& {\displaystyle \frac{1}{s}}\left\{\begin{array}{cc}{\displaystyle {\left(log\frac{t}{y_1}\right)}^{(2-\zeta )(1-\rho )}{\left(log\frac{y_2}{\upsilon }\right)}^{\rho (2-\zeta )}{\widehat{G}}_1(t,\upsilon ),}\hfill & y_1\le \upsilon \le t\le y_2,\hfill \\ {\displaystyle {\left(log\frac{t}{y_1}\right)}^{(2-\zeta )(1-\rho )}{\left(log\frac{y_2}{\upsilon }\right)}^{\rho (2-\zeta )}{\widehat{G}}_2(t,\upsilon ),}\hfill & y_1\le t\le \upsilon \le y_2,\hfill \end{array}\right.\hfill \end{array}$$

where $\widehat{G}(t,\upsilon )$, ${\widehat{G}}_1(t,\upsilon )$, and ${\widehat{G}}_2(t,\upsilon )$ are given in Lemma 14. Then,

$$\begin{array}{c}\hfill |\upsilon \overline{H}(t,\upsilon )|\le {\displaystyle \frac{{\displaystyle \left(log\frac{y_2}{y_1}\right)}}{A}}\left[{\displaystyle \frac{1}{1-(2-\zeta )(1-\rho )}}+{\displaystyle \frac{1}{1-\rho (2-\zeta )}}-{\displaystyle \frac{1}{2}}\right]+{\displaystyle \frac{{\displaystyle \left(log\frac{y_2}{y_1}\right)}}{\mathrm{\Gamma }\left(\zeta \right)}},\end{array}$$

$(t,\upsilon )\in [y_1,y_2]\times [y_1,y_2],$ where $A=2\mathrm{\Gamma }(1-2\rho +\zeta \rho )\mathrm{\Gamma }(1-(2-\zeta \left)\right(1-\rho \left)\right).$

Now, Lyapunov-type inequalities for the BVPs (10) and (11) are presented.

Theorem 9.

Let $l,m,n,p\ge 0$, and ${\displaystyle mn+lp+ln\left(log\frac{y_2}{y_1}\right)>0}$. Also, assume problem (10) has a nontrivial solution; then,

$${\displaystyle {\int }_{y_1}^{y_2}\frac{1}{\upsilon }{\left(log\frac{\upsilon }{y_1}\right)}^{(\zeta -2)(1-\rho )}{\left(log\frac{y_2}{\upsilon }\right)}^{\rho (\zeta -2)}\left|r\left(\upsilon \right)\right|d\upsilon \ge \frac{1}{\mathrm{\Lambda }},}$$

where

$$\mathrm{\Lambda }=max\left\{\mathrm{\Omega },{\displaystyle \frac{{\displaystyle \left(log\frac{y_2}{y_1}\right)}}{\mathrm{\Gamma }\left(\zeta \right)}}\right\}.$$

Theorem 10.

If a nontrivial solution exists to problem (11), then

$${\displaystyle {\int }_{y_1}^{y_2}\frac{1}{\upsilon }{\left(log\frac{\upsilon }{y_1}\right)}^{(\zeta -2)(1-\rho )}{\left(log\frac{y_2}{\upsilon }\right)}^{\rho (\zeta -2)}\left|r\left(\upsilon \right)\right|d\upsilon \ge \frac{1}{\mathrm{\Theta }},}$$

where

$$\mathrm{\Theta }={\displaystyle \frac{{\displaystyle \left(log\frac{y_2}{y_1}\right)}}{A}}\left[{\displaystyle \frac{1}{1-(2-\zeta )(1-\rho )}}+{\displaystyle \frac{1}{1-\rho (2-\zeta )}}-{\displaystyle \frac{1}{2}}\right]+{\displaystyle \frac{{\displaystyle \left(log\frac{y_2}{y_1}\right)}}{\mathrm{\Gamma }\left(\zeta \right)}}.$$

6. Sequential $\psi$-Riemann–Liouville Fractional Differential Equations and Lyapunov-Type Inequalities

In [9], Haddouchi and Samei studied Lyapunov-type inequalities for a sequential $\psi$-Riemann–Liouville fractional differential equation with nonlocal boundary conditions:

$$\left\{\begin{array}{c}{(}^{RL}{D}_{a_1+}^{\zeta ,\psi }+\lambda {(}^{RL}{D}_{a_1+}^{\zeta -1,\psi })z\left(t\right)+f(t,z\left(t\right))=0,a_1<t<a_2,\hfill \\ z\left(a_1\right)=0,z\left(a_2\right)=\theta \left(z\left(\epsilon \right)\right),\hfill \end{array}\right.$$

(12)

where $1<\zeta \le 2,$ $\lambda \in \mathbb{R},$ $a_1<\epsilon <a_2,$ $f\in C((a_1,a_2)\times \mathbb{R},\mathbb{R}),$ $\psi :[a_1,a_2]\to \mathbb{R}$ is a strictly increasing function such that ${\psi }^{\prime }\left(t\right)\ne 0$ for each $t\in [a_1,a_2].$

Lemma 17.

Let $h\in C([a_1,a_2],\mathbb{R}).$ Then, the solution of the sequential ψ-Riemann–Liouville fractional BVP:

$$\left\{\begin{array}{c}{(}^{RL}{D}_{a_1+}^{\zeta ,\psi }+\lambda {(}^{RL}{D}_{a_1+}^{\zeta -1,\psi })z\left(t\right)+h\left(t\right)=0,a_1<t<a_2,\hfill \\ z\left(a_1\right)=0,z\left(a_2\right)=\theta \left(z\left(\epsilon \right)\right),\hfill \end{array}\right.$$

(13)

has an integral solution given by

$$\begin{array}{ccc}\hfill z\left(t\right)& =& {\int }_{a_1}^{a_2}G(t,\upsilon ){\psi }^{\prime }\left(\upsilon \right)h\left(\upsilon \right)d\upsilon +\lambda [{\left({\displaystyle \frac{\psi \left(t\right)-\psi \left(a_1\right)}{\psi \left(a_2\right)-\psi \left(a_1\right)}}\right)}^{\zeta -1}{\int }_{a_1}^{a_2}{\psi }^{\prime }\left(\upsilon \right)z\left(\upsilon \right)d\upsilon \hfill \\ & & -{\int }_{a_1}^t{\psi }^{\prime }\left(\upsilon \right)z\left(\upsilon \right)d\upsilon ]+{\left({\displaystyle \frac{\psi \left(t\right)-\psi \left(a_1\right)}{\psi \left(a_2\right)-\psi \left(a_1\right)}}\right)}^{\zeta -1}\theta \left(z\left(\epsilon \right)\right),\hfill \end{array}$$

where

$$G(t,\upsilon )={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\zeta \right)}}\left\{\begin{array}{c}{\displaystyle {\left(\frac{\psi \left(t\right)-\psi \left(a_1\right)}{\psi \left(a_2\right)-\psi \left(a_1\right)}\right)}^{\zeta -1}[{(\psi \left(a_2\right)-\psi \left(\upsilon \right))}^{\zeta -1}}\hfill \\ -{(\psi \left(t\right)-\psi \left(\upsilon \right))}^{\zeta -1}],\hfill & a_1\le \upsilon \le t\le a_2,\hfill \\ {\displaystyle {\left(\frac{\psi \left(t\right)-\psi \left(a_1\right)}{\psi \left(a_2\right)-\psi \left(a_1\right)}\right)}^{\zeta -1}\left[{(\psi \left(a_2\right)-\psi \left(\upsilon \right))}^{\zeta -1}\right],}\hfill & a_1\le t\le \upsilon \le a_2.\hfill \end{array}\right.$$

(14)

In the next lemma, we present some properties of Green’s functions $G(t,s)$ defined in (14).

Lemma 18.

The Green’s functions $G(t,s)$ expressed in (14) satisfies the following properties:

(i) 

$G(t,\upsilon )$ is continuous on $[a_1,a_2]\times [a_1,a_2];$

(ii) 

$G(t,\upsilon )\ge 0,$ for each $t,\upsilon \in [a_1,a_2];$

(iii) 

For all $t\in [a_1,a_2]$ and for $1<\zeta \le 2,$ we have

$$\underset{t\in [a_1,a_2]}{max}G(t,\upsilon )\le {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\zeta \right)}}\left[{(\psi \left(a_2\right)-\psi \left(\upsilon \right))}^{\zeta -1}\right],$$

$$\underset{a_1\le t,\upsilon \le a_2}{max}G(t,\upsilon )\le {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\zeta \right)}}\left[{(\psi \left(a_2\right)-\psi \left(a_1\right))}^{\zeta -1}\right].$$

The main results on Lyapunov-type inequalities for the BVP (12) are stated in the following results.

Theorem 11.

Assume that there exists a function $q\in C([a_2,a_2],\mathbb{R})$ and $\delta >0$ such that

$$f(t,z(t\left)\right)=q\left(t\right)z\left(t\right),\left|\theta \right(z\left)\right|\le \delta \left|z\right|,z\in \mathbb{R}$$

and

$$2\left|\lambda \right|[\psi \left(a_2\right)-\psi \left(a_1\right)]+\delta <1.$$

If a nontrivial solution to the sequential ψ-Riemann–Liouville fractional BVP (12) exists on $[a_1,a_2],$ then

$${\int }_{a_1}^{a_2}{\psi }^{\prime }\left(\upsilon \right)\left|q\left(\upsilon \right)\right|d\upsilon \ge {\displaystyle \frac{\mathrm{\Gamma }\left(\zeta \right)}{{(\psi \left(a_2\right)-\psi \left(a_1\right))}^{\zeta -1}}}\left[1-\left(2\left|\lambda \right|[\psi \left(a_2\right)-\psi \left(a_1\right)]+\delta \right)\right].$$

(15)

Corollary 4.

(i) 

For $\psi \left(t\right)=t,$ inequality (15) can be rewritten as

$${\int }_{a_1}^{a_2}\left|q\left(\upsilon \right)\right|d\upsilon \ge {\displaystyle \frac{\mathrm{\Gamma }\left(\zeta \right)}{{(a_2-a_1)}^{\zeta -1}}}\left[1-\left(2\left|\lambda \right|(a_2-a_1)+\delta \right)\right].$$

(ii) 

For $\psi \left(t\right)=lnt,$ if ${\displaystyle {\int }_{a_1}^{a_2}\frac{\left|q\right(\upsilon \left)\right|}{\upsilon }ds<\infty ,}$ inequality (15) becomes

$${\int }_{a_1}^{a_2}{\displaystyle \frac{\left|q\right(\upsilon \left)\right|}{\upsilon }}d\upsilon \ge {\displaystyle \frac{\mathrm{\Gamma }\left(\zeta \right)}{{\displaystyle {\left(ln\frac{a_2}{a_1}\right)}^{\zeta -1}}}}\left[1-\left(2\left|\lambda \right|ln{\displaystyle \frac{a_2}{a_1}}+\delta \right)\right].$$

(iii) 

Let $w_e$ be an eigenvalue to problem (12), that is, $q\left(t\right)=w_e$ for each $t\in [a_1,a_2],$ then

$$|w_e|\ge {\displaystyle \frac{\mathrm{\Gamma }\left(\zeta \right)}{{(\psi \left(a_2\right)-\psi \left(a_1\right))}^{\zeta }}}\left[1-\left(2\left|\lambda \right|[\psi \left(a_2\right)-\psi \left(a_1\right)]+\delta \right)\right].$$

7. Fractional Hybrid BVPs and Lyapunov-Type Inequalities

In [10], Krushna studied the nonlinear fractional hybrid BVP:

$$\left\{\begin{array}{c}{\displaystyle {}^{RL}D_{y_0+}^{\zeta }\left[\frac{z\left(t\right)}{\mathrm{\Psi }(t,z(t\left)\right)}\right]+\ell \left(t\right)Fz\left(t\right)=0,y_0<t<y_1,}\hfill \\ z^{\left(i\right)}\left(y_0\right)=0,i=0,1,2,n-2,z\left(y_1\right)=0,\hfill \end{array}\right.$$

(16)

where $y_1>y_0\ge 0,$ $\zeta \in (n-1,n),$ $n\ge 2$, and ${}^{RL}D_{y_0+}^{(·)}$ is the Riemann–Liouville fractional derivative.

Relative to problem (16), we need the following assumptions:

$\left(H_1\right)$

$\mathrm{\Psi }:[y_0,y_1]\times \mathbb{R}\to \mathbb{R}\setminus \left\{0\right\}$ is a continuous and bounded function;

$\left(H_2\right)$

$F:C[y_0,y_1]\to C[y_0,y_1]$ and there exists $\epsilon >0$ such that $z\left(t\right)\in C[y_0,y_1]$ and $z\left(t\right)\ge 0$ for $y_0\le t\le y_1,$ then

$${\parallel Fz\parallel }_{\infty }\le \epsilon {\parallel z\parallel }_{\infty };$$

$\left(H_3\right)$

$\ell \in C([y_0,y_1],\mathbb{R}).$

Lemma 19.

Suppose that $\left(H_1\right)$ holds and $h\in C([y_0,y_1],\mathbb{R}).$ Then, $z\left(t\right)\in C([y_0,y_1],\mathbb{R})$ is a solution to the hybrid problem

$$\left\{\begin{array}{c}{\displaystyle {}^{RL}D_{y_0+}^{\zeta }\left[\frac{z\left(t\right)}{\mathrm{\Psi }(t,z(t\left)\right)}\right]+h\left(t\right)=0,y_0<t<y_1,}\hfill \\ z^{\left(i\right)}\left(y_0\right)=0,i=0,1,2,n-2,z\left(y_1\right)=0,\hfill \end{array}\right.$$

(17)

if and only if

$$z\left(t\right)=\mathrm{\Psi }(t,z\left(t\right)){\int }_{y_0}^{y_1}{G}_{\zeta }(t,\upsilon )h\left(\upsilon \right)d\upsilon ,$$

where

$$G_{\zeta }(t,\upsilon )=\left\{\begin{array}{cc}{\displaystyle \frac{{(y_1-\upsilon )}^{\zeta -1}{(t-y_0)}^{\zeta -1}}{{(y_1-y_0)}^{\zeta -1}\mathrm{\Gamma }\left(\zeta \right)},}\hfill & y_0\le t\le \upsilon \le y_1,\hfill \\ {\displaystyle \frac{{(y_1-\upsilon )}^{\zeta -1}{(t-y_0)}^{\zeta -1}}{{(y_1-y_0)}^{\zeta -1}\mathrm{\Gamma }\left(\zeta \right)}-\frac{{(t-\upsilon )}^{\zeta -1}}{\mathrm{\Gamma }\left(\zeta \right)},}\hfill & y_1\le \upsilon \le t\le y_1.\hfill \end{array}\right.$$

For $n=2$ and $n\ge 3$ cases, we have the following estimates for $G_{\zeta }$ given in Lemma 19.

Lemma 20.

For $n=2,$ the Green function $G_{\zeta }(t,\upsilon )$ given in Lemma 19 satisfies the following properties:

(a) 

$G_{\zeta }(t,\upsilon )\ge 0$ for all $t,\upsilon \in [y_0,y_1];$

(b) 

${\displaystyle \underset{t\in [y_0,y_1]}{max}{G}_{\zeta }(t,\upsilon )=G_{\zeta }(\upsilon ,\upsilon )=\frac{{(\upsilon -y_0)}^{\zeta -1}{(y_1-\upsilon )}^{\zeta -1}}{\mathrm{\Gamma }\left(\zeta \right){(y_1-y_0)}^{\zeta -1}},y_0\le \upsilon \le y_1;}$

(c) 

${\displaystyle \underset{t\in [y_0,y_1]}{max}{G}_{\zeta }(\upsilon ,\upsilon )=\frac{1}{\mathrm{\Gamma }\left(\zeta \right)}{\left(\frac{y_1-y_0}{4}\right)}^{\zeta -1}.}$

Lemma 21.

For $n\in \mathbb{N},n\ge 3,$ Green’s function $G_{\zeta }(t,s)$ given in Lemma 19 satisfies the following properties:

(a) 

$G_{\zeta }(t,\upsilon )\ge 0$ for all $t,\upsilon \in [y_0,y_1];$

(b) 

For $t\in [y_0,y_1],\upsilon \in [y_0,y_1],$

$$G_{\zeta }(t,\upsilon )\le G_{\zeta }({\upsilon }^{*},\upsilon )={\displaystyle \frac{{(y_1-\upsilon )}^{\zeta -1}{(\upsilon -y_0)}^{\zeta -1}}{{\displaystyle {(y_1-y_0)}^{\zeta -1}\mathrm{\Gamma }\left(\zeta \right){\left[1-{\left(\frac{y_1-\upsilon }{y_1-y_0}\right)}^{\frac{\zeta -1}{\zeta -2}}\right]}^{\zeta -2}}}},$$

where

$${\displaystyle y^{*}=\frac{{\displaystyle \upsilon -y_0{\left(\frac{y_1-\upsilon }{y_1-y_0}\right)}^{\frac{\zeta -1}{\zeta -2}}}}{{\displaystyle 1-{\left(\frac{y_1-\upsilon }{y_1-y_0}\right)}^{\frac{\zeta -1}{\zeta -2}}}}.}$$

Now, we will present Lyapunov-type inequalities for hybrid problem (16) for two cases, namely $n=2$ and $n\ge 3$.

• Case $n=2$.

In this case, problem (16) takes the following form:

$$\left\{\begin{array}{c}{\displaystyle {}^{RL}D_{y_0+}^{\zeta }\left[\frac{z\left(t\right)}{\mathrm{\Psi }(t,z(t\left)\right)}\right]+\ell \left(t\right)Fz\left(t\right)=0,y_0<t<y_1,}\hfill \\ z\left(y_0\right)=0,z\left(y_1\right)=0,\hfill \end{array}\right.$$

(18)

where $y_1>y_0\ge 0,$$\zeta \in (1,2).$

Theorem 12.

Assume that $\left(H_1\right)-\left(H_3\right)$ hold. If z represents a non-zero solution to BVP (18), then

$${\int }_{y_0}^{y_1}|\ell \left(\upsilon \right)|d\upsilon \ge {\displaystyle \frac{\mathrm{\Gamma }\left(\zeta \right)}{\mathcal{R}\epsilon }}{\left({\displaystyle \frac{4}{y_1-y_0}}\right)}^{\zeta -1},$$

where $\mathcal{R}=sup\left\{\right|\mathrm{\Psi }(t,z):t\in [y_0,y_1],z\in \mathbb{R}\}.$

• Case $n\ge 3$.

Theorem 13.

Assume that $\left(H_1\right)$–$\left(H_3\right)$ hold. If z represents a non-zero solution to BVP (16), then

$${\int }_{y_0}^{y_1}|\ell \left(\upsilon \right)|d\upsilon \ge {\displaystyle \frac{\mathrm{\Gamma }\left(\zeta \right){\left(1-{\varpi }_{\zeta }^{\frac{\zeta -1}{\zeta -2}}\right)}^{\zeta -2}}{\mathcal{R}\epsilon {(y_1-y_0)}^{\zeta -1}{\varpi }_{\zeta }^{\zeta -1}{(1-{\varpi }_{\zeta })}^{\zeta -1}}},$$

where ${\varpi }_{\zeta }$ is the unique zero of ${\varpi }^{{\displaystyle \frac{2\zeta -3}{\zeta -2}}}-2\varpi +1=0$ in ${\displaystyle \left(0,{\left(\frac{2\zeta -4}{2\zeta -3}\right)}^{\frac{\zeta -2}{\zeta -1}}\right).}$

Corollary 5.

If $\left(H_1\right)$–$\left(H_3\right)$ are satisfied, then the estimate

$${\int }_{y_0}^{y_1}|\ell \left(\upsilon \right)|d\upsilon <{\displaystyle \frac{\mathrm{\Gamma }\left(\zeta \right)}{\mathcal{R}\epsilon }}{\left({\displaystyle \frac{4}{y_1-y_0}}\right)}^{\zeta -1},$$

implies that the problem

$$\left\{\begin{array}{c}{\displaystyle {}^{RL}D_{y_0+}^{\zeta }z\left(t\right)+\ell \left(t\right)Fz\left(t\right)=0,y_0<t<y_1,}\hfill \\ z^{\left(i\right)}\left(y_0\right)=0,i=0,1,2,n-2,z\left(y_1\right)=0,\hfill \end{array}\right.$$

has only the trivial solution $z\left(t\right)\equiv 0,$ where $y_1>y_0\ge 0,\zeta \in (1,2).$

Corollary 6.

If $\left(H_1\right)$–$\left(H_3\right)$ are satisfied, then the estimate

$${\int }_{y_0}^{y_1}|\ell \left(\upsilon \right)|d\upsilon <{\displaystyle \frac{\mathrm{\Gamma }\left(\zeta \right){\left(1-{\varpi }_{\zeta }^{\frac{\zeta -1}{\zeta -2}}\right)}^{\zeta -2}}{\mathcal{R}\epsilon {(y_1-y_0)}^{\zeta -1}{\varpi }_{\zeta }^{\zeta -1}{(1-{\varpi }_{\zeta })}^{\zeta -1}}},$$

implies that the following problem has only the trivial solution $z\left(t\right)\equiv 0$:

$$\left\{\begin{array}{c}{\displaystyle {}^{RL}D_{y_0+}^{\zeta }z\left(t\right)+\ell \left(t\right)Fz\left(t\right)=0,y_0<t<y_1,}\hfill \\ z^{\left(i\right)}\left(y_0\right)=0,i=0,1,2,n-2,z\left(y_1\right)=0,\hfill \end{array}\right.$$

where $y_1>y_0\ge 0,$ $\zeta \in (n-1,n),n\ge 3,$ ${\varpi }_{\zeta }$ is the unique zero of ${\varpi }^{{\displaystyle \frac{2\zeta -3}{\zeta -2}}}-2\varpi +1=0$ in ${\displaystyle \left(0,{\left(\frac{2\zeta -4}{2\zeta -3}\right)}^{\frac{\zeta -2}{\zeta -1}}\right).}$

Corollary 7.

Suppose that $\left(H_1\right)$ and $\left(H_3\right)$ hold. Then, any eigenvalue λ of the fractional hybrid BVP:

$$\left\{\begin{array}{c}{\displaystyle {}^{RL}D_{y_0+}^{\zeta }\left[\frac{z\left(t\right)}{\mathrm{\Psi }(t,z(t\left)\right)}\right]+\lambda Fz\left(t\right)=0,y_0<t<y_1,}\hfill \\ z^{\left(i\right)}\left(y_0\right)=0,i=0,1,2,n-2,z\left(y_1\right)=0,\hfill \end{array}\right.$$

where $y_1>y_0\ge 0,$ $\zeta \in (n-1,n),n\ge 3,$ ${\varpi }_{\zeta }$ is the unique zero of ${\varpi }^{{\displaystyle \frac{2\zeta -3}{\zeta -2}}}-2\varpi +1=0$ in ${\displaystyle \left(0,{\left(\frac{2\zeta -4}{2\zeta -3}\right)}^{\frac{\zeta -2}{\zeta -1}}\right),}$ satisfies

$$\left|\lambda \right|\ge {\displaystyle \frac{\mathrm{\Gamma }\left(\zeta \right){\left(1-{\varpi }_{\zeta }^{\frac{\zeta -1}{\zeta -2}}\right)}^{\zeta -2}}{\mathcal{R}\epsilon {(y_1-y_0)}^{\zeta }{\varpi }_{\zeta }^{\zeta -1}{(1-{\varpi }_{\zeta })}^{\zeta -1}}}.$$

8. Lyapunov-Type Inequalities for Atangana–Baleanu Fractional Differential Equations

In [11], Hamiaz studied a nonlinear anti-periodic fractional BVP:

$$\left\{\begin{array}{c}{(}^{ABC}{D}_{y_1}^{\zeta }z)\left(t\right)+r\left(t\right)z\left(t\right)=0,t\in (y_1,y_2),y_1,y_2\in \mathbb{R},y_1<y_2,\hfill \\ z\left(y_1\right)+z\left(y_2\right)=0,z^{\prime }\left(y_1\right)+z^{\prime }\left(y_2\right)=0,\hfill \end{array}\right.$$

(19)

where ${}^{ABC}D^{\zeta }$ represents the Atangana–Baleanu Caputo fractional derivative of order $\zeta \in (1,2).$

Definition 5

([12]). Let $\zeta ,y_1,y_2\in \mathbb{R}$ such that $y_1<y_2,$ $\zeta \in (0,1).$ Let $z:[y_1,y_2]\to \mathbb{R}$ be a function such that $z\in H^1(y_1,y_2).$ The left Atangana–Baleanu fractional derivative in the Caputo sense is defined by

$${}^{ABC}D_{y_1}^{\zeta }z\left(t\right)={\displaystyle \frac{M\left(\zeta \right)}{1-\zeta }}{\int }_{y_1}^t{\displaystyle \frac{d}{d\xi }}z\left(\xi \right)E_{\zeta }\left[-{\displaystyle \frac{\zeta }{1-\zeta }}{(t-\xi )}^{\zeta }\right]d\xi ,$$

where $M\left(\zeta \right)>0$ with $M\left(0\right)=M\left(1\right)=1$ is a normalization function and $E_{\zeta }(·)$ is the Mittag–Leffler function given by

$$E_{\nu }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathrm{\Gamma }(k\nu +1)}},\Re \left(\nu \right)>0.$$

For $m<\zeta \le m+1,m\in \mathbb{N}$, $\mu =\zeta -m\in (0,1]$, we have ${}^{ABC}D_{y_1}^{\zeta }z\left(t\right)={}^{ABC}D_{y_1}^{\mu }{z}^{\left(m\right)}\left(t\right).$

The left Atangana–Baleanu fractional integral is defined by

$${}^{AB}I_{y_1}^{\zeta }z\left(t\right)={\displaystyle \frac{1-\zeta }{M\left(\zeta \right)}}z\left(t\right)+{\displaystyle \frac{\zeta }{M\left(\zeta \right)}}{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\zeta \right)}}{\int }_{y_1}^t{(t-\xi )}^{\zeta -1}z\left(\xi \right)d\xi .$$

Lemma 22.

Let $r\in C\left([y_1,y_2]\right)$ be such that $r\left(y_1\right)=r\left(y_2\right).$ Then, z is a solution to BVP (19) if and only if z satisfies the integral equation

$$z\left(t\right)={\int }_{y_1}^{y_2}{(y_2-\xi )}^{\zeta -2}G(t,\xi )r\left(\xi \right)z\left(\xi \right)d\xi ,t\in [y_1,y_2],$$

where $G(t,\xi )$ is defined by

$$G(t,\xi )={\displaystyle \frac{1}{M(\zeta -1)\mathrm{\Gamma }(\zeta -1)}}\left\{\begin{array}{cc}{G}_1(t,\xi ),\hfill & y_1\le t<\xi \le y_2,\hfill \\ G_2(t,\xi ),\hfill & y_1\le \xi <t\le y_2,\hfill \end{array}\right.$$

with

$$\begin{array}{ccc}\hfill G_1(t,\xi )& =& (\zeta -1)\left({\displaystyle \frac{t-y_1}{2}}-{\displaystyle \frac{y_2-y_1}{4}}\right)+{\displaystyle \frac{y_2-\xi }{2}}+{\displaystyle \frac{(2-\zeta )\mathrm{\Gamma }(\zeta -1)}{2}}{(y_2-\xi )}^{2-\zeta },\hfill \\ \hfill G_2(t,\xi )& =& (\zeta -1)\left({\displaystyle \frac{t-y_1}{2}}-{\displaystyle \frac{y_2-y_1}{4}}\right)+{\displaystyle \frac{y_2-\xi }{2}}-{\displaystyle \frac{{(t-\xi )}^{\zeta -1}}{{(y_2-\xi )}^{\zeta -2}}}\hfill \\ & & -{\displaystyle \frac{(2-\zeta )\mathrm{\Gamma }(\zeta -1)}{2}}{(y_2-\xi )}^{2-\zeta }.\hfill \end{array}$$

Lemma 23.

Let $G(t,\xi )$ be defined in Lemma 22. Then,

$$\underset{t,\xi \in [y_1,y_2]}{max}\left|G(t,\xi )\right|={\displaystyle \frac{(y_2-y_1)(3-\zeta )+2(2-\zeta )\mathrm{\Gamma }(\zeta -1){(y_2-y_1)}^{2-\zeta }}{4M(\zeta -1)\mathrm{\Gamma }(\zeta -1)}}.$$

Now, we provide for the BVP (19) the Lyapunov-type inequality.

Theorem 14.

Let $\zeta \in (1,2]$ and $y_1,y_2$ be real numbers such that $y_2-y_1\ge 1.$ Let $r\in C\left([y_1,y_2]\right)$ be not identically zero such that $r\left(y_1\right)=r\left(y_2\right)=0.$ Suppose that BVP (19) has a nontrivial continuous solution $z.$ Then,

$${\int }_{y_1}^{y_2}{(y_2-\xi )}^{\zeta -2}\left|r\left(\xi \right)\right|d\xi \ge {\displaystyle \frac{4M(\zeta -1)\mathrm{\Gamma }(\zeta -1)}{(y_2-y_1)(3-\zeta )+2(2-\zeta )\mathrm{\Gamma }(\zeta -1){(y_2-y_1)}^{2-\zeta }}}.$$

Corollary 8.

If

$${\int }_{y_1}^{y_2}{(y_2-\xi )}^{\zeta -2}\left|r\left(\xi \right)\right|d\xi <{\displaystyle \frac{4M(\zeta -1)\mathrm{\Gamma }(\zeta -1)}{(y_2-y_1)(3-\zeta )+2(2-\zeta )\mathrm{\Gamma }(\zeta -1){(y_2-y_1)}^{2-\zeta }}},$$

then problem (19) has the trivial solution $z=0.$

An application of the Lyapunov-type inequality to an eigenvalue problem is given in the example below.

Example 2.

Let $M\left(x\right)=1$ for all $x\in [0,1].$ Consider the following eigenvalue problem:

$$\left\{\begin{array}{c}{(}^{ABC}{D}_0^{\zeta }z)\left(t\right)+\lambda sin\left(t\right)z\left(t\right)=0,t\in (0,2\pi ),\hfill \\ z\left(0\right)+z\left(2\pi \right)=0,z^{\prime }\left(0\right)+z^{\prime }\left(2\pi \right)=0.\hfill \end{array}\right.$$

(20)

By Theorem 14, if the fractional BVP (20) has a nontrivial solution, then

$${\int }_0^{2\pi }{(2\pi -t)}^{-{\displaystyle \frac{1}{2}}}|\lambda sin\left(t\right)|dt\ge {\displaystyle \frac{4}{(3+\sqrt{2})\sqrt{\pi }}}\approx 0.511249.$$

Note that the above inequality holds true for $\left|\lambda \right|\ge 0.188062$ as ${\int }_0^{2\pi }{(2\pi -t)}^{-{\displaystyle \frac{1}{2}}}|sin\left(t\right)|dt$ $\approx 2.7185104.$ Hence, if $\left|\lambda \right|<0.188062,$ then we deduce by Corollary 8 that $z\left(t\right)=0$ is the unique solution to problem (20).

9. Tempered Fractional Differential Equations and Lyapunov-Type Inequalities

In [13], Ma and Li studied a multi-point coupled BVP for a system of nonlinear tempered fractional differential equations:

$$\left\{\begin{array}{c}{}_0{}^R\mathbb{D}_t^{{\alpha }_1,\lambda }x\left(t\right)+f(t,x\left(t\right),y\left(t\right))=0,t\in (0,1),\hfill \\ {}_0{}^R\mathbb{D}_t^{{\alpha }_2,\lambda }y\left(t\right)+g(t,x\left(t\right),y\left(t\right))=0,t\in (0,1),\hfill \\ {\displaystyle x\left(0\right)=0,x\left(1\right)=\sum _{i=1}^n{a}_{1i}x\left({\omega }_i\right)+\sum _{j=1}^n{a}_{2j}y\left({\varrho }_j\right),}\hfill \\ {\displaystyle y\left(0\right)=0,y\left(1\right)=\sum _{i=1}^n{a}_{3i}x\left({\omega }_i\right)+\sum _{j=1}^n{a}_{4j}y\left({\varrho }_j\right),}\hfill \end{array}\right.$$

(21)

where ${}_0{}^R\mathbb{D}_t^{{\alpha }_i,\lambda }$ denotes the tempered fractional derivative operator of order ${\alpha }_i\in (1,2]$ with $\lambda \ge 0,$ $f,g:[0,1]\times {\mathbb{R}}^2\to \mathbb{R},$ $0<{\omega }_1<{\omega }_2<\dots <{\omega }_n<1,$ $0<{\varrho }_1<{\varrho }_2<\dots <{\varrho }_n<1,$ $a_{ij}\ge 0(i=1,2,3,4,j=1,2,\dots ,n).$

Definition 6.

Let $f\in L[0,1]$ be a piece-wise continuous function on $(0,1)$ and $\lambda \ge 0.$ We define the tempered fractional derivative of Riemann–Liouville type for f of order $\alpha >0$ as

$$\begin{array}{ccc}\hfill {}_0{}^R\mathbb{D}_t^{\alpha ,\lambda }f\left(t\right)& =& e^{-\lambda t}{\mathcal{D}}_{0+}^{\alpha }\left(e^{\lambda t}f\left(t\right)\right)\hfill \\ & =& {\displaystyle \frac{e^{-\lambda t}}{\mathrm{\Gamma }(n-\alpha )}}{\left({\displaystyle \frac{d}{dt}}\right)}^{\left(n\right)}{\int }_0^t{(t-\upsilon )}^{n-\alpha -1}f\left(\upsilon \right)e^{\lambda \upsilon }d\upsilon ,t\in [0,1],n=\left[\alpha \right]+1,\hfill \end{array}$$

where ${\mathcal{D}}_{0+}^{\alpha }$ is the Riemann–Liouville fractional derivative given by

$${\mathcal{D}}_{0+}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\alpha )}}{\left({\displaystyle \frac{d}{dt}}\right)}^{\left(n\right)}{\int }_0^t{(t-\upsilon )}^{n-\alpha -1}f\left(\upsilon \right)d\upsilon ,t\in [0,1],n=\left[\alpha \right]+1,$$

provided that the right-hand side is defined pointwise on $[0,1]$. Here, $\left[\alpha \right]$ denotes the integer part of the number α and $\mathrm{\Gamma }(.)$ represents the Euler gamma function.

Definition 7.

For $\lambda \ge 0,$ the Riemann–Liouville tempered fractional integral for a piece-wise continuous function $f\in L[0,1]$ of order $\alpha >0$ is given by

$$\begin{array}{ccc}\hfill {}_0{}^R\mathbb{I}^{\alpha ,\lambda }f\left(t\right)& =& e^{-\lambda t}{I}_{0+}^{\alpha }\left(e^{\lambda t}f\left(t\right)\right)\hfill \\ & =& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\upsilon )}^{\alpha -1}f\left(\upsilon \right)e^{\lambda (t-\upsilon )}d\upsilon ,t\in [0,1],\hfill \end{array}$$

where

$$I_{0+}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\upsilon )}^{\alpha -1}f\left(\upsilon \right)d\upsilon ,t\in [0,1],$$

is the Riemann–Liouville fractional integral.

In the subsequent results, we need the following hypotheses:

$\left(H_1\right)$

$a_{ij}\ge 0,i=1,2,3,4,j=1,2,\dots ,n,$ ${\omega }_i\in (0,1),i=1,2,\dots ,n,$ ${\varrho }_j\in (0,1),\hspace{0.33em}j=1,2,\dots ,n,$ ${\gamma }_{ij}\ge 0,i,j=1,2,$ and $\gamma ={\gamma }_{11}{\gamma }_{22}-{\gamma }_{12}{\gamma }_{21}>0,$ where

$$\begin{array}{ccc}\hfill {\gamma }_{11}& =& 1-\sum _{i=1}^n{a}_{1i}{\omega }_i^{{\alpha }_1-1}{e}^{\lambda (1-{\omega }_i)},{\gamma }_{12}=\sum _{j=1}^n{a}_{2j}{\varrho }_j^{{\alpha }_2-1}{e}^{\lambda (1-{\varrho }_j)},\hfill \\ \hfill {\gamma }_{21}& =& \sum _{i=1}^n{a}_{3i}{\omega }_i^{{\alpha }_1-1}{e}^{\lambda (1-{\omega }_i)},{\gamma }_{22}=1-\sum _{j=1}^n{a}_{4j}{\varrho }_j^{{\alpha }_2-1}{e}^{\lambda (1-{\varrho }_j)};\hfill \end{array}$$

$\left(H_2\right)$

$f,g:[0,1]\times {\mathbb{R}}^2\to \mathbb{R}$ are continuous functions;

$\left(H_3\right)$

$0\le \lambda \le {\displaystyle \frac{\alpha -1}{1-{\xi }_1}},$ where $\alpha =min\{{\alpha }_1,{\alpha }_2\};$

$\left(H_4\right)$

There exist two positive functions $m_{11}\left(t\right)$ and $m_{12}\left(t\right)$ such that

$$\left|f(t,x,y)\right|\le m_{11}\left(t\right)\left|x\right|+m_{12}\left(t\right)\left|y\right|,t\in [0,1],x,y\in \mathbb{R};$$

$\left(H_5\right)$

There exist two positive functions $m_{21}\left(t\right)$ and $m_{22}\left(t\right)$ such that

$$\left|g(t,x,y)\right|\le m_{21}\left(t\right)\left|x\right|+m_{22}\left(t\right)\left|y\right|,t\in [0,1],x,y\in \mathbb{R}.$$

Lemma 24.

Let $h_1,h_2\in C\left([0,1]\right)$; then, $(x,y)$ is a solution of the system

$$\left\{\begin{array}{c}{}_0{}^R\mathbb{D}_t^{{\alpha }_1,\lambda }x\left(t\right)+h_1\left(t\right)=0,\hfill \\ {}_0{}^R\mathbb{D}_t^{{\alpha }_2,\lambda }y\left(t\right)+h_2\left(t\right)=0,\hfill \\ {\displaystyle x\left(0\right)=0,x\left(1\right)=\sum _{i=1}^n{a}_{1i}x\left({\omega }_i\right)+\sum _{j=1}^n{a}_{2j}y\left({\varrho }_j\right),}\hfill \\ {\displaystyle y\left(0\right)=0,y\left(1\right)=\sum _{i=1}^n{a}_{3i}x\left({\omega }_i\right)+\sum _{j=1}^n{a}_{4j}y\left({\varrho }_j\right),}\hfill \end{array}\right.$$

if and only if $(x,y)$ is a solution of the integral equations

$$\left\{\begin{array}{cc}x\left(t\right)\hfill & {\displaystyle ={\int }_0^1{G}_{11}(t,\upsilon )h_1\left(\upsilon \right)d\upsilon +{\int }_0^1{G}_{12}(t,\upsilon )h_2\left(\upsilon \right)dv,}\hfill \\ y\left(t\right)\hfill & {\displaystyle ={\int }_0^1{G}_{21}(t,\upsilon )h_1\left(\upsilon \right)d\upsilon +{\int }_0^1{G}_{22}(t,\upsilon )h_2\left(\upsilon \right)d\upsilon ,}\hfill \end{array}\right.$$

where

$$\begin{array}{ccc}\hfill G_{11}(t,\upsilon )& =& G_{{\alpha }_1}(t,\upsilon )+{\displaystyle \frac{t^{{\alpha }_1-1}{e}^{\lambda (1-t)}}{\gamma }}\sum _{i=1}^n({\gamma }_{22}{a}_{1i}+{\gamma }_{12}{a}_{3i})G_{{\alpha }_1}({\omega }_i,\upsilon ),\hfill \\ \hfill G_{12}(t,\upsilon )& =& {\displaystyle \frac{t^{{\alpha }_1-1}{e}^{\lambda (1-t)}}{\gamma }}\sum _{j=1}^n({\gamma }_{22}{a}_{2j}+{\gamma }_{12}{a}_{4j})G_{{\alpha }_2}({\varrho }_j,\upsilon ),\hfill \\ \hfill G_{21}(t,\upsilon )& =& {\displaystyle \frac{t^{{\alpha }_2-1}{e}^{\lambda (1-t)}}{\gamma }}\sum _{i=1}^n({\gamma }_{21}{a}_{1i}+{\gamma }_{11}{a}_{3i})G_{{\alpha }_1}({\omega }_i,\upsilon ),\hfill \\ \hfill G_{22}(t,\upsilon )& =& G_{{\alpha }_2}(t,\upsilon )+{\displaystyle \frac{t^{{\alpha }_2-1}{e}^{\lambda (1-t)}}{\gamma }}\sum _{j=1}^n({\gamma }_{21}{a}_{2j}+{\gamma }_{11}{a}_{4j})G_{{\alpha }_2}({\varrho }_j,\upsilon ),\hfill \end{array}$$

and

$$G_{{\alpha }_i}(t,\upsilon )={\displaystyle \frac{1}{\mathrm{\Gamma }\left({\alpha }_i\right)}}\left\{\begin{array}{cc}[t^{{\alpha }_i-1}{(1-\upsilon )}^{{\alpha }_i-1}-{(t-\upsilon )}^{{\alpha }_i-1}]e^{\lambda (\upsilon -t)},\hfill & 0\le \upsilon \le t\le 1,i=1,2,\hfill \\ [t^{{\alpha }_i-1}{(1-\upsilon )}^{{\alpha }_i-1}]e^{\lambda (\upsilon -t)},\hfill & 0\le t\le \upsilon \le 1,i=1,2.\hfill \end{array}\right.$$

Lemma 25.

Green’s function $G_{{\alpha }_i}(t,\upsilon )(i=1,2)$ defined in Lemma 24 has the following properties:

(1) 

For any $t,\upsilon \in [0,1],$ $G_{{\alpha }_i}(t,\upsilon )\ge 0i=1,2;$

(2) 

${max}_{t\in [0,1]}{G}_{{\alpha }_i}(t,\upsilon )=G_{{\alpha }_i}(\upsilon ,\upsilon ),\upsilon \in [0,1];$

(3) 

A unique maximum value of $G_{{\alpha }_i}(\upsilon ,\upsilon )$ is

$$\underset{\upsilon \in [0,1]}{max}{G}_{{\alpha }_i}(\upsilon ,\upsilon )=G_{{\alpha }_i}\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left({\alpha }_i\right)}}{\left({\displaystyle \frac{1}{4}}\right)}^{{\alpha }_i-1},i=1,2;$$

(4) 

For any $t,\upsilon \in [0,1],$ $G_{{\alpha }_i}(t,\upsilon )\le {\displaystyle \frac{1}{\mathrm{\Gamma }\left({\alpha }_i\right)}}\left[t^{{\alpha }_i-1}{(1-\upsilon )}^{{\alpha }_i-1}\right]e^{\lambda (\upsilon -t)},i=1,2.$

Lemma 26.

For $0<{\omega }_i<1$, we have

$$\underset{\upsilon \in [0,1]}{max}{G}_{{\alpha }_i}({\omega }_i,\upsilon )=G_{{\alpha }_i}({\omega }_i,{\omega }_i)={\displaystyle \frac{1}{\mathrm{\Gamma }\left({\alpha }_i\right)}}{\omega }_i^{{\alpha }_i-1}{(1-{\omega }_i)}^{{\alpha }_i-1}.$$

Lemma 27.

For any $t,\upsilon \in [0,1],$ the functions $G_{ij},i,j=1,2$ given in Lemma 24 satisfy the following inequalities:

(1) 

$G_{ij}(t,\upsilon )\le {\lambda }_{ij};$

(2) 

$G_{ij}(t,\upsilon )\le {\mu }_{ij}{t}^{{\alpha }_i-1}{(1-\upsilon )}^{{\alpha }_i-1}{e}^{\lambda (\upsilon -t)},$ where ${\lambda }_{ij},{\mu }_{ij}(i=1,2)$ is defined by

$$\begin{array}{ccc}\hfill {\lambda }_{11}& =& {\displaystyle \frac{1}{\mathrm{\Gamma }\left({\alpha }_1\right)}}{\left({\displaystyle \frac{1}{4}}\right)}^{{\alpha }_1-1}+{\displaystyle \frac{1}{\gamma \mathrm{\Gamma }\left({\alpha }_1\right)}}\sum _{i=1}^n({\gamma }_{22}{a}_{1i}+{\gamma }_{12}{a}_{3i}){\omega }_i^{{\alpha }_1-1}{(1-{\omega }_i)}^{{\alpha }_1-1},\hfill \\ \hfill {\lambda }_{12}& =& {\displaystyle \frac{1}{\gamma \mathrm{\Gamma }\left({\alpha }_2\right)}}\sum _{j=1}^n({\gamma }_{22}{a}_{2j}+{\gamma }_{12}{a}_{4j}){\varrho }_j^{{\alpha }_2-1}{(1-{\varrho }_j)}^{{\alpha }_2-1},\hfill \\ \hfill {\lambda }_{21}& =& {\displaystyle \frac{1}{\mathrm{\Gamma }\left({\alpha }_1\right)}}{\left({\displaystyle \frac{1}{4}}\right)}^{{\alpha }_2-1}+{\displaystyle \frac{1}{\gamma \mathrm{\Gamma }\left({\alpha }_1\right)}}\sum _{i=1}^n({\gamma }_{21}{a}_{1i}+{\gamma }_{11}{a}_{3i}){\omega }_i^{{\alpha }_1-1}{(1-{\omega }_i)}^{{\alpha }_1-1},\hfill \\ \hfill {\lambda }_{22}& =& {\displaystyle \frac{1}{\gamma \mathrm{\Gamma }\left({\alpha }_2\right)}}\sum _{j=1}^n({\gamma }_{21}{a}_{2j}+{\gamma }_{11}{a}_{4j}){\varrho }_j^{{\alpha }_2-1}{(1-{\varrho }_j)}^{{\alpha }_2-1},\hfill \end{array}$$

and

$${\mu }_{11}={\displaystyle \frac{{\gamma }_{22}}{\gamma \mathrm{\Gamma }\left({\alpha }_1\right)}},{\mu }_{12}={\displaystyle \frac{{\gamma }_{12}}{\gamma \mathrm{\Gamma }\left({\alpha }_2\right)}},{\mu }_{21}={\displaystyle \frac{{\gamma }_{21}}{\gamma \mathrm{\Gamma }\left({\alpha }_1\right)}},{\mu }_{22}={\displaystyle \frac{{\gamma }_{11}}{\gamma \mathrm{\Gamma }\left({\alpha }_2\right)}}.$$

Theorem 15.

Assume that the conditions $\left(H_1\right)$–$\left(H_5\right)$ hold. If the system (21) has a nontrivial solution, then

$$\begin{array}{cc}\left(a\right)\hfill & J_{11}(m_{11},m_{21})+J_{22}(m_{12},m_{22})\hfill \\ & +\sqrt{{[J_{11}(m_{11},m_{21})-J_{22}(m_{12},m_{22})]}^2+4J_{12}(m_{12},m_{22})J_{21}(m_{11},m_{21})}\ge 2,\hfill \end{array}$$

where

$$J_{ij}(m_{1j},m_{2j})={\lambda }_{i1}{\int }_0^1{m}_{1j}\left(\upsilon \right)d\upsilon +{\lambda }_{i2}{\int }_0^1{m}_{2j}\left(\upsilon \right)d\upsilon ,i,j=1,2,$$

with $m_{ij}\in C[0,1](i,j=1,2);$

$$\begin{array}{cc}\left(b\right)\hfill & I_{11}(m_{11},m_{21})+I_{22}(m_{12},m_{22})\hfill \\ & +\sqrt{{[I_{11}(m_{11},m_{21})-I_{22}(m_{12},m_{22})]}^2+4I_{12}(m_{12},m_{22})I_{21}(m_{11},m_{21})}\ge 2,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill I_{ij}(m_{1j},m_{2j})& =& {\mu }_{i1}{t}^{{\alpha }_i-1}{e}^{-\lambda t}{\int }_0^1{m}_{1j}\left(\upsilon \right){(1-\upsilon )}^{{\alpha }_1-1}{\upsilon }^{{\alpha }_j-1}d\upsilon \hfill \\ & & +{\mu }_{i2}{t}^{{\alpha }_i-1}{e}^{-\lambda t}{\int }_0^1{m}_{2j}\left(\upsilon \right){(1-\upsilon )}^{{\alpha }_2-1}{\upsilon }^{{\alpha }_j-1}ds,i,j=1,2,\hfill \end{array}$$

with $m_{ij}\in C[0,1](i,j=1,2);$

$$\begin{array}{cc}\left(c\right)\hfill & {\tau }_{11}(m_{11},m_{21})+{\tau }_{22}(m_{12},m_{22})\hfill \\ & +\sqrt{{[{\tau }_{11}(m_{11},m_{21})-{\tau }_{22}(m_{12},m_{22})]}^2+4{\tau }_{12}(m_{12},m_{22}){\tau }_{21}(m_{11},m_{21})}\ge 2,\hfill \end{array}$$

where

$$\tau (m_{1j},m_{2j})={\mu }_{i1}{g}_{1j}\left({\upsilon }_{1j}^{\sigma }\right){\int }_0^1{m}_{1j}\left(\upsilon \right)d\upsilon +{\mu }_{i2}{g}_{2j}\left({\upsilon }_{2j}^{\sigma }\right){\int }_0^1{m}_{2j}\left(\upsilon \right)d\upsilon ,i,j=1,2,$$

with $m_{ij}\in C[0,1](i,j=1,2).$

Now, we provide an example to illustrate Theorem 15 (a).

Example 3.

We examine a multi-point BVP for a nonlinear system of coupled tempered fractional differential equations:

$$\left\{\begin{array}{c}{\displaystyle {-}_0^R{\mathbb{D}}_t^{{\alpha }_1,\lambda }x\left(t\right)=\frac{{\alpha }_1^{{\alpha }_1}\mathrm{\Gamma }\left({\alpha }_1\right)}{{({\alpha }_1-1)}^{({\alpha }_1-1)}}\left[t^{2}arctan\left(x\left(t\right)\right)+\frac{e^{-t}}{2}sin\left(y\left(t\right)\right)\right],t\in (0,1),}\hfill \\ {\displaystyle {-}_0^R{\mathbb{D}}_t^{{\alpha }_2,\lambda }y\left(t\right)=\frac{{\alpha }_2^{{\alpha }_2}\mathrm{\Gamma }\left({\alpha }_2\right)}{{({\alpha }_2-1)}^{({\alpha }_2-1)}}\left[\frac{e^{-t}}{2}x\left(t\right)+t^{2}ln(1+|y\left(t\right)\left|\right)\right],t\in (0,1),}\hfill \\ {\displaystyle x\left(0\right)=0,x\left(1\right)=\frac{1}{4}x\left(\frac{1}{3}\right)+\frac{1}{2}x\left(\frac{1}{9}\right)+\frac{1}{6}y\left(\frac{1}{4}\right)+\frac{1}{3}y\left(\frac{1}{16}\right),}\hfill \\ {\displaystyle y\left(0\right)=0,y\left(1\right)=\frac{1}{2}x\left(\frac{1}{3}\right)+x\left(\frac{1}{9}\right)+\frac{1}{8}y\left(\frac{1}{4}\right)+\frac{1}{4}y\left(\frac{1}{16}\right).}\hfill \end{array}\right.$$

(22)

Fixing $a_{1i}={\displaystyle \frac{i}{4}},$$a_{2j}={\displaystyle \frac{j}{6}},$$a_{3i}={\displaystyle \frac{i}{2}},$$a_{4j}={\displaystyle \frac{j}{8}},$${\omega }_i={\displaystyle \frac{1}{3^i}},$ ${\varrho }_j={\displaystyle \frac{1}{4^j}}(i,j=1,2),$ ${\alpha }_1={\alpha }_2=2$, and $\lambda ={\displaystyle \frac{3}{2}},$ we find that

$$m_{11}\left(t\right)=m_{22}\left(t\right)={\displaystyle \frac{{\alpha }_1^{{\alpha }_1}\mathrm{\Gamma }\left({\alpha }_1\right)}{{({\alpha }_1-1)}^{({\alpha }_1-1)}}}{t}^2,m_{12}\left(t\right)=m_{22}\left(t\right)={\displaystyle \frac{{\alpha }_2^{{\alpha }_2}\mathrm{\Gamma }\left({\alpha }_2\right)}{{({\alpha }_2-1)}^{({\alpha }_2-1)}}}{\displaystyle \frac{e^{-t}}{2}}.$$

Moreover, we have

$${\gamma }_{11}\approx 0.563,{\gamma }_{12}\approx 0.213,{\gamma }_{21}\approx 0.875,{\gamma }_{22}\approx 0.84$$

and

$$\gamma ={\gamma }_{11}{\gamma }_{22}-{\gamma }_{12}{\gamma }_{21}\approx 0.287>0.$$

We also find

$${\lambda }_{11}\approx 0.713,{\lambda }_{12}\approx 0.079,{\lambda }_{21}\approx 0.982,{\lambda }_{22}\approx 0.23.$$

Observe that assumptions (H1)–(H5) are satisfied and

$$\begin{array}{cc}& J_{11}(m_{11},m_{21})+J_{22}(m_{12},m_{22})\hfill \\ & +\sqrt{{[J_{11}(m_{11},m_{21})-J_{22}(m_{12},m_{22})]}^2+4J_{12}(m_{12},m_{22})J_{21}(m_{11},m_{21})}\hfill \\ & =0.098<2.\hfill \end{array}$$

Therefore, $(x,y)=(0,0)$ is the unique solution to (22), which confirms the Lyapunov inequality according to Theorem 15 (a).

10. Half-Linear Local Fractional Differential Equations and Lyapunov-Type Inequality

Liu and Wang in [14] considered the following half-linear local fractional BVP:

$$\left\{\begin{array}{c}{D}^{\left(\zeta \right)}\left(h\left(t\right)\right|D^{\left(\zeta \right)}{z\left(t\right)|}^{p-2}{D}^{\left(\zeta \right)}{z\left(t\right))+r\left(t\right)|u|}^{p-2}z=0,y_1<t<y_2,\hfill \\ z\left(y_1\right)=0,z\left(y_2\right)=0,z\left(t\right)\ne 0,t\in (y_1,y_2),\hfill \end{array}\right.$$

(23)

where $h,r\in C([y_1,y_2],{\mathbb{R}}^{+})$.

Definition 8

([15]). A non-differentiable function $f:\mathbb{R}\to {\mathbb{R}}^{\zeta },$ $t\to f\left(t\right)$ is said to be Yang’s local fractional continuous at $t=t_0$ if there exists $\delta >0$ for any $ϵ>0$ such that

$$|f\left(t\right)-f\left(t_0\right)| <{ϵ}^{\zeta },$$

holds for $|t-t_0| <\delta ,$ where $ϵ,\delta \in \mathbb{R}.$ We denote by $C_{\zeta }(y_1,y_2)$ the set of all of Yang’s local fractional continuous functions.

Definition 9

([15]). The Yang’s local fractional derivative for a function $f\in C_{\zeta }(y_1,y_2)$ of order ζ at $t=t_0$ is defined as

$$\begin{array}{ccc}\hfill D^{\left(\zeta \right)}f\left(t_0\right)& =& f^{\left(\zeta \right)}\left(t_0\right)={\displaystyle \frac{d^{\zeta }f\left(t\right)}{d{t}^{\zeta }}}{|}_{t=t_0}\hfill \\ & =& \underset{t\to t_0}{lim}{\displaystyle \frac{{\Delta }^{\zeta }(f\left(t\right)-f\left(t_0\right))}{{(t-t_0)}^{\zeta }}},\hfill \end{array}$$

where ${\Delta }^{\zeta }(f\left(t\right)-f\left(t_0\right))\approx \mathrm{\Gamma }(1+\zeta )[f\left(t\right)-f\left(t_0\right)].$

Definition 10

([15]). The Yang’s local fractional integral for a function $f\in C_{\zeta }(y_1,y_2)$ of order ζ is given by

$$\begin{array}{ccc}\hfill {}_{y_1}I_{y_2}^{\zeta }f\left(t\right)& =& {\displaystyle \frac{1}{\mathrm{\Gamma }(1+\zeta )}}{\int }_{y_1}^{y_2}f\left(\upsilon \right){\left(d\upsilon \right)}^{\zeta }\hfill \\ & =& {\displaystyle \frac{1}{\mathrm{\Gamma }(1+\zeta )}}\underset{\Delta t\to 0}{lim}\sum _{j=0}^{N-1}f\left(t_j\right){(\Delta t_j)}^{\zeta },\hfill \end{array}$$

where $\Delta t_j=t_{j+1}-t_j,$ $\Delta t=max\{\Delta t_1,\Delta t_2,\dots ,\Delta t_j,\dots \Delta t_{N-1}\}$ and $y_1<t_0<t_1<\dots <t_{N-1}<t_N=y_2$ represents a partition of $[y_1,y_2].$

Lemma 28.

If $z\left(t\right)$ is a solution of the half-linear local fractional BVP (23), then

$$\underset{y_1\le t\le y_2}{sup}\left|z\left(t\right)\right|\le {\displaystyle \frac{{(y_2^{\zeta }-y_1^{\zeta })}^{\frac{1}{q}}}{2{\left[\mathrm{\Gamma }(1+\zeta )\right]}^{\frac{q+1}{q}}}}{\left({\int }_{y_1}^{y_2}{\left|z^{\left(\zeta \right)}\left(\upsilon \right)\right|}^p{\left(d\upsilon \right)}^{\zeta }\right)}^{{\displaystyle \frac{1}{p}}},$$

where ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1.$

The next theorem provides a Lyapunov-type inequality for the half-linear local fractional BVP (23).

Theorem 16.

Let $z\left(t\right)$ be a non-zero solution of the BVP (23); then,

$${\int }_{y_1}^{y_2}\left|h\left(\upsilon \right)\right|{\left(d\upsilon \right)}^{\zeta }\ge {\displaystyle \frac{2^p{\left[\mathrm{\Gamma }(1+\zeta )\right]}^{\frac{q+2p}{q}}}{{(y_2^{\zeta }-y_1^{\zeta })}^{\frac{p}{q}}}},$$

where ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1.$

11. Lyapunov-Type Inequality for Discrete Fractional Equations

Definition 11.

Let $g:{\mathbb{N}}_a\to \mathbb{R}$. Then, for a given $\varpi >0$, the fractional ϖ sum of g is given by

$${\Delta }^{-\varpi }g\left(\upsilon \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\varpi \right)}}\sum _{\rho =a}^{\upsilon -\varpi }{(\upsilon -\sigma \left(\rho \right))}^{(\varpi -1)}g\left(\rho \right)$$

for all $\upsilon \in {\mathbb{N}}_{a+\varpi },$$\sigma \left(\rho \right)=\rho +1$ and ${\displaystyle {\upsilon }^{\left(\varpi \right)}=\frac{\mathrm{\Gamma }(\upsilon +1)}{\mathrm{\Gamma }(\upsilon +1-\varpi )}.}$

In [16], Alzabut et al. obtained a Lyapunov-type inequality for a discrete Riemann–Liouville fractional BVP:

$$\left\{\begin{array}{c}{}^{RL}\Delta ^{\varpi }\phi \left(\upsilon \right)+r(\upsilon +\varpi -1)\phi (\upsilon +\varpi -1)=0,\upsilon \in {\mathbb{N}}_0^{\ell },\hfill \\ \phi (\varpi -2)=0,\phi (\varpi +\ell )=0,\hfill \end{array}\right.$$

(24)

where $\varpi \in (1,2]$, $r:{\mathbb{N}}_{\varpi -1}^{\varpi +\ell -1}\to [0,\infty )$, and $\ell \in {\mathbb{N}}_1.$

Lemma 29.

Assume $\varpi \in (1,2]$ and $\epsilon ={(\varpi +\ell )}^{(\varpi -1)}.$ Then, the solution to BVP (24) is

$$\phi \left(\upsilon \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\varpi \right)}}\sum _{\rho =0}^{\ell }G(\upsilon ,\rho )r(\rho +\varpi -1)\phi (\rho +\varpi -1),$$

where $G(\upsilon ,\rho ):{\mathbb{N}}_{\varpi -2}^{\varpi +\ell }\times {\mathbb{N}}_0^{\ell }\to \mathbb{R}$ is defined by

$$G(\upsilon ,\rho )=\left\{\begin{array}{cc}{\displaystyle \frac{{(\varpi +\ell -\sigma \left(\rho \right))}^{(\varpi -1)}{\upsilon }^{(\varpi -1)}}{\epsilon }-{(\upsilon -\sigma \left(\rho \right))}^{(\varpi -1)},}\hfill & 0\le \rho <\upsilon -\varpi +1\le \ell ,\hfill \\ {\displaystyle \frac{{(\varpi +\ell -\sigma \left(\rho \right))}^{(\varpi -1)}{\upsilon }^{(\varpi -1)}}{\epsilon },}\hfill & 0\le \upsilon -\varpi +1\le \rho \le \ell .\hfill \end{array}\right.$$

Lemma 30.

The function G given in Lemma 29 satisfies the following properties:

(i) 

$G(\upsilon ,\rho )\ge 0,\upsilon \in {\mathbb{N}}_{\varpi -2}^{\varpi +\ell },\rho \in {\mathbb{N}}_0^{\ell };$

(ii) 

$maxG(\upsilon ,\rho )=G(\rho +\varpi -1,\rho ),\upsilon \in {\mathbb{N}}_{\varpi -2}^{\varpi +\ell },\rho \in {\mathbb{N}}_0^{\ell };$

(iii) 

${\displaystyle \underset{\rho \in {\mathbb{N}}_0^{\ell }}{max}G(\rho +\varpi -1,\rho )=\frac{\mathrm{\Gamma }\left(\varpi \right)(\ell +1)}{(\varpi +\ell )}.}$

We express now for the discrete Riemann–Liouville fractional BVP (24) the Lyapunov-type inequality.

Theorem 17.

Let $r:{\mathbb{N}}_{\varpi -1}^{\varpi +\ell -1}\to [0,\infty )$ be a non-zero function. If a non-zero solution to the BVP (24) exists, then

$$\sum _{\rho =0}^{\ell }\left|r(\varpi +\rho -1)\right|\ge {\displaystyle \frac{(\varpi +\ell )}{(\ell +1)}}.$$

An application of the above Lyapunov-type inequality to an eigenvalue problem for the discrete fractional eigenvalue problem is discussed in the next corollary.

Corollary 9.

If the discrete fractional eigenvalue problem

$$\left\{\begin{array}{c}{}^{RL}\Delta ^{\varpi }\phi \left(\upsilon \right)+\lambda \phi (\upsilon +\varpi -1)=0,\varpi \in (1,2],\upsilon \in {\mathbb{N}}_0^{\ell },\hfill \\ \phi (\varpi -2)=0,\phi (\varpi +\ell )=0,\hfill \end{array}\right.$$

(25)

has a non-zero solution for any $\lambda \in \mathbb{R},$ then

$$\left|\lambda \right|\ge {\displaystyle \frac{(\varpi +\ell )}{{(\ell +1)}^2}}.$$

Example 4.

Consider the discrete Riemann–Liouville fractional BVP:

$$\left\{\begin{array}{c}{}^{RL}\Delta ^{1.6}\phi \left(\upsilon \right)+\left({\displaystyle \frac{5(\upsilon +0.6)+1}{2}}\right)\phi (\upsilon +0.6)=0,\upsilon \in {\mathbb{N}}_0^6,\hfill \\ \phi (-0.4)=0,\phi \left(7.6\right)=0.\hfill \end{array}\right.$$

Here, $\varpi =1.6,\ell =6$, and $r\left(\upsilon \right)={\displaystyle \frac{5\upsilon +1}{2}}.$ For $r(\rho +0.6):{\mathbb{N}}_{0.6}^{6.6}\to [0,\infty )$, we have

$$\sum _{\rho =0}^6\left|r(\rho +0.6)\right|=\sum _{\rho =0}^6\left|{\displaystyle \frac{5\rho +4}{2}}\right|>66.5>0.$$

By Theorem 17, we obtain

$$\sum _{\rho =0}^6\left|{\displaystyle \frac{5\rho +4}{2}}\right|=66.5>{\displaystyle \frac{7.6}{7}}\approx 1.086.$$

12. Conclusions

In this article, we have described the recent development on Lyapunov-type inequalities for fractional differential equations complemented with different boundary conditions. The fractional BVPs addressed in this review encompass various types of derivative operators of fractional order, such as Riemann–Liouville, Caputo, Hilfer–Hadamard, $\psi$-Riemann–Liouville, Atangana–Baleanu, tempered, half-linear, and discrete fractional, together with a wide range of boundary conditions such as Dirichlet, nonlocal, multi-point, anti-periodic, and discrete conditions. We believe this survey will serve as a useful reference point for researchers working in the area of Lyapunov-type inequalities, helping them update themselves on the existing literature before pursuing their new research for emerging fractional BVPs.