Limits of sub-bifractional Brownian noises

Nenghui Kuang; Nenghui Kuang

doi:10.3934/era.2023063

Electronic Research Archive

2023, Volume 31, Issue 3: 1240-1252. doi: 10.3934/era.2023063

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Research article

Limits of sub-bifractional Brownian noises

Nenghui Kuang ^,

School of Mathematics and Computing Science, Hunan University of Science and Technology, Hunan 411201, China

Received: 19 July 2022 Revised: 12 November 2022 Accepted: 21 December 2022 Published: 03 January 2023

Let $S^{H, K} = \{S^{H, K}_t, t\geq 0\}$ be the sub-bifractional Brownian motion (sbfBm) of dimension 1, with indices $H\in (0, 1)$ and $K\in (0, 1].$ We primarily prove that the increment process generated by the sbfBm $\left\{S^{H, K}_{h+t}-S^{H, K}_h, t\geq 0\right\}$ converges to $\left\{B^{HK}_t, t\geq 0\right\}$ as $h\rightarrow \infty$ , where $\left\{B^{HK}_t, t\geq 0\right\}$ is the fractional Brownian motion with Hurst index $HK$ . Moreover, we study the behavior of the noise associated to the sbfBm and limit theorems to $S^{H, K}$ and the behavior of the tangent process of sbfBm.

Keywords:

Citation: Nenghui Kuang. Limits of sub-bifractional Brownian noises[J]. Electronic Research Archive, 2023, 31(3): 1240-1252. doi: 10.3934/era.2023063

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Abstract

1. Introduction

The well-known classical Lyapunov inequality ^[15] states that, if $u$ is a nontrivial solution of the Hill's equation

$\begin{equation} u^{\prime \prime }\left( t\right) +q\left( t\right) u\left( t\right) = 0, \text{ }a \lt t \lt b, \end{equation}$

(1.1)

subject to Dirichlet-type boundary conditions:

$\begin{equation} u\left( a\right) = u\left( b\right) = 0, \end{equation}$

(1.2)

then

$\begin{equation} \int_{a}^{b}\left\vert q\left( t\right) \right\vert dt \gt \frac{4}{b-a}, \end{equation}$

(1.3)

where $q:[a, b]\rightarrow \mathbb{R}$ is a real and continuous function.

Later, in 1951, Wintner ^[24], obtained the following inequality:

$\begin{equation} \int_{a}^{b}q^{+}\left( t\right) dt \gt \frac{4}{b-a}, \end{equation}$

(1.4)

where $q^{+}(t) = \max \left\{ q\left(t\right), 0\right\}.$

A more general inequality was given by Hartman and Wintner in ^[12], that is known as Hartman Wintner-type inequality:

$\begin{equation} \int_{a}^{b}\left( t-a\right) \left( b-t\right) q^{+}\left( t\right) dt \gt b-a, \end{equation}$

(1.5)

Since $\underset{t\in \lbrack a, b]}{\max }\left(t-a\right) \left(b-t\right) = \frac{\left(b-a\right) ^{2}}{4}$ , then, (1.5) implies (1.4).

The Lyapunov inequality and its generalizations have many applications in different fields such in oscillation theory, asymptotic theory, disconjugacy, eigenvalue problems.

Recently, many authors have extended the Lyapunov inequality (1.3) for fractional differential equations ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,15,18,20,22,23,24]}. For this end, they substituted the ordinary second order derivative in (1.1) by a fractional derivative or a conformable derivative. The first result in which a fractional derivative is used instead of the ordinary derivative in equation (1.1), is the work of Ferreira ^[6]. He considered the following two-point Riemann-Liouville fractional boundary value problem

$\begin{equation*} D_{a^{+}}^{\alpha }u\left( t\right) +q(t)u\left( t\right) = 0, \text{ }a \lt t \lt b, \text{ }1 \lt \alpha \leq 2 \end{equation*}$

$\begin{equation*} u\left( a\right) = u\left( b\right) = 0. \end{equation*}$

And obtained the Lyapunov inequality:

$\begin{equation*} \int_{a}^{b}\left\vert q\left( t\right) \right\vert dt \gt \Gamma \left( \alpha \right) \left( \frac{4}{b-a}\right) ^{\alpha -1}. \end{equation*}$

Then, he studied in ^[7], the Caputo fractional differential equation

$\begin{equation*} ^{C}D_{a^{+}}^{\alpha }u\left( t\right) +q(t)u\left( t\right) = 0, \text{ } a \lt t \lt b, \text{ }1 \lt \alpha \leq 2 \end{equation*}$

under Dirichlet boundary conditions (1.2). In this case, the corresponding Lyapunov inequality has the form

$\begin{equation*} \int_{a}^{b}\left\vert q\left( t\right) \right\vert dt \gt \frac{\alpha ^{\alpha }\Gamma \left( \alpha \right) }{\left( \left( \alpha -1\right) \left( b-a\right) \right) ^{\alpha -1}}. \end{equation*}$

Later Agarwal and Özbekler in ^[1], complimented and improved the work of Ferreira ^[6]. More precisely, they proved that if $u$ is a nontrivial solution of the Riemann-Liouville fractional forced nonlinear differential equations of order $\alpha \in (0, 2]$ :

$\begin{equation*} D_{a^{+}}^{\alpha }u\left( t\right) +p\left( t\right) \left\vert u\left( t\right) \right\vert ^{\mu -1}u\left( t\right) +q\left( t\right) \left\vert u\left( t\right) \right\vert ^{\gamma -1}u\left( t\right) = f\left( t\right) , \text{ }a \lt t \lt b, \end{equation*}$

satisfying the Dirichlet boundary conditions (1.2), then the following Lyapunov type inequality

$\begin{equation*} \left( \int_{a}^{b}\left[ p^{+}\left( t\right) +q^{+}\left( t\right) \right] dt\right) \left( \int_{a}^{b}\left[ \mu _{0}p^{+}\left( t\right) +\gamma _{0}q^{+}\left( t\right) +\left\vert f\left( t\right) \right\vert \right] dt\right) \gt \frac{4^{2\alpha -3}\Gamma ^{2}\left( \alpha \right) }{\left( b-a\right) ^{2\alpha -2}}. \end{equation*}$

holds, where $p,$ $q,$ $f$ are real-valued functions, $0 < \gamma < 1 < \mu < 2$ , $\mu _{0} = \left(2-\mu \right) \mu ^{\mu /\left(2-\mu \right) }2^{2/\left(\mu -2\right) }$ and $\gamma _{0} = \left(2-\gamma \right) \gamma ^{\gamma /\left(2-\gamma \right) }2^{2/\left(\gamma -2\right) }$ .

In 2017, Guezane-Lakoud et al. ^[11], derived a new Lyapunov type inequality for a boundary value problem involving both left Riemann-Liouville and right Caputo fractional derivatives in presence of natural conditions

$\begin{equation*} -^{C}D_{b^{-}}^{\alpha }D_{a^{+}}^{\beta }u\left( t\right) +q(t)u\left( t\right) = 0, \text{ }a \lt t \lt b, \text{ }0 \lt \alpha , \beta \leq 1 \end{equation*}$

$\begin{equation*} u\left( a\right) = D_{a^{+}}^{\beta }u\left( b\right) = 0, \text{ } \end{equation*}$

then, they obtained the following Lyapunov inequality:

$\begin{equation*} \int_{a}^{b}\left\vert q\left( t\right) \right\vert dt \gt \frac{\left( \alpha +\beta -1\right) \Gamma \left( \alpha \right) \Gamma \left( \beta \right) }{ \left( b-a\right) ^{\alpha +\beta -1}}. \end{equation*}$

Recently, Ferreira in ^[9], derived a Lyapunov-type inequality for a sequential fractional right-focal boundary value problem

$\begin{equation*} ^{C}D_{a^{+}}^{\alpha }D_{a^{+}}^{\beta }u\left( t\right) +q(t)u\left( t\right) = 0, \text{ }a \lt t \lt b \end{equation*}$

$\begin{equation*} u\left( a\right) = D_{a^{+}}^{\gamma }u\left( b\right) = 0, \text{ } \end{equation*}$

where $0 < \alpha, \beta, \gamma \leq 1,$ $1 < \alpha +\beta \leq 2,$ then, they obtained the following Lyapunov inequality:

$\begin{equation*} \int_{a}^{b}\left( b-s\right) ^{\alpha +\beta -\gamma -1}\left\vert q\left( t\right) \right\vert dt \gt \frac{1}{C}, \end{equation*}$

where

$\begin{eqnarray*} C & = &\left( b-a\right) ^{\gamma }\max \left\{ \frac{\Gamma \left( \beta -\gamma +1\right) }{\Gamma \left( \alpha +\beta -\gamma \right) \Gamma \left( \beta +1\right) }, \right. \\ &&\left. \frac{1-\alpha }{\beta \Gamma \left( \alpha +\beta \right) }\left( \frac{\Gamma \left( \beta -\gamma +1\right) \Gamma \left( \alpha +\beta -1\right) }{\Gamma \left( \alpha +\beta -\gamma \right) \Gamma \left( \beta \right) }\right) ^{\frac{\alpha +\beta -1}{\alpha -1}}, \text{ with }\alpha \lt 1\right\} \end{eqnarray*}$

Note that more generalized Lyapunov type inequalities have been obtained for conformable derivative differential equations in ^[13]. For more results on Lyapunov-type inequalities for fractional differential equations, we refer to the recent survey of Ntouyas et al. ^[18].

In this work, we obtain Lyapunov type inequality for the following mixed fractional differential equation involving both right Caputo and left Riemann-Liouville fractional derivatives

$\begin{equation} -^{C}D_{b^{-}}^{\alpha }D_{a^{+}}^{\beta }u\left( t\right) +q(t)u\left( t\right) = 0, \text{ }a \lt t \lt b, \end{equation}$

(1.6)

satisfying the Dirichlet boundary conditions (1.2), here $0 < \beta \leq \alpha \leq 1,$ $1 < \alpha +\beta \leq 2,$ $^{C}D_{b^{-}}^{\alpha }$ denotes right Caputo derivative, $D_{a^{+}}^{\beta }$ denotes the left Riemann-Liouville and $q$ is a continuous function on $\left[a, b\right].$

So far, few authors have considered sequential fractional derivatives, and some Lyapunov type inequalities have been obtained. In this study, we place ourselves in a very general context, in that in each fractional operator, the order of the derivative can be different. Such problems, with both left and right fractional derivatives arise in the study of Euler-Lagrange equations for fractional problems of the calculus of variations ^[2,16,17]. However, the presence of a mixed left and right Caputo or Riemann-Liouville derivatives of order $0 < \alpha < 1$ leads to great difficulties in the study of the properties of the Green function since in this case it's given as a fractional integral operator.

We recall the concept of fractional integral and derivative of order $p > 0$ . For details, we refer the reader to ^[14,19,21]

The left and right Riemann-Liouville fractional integral of a function $g$ are defined respectively by

$\begin{eqnarray*} I_{a^{+}}^{p}g(t) & = &\frac{1}{\Gamma \left( p\right) }\int_{a}^{t}\frac{g(s) }{(t-s)^{1-p}}ds, \\ I_{b^{-}}^{p}g(t) & = &\frac{1}{\Gamma \left( p\right) }\int_{t}^{b}\frac{g(s) }{(s-t)^{1-p}}ds. \end{eqnarray*}$

The left and right Caputo derivatives of order $p > 0,$ of a function $g$ are respectively defined as follows:

$\begin{eqnarray*} ^{C}D_{a^{+}}^{p}g(t) & = &I_{a^{+}}^{n-p}g^{\left( n\right) }(t), \\ ^{C}D_{b^{-}}^{p}g(t) & = &\left( -1\right) ^{n}I_{b^{-}}^{n-p}g^{\left( n\right) }(t), \end{eqnarray*}$

and the left and right Riemann-Liouville fractional derivatives of order $p > 0,$ of a function $g$ \ are respectively defined as follows:

$\begin{eqnarray*} D_{a^{+}}^{p}g(t) & = &\frac{d^{n}}{dt^{n}}\left( I_{a^{+}}^{n-p}g\right) (t), \\ D_{b^{-}}^{p}g(t) & = &\left( -1\right) ^{n}\frac{d^{n}}{dt^{n}} I_{b^{-}}^{n-p}g(t), \end{eqnarray*}$

where $n$ is the smallest integer greater or equal than $p$ .

We also recall the following properties of fractional operators. Let $0 < p < 1$ , then:

1- $I_{a^{+}}^{pC}D_{a^{+}}^{p}f\left(t\right) = f\left(t\right) -f(a).$

2- $I_{b^{-}}^{pC}D_{b^{-}}^{p}f\left(t\right) = f\left(t\right) -f\left(b\right).$

3- $\left(I_{a^{+}}^{p}c\right) \left(t\right) = \frac{c\left(t-a\right) ^{p}}{\Gamma \left(p+1\right) }, c\in \mathbb{R}$

4- $D_{a^{+}}^{p}u(t) = ^{C}D_{a^{+}}^{p}u(t)$ , when $u\left(a\right) = 0.$

5- $D_{b^{-}}^{p}u(t) = ^{C}D_{b^{-}}^{p}u(t)$ , when $u\left(b\right) = 0.$

2. Lyapunov inequality

Next we transform the problem (1.6) with (1.2) to an equivalent integral equation.

Lemma 1. Assume that $0 < \alpha, \beta \leq 1.$ The function $u$ is a solution to the boundary value problem (1.6) with (1.2) if and only if $u$ satisfies the integral equation

$\begin{equation} u\left( t\right) = \int_{a}^{b}G(t, r)q(r)u(r)dr, \end{equation}$

(2.1)

where

$\begin{equation} G(t, r) = \frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \left( \int_{a}^{\inf \left\{ r, t\right\} }\left( t-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right. \notag \end{equation}$

$\begin{equation} \left. -\frac{\left( t-a\right) ^{\beta }}{\left( b-a\right) ^{\beta }} \int_{a}^{r}\left( b-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right) \end{equation}$

(2.2)

is the Green's function of problem (1.6) with (1.2).

Proof. Firstly, we apply the right side fractional integral $I_{b^{-}}^{\alpha }$ to equation (1.6), then the left side fractional integral $I_{a^{+}}^{\beta }$ to the resulting equation and taking into account the properties of Caputo and\Riemann-Liouville fractional derivatives and the fact that $D_{a^{+}}^{\beta }u(t) = ^{C}D_{a^{+}}^{\beta }u(t)$ , we get

$\begin{equation} u\left( t\right) = I_{a^{+}}^{\beta }I_{b^{-}}^{\alpha }q(t)u(t)+\frac{ c\left( t-a\right) ^{\beta }}{\Gamma \left( \beta +1\right) }. \end{equation}$

(2.3)

In view of the boundary condition $u\left(b\right) = 0,$ we get

$\begin{equation*} c = \frac{-\Gamma \left( \beta +1\right) }{\left( b-a\right) ^{\beta }} I_{a^{+}}^{\beta }I_{b^{-}}^{\alpha }q(t)u(t)\mid _{t = b}. \end{equation*}$

Substituting $c$ in (2.3), it yields

$\begin{eqnarray*} u\left( t\right) & = &I_{a^{+}}^{\beta }I_{b^{-}}^{\alpha }q(t)u(t)-\frac{ \left( t-a\right) ^{\beta }}{\left( b-a\right) ^{\beta }}I_{a^{+}}^{\beta }I_{b^{-}}^{\alpha }q(t)u(t)\mid _{t = b} \\ & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \int_{a}^{t}\left( t-s\right) ^{\beta -1}\left( \int_{s}^{b}\left( r-s\right) ^{\alpha -1}q(r)u(r)dr\right) ds \\ &&-\frac{\left( t-a\right) ^{\beta }}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) \left( b-a\right) ^{\beta }}\int_{a}^{b}\left( b-s\right) ^{\beta -1}\left( \int_{s}^{b}\left( r-s\right) ^{\alpha -1}q(r)u(r)dr\right) ds. \end{eqnarray*}$

Finally, by exchanging the order of integration, we get

$\begin{eqnarray*} u\left( t\right) & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\int_{a}^{t}\left( \int_{a}^{r}\left( t-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right) q(r)u(r)dr \\ &&+\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \int_{t}^{b}\left( \int_{a}^{t}\left( t-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right) q(r)u(r)dr \\ &&-\frac{\left( t-a\right) ^{\beta }}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) \left( b-a\right) ^{\beta }}\int_{a}^{b}\left( \int_{a}^{r}\left( b-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right) q(r)u(r)dr, \end{eqnarray*}$

thus

$\begin{equation*} u\left( t\right) = \int_{a}^{b}G(t, r)q(r)u(r)dr, \end{equation*}$

with

$\begin{equation*} G(t, r) = \frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \left\{ \begin{array}{c} \int_{a}^{r}\left( t-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds \\ -\frac{\left( t-a\right) ^{\beta }}{\left( b-a\right) ^{\beta }} \int_{a}^{r}\left( b-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds, a\leq r\leq t\leq b, \\ \int_{a}^{t}\left( t-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds \\ -\frac{\left( t-a\right) ^{\beta }}{\left( b-a\right) ^{\beta }} \int_{a}^{r}\left( b-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds, a\leq t\leq r\leq b. \end{array} \right. \end{equation*}$

that can be written as

$\begin{eqnarray*} G(t, r) & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \left( \int_{a}^{\inf \left\{ r, t\right\} }\left( t-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right. \\ &&\left. -\frac{\left( t-a\right) ^{\beta }}{\left( b-a\right) ^{\beta }} \int_{a}^{r}\left( b-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right) . \end{eqnarray*}$

Conversely, we can verify that if $u$ satisfies the integral equation (2.1), then $u$ is a solution to the boundary value problem (1.6) with (1.2). The proof is completed.

In the next Lemma we give the property of the Green function $G$ that will be needed in the sequel.

Lemma 2. Assume that $0 < \beta \leq \alpha \leq 1, 1 < \alpha +\beta \leq 2,$ then the Green function $G\left(t, r\right)$ given in (2.2) of problem (1.6) with (1.2) satisfies the following property:

$\begin{equation*} \left\vert G(t, r)\right\vert \leq \frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) \left( \alpha +\beta -1\right) \left( \alpha +\beta \right) }\left( \frac{\alpha \left( b-a\right) }{\left( \beta +\alpha \right) }\right) ^{\alpha +\beta -1}, \end{equation*}$

for all $a\leq r\leq t\leq b$ .

Proof. Firstly, for $a\leq r\leq t\leq b$ , we have $G(t, r)\geq 0$ . In fact, we have

$\begin{eqnarray*} G(t, r) & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \left( \int_{a}^{r}\left( t-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right. \\ &&-\frac{\left( t-a\right) ^{\beta }}{\left( b-a\right) ^{\beta }}\left. \int_{a}^{r}\left( b-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right) \\ &\geq &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \left( \int_{a}^{r}\left( b-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right. \\ &&-\frac{\left( t-a\right) ^{\beta }}{\left( b-a\right) ^{\beta }}\left. \int_{a}^{r}\left( b-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right) \end{eqnarray*}$

$\begin{equation} = \frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\left( 1- \frac{\left( t-a\right) ^{\beta }}{\left( b-a\right) ^{\beta }}\right) \int_{a}^{r}\left( b-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\geq 0 \end{equation}$

(2.4)

in addition,

$\begin{eqnarray*} G(t, r) &\leq &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\left( \int_{a}^{r}\left( r-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right. \\ &&-\frac{\left( r-a\right) ^{\beta }}{\left( b-a\right) ^{\beta }}\left. \int_{a}^{r}\left( b-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right) \\ &\leq &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \left( \frac{(r-a)^{\alpha +\beta -1}}{\left( \alpha +\beta -1\right) } \right. \\ &&\left. -\frac{\left( r-a\right) ^{\beta }}{\left( b-a\right) ^{\beta }} \int_{a}^{r}\left( b-a\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right) \end{eqnarray*}$

$\begin{equation} = \frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\left( \frac{(r-a)^{\alpha +\beta -1}}{\left( \alpha +\beta -1\right) }-\frac{ \left( r-a\right) ^{\beta +\alpha }}{\alpha \left( b-a\right) }\right) . \end{equation}$

(2.5)

Thus, from (2.4) and (2.5), we get

$\begin{equation} 0\leq G(t, r)\leq h\left( r\right) , \text{ }a\leq r\leq t\leq b, \end{equation}$

(2.6)

where

$\begin{equation*} h\left( s\right) : = \frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\left( \frac{(s-a)^{\alpha +\beta -1}}{\left( \alpha +\beta -1\right) }-\frac{\left( s-a\right) ^{\beta +\alpha }}{\alpha \left( b-a\right) }\right) , \end{equation*}$

it is clear that $h\left(s\right) \geq 0,$ for all $s\in \left[a, b\right]$ .

Now, for $a\leq t\leq r\leq b$ , we have

$\begin{eqnarray*} G(t, r) & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \left( \int_{a}^{t}\left( t-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right. \\ &&\left. -\frac{\left( t-a\right) ^{\beta }}{\left( b-a\right) ^{\beta }} \int_{a}^{r}\left( b-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right) \\ &\leq &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \left( \int_{a}^{t}\left( t-s\right) ^{\beta -1}\left( t-s\right) ^{\alpha -1}ds\right. \\ &&\left. -\frac{\left( t-a\right) ^{\beta }}{\left( b-a\right) } \int_{a}^{r}\left( r-s\right) ^{\alpha -1}ds\right) \\ & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\left( \frac{(t-a)^{\alpha +\beta -1}}{\left( \alpha +\beta -1\right) }-\frac{ \left( t-a\right) ^{\beta }\left( r-a\right) ^{\alpha }}{\alpha \left( b-a\right) }\right) \end{eqnarray*}$

$\begin{equation} \leq \frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \left( \frac{(t-a)^{\alpha +\beta -1}}{\left( \alpha +\beta -1\right) }- \frac{\left( t-a\right) ^{\beta +\alpha }}{\alpha \left( b-a\right) }\right) = h(t). \end{equation}$

(2.7)

On the other hand,

$\begin{eqnarray*} G(t, r) &\geq &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\left( r-a\right) ^{\alpha -1}\int_{a}^{t}\left( t-s\right) ^{\beta -1}ds \\ &&\left. -\frac{\left( t-a\right) ^{\beta }}{\left( b-a\right) ^{\beta }} \int_{a}^{r}\left( r-s\right) ^{\beta -1}\left( r-s\right) ^{\alpha -1}ds\right) \\ &\geq &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \left( \frac{\left( t-a\right) ^{\alpha }\left( t-a\right) ^{\beta }}{\beta \left( b-a\right) }-\frac{\left( t-a\right) ^{\beta }}{\left( b-a\right) ^{\beta }}\frac{(r-a)^{\alpha +\beta -1}}{\left( \alpha +\beta -1\right) } \right) \\ &\geq &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \left( \frac{\left( t-a\right) ^{\alpha +\beta }}{\beta \left( b-a\right) }- \frac{\left( t-a\right) ^{\beta }(r-a)^{\alpha -1}}{\left( \alpha +\beta -1\right) }\right) \\ &\geq &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \left( \frac{\left( t-a\right) ^{\alpha +\beta }}{\beta \left( b-a\right) }- \frac{(t-a)^{\alpha +\beta -1}}{\left( \alpha +\beta -1\right) }\right) , \end{eqnarray*}$

since $\beta \leq \alpha$ , we get

$\begin{equation} G(t, r)\geq -h\left( t\right) , \text{ }a\leq t\leq r\leq b. \end{equation}$

(2.8)

From (2.7) and (2.8) we obtain

$\begin{equation} \left\vert G(t, r)\right\vert \leq h\left( t\right) , \text{ }a\leq t\leq r\leq b. \end{equation}$

(2.9)

Finally, by differentiating the function $h,$ it yields

$\begin{equation*} h^{\prime }(s) = \frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }(s-a)^{\alpha +\beta -2}\left( 1-\frac{\left( \beta +\alpha \right) \left( s-a\right) }{\alpha \left( b-a\right) }\right) . \end{equation*}$

We can see that $h^{\prime }(s) = 0$ for $s_{0} = a+\frac{\alpha \left(b-a\right) }{\left(\beta +\alpha \right) }\in \left(a, b\right),$ $h^{\prime }(s) < 0$ for $s > s_{0}$ and $h^{\prime }(s) > 0$ for $s < s_{0}.$ Hence, the function $h(s)$ has a unique maximum given by

$\begin{eqnarray*} \underset{s\in \left[ a, b\right] }{\max }h\left( s\right) & = &h\left( s_{0}\right) \\ & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\left( \frac{\left( \frac{\alpha \left( b-a\right) }{\left( \beta +\alpha \right) } \right) ^{\alpha +\beta -1}}{\left( \alpha +\beta -1\right) }-\frac{\left( \frac{\alpha \left( b-a\right) }{\left( \beta +\alpha \right) }\right) ^{\beta +\alpha }}{\alpha \left( b-a\right) }\right) \\ & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) \left( \alpha +\beta -1\right) \left( \alpha +\beta \right) }\left( \frac{\alpha \left( b-a\right) }{\left( \beta +\alpha \right) }\right) ^{\alpha +\beta -1}. \end{eqnarray*}$

From (2.6) and (2.9), we get $\left\vert G(t, r)\right\vert \leq h(s_{0})$ , from which the intended result follows.

Next, we state and prove the Lyapunov type inequality for problem (1.6) with (1.2).

Theorem 3. Assume that $0 < \beta \leq \alpha \leq 1$ and $1 < \alpha +\beta \leq 2.$ If the fractional boundary value problem (1.6) with (1.2) has a nontrivial continuous solution, then

$\begin{equation} \int_{a}^{b}\left\vert q\left( r\right) \right\vert dr\geq \frac{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) \left( \alpha +\beta -1\right) \left( \alpha +\beta \right) ^{\alpha +\beta }}{\left( \alpha \left( b-a\right) \right) ^{\alpha +\beta -1}}. \end{equation}$

(2.10)

Proof. Let $X = C[a, b]$ be the Banach space endowed with norm $\left\vert \left\vert u\right\vert \right\vert = \underset{t\in \left[a, b\right] }{\max } \left\vert u\left(t\right) \right\vert$ . It follows from Lemma 1 that a solution $u\in X$ to the boundary value problem (1.6) with (1.2) satisfies

$\begin{eqnarray*} \left\vert u\left( t\right) \right\vert &\leq &\int_{a}^{b}\left\vert G(t, r)\right\vert \left\vert q\left( r\right) \right\vert \left\vert u\left( r\right) \right\vert dr \\ &\leq &\left\Vert u\right\Vert \int_{a}^{b}\left\vert G(t, r)\right\vert q\left( r\right) dr, \end{eqnarray*}$

Now, applying Lemma 2 to equation (2.1), it yields

$\begin{equation*} \left\vert u\left( t\right) \right\vert \leq \frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) \left( \alpha +\beta -1\right) \left( \alpha +\beta \right) }\left( \frac{\alpha \left( b-a\right) }{\left( \beta +\alpha \right) }\right) ^{\alpha +\beta -1}\left\Vert u\right\Vert \int_{a}^{b}\left\vert q\left( r\right) \right\vert dr \end{equation*}$

Hence,

$\begin{equation*} \left\Vert u\right\Vert \leq \frac{\left( \alpha \left( b-a\right) \right) ^{\alpha +\beta -1}}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) \left( \alpha +\beta -1\right) \left( \alpha +\beta \right) ^{\alpha +\beta }}\left\Vert u\right\Vert \int_{a}^{b}\left\vert q\left( r\right) \right\vert dr, \end{equation*}$

from which the inequality (2.10) follows. Note that the constant in (2.10) is not sharp. The proof is completed.

Remark 4. Note that, according to boundary conditions (1.2), the Caputo derivatives $^{C}D_{b^{-}}^{\alpha }$ and $\ ^{C}D_{a^{+}}^{\beta }$ coincide respectively with the Riemann-Liouville derivatives $D_{b^{-}}^{\alpha }$ and $D_{a^{+}}^{\beta }$ . So, equation (1.6) is reduced to the one containing only Caputo derivatives or only Riemann-Liouville derivatives, i.e.,

$\begin{equation*} -^{C}D_{b^{-}}^{\alpha C}D_{a^{+}}^{\beta }u\left( t\right) +q(t)u\left( t\right) = 0, \text{ }a \lt t \lt b \end{equation*}$

$\begin{equation*} -D_{b^{-}}^{\alpha }D_{a^{+}}^{\beta }u\left( t\right) +q(t)u\left( t\right) = 0, \text{ }a \lt t \lt b \end{equation*}$

Furthermore, by applying the reflection operator $(Qf)(t) = f(a+b-t)$ and taking into account that $Q^{C}D_{a+}^{\alpha } = ^{C}D_{b^{-}}^{\alpha }Q$ and $Q^{C}D_{b^{-}}^{\beta } = ^{C}D_{a^{+}}^{\beta }Q$ (see ^[21]), we can see that, the boundary value problem (1.6) with (1.2) is equivalent to the following problem

$\begin{equation*} -^{C}D_{a^{+}}^{\alpha }D_{b^{-}}^{\beta }u\left( t\right) +q(t)u\left( t\right) = 0, \text{ }a \lt t \lt b, \end{equation*}$

$\begin{equation*} u\left( a\right) = u\left( b\right) = 0. \end{equation*}$

Remark 5. If we take $\alpha = \beta = 1,$ then the Lyapunov type inequality (2.3) is reduced to

$\begin{equation*} \int_{a}^{b}\left\vert q\left( t\right) \right\vert dt\geq \frac{4}{b-a}. \end{equation*}$

Acknowledgements

The authors thank the anonymous referees for their valuable comments and suggestions that improved this paper.

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	C. E. Nouty, J. L. Journé, The sub-bifractional Brownian motion, Stud. Sci. Math. Hung., 50 (2013), 67–121. https://dx.doi.org/10.1556/SScMath.50.2013.1.1231 doi: 10.1556/SScMath.50.2013.1.1231
[2]	N. Kuang, B. Liu, Parameter estimations for the sub-fractional Brownian motion with drift at discrete observation, Braz. J. Probab. Stat., 29 (2015), 778–789. https://dx.doi.org/10.1214/14-BJPS246 doi: 10.1214/14-BJPS246
[3]	N. Kuang, B. Liu, Least squares estimator for $\alpha$ -sub-fractional bridges, Stat. Papers, 59 (2018), 893–912. https://dx.doi.org/10.1007/s00362-016-0795-2 doi: 10.1007/s00362-016-0795-2
[4]	N. Kuang, H. Xie, Maximum likelihood estimator for the sub-fractional Brownian motion approximated by a random walk, Ann. Inst. Stat. Math., 67 (2015), 75–91. https://dx.doi.org/10.1007/s10463-013-0439-4 doi: 10.1007/s10463-013-0439-4
[5]	N. Kuang, H. Xie, Asymptotic behavior of weighted cubic variation of sub-fractional brownian motion, Commun. Stat. Simul. Comput., 46 (2017), 215–229. https://dx.doi.org/10.1080/03610918.2014.957849 doi: 10.1080/03610918.2014.957849
[6]	N. Kuang, On the collision local time of sub-bifractional Brownian motions, Adv. Math. (China), 48 (2019), 627–640. https://dx.doi.org/10.11845/sxjz.2018023b doi: 10.11845/sxjz.2018023b
[7]	N. Kuang, Y. Li, Berry-Esséen bounds and almost sure CLT for the quadratic variation of the sub-bifractional Brownian motion, Commun. Stat. Simul. Comput., 51 (2022), 4257–4275. https://dx.doi.org/10.1080/03610918.2020.1740265 doi: 10.1080/03610918.2020.1740265
[8]	N. Kuang, H. Xie, Derivative of self-intersection local time for the sub-bifractional Brownian motion, AIMS Math., 7(2022), 10286–10302. https://dx.doi.org/10.3934/math.2022573 doi: 10.3934/math.2022573
[9]	N. Kuang, B. Liu, Renormalized self-intersection local time for sub-bifractional Brownian motion, Filomat, 36 (2022), 4023–4040. https://doi.org/10.2298/FIL2212023K doi: 10.2298/FIL2212023K
[10]	H. Xie, N. Kuang, Least squares type estimations for discretely observed nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind, AIMS Math., 7 (2022), 1095–1114. https://dx.doi.org/10.3934/math.2022065 doi: 10.3934/math.2022065
[11]	M. Maejima, C. Tudor, Limits of bifractional Brownian noises, preprint, arXiv: 0810.4764v1.
[12]	M. A. Ouahra, S. Moussaten, A. Sghir, On limit theorems of some extensions of fractional Brownian motion and their additive functionals, Stoch. Dynam., 17 (2017), 1750022. https://dx.doi.org/10.1142/S0219493717500228 doi: 10.1142/S0219493717500228
[13]	P. Lei, D. Nualart, A decomposition of the bifractional Brownian motion and some applications, Stat. Probabil. Lett., 79 (2009), 619–624. https://doi.org/10.1016/j.spl.2008.10.009 doi: 10.1016/j.spl.2008.10.009
[14]	J. R. de Chávez, C. Tudor, A decomposition of sub-fractional Brownian motion, Math. Rep., 61 (2009), 67–74.
[15]	X. Bardina, D. Bascompte, Weak convergence towards two independent Gaussian processes from a unique Poisson process, Collect. Math., 61 (2010), 191–204.
[16]	R. Dobrushin, P. Major, Non-central limit theorems for non-linear functionals of Gaussian fields, Probabil. Theory Rel. Fields, 50 (1979), 27–52. https://doi.org/10.1007/BF00535673 doi: 10.1007/BF00535673

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