On a Nonlinear Fractional Langevin Equation of Two Fractional Orders with a Multiplicative Noise

Abstract

We consider a stochastic nonlinear fractional Langevin equation of two fractional orders $D^{\beta }(D^{\alpha }+\gamma )\psi \left(t\right)=\lambda \vartheta (t,\psi \left(t\right))\dot{w}\left(t\right),0<t\le 1.$ Given some suitable conditions on the above parameters, we prove the existence and uniqueness of the mild solution to the initial value problem for the stochastic nonlinear fractional Langevin equation using Banach fixed-point theorem (Contraction mapping theorem). The upper bound estimate for the second moment of the mild solution is given, which shows exponential growth in time t at a precise rate of $3c_{1}exp\left(c_3{t}^{2(\alpha +\beta )-1}+c_4{t}^{2\alpha -1}\right)$ on the parameters $\alpha >1$ and $\alpha +\beta >1$ for some positive constants $c_1,c_3$ and $c_4$.

1. Introduction

The standard Langevin equation for one-dimensional stochastic process $x\left(t\right)$ is of the form (see [1])

$$\frac{dx\left(t\right)}{dt}=a\left(x\left(t\right)\right)+\sqrt{b\left(x\right(t\left)\right)}F\left(t\right),$$

(1)

where $F\left(t\right)$ is the random force known as Gaussian white noise.

The Langevin equation was first formulated in 1908 by Langevin [2]. The equation has been found to be an effective and useful tool for describing the evolution of physical phenomena in a fluctuating environment, see [2,3]. Thus, the Langevin equation best describes stochastic (random movement of particles-Brownian motion) concepts and problems appearing in fluctuating environments. While the Langevin equation describes Brownian motion well when the fluctuation force is assumed to be white noise, the generalized Langevin equation describes the motion of the particle when the random fluctuation is a non-white noise.

The second-order Langevin equation in a thermally fluctuating environment is (see [4])

$$m\ddot{x}+\gamma \dot{x}=F(x,t)+\eta \left(t\right),$$

with an external driving force $F(x,t)$, an internal noise $\eta \left(t\right)$, and damping $\gamma \dot{x}$. The above equation is known to model the motion of Brownian particles in an ideal liquid. However, when we have an inhomogeneous solution (heterogeneous media), then the damping force on the particles changes with time and is expressed by the generalized Langevin equation:

$${\displaystyle m\ddot{x}+\gamma {\int }_0^t\beta (t-s)\dot{x}\left(s\right)ds=F(x,t)+\eta \left(t\right),}$$

with damping coefficient $\gamma$ and time-dependent damping kernel function $\beta \left(t\right)$. Consequently, the generalized Langevin equation was introduced to model anomalous diffusive processes and phenomena in a complex and viscoelastic environment [5].

Nevertheless, the ordinary (integer order) Langevin equation does not completely and correctly depict or describe the dynamics of systems in complex media, hence, the need for a fractional order Langevin equation. The fractional Langevin equation models fractal processes [6] and is also a model for the characterization of anomalous Brownian motion from NMR signals [7].

In 1996, the fractional Langevin equation was introduced by Mainardi and Pironi [8] to further develop the study of the Lanvegin equation.

Ahmed and Nieto [9] studied the existence of the following Dirichlet boundary value problem of Langevin equation with two fractional orders in 2010,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c{D}^{\beta }{(}^c{D}^{\alpha }+\lambda )x\left(t\right)=f(t,x\left(t\right)),0<t<1,0<\alpha ,\beta \le 1,\hfill \\ x\left(0\right)={\gamma }_1,x\left(1\right)={\gamma }_2,\hfill \end{array}\right.\end{array}$$

where ${}^{c}D$ is the Caputo fractional derivative, $f:[0,1]\times X\to X,\lambda \in \mathbb{R}$, and ${\gamma }_1,{\gamma }_2\in X$.

In 2011, the authors of [10] studied the existence of solutions to the nonlinear Langevin equation involving fractional order with boundary conditions,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c{D}^{\beta }{(}^c{D}^{\alpha }+\lambda )u\left(t\right)=f(t,u\left(t\right)),0<t<1,t\in [0,T],\hfill \\ u\left(0\right)=-u\left(T\right),u^{\prime }\left(0\right)=u^{\prime }\left(T\right)=0,\hfill \end{array}\right.\end{array}$$

where $T>0,1<\alpha \le 2,0<\beta \le 1,{}^c{D}^{\alpha }$ and ${}^c{D}^{\beta }$ are Caputo fractional derivatives, $f:[0,T]\times \mathbb{R}\to \mathbb{R}$ is continuous and $\lambda \in \mathbb{R}.$

Ahmad et al. [11], in 2012, studied the existence of solutions for a three-point boundary value problem of the Langevin equation with two different fractional orders,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c{D}^{\beta }{(}^c{D}^{\alpha }+\lambda )x\left(t\right)=f(t,x\left(t\right)),0<t<1,0<\alpha \le 1,1<\beta \le 2,\hfill \\ x\left(0\right)={\gamma }_0,x\left(\eta \right)=0,x\left(1\right)=0,0<\eta <1,\hfill \end{array}\right.\end{array}$$

where ${}^{c}D$ is the Caputo fractional derivative, $f:[0,1]\times \mathbb{R}\to \mathbb{R},$ is a given continuous function, and $\lambda \in \mathbb{R}$.

Recently, in 2022, Kou and Kosari [12] studied the existence and uniqueness of solutions to a generalization of the nonlinear Langevin equation of fractional order with four boundary conditions

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c{D}_{0^{+}}^{\beta }{(}^c{D}_{0^{+}}^{\alpha }+\gamma )x\left(t\right)=f(t,x\left(t\right),x^{\prime }\left(t\right)),t\in (0,1),0<\alpha \le 1,2<\beta \le 3,\hfill \\ x\left(0\right)=x\left(1\right)=x^{\prime }\left(0\right)=x^{\prime }\left(1\right)=0,\hfill \end{array}\right.\end{array}$$

where ${}^{c}D$ is the Caputo fractional derivative, $f:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a continuous function, and $\gamma \in \mathbb{R}$.

See [11,13,14,15] and their references for further studies on the fractional Langevin equation of two fractional orders with different boundary conditions.

Motivated by Baghani [16] and the authors of the papers [6,13], we consider the initial value problem of the stochastic nonlinear fractional Langevin equation of two fractional orders:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{D}^{\beta }(D^{\alpha }+\gamma )\psi \left(t\right)=\lambda \vartheta (t,\psi \left(t\right))\dot{w}\left(t\right),0<t\le 1,\hfill \\ \\ \psi \left(0\right)={\mu }_0,{\psi }^{\left(\alpha \right)}\left(0\right)={\nu }_0,\hfill \end{array}\right.\end{array}$$

(2)

where $\gamma \in \mathbb{R},0<\alpha ,\beta \le 1,$ $D^{\alpha }$ and $D^{\beta }$ are the Caputo fractional derivatives, $\vartheta :[0,1]\times \mathbb{R}\to \mathbb{R}$ is a Lipschitz continuous function on the second variable, $\dot{w}\left(t\right)$ is a generalized derivative of the Wiener process $w\left(t\right)$ (Gaussian white noise), and $\lambda >0$ is the noise level.

Remark 1.

1. 

The above stochastic Langevin equation of two fractional orders in (2) can find its application in modeling dynamical processes that exhibit both fractal (phenomena in hierarchical or porous media) and fractional (systems with long-term memory and long-range interaction) properties.

2. 

The addition of randomness (noise term) to the particles’ dynamics takes care of the challenges due to the complexity of a medium environment that particles may encounter.

3. 

The white noise force $F\left(t\right)$ used by the Langevin in Equation (1) is related to the Wiener process $w\left(t\right)$ by ${\displaystyle w\left(t\right)={\int }_0^{t}F\left(s\right)ds.}$ It is known that the Wiener process smoothes the white noise process [2].

4. 

Moreso, the main difficulty when considering the Wiener process is the use of Itô isometry in computing the second moment (energy growth) of the mild solution.

5. 

When $\alpha =\beta =1$, Equation (2) converts to a second-order nonlinear stochastic Langevin equation as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\ddot{\psi }\left(t\right)+\gamma \dot{\psi }\left(t\right)=\lambda \vartheta (t,\psi \left(t\right))\dot{w}\left(t\right),0<t\le 1,\hfill \\ \psi \left(0\right)={\mu }_0,\dot{\psi }\left(0\right)={\nu }_0.\hfill \end{array}\right.\end{array}$$

Formulation of Solution

Now, we make sense of the solution to Equation (2).

Definition 1.

The function ${\left\{\psi \left(t\right)\right\}}_{0\le t\le 1}$ is called a mild solution of Equation (2) if, almost surely, it satisfies the following,

$$\begin{array}{cc}\hfill \psi \left(t\right)& ={\varphi }_0\left(t\right)+\frac{\lambda }{\Gamma (\alpha +\beta )}{\int }_0^t{(t-s)}^{\alpha +\beta -1}\vartheta (s,\psi \left(s\right))\dot{w}\left(s\right)ds\hfill \\ \hfill & -\frac{\gamma }{\Gamma \left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}\psi \left(s\right)ds\hfill \\ \hfill & ={\varphi }_0\left(t\right)+\frac{\delta }{\Gamma (\alpha +\beta )}{\int }_0^t{(t-s)}^{\alpha +\beta -1}\vartheta (s,\psi \left(s\right))dw\left(s\right)\hfill \\ \hfill & -\frac{\gamma }{\Gamma \left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}\psi \left(s\right)ds,\hfill \end{array}$$

where ${\varphi }_0\left(t\right)=\frac{{\nu }_0+\gamma {\mu }_0}{\Gamma (\alpha +\beta )}{t}^{\alpha }+{\mu }_0$.

If ${\left\{\psi \left(t\right)\right\}}_{t\in [0,1]}$ satisfies the additional condition,

$${\displaystyle \underset{t\in [0,1]}{sup}\mathbf{E}{\left|\psi \left(t\right)\right|}^2<\infty ,}$$

(3)

then ${\left\{\psi \left(t\right)\right\}}_{t\in [0,1]}$ is said to be a random field solution to Equation (2).

We will make the following assumptions:

• ${\displaystyle \underset{0\le t\le 1}{sup}{\left|{\varphi }_0\left(t\right)\right|}^2\le c_1,c_1>0}$ and
• For $t\in [0,1]$, ${\displaystyle \mathbf{E}\left[{\int }_0^t{\left|\psi \left(s\right)\right|}^{2}ds\right]={\int }_0^t\mathbf{E}{\left|\psi \left(s\right)\right|}^{2}ds.}$

This last assumption is based on the following:

Remark 2

([17], Page 3). Suppose X takes value in $\mathbb{R}$ and I is a subinterval of $[0,\infty )$ such that ${\displaystyle {\int }_I\mathbf{E}\left|X_t\right|dt<\infty }$, then ${\displaystyle {\int }_I\left|X_t\right|dt<\infty a.s\mathbf{P}}$, and

$${\displaystyle {\int }_I\mathbf{E}\left[X_t\right]dt=\mathbf{E}{\int }_I{X}_{t}dt.}$$

Now, we define the following $L^2\left(\mathbf{P}\right)$ norm of $\psi$ by

$${\parallel \psi \parallel }_2^2:=\underset{0\le t\le 1}{sup}\mathbf{E}{\left|\psi \left(t\right)\right|}^2.$$

The paper is organized as follows. Section 2 contains the preliminaries, and in Section 3, we give the main results of the paper. Section 4 contains a concise summary of the paper.

2. Preliminaries

See the following papers [18,19] for more studies on the Langevin models and its fractional equation. It is known, see [19], that the solution to a fractional Langevin equation

$$\left(\frac{d^{\alpha }}{d{t}^{\alpha }}+\gamma \right){x\left(t\right)=F\left(t\right),x\left(t\right)|}_{t=0}=x_0,$$

is given by ${\displaystyle x\left(t\right)=x_0{E}_{1,\alpha }(-\gamma t^{\alpha })+{\int }_0^{t}F\left(s\right){(t-s)}^{\alpha -1}{E}_{\alpha ,\alpha }(-\gamma {(t-s)}^{\alpha })ds,}$ where ${\displaystyle E_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{\Gamma (\alpha +\beta k)}}$ is the Mittag-Leffler function.

There are many results that exist for the solution to fractional Langevin equations, worthy of note are the following:

Lemma 1

([13]). The unique solution of the fractional Langevin equation

$$D^{\alpha }(D^2+{\gamma }^2)x\left(t\right)=\theta \left(t\right),n-1<\alpha \le n,n\in \mathbb{N},$$

where θ is a continuous function on $[0,1]$, is given by

$$\begin{array}{cc}\hfill x\left(t\right)& =\frac{1}{\gamma }{\int }_0^{t}sin\gamma (t-s)\left(\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^s{(s-r)}^{\alpha -1}\theta \left(r\right)ds+\sum _{i=1}^{n-1}{c}_i{s}^i\right)ds\hfill \\ & +c_{n}cos\left(\gamma t\right)+c_{n+1}sin\left(\gamma t\right),\hfill \end{array}$$

with $c_i\in \mathbb{R}$ for $i=1,2,\dots ,n+1.$

The authors of [6,13,16] considered the fractional Langevin equation of two fractional orders:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{D}^{\beta }(D^{\alpha }+\gamma )x\left(t\right)=f(t,x\left(t\right)),0<t\le 1\hfill \\ \\ x^{\left(k\right)}\left(0\right)={\mu }_k,0\le k<l\hfill \\ \\ x^{(\alpha +k)}\left(0\right)={\nu }_k,0\le k<n,\hfill \end{array}\right.\end{array}$$

(4)

where $\gamma \in \mathbb{R},m-1<\alpha \le m,n-1<\beta \le n,l=max\{m,n\},m,n\in \mathbb{N},D^{\alpha }$, and $D^{\beta }$ are the Caputo fractional derivatives, $x^{(\alpha +k)}\left(0\right)$ equals $D^{\alpha }{D}^{k}x\left(0\right)$, and $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a Lebesgue measurable function (or continuously differentiable function). The unknown function $x\left(t\right)$ represents the particle displacement, $\gamma$ is the viscous friction coefficient, and the given function f stands for a noise term.

Lemma 2

([16]). The function $x\left(t\right)$ is a solution of Equation (4) if and only if it is a solution of the integral equation

$$x\left(t\right)=\frac{1}{\Gamma (\alpha +\beta )}{\int }_0^t{(t-s)}^{\alpha +\beta -1}f(s,x\left(s\right))ds-\frac{\gamma }{\Gamma \left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}x\left(s\right)ds+\varphi \left(t\right)$$

with ${\displaystyle \varphi \left(t\right):=\sum _{i=0}^{n-1}\frac{{\nu }_i+\gamma {\mu }_i}{\Gamma (\alpha +i+1)}{t}^{\alpha +i}+\sum _{j=0}^{m-1}\frac{{\mu }_j}{\Gamma (j+1)}{t}^j}$.

Lemma 3

([6]). The general solution of Equation (4) is given by

$$\begin{array}{cc}\hfill x\left(t\right)& =\sum _{j=0}^{m-1}{\mu }_j{t}^j{E}_{\alpha +j}(-\gamma t^{\alpha })+\sum _{i=0}^{n-1}{\nu }_i{t}^{\alpha +i}{E}_{\alpha ,\alpha }(-\gamma t^{\alpha })\hfill \\ \hfill & +\sum _{i=0}^{n-1}{\mu }_i{t}^i\left(\frac{1}{\Gamma (i+1)}-E_{\alpha }(-\gamma t^{\alpha })\right)\hfill \\ \hfill & +{\int }_0^t{(t-s)}^{\alpha +\beta -1}{E}_{\alpha ,\alpha +\beta }(-\gamma {(t-s)}^{\alpha })f(s,x\left(s\right))ds.\hfill \end{array}$$

Definition 2.

The Caputo fractional derivative of order $\alpha >0$ of a function $x:[0,1]\to \mathbb{R}$ is given by

$$D^{\alpha }x\left(t\right)=\frac{1}{\Gamma (n-\alpha )}{\int }_0^t{(t-s)}^{n-\alpha -1}{x}^{\left(n\right)}\left(s\right)ds,$$

where $n-1<\alpha \le n,n\in \mathbb{N}$, provided that the integral exists and finite.

The definition of a generalized derivative of a deterministic function w is as follows:

Definition 3.

Let $\Phi \left(t\right)$ be a smooth and compactly supported function. Then the generalized derivative $\dot{w}\left(t\right)$ of $w\left(t\right)$ (not necessarily differentiable) is

$${\int }_0^{\infty }\Phi \left(t\right)\dot{w}\left(t\right)dt=-{\int }_0^{\infty }\dot{\Phi }\left(t\right)w\left(t\right)dt.$$

In what follows,

$${\int }_0^t\Phi \left(s\right)\dot{w}\left(s\right)ds=\Phi \left(t\right)w\left(t\right)-{\int }_0^t\dot{g}\left(s\right)w\left(s\right)ds.$$

3. Main Results

Here, one makes a global Lipschitz continuity condition on $\vartheta (.,\psi )$ as follows:

Condition 1.

Let $0<{\mathrm{Lip}}_{\vartheta }<\infty ,$ and for all $x,y\in \mathbb{R}$, and $t\in [0,1]$, we have

$$|\vartheta (t,x)-\vartheta (t,y)|\le {\mathrm{Lip}}_{\vartheta }|x-y|.$$

We set $\vartheta (t,0)=0$ for convenience only.

Theorem 1.

Let $\alpha >\frac{1}{2}$ and $\alpha +\beta >\frac{1}{2}$, and suppose Condition 1 holds. Then there exists a positive constant λ with $\lambda <\frac{\Gamma (\alpha +\beta )\sqrt{\left(2(\alpha +\beta )-1\right)\left((2\alpha -1){\Gamma }^2\left(\alpha \right)-3{\gamma }^2\right)}}{{\mathrm{Lip}}_{\sigma }\Gamma \left(\alpha \right)\sqrt{3(2\alpha -1)}}$, ${\mathrm{Lip}}_{\sigma }>0$, such that Equation (2) has a unique solution.

The above result will be proved by Banach fixed point theorem. To begin, we define the operator

$$\begin{array}{cc}\hfill \mathcal{A}\psi \left(t\right)& ={\varphi }_0\left(t\right)+\frac{\lambda }{\Gamma (\alpha +\beta )}{\int }_0^t{(t-s)}^{\alpha +\beta -1}\vartheta (s,\psi \left(s\right))dw\left(s\right)\hfill \\ \hfill & -\frac{\gamma }{\Gamma \left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}\psi \left(s\right)ds,\hfill \end{array}$$

and show that the fixed point of operator $\mathcal{A}$ gives the solution to Equation (2).

Lemma 4.

Let ψ be a mild solution satisfying Equation (3) and suppose that Condition 1 holds. Then for $\alpha >\frac{1}{2}$ and $\alpha +\beta >\frac{1}{2}$

$${\parallel \mathcal{A}\psi \parallel }_2^2\le {\tilde{c}}_1+c_2{\parallel \psi \parallel }_2^2,$$

where $c_2:=\frac{3{\lambda }^2{\mathrm{Lip}}_{\vartheta }^2}{[2(\alpha +\beta )-1]{\Gamma }^2(\alpha +\beta )}+\frac{3{\gamma }^2}{(2\alpha -1){\Gamma }^2\left(\alpha \right)}>0$ and ${\tilde{c}}_1:=3c_1$.

Proof. 

Applying Itô isometry, Condition 1, and H$\ddot{o}$lder’s inequality, we have

$$\begin{array}{cc}\hfill {\mathbf{E}\left|\mathcal{A}\psi \left(t\right)\right|}^2& \le 3|{\varphi }_0{\left(t\right)|}^2+\frac{3{\lambda }^2}{{\Gamma }^2(\alpha +\beta )}{\int }_0^t{(t-s)}^{2(\alpha +\beta -1)}\mathbf{E}{\left|\vartheta (s,\psi \left(s\right))\right|}^{2}ds\hfill \\ & +3\mathbf{E}|\frac{\gamma }{\Gamma \left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}\psi \left(s\right)ds{|}^2\hfill \\ \hfill & \le 3|{\varphi }_0{\left(t\right)|}^2+\frac{3{\lambda }^2{\mathrm{Lip}}_{\vartheta }^2}{{\Gamma }^2(\alpha +\beta )}{\int }_0^t{(t-s)}^{2(\alpha +\beta -1)}\mathbf{E}{\left|\psi \left(s\right)\right|}^{2}ds\hfill \\ \hfill & +\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}\mathbf{E}{\left[{\int }_0^t{(t-s)}^{\alpha -1}\left|\psi \left(s\right)\right|ds\right]}^2\hfill \\ \hfill & \le 3|{\varphi }_0{\left(t\right)|}^2+\frac{3{\lambda }^2{\mathrm{Lip}}_{\vartheta }^2}{{\Gamma }^2(\alpha +\beta )}{\int }_0^t{(t-s)}^{2(\alpha +\beta -1)}\mathbf{E}{\left|\psi \left(s\right)\right|}^{2}ds\hfill \\ \hfill & +\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}\mathbf{E}{\left[{\left({\int }_0^t{(t-s)}^{2(\alpha -1)}ds\right)}^{1/2}{\left({\int }_0^t{\left|\psi \left(s\right)\right|}^{2}ds\right)}^{1/2}\right]}^2\hfill \\ \hfill & =3|{\varphi }_0{\left(t\right)|}^2+\frac{3{\lambda }^2{\mathrm{Lip}}_{\vartheta }^2}{{\Gamma }^2(\alpha +\beta )}{\int }_0^t{(t-s)}^{2(\alpha +\beta -1)}\mathbf{E}{\left|\psi \left(s\right)\right|}^{2}ds\hfill \\ \hfill & +\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}{\int }_0^t{(t-s)}^{2(\alpha -1)}ds\mathbf{E}\left[{\int }_0^t{\left|\psi \left(s\right)\right|}^{2}ds\right]\hfill \\ \hfill & \le 3|{\varphi }_0{\left(t\right)|}^2+\frac{3{\lambda }^2{\mathrm{Lip}}_{\vartheta }^2}{{\Gamma }^2(\alpha +\beta )}\underset{0<s\le t}{sup}\mathbf{E}{\left|\psi \left(s\right)\right|}^2{\int }_0^t{(t-s)}^{2(\alpha +\beta -1)}ds\hfill \\ \hfill & +\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}\left[\underset{0<s\le t}{sup}\mathbf{E}{\left|\psi \left(s\right)\right|}^2\right]t{\int }_0^t{(t-s)}^{2(\alpha -1)}ds\hfill \\ \hfill & =3|{\varphi }_0{\left(t\right)|}^2+\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}\underset{0<s\le t}{sup}\mathbf{E}{\left|\psi \left(s\right)\right|}^2\frac{t^{2(\alpha +\beta )-1}}{2(\alpha +\beta )-1}\hfill \\ \hfill & +\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}\underset{0<s\le t}{sup}\mathbf{E}{\left|\psi \left(s\right)\right|}^2\frac{t^{2\alpha }}{2\alpha -1}.\hfill \end{array}$$

Taking supremum over $t\in [0,1]$ on both sides, we get

$${\parallel \mathcal{A}\psi \parallel }_2^2\le 3c_1+\left[\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{[2(\alpha +\beta )-1]{\Gamma }^2(\alpha +\beta )}+\frac{3{\gamma }^2}{(2\alpha -1){\Gamma }^2\left(\alpha \right)}\right]{\parallel \psi \parallel }_2^2,$$

and the result follows. □

Lemma 5.

Let ψ and φ be mild solutions satisfying Equation (3) and suppose that Condition 1 holds. Then for $\alpha >\frac{1}{2}$ and $\alpha +\beta >\frac{1}{2}$

$${\parallel \mathcal{A}\psi -\mathcal{A}\phi \parallel }_2^2\le c_2{\parallel \psi -\phi \parallel }_2^2.$$

Proof. 

Since the proof follows the same steps as the proof of Lemma 4, we omit the details to avoid repetition. □

Proof of Theorem 1.

Employing the fixed-point theorem, one has $\psi (x,t)=\mathcal{A}\psi (x,t)$. Therefore, using Lemma 4,

$$\begin{array}{c}\hfill {\parallel \psi \parallel }_2^2={\parallel \mathcal{A}\psi \parallel }_2^2\le {\tilde{c}}_1+c_2{\parallel \psi \parallel }_2^2.\end{array}$$

This gives ${\parallel \psi \parallel }_2^2\left[1-c_2\right]\le {\tilde{c}}_1$ and, therefore, ${\parallel \psi \parallel }_2<\infty$ if and only if $c_2<1.$

On the other hand, suppose $\psi \ne \phi$ are two solutions to Equation (2). Then, from Lemma 5, one obtains

$${\parallel \psi -\phi \parallel }_2^2={\parallel \mathcal{A}\psi -\mathcal{A}\vartheta \parallel }_2^2\le c_2{\parallel \psi -\phi \parallel }_2^2.$$

Thus, ${\parallel \psi -\phi \parallel }_2^2\left[1-c_2\right]\le 0.$ However, $1-c_2>0$, it follows that ${\parallel \psi -\phi \parallel }_2<0$, which is a contradiction, and therefore, ${\parallel \psi -\phi \parallel }_2=0$. Hence, the existence and uniqueness result follows the Banach contraction principle. □

Upper Moment Bound

For the growth moment result, we present the following retarded Gronwall-type inequality:

Proposition 1

([20]). Let $x,g,h\in C([t_0,T),{\mathbb{R}}_{+})$, and $w\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ be nondecreasing with $w\left(x\right)>0$ for $x>0$, and $b\in C^1([t_0,T),[t_0,T))$ be nondecreasing with $b\left(t\right)\le t$ on $[t_0,T)$. If

$$x\left(t\right)\le k+{\int }_{t_0}^{t}g\left(s\right)w\left(x\left(s\right)\right)ds+{\int }_{b\left(t_0\right)}^{b\left(t\right)}h\left(s\right)w\left(x\left(s\right)\right)ds,t_0\le t<T,$$

where k is a nonnegative constant, then for $t_0\le t<t_1$,

$$x\left(t\right)\le G^{-1}\left(G\left(k\right)+{\int }_{t_0}^{t}g\left(s\right)ds+{\int }_{b\left(t_0\right)}^{b\left(t\right)}h\left(s\right)ds\right),$$

with ${\displaystyle G\left(r\right)={\int }_1^r\frac{ds}{w\left(s\right)},r>0}$ and $t_1\in (t_0,T)$ chosen so that the right-hand side is well-defined.

Unfortunately, the above proposition is not sufficient for our problem since g and h do not depend on the variable t. Thus, there is a need for a new form of the above inequality. In 2005, Agarwal et al. in [21], generalized the above retarded Gronwall-type inequality to:

$$x\left(t\right)\le a\left(t\right)+\sum _{i=1}^n{\int }_{b_i\left(t_0\right)}^{b_i\left(t\right)}{g}_i(t,s)w_i\left(x\left(s\right)\right)ds,t_0\le t<t_1.$$

(5)

Theorem 2

(Theorem 2.1 of [21]). Suppose that the hypotheses of (Theorem 2.1 of [21]) hold and $x\left(t\right)$ is a continuous and nonnegative function on $[t_0,t_1)$ satisfying (5). Then

$$x\left(t\right)\le W_n^{-1}\left[W_n\left(r_n\left(t\right)\right)+{\int }_{b_n\left(t_0\right)}^{b_n\left(t\right)}\underset{t_0\le \tau \le t}{max}{g}_n(\tau ,s)ds\right],t_0\le t\le T_1,$$

where $r_n\left(t\right)$ is determined recursively by

$$r_1\left(t\right):=a\left(t_0\right)+{\int }_{t_0}^t\left|a^{\prime }\left(s\right)\right|ds,$$

$$r_{i+1}:=W_i^{-1}\left[W_i\left(r_i\left(t\right)\right)+{\int }_{b_i\left(t_0\right)}^{b_i\left(t\right)}\underset{t_0\le \tau \le t}{max}{g}_i(\tau ,s)ds\right],i=1,\dots ,n-1,$$

and ${\displaystyle W_i(x,x_i):={\int }_{x_i}^x\frac{dz}{w_i\left(z\right)}}$.

Remark 3.

We will consider the case where $n=2$. That is, if

$$x\left(t\right)\le a\left(t\right)+{\int }_{b_1\left(t_0\right)}^{b_1\left(t\right)}{g}_1(t,s)w_1\left(x\left(s\right)\right)ds+{\int }_{b_2\left(t_0\right)}^{b_2\left(t\right)}{g}_2(t,s)w_2\left(x\left(s\right)\right)ds,$$

then

$${\displaystyle x\left(t\right)\le W_2^{-1}\left[W_2\left(r_2\left(t\right)\right)+{\int }_{b_2\left(t_0\right)}^{b_2\left(t\right)}\underset{t_0\le \tau \le t}{max}{g}_2(\tau ,s)ds\right],}$$

with ${\displaystyle r_2\left(t\right)=W_1^{-1}\left[W_1\left(r_1\left(t\right)\right)+{\int }_{b_1\left(t_0\right)}^{b_1\left(t\right)}\underset{t_0\le \tau \le t}{max}{g}_1(\tau ,s)ds\right]}$.

Here, take $w_1\left(x\left(s\right)\right)=w_2\left(x\left(s\right)\right)=x\left(s\right),b_1\left(t_0\right)=b_2\left(t_0\right)=t_0=0$ and $b_1\left(t\right)=b_2\left(t\right)=t.$

We are able to give only the upper bound moment growth of the mild solution. Assume that the function ${\varphi }_0\left(t\right)$ is bounded above to obtain:

Theorem 3.

Given that Condition 1 holds. Then for all $t\in [0,1]$ and $c_1,c_3,c_4>0$, we have

$${\mathbf{E}\left|\psi \left(t\right)\right|}^2\le 3c_{1}exp\left(c_3{t}^{2(\alpha +\beta )-1}+c_4{t}^{2\alpha -1}\right),$$

with $c_3=\frac{3{\lambda }^2{\mathrm{Lip}}_{\vartheta }^2}{{\Gamma }^2(\alpha +\beta )}\frac{1}{2(\alpha +\beta )-1},c_4=\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}\frac{1}{2\alpha -1},\alpha >1$, and $\alpha +\beta >1.$

Proof. 

Since it is assumed that ${\displaystyle \underset{t\in [0,1]}{sup}{\left|{\varphi }_0\left(t\right)\right|}^2\le c_1}$, then from the proof of Lemma 4, one obtains

$$\begin{array}{cc}\hfill {\mathbf{E}\left|\psi \left(t\right)\right|}^2& \le 3c_1+\frac{3{\lambda }^2{\mathrm{Lip}}_{\vartheta }^2}{{\Gamma }^2(\alpha +\beta )}{\int }_0^t{(t-s)}^{2(\alpha +\beta )-2}\mathbf{E}{\left|\psi \left(s\right)\right|}^{2}ds\hfill \\ \hfill & +\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}\mathbf{E}{\left[{\left({\int }_0^t{(t-s)}^{2(\alpha -1)}{\left|\psi \left(s\right)\right|}^{2}ds\right)}^{1/2}{\left({\int }_0^{t}1ds\right)}^{1/2}\right]}^2\hfill \\ \hfill & =3c_1+\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}{\int }_a^t{(t-s)}^{2(\alpha +\beta )-2}\mathbf{E}{\left|\psi \left(s\right)\right|}^{2}ds\hfill \\ \hfill & +\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}t{\int }_0^t{(t-s)}^{2(\alpha -1)}\mathbf{E}{\left|\psi \left(s\right)\right|}^{2}ds.\hfill \end{array}$$

Let $f\left(t\right):={\mathbf{E}\left|\psi \left(t\right)\right|}^2$, and since $t\le 1$, we obtain

$$\begin{array}{cc}\hfill f\left(t\right)& \le {\tilde{c}}_1+\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}{\int }_0^t{(t-s)}^{2(\alpha +\beta )-2}f\left(s\right)ds\hfill \\ \hfill & +\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}{\int }_0^t{(t-s)}^{2\alpha -2}f\left(s\right)ds.\hfill \end{array}$$

(6)

Now, we apply Theorem 2 to (6). For $W_2$, we have

$${\displaystyle W_2(x,x_2)={\int }_{x_2}^x\frac{dz}{z}=lnx-ln{x}_2.}$$

For convenience, we take $x_2=1$ and $W_2\left(x\right)=lnx$ with the inverse $W_2^{-1}\left(x\right)=e^x.$ Similarly, $W_1\left(x\right)=lnx$ with its inverse $W_1^{-1}\left(x\right)=e^x.$

Further, $a\left(t\right)={\tilde{c}}_1$ and $a^{\prime }\left(t\right)=0$, so $r_1\left(t\right)={\tilde{c}}_1$. Next, we define non-negative functions $g_1,g_2:[0,1]\times [0,1]\to {\mathbb{R}}_{+}$ as follows:

$$\begin{array}{c}\hfill g_1(\tau ,s):=\left\{\begin{array}{c}\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}{(\tau -s)}^{2(\alpha +\beta )-2},0\le s<\tau \hfill \\ \\ \frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}{(s-\tau )}^{2(\alpha +\beta )-2},0\le \tau <s,\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill g_2(\tau ,s):=\left\{\begin{array}{c}\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}{(\tau -s)}^{2\alpha -2},0\le s<\tau \hfill \\ \\ \frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}{(s-\tau )}^{2\alpha -2},0\le \tau <s.\hfill \end{array}\right.\end{array}$$

We consider two cases.

• Case 1: For $0\le s<\tau$. Given that $\alpha +\beta >1$, then $g_1$ is continuous and increasing (continuously increasing), hence,

  $${\displaystyle \underset{0\le \tau \le t}{max}{g}_1(\tau ,s)=\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}{(t-s)}^{2(\alpha +\beta )-2},}$$

  and we have

  $$\begin{array}{ccc}\hfill r_2\left(t\right)& =& exp\left[ln\left({\tilde{c}}_1\right)+\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}{\int }_0^t{(t-s)}^{2(\alpha +\beta )-2}ds\right]\hfill \\ & =& exp\left[ln\left({\tilde{c}}_1\right)+\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}\frac{t^{2(\alpha +\beta )-1}}{2(\alpha +\beta )-1}\right].\hfill \end{array}$$
  Further, for $0\le s<\tau ,$ and for all $\alpha >1$, $g_2$ is continuously increasing, and

  $$\underset{0\le \tau \le t}{max}{g}_2(\tau ,s)=\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}{(t-s)}^{2\alpha -2}.$$
  Thus,

  $$\begin{array}{ccc}\hfill f\left(t\right)& \le & exp\left[ln\left(r_2\left(t\right)\right)+\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}{\int }_0^t{(t-s)}^{2\alpha -2}ds\right]\hfill \\ & =& exp\left[ln\left({\tilde{c}}_1\right)+\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}\frac{t^{2(\alpha +\beta )-1}}{2(\alpha +\beta )-1}+\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}\frac{t^{2\alpha -1}}{2\alpha -1}\right]\hfill \\ & =& {\tilde{c}}_{1}exp\left[\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}\frac{t^{2(\alpha +\beta )-1}}{2(\alpha +\beta )-1}+\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}\frac{t^{2\alpha -1}}{2\alpha -1}\right].\hfill \end{array}$$
• Case 2: For $0\le \tau <s$, we follow similar steps in case 1, which shows that $g_1$ is continuously increasing and

  $${\displaystyle \underset{0\le s\le t}{max}{g}_1(\tau ,s)=\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}{(t-\tau )}^{2(\alpha +\beta )-2},}$$

  thus,

  $$\begin{array}{ccc}\hfill r_2\left(t\right)& =& exp\left[ln\left({\tilde{c}}_1\right)+\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}{\int }_0^t{(t-\tau )}^{2(\alpha +\beta )-2}d\tau \right]\hfill \\ & =& exp\left[ln\left({\tilde{c}}_1\right)+\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}\frac{t^{2(\alpha +\beta )-1}}{2(\alpha +\beta )-1}\right].\hfill \end{array}$$
  On the other hand, $g_2$ is continuously increasing, and

  $$\underset{0\le s\le t}{max}{g}_2(\tau ,s)=\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}{(t-\tau )}^{2\alpha -2}.$$
  Therefore,

  $$\begin{array}{ccc}\hfill f\left(t\right)& \le & exp\left[ln\left(r_2\left(t\right)\right)+\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}{\int }_0^t{(t-\tau )}^{2\alpha -2}d\tau \right]\hfill \\ & =& exp\left[ln\left({\tilde{c}}_1\right)+\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}\frac{t^{2(\alpha +\beta )-1}}{2(\alpha +\beta )-1}+\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}\frac{t^{2\alpha -1}}{2\alpha -1}\right]\hfill \\ & =& {\tilde{c}}_{1}exp\left[\frac{3{\lambda }^2{\mathrm{Lip}}_{\sigma }^2}{{\Gamma }^2(\alpha +\beta )}\frac{t^{2(\alpha +\beta )-1}}{2(\alpha +\beta )-1}+\frac{3{\gamma }^2}{{\Gamma }^2\left(\alpha \right)}\frac{t^{2\alpha -1}}{2\alpha -1}\right],\hfill \end{array}$$

  and this completes the proof.

□

4. Conclusions

The Langevin equation is an important equation in modeling dynamical processes in fluctuating environments, which includes investigating the stochastic modeling of surface reactions by using reflected chemical Langevin equations [22], modeling the structural and thermal properties of matter using molecular dynamics simulation [23], and the modeling of persistent time series [1]. Due to the fractal and complex nature of some fluctuating environments, it became necessary to model using the fractional Langevin equation in order to completely capture the dynamics of particles in the complex media. Our result showed that particles in a complex-chaotic media can still exhibit some exponential growth but for a short time $t\in [0,1]$, which is subject to the viscous friction $\gamma$ and the noise level $\lambda$. Simply put, the second-moment upper-growth bound was obtained, and it shows exponential growth in time at a precise rate in terms of the parameters $\gamma$ and $\lambda$. The existence and uniqueness of the result for the solution of a stochastic nonlinear fractional Langevin equation of two fractional orders were given, applying Banach’s fixed-point theorem. Future research includes the long-term asymptotic behavior of our solution. That is to say, “what happens to the particles in the complex-chaotic environment after a long period of time?” One can also study the above problem for colored noise. The lower bound estimate is still open for further research.