Existence, uniqueness and Ulam's stabilities for a class of implicit impulsive Langevin equation with Hilfer fractional derivatives

1.   Introduction

Fractional differential equations is the generalized form of classical differential equations of integer order. Fractional calculus is now a developed area and it has many applications in porous media, electrochemistry, economics, electromagnetics, physical sciences, medicine etc., Progressively, the role of fractional differential equations is very important in viscoelasticity, statistical physics, optics, signal processing, control, defence, electrical circuits, astronomy etc. Some interesting articles provide the main theoretical tools for the qualitative analysis of this area and also shows the interconnection as well as the distinction between classical, integral models and fractional differential equations, see ^{[1,17,19,22,23,24,25,26,29,34,35]}.

The Langevin equation is an excellent technique to describe some phenomena which can help physicians, engineers, economists, etc., effectively to describe processes. The Langevin equation (drafted for first by Langevin in $1908$) is obtained to be an accurate tool to describe the development of physical phenomena. These equations are used to described stochastic problems in physics, defence system, image processing, chemistry, astronomy, mechanical and electrical engineering. They are also used to describe Brownian motion when the random oscillation force is supposed to be Gaussian noise. Fractional order differential equations are utilized for the removal of noise. For more details, see ^{[2,12,20,21,28]}.

Recently impulsive differential equations have been considered by many authors due to their significant applications in various fields of science and technology. These equations describe the evolution processes that are subjected to abrupt changes and discontinuous jumps in their states. Many physical systems like the function of pendulum clock, the impact of mechanical systems, preservation of species by means of periodic stocking or harvesting and the heart's function, etc. naturally experience the impulsive phenomena. Similarly in many other situations, the evolutional processes have the impulsive behavior. For example, the interruptions in cellular neural networks, the damper's operation with percussive effects, electromechanical systems subject to relaxational oscillations, dynamical systems having automatic regulations, etc., have the impulsive phenomena. For detail study, see ^{[10,13,16,38,18,42,45,5,40,30]}. Due to its large number of applications, this area has been received great importance and remarkable attention from the researchers.

At Wisconsin university, Ulam raised a question about the stability of functional equations in the year 1940. The question of Ulam was: under what conditions does there exist an additive mapping near an approximately additive mapping ^[36]. In 1941, Hyers was the first mathematician who gave partial answer to Ulam's question ^[14], over Banach space. Afterwards, stability of such form is known as Ulam-Hyers stability. In 1978, Rassias ^[27], provided a remarkable generalization of the Ulam-Hyers stability of mappings by considering variables. For more information about the topic, we refer the reader to ^{[6,15,31,33,37,43,44,46]}.

Recently, the existence, uniqueness and different types of fractional nonlinear differential equations with Caputo fractional derivative have received a considerable attention, see ^{[3,7,8,9,32,33]}.

Wang et al. ^[39], studied generalized Ulam-Hyers-Rassias stability of the following fractional differential equation:

Zada et al. ^[41], studied existence, uniqueness of solutions by using Diaz-Margolis's fixed point theorem ^[11] and presented different types of Ulam-Hyers stability for a class of nonlinear implicit fractional differential equation with non-instantaneous integral impulses and nonlinear integral boundary conditions:

Motivated by the aforesaid work, in this manuscript, we investigate the existence, uniqueness, Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias and generalized Ulam-Hyers-Rassias stability results for the following nonlinear implicit impulsive Langevin equation with two Hilfer fractional derivatives:

where $\mathcal{D}^{\alpha_1, \beta}$ and $\mathcal{D}^{\alpha_2, \beta}$ represents two Hilfer fractional derivatives, of order $\alpha_1$ and $\alpha_2$ respectively, $\beta$ determines to the type of initial condition used in the problem. Further $f:J\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ is continuous and $I_i:\mathbb{R}\rightarrow\mathbb{R}$ for all $i = 1, 2, \dots, m,$ represents impulsive nonlinear mapping and $\Delta x(\upsilon_i) = x(\upsilon_i^+)-x(\upsilon_i^-)$, where $x(\upsilon_i^+)$ and $x(\upsilon_i^-)$ represent the right and the lift limits, respectively, at $\upsilon = \upsilon_i$ for $i = 1, 2, \dots, m.$

In the second section of this paper, we introduce some notations, definitions and auxiliary results. In section $3$, we give the existence, uniqueness results for the proposed model (1.1) obtained via the Banach's contraction. In Section $4$, we investigate the Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability of our proposed model (1.1). Finally, we give an example which supports our main result.

2.   Preliminaries

We recall some definitions of fractional calculus from ^[17,26] as follows.

Definition 2.1. The fractional integral of order $\alpha$ from $0$ to $x$ for the function $f$ is

where $\Gamma(\cdot)$ is the Gamma function.

Definition 2.2. The Riemman-Liouville fractional derivative of fractional order $\alpha$ for $f$ is

Definition 2.3. The Caputo derivative of fractional order $\alpha$ for $f$ is

Definition 2.4. The classical Caputo derivative of order $\alpha$ of $f$ is

Definition 2.5. The Hilfer fractional derivative of order $0 < \alpha < 1$ and $0\leq\beta\leq1$ of function $f(x)$ is

The Hilfer fractional derivative is used as an interpolator between the Riemman-Liouville and Caputo derivative.

Remark 2.1. (a) Operator $\mathcal{D}^{\alpha, \beta}$ also can be written as

(b) If $\beta = 0$, then $\mathcal{D}^{\alpha, \beta} = \mathcal{D}^{\alpha, 0}$ is called Riemman–Liouville fractional derivative.

(c) If $\beta = 1,$ then $\mathcal{D}^{\alpha, \beta} = I^{1-\alpha}\mathcal{D}$ is called Caputo fractional derivative.

Remark 2.2. (ⅰ) If $f(\cdot)\in C^m([0, \infty), \mathbb{R}),$ then

(ⅱ) In Definition 2.4, the integrable function $f$ can be discontinuous. This fact can support us to consider impulsive fractional problems in the sequel.

Lemma 2.1. ^[17] The fractional differential equation $^cD^\alpha f(x) = 0$ with $\alpha > 0,$ involving Caputo differential operator $^cD^\alpha$ have a solution in the following form:

where $c_i\in\mathbb{R},$ $i = 0, 1, \dots, m-1$ and $m = [\alpha]+1.$

Lemma 2.2. ^[17] For arbitrary $\alpha > 0,$ we have

where $c_i\in\mathbb{R},$ $i = 0, 1, \dots, m-1$ and $m = [\alpha]+1.$

Lemma 2.3. ^[26] Let $\alpha > 0$ and $\beta > 0$, $f\in L^1\big([a, b]\big).$

Let $J = [0, T], \ J_0 = [0, \upsilon_1], \ J_1 = (\upsilon_1, \upsilon_2], \ J_2 = (\upsilon_2, \upsilon_3], \dots, \ J_{m-1} = (\upsilon_{m-1}, \upsilon_m], \ J_m = (\upsilon_m, T],$ $J^{'} = J-\{\upsilon_0, \upsilon_1, \upsilon_2, \dots, \upsilon_m\}.$ Also for convenience use the notation $J_{i} = (\upsilon_{i}, \upsilon_{i+1}].$

Theorem 2.1. [^[4](Banach's fixed point theorem)]. Let $B$ be a Banach space. Then any contraction mapping $N:B\rightarrow B$ has a unique fixed point.

3.   Existence and uniqueness

In this section, we investigate the existence, uniqueness of solutions to the proposed Langevin equation using two Hilfer fractional derivatives.

Lemma 3.1. Let $f : J\times \mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R}$ is a function such that $f(\cdot, x(\cdot), \mathcal{D}^{\alpha_1, \beta}x(\cdot))\in C_{1-\gamma}[0, T]$ for all $x\in C_{1-\gamma}[0, T].$ A function $x\in C_{1-\gamma}^{\gamma}[0, T]$ is equivalent to the integral equation

is the only solution of the problem (1.1)

Proof. Let $x$ satisfies (1.1), then for any $\upsilon\in J_0$, there exists a constant $c\in\mathbb{R},$ such that

Using the condition $I^{1-\gamma}x(0) = x_0,$ Eq (3.2) yields that

Similarly for $\upsilon\in J_1$, there exists a constant $d_1\in\mathbb{R}$, such that

Using the condition, we get

In view of

we get

For this value of $d_1$, we have

Similarly for $\upsilon\in J_i$, we get

Conversely, let that $x$ satisfies (3.1), then it can be easily proved that the solution $x(\upsilon)$ given by (3.1) satisfies (1.1).

Consider some assumptions as follows:

$(H_1)$ $f\in C(J\times \mathbb{R}\times \mathbb{R}, \mathbb{R})$ is continuous.

$(H_2)$ There exists positive constants $Ł_{f}$ and $Ł_{g},$ such that $\big|f(w, u, m)-f(w, v, n)\big|\leq Ł_{f}|u-v|+Ł_{g}|m-n|,$ for each $w\in J$ and all $u, v, m, n\in \mathbb{R}.$

$(H_3)$ There exists $Ł_{k} > 0,$ such that $\big|I_i(u)-I_i(v)\big|\leq Ł_{k}|u-v|$, for each $\upsilon\in J_i, \ i = 1, 2, \dots, m,$ and for all $u, v\in\mathbb{R}.$

$(H_4)$ There exists $\varphi\in PC(J, \mathbb{R}^+)$ and $\lambda_\varphi > 0$ $\ni$ $I^\alpha\varphi(\upsilon)\leq\lambda_\varphi\varphi(\upsilon)\ \ \ \rm{for each}\ \ \upsilon\in J.$

Theorem 3.1. Let assumptions $(H_1)-(H_3)$ be satisfied and if

then (1.1) has a unique solution $x$ in $C_{1-\gamma} [0, T].$

Proof. We define a mapping ${N}:C_{1-\gamma} [0, T]\rightarrow C_{1-\gamma} [0, T]$

For any $x, y\in C_{1-\gamma} [0, T]$ and $\upsilon\in J_i$, consider the following

Now since

Hence $x$ is a contraction according to Banach's contraction theorem and so it has only one fixed point, which is the only one solution of (1.1).

4.   Ulam-Hyers stability analysis

Let $\varepsilon > 0$ and $\varphi : J \rightarrow \mathbb{R}^+$ be a continuous function. Consider

and

Definition 4.1. The problem (1.1) is Ulam-Hyers stable if there exists a real number $C_{f, i, q, \sigma}$ such that for each solution $\varepsilon > 0$ and for each solution $z\in C_{1-\gamma} [0, T]$ of the inequality (4.1), there exists a solution $x\in C_{1-\gamma} [0, T]$ of the problem (1.1) such that

Definition 4.2. The problem (1.1) is generalized Ulam-Hyers stable if there exists $\phi_{f, i, q, \sigma}\in C_{1-\gamma} [0, T],$ $\phi_{f, i, q, \sigma}(0) = 0$ and $\varepsilon > 0$ such that for each solution $z\in C_{1-\gamma [0, T]}$ of the inequality (4.1), there exists a solution $x\in C_{1-\gamma} [0, T]$ of the problem (1.1) such that

Remark 4.1. Keep in mind that Definition 4.1 $\Rightarrow$ Definition 4.2.

Definition 4.3. The problem (1.1) is Ulam-Hyers-Rassias stable with respect to $(\varphi, \psi)$ if there exists $C_{f, i, q, \sigma, \varphi} > 0$ such that for each $\varepsilon > 0$ and for each solution $z\in C_{1-\gamma} [0, T]$ of inequality (4.3) there is a solution $x\in C_{1-\gamma} [0, T]$ of the problem (1.1) with

Definition 4.4. The problem (1.1) is generalized Ulam-Hyers-Rassias stable with respect to $(\varphi, \psi)$ if there exists $C_{f, i, q, \sigma, \varphi} > 0$ such that for each solution $z\in C_{1-\gamma} [0, T]$ of inequality (4.2) there is a solution $x\in C_{1-\gamma} [0, T]$ of the problem (1.1) with

Remark 4.2. It should be noted that Definition 4.3 implies Definition 4.4.

Remark 4.3. A function $z\in C_{1-\gamma} [0, T]$ is a solution of the inequality (4.1) $\Leftrightarrow$ there exists a function $g\in C_{1-\gamma} [0, T]$ and a sequence $g_i, i = 1, 2, \dots, m,$ depending on $g$, such that

(a) $|g(\upsilon)|\leq \varepsilon, \ |g_i|\leq \varepsilon \ \ \upsilon\in J_i, \ i = 1, 2, \dots, m,$

(b) $\mathcal{D}^{\alpha_1, \beta}(\mathcal{D}^{\alpha_2, \beta}+\lambda)z(\upsilon) = f(\upsilon, z(\upsilon), \mathcal{D}^{\alpha_1, \beta}z(\upsilon))+g(\upsilon), \ \ \upsilon\in J_i, \ i = 1, 2, \dots, m,$

(c) $\Delta\ x(\upsilon_i) = I_i(x(\upsilon_i))+g_i, \ \ \upsilon\in J_i, \ i = 1, 2, \dots, m.$

Remark 4.4. A function $z\in C_{1-\gamma} [0, T]$ satisfies (4.2) $\Leftrightarrow$ there exists $g\in C_{1-\gamma} [0, T]$ and a sequence $g_i, i = 1, 2, \dots, m,$ depending on $g$, such that

(a) $|g(\upsilon)|\leq \varphi(\upsilon), \ |g_i|\leq \psi \ \ \upsilon\in J_i, \ i = 1, 2, \dots, m,$

(b) $\mathcal{D}^{\alpha_1, \beta}(\mathcal{D}^{\alpha_2, \beta}+\lambda)z(\upsilon) = f(\upsilon, z(\upsilon), \mathcal{D}^{\alpha_1, \beta}z(\upsilon))+g(\upsilon), \ \ \upsilon\in J_i, \ i = 1, 2, \dots, m,$

(c) $\Delta\ x(\upsilon_i) = I_i(x(\upsilon_i))+g_i, \ \ \upsilon\in J_i, \ i = 1, 2, \dots, m.$

Remark 4.5. A function $z\in C_{1-\gamma} [0, T]$ satisfies (4.2) $\Leftrightarrow$ there exists $g\in C_{1-\gamma} [0, T]$ and a sequence $g_i, i = 1, 2, \dots, m,$ depending on $g$, such that

(a) $|g(\upsilon)|\leq \varepsilon\varphi(\upsilon), \ |g_i|\leq \varepsilon\psi \ \ \upsilon\in J_i, \ i = 1, 2, \dots, m,$

(b) $\mathcal{D}^{\alpha_1, \beta}(\mathcal{D}^{\alpha_2, \beta}+\lambda)z(\upsilon) = f(\upsilon, z(\upsilon), \mathcal{D}^{\alpha_1, \beta}z(\upsilon))+g(\upsilon), \ \ \upsilon\in J_i, \ i = 1, 2, \dots, m,$

(c) $\Delta\ x(\upsilon_i) = I_i(x(\upsilon_i))+g_i, \ \ \upsilon\in J_i, \ i = 1, 2, \dots, m.$

Theorem 4.1. If the assumptions $(H1)-(H3)$ and the inequality (3.3) hold, then Eq (1.1) is Ulam–Hyers stable and consequently generalized Ulam–Hyers stable.

Proof. Let $y\in C_{1-\gamma} [0, T]$ satisfies (4.1) and let $x$ be the only one solution of

By Lemma 3.1, we have for each $\upsilon\in J_i$

Since $y$ satisfies inequality (4.1), so by Remark 4.3., we get

Obviously the solution of (4.8), will be

Therefore, for each $\upsilon\in J_i$, we have the following

which implies that

Thus

where

So Eq (1.1) is Ulam-Hyers stable and if we set $\phi(\varepsilon) = \varepsilon C_{f, g, \alpha_1, \alpha_2},$ $\phi(0) = 0$, then Eq (1.1) is generalized Ulam-Hyers stable.

Theorem 4.2. If the assumptions $(H_1)-(H_4)$ and the inequality (3.3) are satisfied, then the problem (1.1) is Ulam-Hyers-Rassias stable with respect to $(\varphi, \psi)$, consequently generalized Ulam-Hyers-Rassias stable.

Proof. Let $y\in C_{1-\gamma} [0, T]$ be a solution of the inequality (4.3) and let $x$ be the only one solution of the following problem

From Theorem 4.1, $\forall\ \upsilon\in J_i$, we get

which implies that

Thus

where

Hence (1.1) is Ulam-Hyers-Rassias stable and is obviously generalized Ulam-Hyers-Rassias stable. Finally we give an example to illustrate our main result.

Example 4.1.

Let $J_0 = [0, \frac{1}{2}]$, $J_1 = [\frac{1}{2}, 1]$ $\alpha_1 = \frac{1}{2}$, $\alpha_2 = \frac{1}{3}$, $\lambda = \lambda_\varphi = \frac{1}{2}$, $Ł_f = Ł_{k} = \frac{1}{90e^2}$ and $m = T = 1.$

Obviously

Thus, thanks to Theorem 3.1, the given problem (4.9) has a unique solution. Further the conditions of Theorem 4.1 are satisfied so the solution of the given problem (4.9) is Ulam-Hyers stable and generalized Ulam-Hyers stable. Further it is also easy to check the conditions of Theorem 4.2 hold and thus the problem (4.9) is Ulam-Hyers-Rassias stable and consequently generalized Ulam-Hyers-Rassias stable.

5.   Conclusions

In this article, we consider a class of implicit impulsive Langevin equation with Hilfer fractional derivative. Some conditions are made to beat the hurdles to investigate the existence, uniqueness and to discuss different types of Ulam-Hyers stability of our considered model, using Banach's fixed point theorem. We give an example which supports our main result.

Acknowledgments

The authors wish to thank the anonymous referees for their kind comments, correcting errors, improving written language and constructive suggestions. This work was supported by the Natural Science Foundation of Jiangxi Province (Grant No. 20192BAB201011) and the National Natural Science Foundation of China (Grant No. 11861053).

Conflict of interest

The authors declare no conflict of interest in this paper.