Switched coupled system of nonlinear impulsive Langevin equations with mixed derivatives

1.   Introduction

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 ^[28]? In 1941, Hyers was the first mathematician who gave partial answer to Ulam's question ^[11], over Banach space. Afterwards, stability of such form is known as Ulam-Hyers stability. In 1978, Rassias ^[18], 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 ^{[21,25,26,29,39,40,42]}.

Fractional Langevin differential equations have been one of the important subject in physics, chemistry and electrical engineering. The Langevin equation (first formulated by Langevin in $1908$) is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments. For instance, Brownian motion is well described by the Langevin equation when the random fluctuation force is assumed to be white noise. In case the random fluctuation force is not white noise, the motion of the particle is described by the generalized Langevin equation. For systems in complex media, ordinary Langevin equation does not provide the correct description of the dynamics. Various generalizations of Langevin equations have been proposed to describe dynamical processes in a fractal medium. One such generalization is the generalized Langevin equation which incorporates the fractal and memory properties with a dissipative memory kernel into the Langevin equation. Another possible extension requires the replacement of ordinary derivative by a fractional derivative in the Langevin equation to give the fractional Langevin equation. For more details, see ^{[2,10,15,16,19,33,34,35]}.

Fractional order differential equations are the generalizations of the classical integer order differential equations. Fractional calculus has become a speedily developing area and its applications can be found in diverse fields ranging from physical sciences, porous media, electrochemistry, economics, electromagnetics, medicine and engineering to biological sciences. Progressively, fractional differential equations play a very important role in fields such as thermodynamics, statistical physics, nonlinear oscillation of earthquakes, viscoelasticity, defence, optics, control, signal processing, electrical circuits, astronomy etc. There are some outstanding articles which provide the main theoretical tools for the qualitative analysis of this research field, and at the same time, shows the interconnection as well as the distinction between integral models, classical and fractional differential equations, see ^{[1,12,17,20,23,27,32]}.

Impulsive fractional differential equations are used to describe both physical, social sciences and many dynamical systems such as evolution processes pharmacotherpy. There are two types of impulsive fractional differential equations the first one is instantaneous impulsive fractional differential equations while the other one is non-instantaneous impulsive fractional differential equations. In last few decades, the theory of impulsive fractional differential equations are well utilized in medicine, mechanical engineering, ecology, biology and astronomy etc. see ^{[3,8,13,24,30,36,38,41]}.

Recently, many mathematicians received a considerable attention to the existence, uniqueness and different types of Hyres-Ulam stability of the solutions of nonlinear implicit fractional differential equations with Caputo fractional derivative, see ^[5,7,22,26].

Wang et al. ^[31], studied generalized Ulam-Hyers-Rassias stability of the following fractional differential equation:

Zada et al. ^[37], studied existence and uniqueness of solutions by using Diaz Margolis's fixed point theorem 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:

In recent years, many researchers paid much attention to the coupled system of fractional differential equations due to its applications in different fields ^[3,6,32].

Ali et al. ^[4], studied the existence, uniqueness of solutions by using using the classical fixed point theorems such as Banach contraction principle and Leray-Schauder of cone type and presented various kinds of Ulam stability including Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability of the solutions to a nonlinear coupled systems of implicit fractional differential equations involving Caputo derivative of the form:

In this paper, we study switched coupled system of nonlinear impulsive Langevin equations with mixed derivatives of the form:

where ${^c}\mathcal{D}^{\nu}_{0, w}$ and ${^c}\mathcal{D}^{\mu}_{0, w}$ represents classical Caputo derivative ^[12], of order $\nu$ and $\mu$ respectively with the lower bound zero, $0 = w_0 < s_0 < w_1 < s_1 < \dots < w_m < s_m = T,$ $T$ is the pre–fixed number and $\lambda_1, \lambda_2\in\mathbb{R}\setminus\{0\}, \ 0 < \nu < 1, \ p_1, p_2 > 0, \ u_0, v_0$ are constants, $f_1, f_2:[0, T]\times\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ is continuous and $g_k: [s_{k-1}, w_k]\times\mathbb{R}\rightarrow \mathbb{R}$ is continuous for all $k = 1, 2, \dots, m.$

In the second section of this paper, we create a uniform framework to originate appropriate formula of solutions for our proposed model. In section 3, we implement the concept of generalized Ulam-Hyers-Rassias stability of Eq (1.1). Finally, we give an example which supports our main result.

2.   Preliminaries

Let $J = [0, T]$ and $C(J, \mathbb{R})$ be the space of all continuous functions from $J$ to $\mathbb{R}$, and the piecewise continuous function space $PC(J, \mathbb{R}) = \{z:J\rightarrow \mathbb{R}: z\in C ((w_k, w_{k-1}], \mathbb{R}), k = 0, \dots, m$ and there exist $z(w_k^-) \text{ and } z(w_k^+), \ k = 1, 2, \dots, m \text{ with } z(w_k^-) = z(w_k)\}$.

In the current section, we create a uniform framework to originate appropriate formula for the solution of impulsive fractional differential equation of the form:

We recall some definitions of fractional calculus from ^[12] as follows:

Definition 2.1. The fractional integral of order $\nu$ from $0$ to $w$ for the function $f$ is defined by

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

Definition 2.2. The Riemman-Liouville fractional derivative of fractional order $\nu$ from $0$ to $w$ for a function $f$ can be written as

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

Definition 2.3. The Caputo derivative of fractional order $\nu$ from $0$ to $w$ for a function $f$ can be defined as

Definition 2.4. The general form of classical Caputo derivative of order $\nu$ of a function $f$ can be given as

Remark 2.5. (i) If $f(\cdot)\in C^m([0, \infty), \mathbb{R}),$ then

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

Definition 2.6. The Hilfer fractional derivative of order $0 < \alpha < 1$ and $0\leq\gamma\leq1$ of function $f(x)$ is

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

Remark 2.7. (a) Operator $D^{\alpha, \gamma}$ also can be written as

(b) If $\gamma = 0$, then $D^{\alpha, \gamma} = D^{\alpha, 0}$ is called the Riemman-Liouville fractional derivative.

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

Lemma 2.8. ^[12] 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 $c_k\in\mathbb{R},$ $k = 0, 1, \dots, m-1$ and $m = [\alpha]+1.$

Lemma 2.9. ^[17] Let $\alpha > 0$ and $\gamma > 0$, $f\in L^1\big([a, b]\big).$

Lemma 2.10. ^[24] The function $u\in PC(J, \mathbb{R})$ is a solution of $(2.1)$ if and only if

where

In view of Lemma 2.10 the solution form of proposed system (1.1) is given by

3.   Generalized Ulam-Hyers-Rassias stability concept

By the ideas of stability in ^[25,29], we can generate a generalized Ulam-Hyers-Rassias stability concept for Eq (1.1).

Definition 3.1. Equation (1.1) is generalized Ulam-Hyers-Rassias stable with respect to $(\varphi_u, \varphi_v, \psi)$ if there exists $C_u, C_v > 0$ such that for each solution $(u, v)\in PC(J, \mathbb{R})\times PC(J, \mathbb{R})$ of inequality (3.1) there is a solution $(u_0, v_0)\in PC(J, \mathbb{R})\times PC(J, \mathbb{R})$ of Eq (1.1) with

Remark 3.2. A function $u, v\in PC(J, \mathbb{R})$ is a solution of the inequality (3.1) if and only if there is $G\in PC(J, \mathbb{R})$ and a sequence $G_k, \ k = 1, 2, \dots, m$ (which depends on $z$) such that

(i) $|G(w)|\leq \varphi(w), \ w\in J$ and $|G_k|\leq \psi, \quad k = 1, 2, \dots, m,$

(ii) ${^c}\mathcal{D}_{0, w}^\nu(\mathcal{D}+{\lambda_1})u(w) = f_1(w, v(w), u(w))+G(w), \ w\in (w_k, s_k], \quad k = 0, 1, \dots, m$,

(iii) ${^c}\mathcal{D}_{0, w}^\mu(\mathcal{D}+\lambda_2)v(w) = f_2(w, u(w), v(w))+G(w), \ w\in (w_k, s_k], \quad k = 0, 1, \dots, m$,

(iv) $u(w) = {N_{k_1}} g_k(w, u(w))+G_k, \ w\in (s_{k-1}, w_k], \quad k = 1, 2, \ \dots, m$.

Remark 3.3. (1) If $u\in PC(J, \mathbb{R})$ is a solution of the inequality (3.1) then $u$ is a solution of the following integral inequality,

In fact by Remark 3.2, we get

Clearly, the solution of Eq (3.5) is given by

For $w\in(w_k, s_k], \quad k = 0, 1, \dots, m,$ we get

Proceeding the above, we derive that

and

Similarly

(2) If $v\in PC(J, \mathbb{R})$ is a solution of the inequality (3.1) then $v$ is a solution of the following integral inequality,

In fact by Remark 3.2, we get

Clearly, the solution of Eq (3.5) is given by

For $w\in(w_k, s_k], \quad k = 0, 1, \dots, m,$ we get

Proceeding the above, we derive that

and

4.   Main results via fixed point methods

In order to apply a fixed point theorem of the alternative, for contractions on a generalized complete metric space to achieve our main result, we want to collect the following realities.

Definition 4.1. For a non empty set $V$, a function $d:V\times V\rightarrow [0, \infty]$ is called a generalized metric on $V$ if and only if $d$ satisfies:

$\diamond$ $d(\upsilon_1, \upsilon_2) = 0$ if and only if $\upsilon_1 = \upsilon_2$;

$\diamond$ $d(\upsilon_1, \upsilon_2) = d(\upsilon_2, \upsilon_1)$ for all $\upsilon_1, \upsilon_2\in V$;

$\diamond$ $d(\upsilon_1, \upsilon_3)\leq d(\upsilon_1, \upsilon_2)+d(\upsilon_2, \upsilon_3)$ for all $\upsilon_1, \upsilon_2, \upsilon_3\in V.$

Lemma 4.2. (see ^[9] (Generalized Diaz-Margolis's fixed point theorem)). Let $(V, d)$ be a generalized complete metric space. Assume that $T:V\rightarrow V$ is a strictly contractive operator with the Lipschitz constant $L < 1$. If there exists a $k\geq 0$ such that $d(T^{k+1} v, T^k v) < \infty$ for some $v$ in $V$, then the followings statements are true:

$(B_1)$ The sequence $\{T^n v\}$ converges to a fixed point $v^*$ of $T$;

$(B_2)$ The unique fixed point of $T$ is $v^*\in V^* = \bigg\{u\in V { such~ ~that } d(T^k v, u) < \infty\bigg\};$

$(B_3)$ $u\in V^*,$ then $d(u, v^*)\leq \frac{1}{1-L}d(Tu, u).$

We introduced some assumptions as follows:

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

$(H_2)$ ● There exists positive constants $0 < L_{f_{u_1}} < 1$ and $0 < L_{f_{u_2}} < 1,$ such that $\big|f_1(w, u, m)-f_1(w, v, n)\big|\leq L_{f_{u_1}}|u-v|+L_{f_{u_2}}|m-n|,$ for each $w\in J$ and all $u, v, m, n\in \mathbb{R}.$

● There exists positive constants $0 < L_{f_{v_1}} < 1$ and $0 < L_{f_{v_2}} < 1,$ such that $\big|f_2(w, u, m)-f_2(w, v, n)\big|\leq L_{f_{v_1}}|u-v|+L_{f_{v_2}}|m-n|,$ for each $w\in J$ and all $u, v, m, n\in \mathbb{R}.$

$(H_3)$ ● $g_k\in C\big((s_{k-1}, w_k]\times\mathbb{R}, \mathbb{R}\big)$ and there are positive constant $L_{gk_1}, \ k = 1, 2, \dots, m$ such that $\big|g_k (w, v)-g_k (w, v)\big|\leq L_{gk_1}|u-v|$, for each $w\in (s_{k-1}, w_k],$ and all $u, v\in\mathbb{R}.$

● $g_k\in C\big((s_{k-1}, w_k]\times\mathbb{R}, \mathbb{R}\big)$ and there are positive constant $L_{gk_2}, \ k = 1, 2, \dots, m$ such that $\big|g_k (w, v)-g_k (w, v)\big|\leq L_{gk_2}|u-v|$, for each $w\in (s_{k-1}, w_k],$ and all $u, v\in\mathbb{R}.$

$(H_4)$ ● Let $\varphi_u\in C(J, \mathbb{R}_+)$ be a nondecreasing function, there exists $c_\varphi > 0$ such that

● Let $\varphi_v\in C(J, \mathbb{R}_+)$ be a nondecreasing function, there exists $c_\varphi > 0$ such that

Theorem 4.3. Suppose that $(H_1)$–$(H_4)$ are satisfied and also a function $u, v\in PC(J, \mathbb{R})$ satisfies $(3.1)$. Then there exists unique solutions $u_0, v_0$ of Eq $(1.1)$ such that

similarly

and

for all $w\in J$, if $0 < \nu < 1$ and

where

Proof. Consider the space of piecewise continuous functions

endowed with the generalized metric on $V,$ defined by

where

and

It is easy to verify that $(V, d)$ is a complete generalized metric space ^[14].

Define an operator $\Lambda:V\rightarrow V$ by

for all $u$ belongs to $V$ and $w\in J$. Obviously, according to $(H_1)$, $\Lambda$ is well defined operator.

Next we shall verify that $\Lambda$ is strictly contractive on $V$. Note that according to definition of $(V, d)$, for any $\mu, \upsilon\in V$, it is possible to find $C_1, C_2\in [0, \infty]$ such that

From the definition of $\Lambda$ in Eq (4.6), $(H_2)$, $(H_3)$ and (4.7), we obtain that

Case 1: For $w\in[0, s_0],$

where, we use the following notations for convenience

which further gives

similarly

Put (4.9) in (4.8), we obtain

On the similar way, we can obtain

Therefore from (4.10) and (4.11), we get the following result

Suppose that $\max\{\lambda_1, \lambda_2\} = \lambda,$ $\max\{p_1, p_2\} = p,$ $\max\{A_{11}, A_{22}\} = A$ and $\max\{\eta_1, \eta_2\} = \eta$

Case 2: For $w\in (s_{k-1}, w_k],$ we have

On the similar way, we can obtain

Therefore from (4.12) and (4.13), we get the result given as

Case 3: For $w\in (w_k, s_k],$ and $s\in (w_k, s_k],$

that is,

On the similar way, we can obtain

Therefore from (4.14) and (4.15), we get the result given as

Suppose that $\max\{\lambda_1, \lambda_2\} = \lambda,$ $\max\{p_1, p_2\} = p,$ $\max\{M_{K_1}, M_{K_2}\} = M_{K},$ $\max\{N_{K_1}, N_{K_2}\} = N_{K}$ $\max\{L_{gk_1}, L_{gk_2}\} = L_{gk}$ and $\max\{\eta_1, \eta_2\} = \eta$

Also, for $w\in (w_k, s_k],$ and $s\in (s_{k-1}, w_k],$

that is

On the similar way, we can obtain

Therefore from (4.16) and (4.17), we get the following result

Suppose that $\max\{\lambda_1, \lambda_2\} = \lambda,$ $\max\{p_1, p_2\} = p,$ $\max\{M_{K_1}, M_{K_2}\} = M_{K},$ $\max\{N_{K_1}, N_{K_2}\} = N_{K}$ $\max\{L_{gk_1}, L_{gk_2}\} = L_{gk}$ and $\max\{\eta_1, \eta_2\} = \eta$

From above, we have

that is,

Hence, we conclude that

for any $(u, v), (\bar{u}, \bar{v})\in V$, since the condition (4.4) is strictly contraction property is shown.

Now we take $(u_0, v_0)\in V$. From the piecewise continuous property of $(u_0, v_0)$ and $\Lambda (u_0, v_0),$ it follows that there exists a constant $0 < G_1 < \infty,$ such that

There exists a constant $0 < G_2 < \infty,$ such that

where $w\in (s_{k-1}, w_k], \ k = 1, 2, \dots, m.$

Also we can find a constant $0 < G_3 < \infty,$ such that

Since $f, \ (u_k, v_k)$ and $(u_0, v_0)$ are bounded on $J$ and $\big(C_\varphi \big(\varphi_u(w)+\varphi_v(w)\big)+L_{gk} (C_u+C_v)\psi\big) > 0.$ Thus (4.5) implies that $d(\Lambda (u_0, v_0), (u_0, v_0)) < \infty.$

By using Banach fixed point theorem, there exists a continuous function $u_0, v_0:J\rightarrow \mathbb{R}$ such that $\Lambda^n(u_0, v_0)\rightarrow (u_0, v_0)$ in $(V, d)$ as $n\rightarrow \infty$ and $\Lambda (u_0, v_0) = (u_0, v_0),$ that is, $u_0, \ v_0$ satisfies Eqs (4.1) and (4.2) for every $w\in J.$

Now we show that $\{u, v\in V \text{ such that } d((u_0, v_0), (u_0, v_0)) < \infty\} = V.$ For any $u, v\in V,$ since $u, v$ and $u_0, v_0$ are bounded on $J$ and $\min_{w\in J}\big(C_\varphi \big(\varphi_u(w)+\varphi_v(w)\big)+L_{gk} (C_u+C_v)\psi\big) > 0,$ there exists a constant $0 < C_{(u, v)} < \infty$ such that $|(u_0, v_0)(w)-(u, v)(w)|\leq C_{(u, v)}\big(C_\varphi \big(\varphi_u(w)+\varphi_v(w)\big)+L_{gk} (C_u+C_v)\psi\big),$ for any $w\in J.$ Hence, we have $d((u_0, v_0), (u, v)) < \infty$ for all $u, v\in V,$ that is $\{(u, v)\in V~~ \text{ such that } ~~d((u_0, v_0), (u, v)) < \infty\} = V$. Thus, we determine that $u, v$ are the unique continuous functions with the Eqs (4.1), and (4.2) respectively. From (3.2), (3.4) and $(H_4),$ we can write

Summarizing we have

This shows that (4.19) is true for $w\in J$.

Finally we give an example to illustrate our main result.

Example 4.4.

and

Let $J = [0, 3]$, $\mu = \nu = \frac{1}{2}$, $p_1 = p_2 = p = \frac{4}{3}$, $\eta_1 = \eta_2 = \eta = \frac{1}{4}$ and $0 = w_0 < s_0 = 1 < w_1 = 2 < s_1 = \tau = T = 3.$ Denote $f_1(w, u(w)) = f_2(w, v(w)) = \frac{|z(w)|}{8+e^w+w^2}$ with $L_{f_{u_1}} = L_{f_{u_2}} = = L_{f_{v_1}} = L_{f_{v_2}} = \frac{1}{4}$ for $w\in (0, 1]\cup(2, 3]$ and $g_k(w, u(w)) = \frac{u(w)}{(3+w^2)(1+|u(w)|)}$, $g_k(w, v(w)) = \frac{v(w)}{(3+w^2)(1+|v(w)|)}$ with $L_{g_k} = 1$ for $w\in(1, 2].$ Putting $Ł_{f} = \frac{1}{4},$ $\varphi_u(w) = \varphi_v(w) = e^w$ and $C_\varphi = C_u = C_v = 1,$ we have $\int_0^w I^{\frac{1}{2}}e^sds\leq e^w$ and $L_1\approx 0.1012$, $L_2\approx 0.9501$, so $L\approx0.9501 < 1.$

By Theorem 4.3, there exists a unique solution $(u, v):[0, 3]\rightarrow \mathbb{R}$ such that

putting maximum of $w = w_k = T = \eta = \tau$

Now putting the values, we get

Thus the problem (4.18) is Ulam-Hyers-Rassias stability.

5.   Conclusions

In this article, we considered switched coupled system of nonlinear impulsive Langevin equations with mixed derivatives and Some sufficient conditions are constructed to observe the existence, uniqueness and generalized Ulam-Hyers-Rassias stability. After introduction we built a uniform structure to originate a formula of solutions for our proposed model. We implemented the new concept of generalized Ulam-Hyers-Rassias stability to our proposed model, finally we solved a particular example for our proposed model.

Conflict of interest

The authors declare that they have no competing interest regarding this research work.