Dynamics and stability for Katugampola random fractional differential equations

1.   Introduction

The history of fractional calculus dates back to the 17th century. So many mathematicians define the most used fractional derivatives, Riemann-Liouville in 1832, Hadamard in 1891 and Caputo in 1997 ^[24,28,34]. Fractional calculus plays a very important role in several fields such as physics, chemical technology, economics, biology; see ^[2,24] and the references therein. In 2011, Katugampola introduced a derivative that is a generalization of the Riemann-Liouville fractional operators and the fractional integral of Hadamard in a single form ^[21,22].

There are several articles dealing with different types of fractional operators; see ^{[1,3,9,10,11,12,13,16,32]}. Various results about existence of solutions as well as Ulam stability are provided in ^{[6,7,8,14,15,17,19,20,23,25,26,27,28,29,30,31,33]}. In this article we investigate the following class of Katugampola random fractional differential equation

with the terminal condition

where $x_T :\Omega\to E$ is a measurable function, $\varsigma\in(0, 1], \ T > 0, \ f:I\times E \times \Omega \to E, \ ^{\rho}D_{0}^{\varsigma}$ is the Katugampola operator of order $\varsigma,$ and $\Omega$ is the sample space in a probability space, and $(E, \|\cdot\|)$ is a Banach space.

2.   Preliminaries

By $C(I): = C(I, E)$ we denote the Banach space of all continuous functions $x:I\to E$ with the norm

and $L^{1}(I, E)$ denotes the Banach space of measurable function $x:I\to E$ with are Bochner integrable, equipped with the norm

Let $C_{\varsigma, \rho}(I)$ be the weighted space of continuous functions defined by

with the norm

Definition 2.1. ^[2]. The Riemann-Liouville fractional integral operator of the function $h\in L^{1}(I, E)$ of order $\varsigma\in \mathbb{R}_{+}$ is defined by

Definition 2.2. ^[2]. The Riemann-Liouville fractional operator of order $\varsigma \in \mathbb{R}_{+}$ is defined by

Definition 2.3. (Hadamard fractional integral)^[4]. The Hadamard fractional integral of order $r$ is defined as

provided that the left-hand side is well defined for almost every $\xi \in (0, T)$.

Definition 2.4. (Hadamard fractional derivative)^[4]. The Hadamard fractional derivative of order $r$ is defined as

provided that the left-hand side is well defined for almost every $\xi \in (0, T)$.

Definition 2.5. (Katugampola fractional integral)^[21]. The Katugampola fractional integrals of order $(\varsigma > 0)$ is defined by

for $\rho > 0$ and $\xi\in I,$ provided that the left-hand side is well defined for almost every $\xi \in (0, T)$.

Definition 2.6. (Katugampola fractional derivative)^[21]. The Katugampola fractional derivative of order $\varsigma > 0$ is defined by:

provided that the left-hand side is well defined for almost every $\xi \in (0, T)$.

We present in the following theorem some properties of Katugampola fractional integrals and derivatives.

Theorem 2.7. ^[21] Let $0 < Re(\varsigma) < 1$ and $0 < Re(\eta) < 1$ and $\rho > 0$, for $a > 0$:

● Index property:

● Linearity property:

and we have

Theorem 2.8. ^[21] Let $r$ be a complex number, $Re(r)\geq 0$, $n = [Re(r)]$ and $\rho > 0.$ Then, for $t > a;$

(1) $\lim_{\rho\to 1}(^{\rho}I^{r}_ah)(t) = \frac{1}{\Gamma(r)}\int^{t}_{a}(t-\tau)^{r-1}h(\tau)d\tau$.

(2) $\lim_{\rho\to 0^{+}}(^{\rho}I^{r}_ah)(t) = \frac{1}{\Gamma(r)}\int^{t}_{a}(\log\frac{t}{\tau})^{r-1}h(\tau)\frac{d\tau}{\tau}.$

(3) $\lim_{\rho\to1}(^{\rho}D^{r}_ah)(t) = (\frac{d}{dt})^n\frac{1}{\Gamma(n-r)}\int^{t}_{a}\frac{h(\tau)}{(t-\tau)^{r-n+1}}d\tau.$

(4) $\lim_{\rho\to 0^{+}}(^{\rho}D^{r}_ah)(t) = \frac{1}{\Gamma(n-r)}(t\frac{d}{dt})^n\int^{t}_{a}(\log\frac{t}{\tau})^{n-r-1}h(\tau)\frac{d\tau}{\tau}.$

Remark 2.9.

(1) $\lim_{\rho\to 1}(^{\rho}I^{r}_ah)(t) = (^{RL}I^{r}_ah)(t)$.

(2) $\lim_{\rho\to 0^{+}}(^{\rho}I^{r}_ah)(t) = (^{H}I^{r}_ah)(t)$.

(3) $\lim_{\rho\to1}(^{\rho}D^{r}_ah)(t) = (^{RL}D^{r}_ah)(t)$.

(4) $\lim_{\rho\to 0^{+}}(^{\rho}D^{r}_ah)(t) = (^{H}D^{r}_ah)(t)$.

Lemma 2.10. Let $0 < r < 1.$ The fractional equation $(^{\rho}D^{r}_0 v)(t) = 0,$ has as solution

with $c\in{\Bbb R}.$

Lemma 2.11. Let $0 < r < 1.$ Then

Proof. We have

Thus, $I_0^rD_0^ru(t) = I_1+I_2,$ with

and

Hence, we get

and

Finally we obtain

Lemma 2.12. The problem

has the following solution

where

Proof. Solving the equation

we get

From the condition, we get

hence, we obtain (2.4).

Definition 2.13. By a random solution of problem (1.1) and (1.2), we mean a measurable function $x(w, \cdot)\in C_{\varsigma, \rho}(I)$ which satisfies (1.1) and (1.2).

Lemma 2.14. $u$ is a random solution of (1.1) and (1.2), if and only if it satisfies

where

Lemma 2.15. ^[4,13] Let $T:\Omega\times E \to E$ be a mapping such that $T(\cdot, v)$ is measurable for all $v\subset E,$ and $T(w, \cdot)$ is continuous for all $w\subset\Omega$. Then the map $(w, v)\to T(w, v)$ is jointly measurable.

Definition 2.16. A function $f:I\times E \times\Omega\to E$ is called random Carathéodory if the following conditions are satisfied:

● (i) The map $(s, w)\to f(s, x, w)$ is jointly measurable for all $x\in E$, and

● (ii) The map $x\to f(s, x, w)$ is continuous for almost all $s\in I$ and $w\subset\Omega.$

Let $\epsilon > 0$ and $\Phi:\Omega\times I\to{\Bbb R}_+$ be a jointly measurable function. We consider the following inequality

Definition 2.17. ^[5] The problem (1.1) and (1.2) is generalized Ulam-Hyers-Rassias stable with respect to $\Phi$ if there exists $c_{f, \phi} > 0$ such that for each solution $x(\cdot, w)\in C_{\varsigma, \rho}(I)$ of the inequality (2.6), there exists $y(\cdot, w)\in C_{\varsigma, \rho}(I)$ satisfies (1.1) and (1.2) with

Theorem 2.18. ^[18] Let $X$ be a nonempty, closed convex bounded subset of the separable Banach space $E$ and let $G:\Omega\times X\to X$ be a compact and continuous random operator. Then the random equation $G(w)u = u$ has a random solution.

3.   Existence and Ulam stability results

We shall make use of the following hypotheses:

$(H_1)$ $f$ is a random Carathéodory function.

$(H_2)$ There exist measurable and essentially bounded functions $l_{i}:\Omega\to C(I); \ i = 1, 2$ such that

for all $x\in E$ and $t\in I$ with

Theorem 3.1. If $(H_{1})$ and $(H_{2})$ hold, and

then there exists a random solution for (1.1) and (1.2).

Proof. Let $N:\Omega\times C_{\varsigma, \rho}(I)\to C_{\varsigma, \rho}(I)$ be the operator defined by

and set

and define the ball

For any $w\in\Omega$ and each $t\in I,$ we have

Thus

Hence $N(w)(B_R)\subset B_R.$ We shall prove that $N:\Omega\times B_{R}\to B_{R}$ satisfies the assumptions of Theorem 2.18.

Step 1. $N(w)$ is a random operator.

From $(H_1),$ the map $w\longrightarrow f(t, x, w)$ is measurable and further the integral is a limit of a finite sum of measurable functions therefore the map

is measurable.

Step 2. $N(w)$ is continuous.

Consider the sequence $({x_{n}})_n$ such that $x_{n}\rightarrow u$ in $C_{\varsigma, \rho}.$

Set

Then

By $(H_1)$ we obtain

Consequently, $N(w):B_{R}\subset B_{R}$ is continuous.

Step 3. $N(w)B_R$ is equicontinuous. For $1\leq t_1\leq t_2\leq T,$ and $x\in B_R,$ we have

Arzelá-Ascoli theorem implies that $N:\Omega\times B_{R}\to B_{R}$ is continuous and compact. Hence; from Theorem 2.18, we deduce the existence of random solution to problem (1.1) and (1.2).

Now, we prove a result concerning the generalized Ulam-Hyers-Rassias stability of (1.1) and (1.2).

We introduce the following additional hypotheses:

$(H_3)$ For any $w\in\Omega,$ $\Phi(t, \cdot)\subset L^{1}[0, \infty),$ and there exists a measurable and essentially bounded function $q:\Omega\to C(I, [0, \infty));$ such that

$(H_4)$ There exists $\lambda_{\Phi} > 0$ such that

Remark 3.2. Hypothesis $(H_3)$ implies $(H_2)$ with

Set

Theorem 3.3. If $(H_{1}), \ (H_{3}), \ (H_{4})$ and

hold. Then the problem (1.1) and (1.2) has random solutions defined on $I,$ and it is generalized Ulam-Hyers-Rassias stable.

Proof. From $(H_{1}), \ (H_{3})$ and Remark 3.2; the problem (1.1) and (1.2) has at least one random solution $y.$ Then, we have

Assume $x$ be a random solution of (2.6). We obtain

From hypotheses $(H_3)$ and $(H_4),$ we have

Thus, we get

Hence, problem (1.1) and (1.2) is generalized Ulam-Hyers-Rassias stable.

4.   An example

Let $\Omega = (-\infty, 0)$ be equipped with the usual $\sigma$-algebra consisting of Lebesgue measurable subsets of $(-\infty, 0),$ and let

be the Banach space with the norm

Consider the Katugampola random fractional differential equation

with the terminal condition

with $x = (x_{1}, x_{2}, \ldots, x_{n}, \ldots), \ f = (f_{1}, f_{2}, \ldots, f_{n}, \ldots),$

and

We have

Hence, hypotheses $(H_3)$ and $(H_4)$ are satisfied with

Hence by theorems 3.1 and 3.3, problem (4.1) and (4.2) admits a random solution, and is generalized Ulam-Hyers-Rassias stable.

5.   Conclusions

In this paper, we provided some sufficient conditions ensuring the existence of random solutions and the Ulam stability for a class of fractional differential equations involving the Katugampola fractional derivative in Banach spaces. The techniques used are the random fixed point theory and the notion of Ulam-Hyers-Rassias stability.

Acknowledgments

The work of Juan J. Nieto has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain, co-financed by the European Fund for Regional Development (FEDER) corresponding to the 2014-2020 multiyear financial framework, project MTM2016-75140-P and by Xunta de Galicia under grant ED431C 2019/02.

Conflict of interest

The authors declare no conflict of interests.