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Coupled Hilfer fractional differential systems with random effects
Advances in Difference Equations volume 2018, Article number: 369 (2018)
Abstract
This paper deals with some existence results for two classes of coupled systems of Hilfer and Hilfer–Hadamard random fractional differential equations. The main tool used to carry out our results is Itoh’s random fixed point theorem.
Introduction
Fractional differential equations have recently been applied in various areas of engineering, mathematics, physics and bioengineering, and other applied sciences [13, 23]. For some fundamental results in the theory of fractional calculus and fractional differential equations we refer the reader to the monographs of Abbas et al. [5, 6], Samko et al. [22], and Kilbas et al. [18], and a series of papers [2–4, 7, 15, 26, 27, 29–33] and the references cited therein. Coupled systems of Hadamard type sequential fractional differential equations were considered in [8, 9]. Recently, considerable attention has been given to the existence of solutions of initial and boundary value problems for fractional differential equations with Hilfer fractional derivative; see [1, 11–13, 16, 24, 28].
The nature of a dynamic system in engineering or natural sciences depends on the accuracy of the information we have concerning the parameters that describe that system. If the knowledge about a dynamic system is precise then a deterministic dynamical system arises. Unfortunately in most cases the available data for the description and evaluation of parameters of a dynamic system are inaccurate, imprecise or confusing. In other words, evaluation of parameters of a dynamical system is not without uncertainties. When our knowledge about the parameters of a dynamic system are of statistical nature, that is, the information is probabilistic, the common approach in mathematical modeling of such systems is the use of random differential equations or stochastic differential equations. Random differential equations, as natural extensions of deterministic ones, arise in many applications and have been investigated by many mathematicians. We refer the reader to Refs. [10, 19, 25].
In this paper we discuss the existence of solutions for the following coupled random Hilfer fractional differential system:
where \(I:=[0,T]\), \(T>0\), \(\alpha_{i}\in(0,1)\), \(\beta_{i}\in[0,1]\), \(\gamma _{i}=\alpha _{i}+\beta_{i}\alpha_{i}\beta_{i}\); \(i=1,2\); \((\Omega,\mathcal{A})\) is a measurable space, \(\phi_{i}:\Omega\to {\Bbb {R}}\) is a measurable function, \(f_{i}:I\times{\Bbb {R}}\times{\Bbb {R}}\times\Omega\to{\Bbb {R}}\) is a given function, \(I_{0}^{1\gamma_{i}}\) is the leftsided mixed Riemann–Liouville integral of order \(1\gamma_{i}\), and \(D_{0}^{\alpha_{i},\beta_{i}}\) is the Hilfer fractional derivative of order \(\alpha_{i}\) and type \(\beta_{i}\).
Next, we consider the following coupled system of random Hilfer–Hadamard fractional differential equations:
where \(T>1\), \(\alpha_{i}\in(0,1)\), \(\beta_{i}\in[0,1]\), \(\gamma_{i}=\alpha _{i}+\beta _{i}\alpha_{i}\beta_{i}\), \(\psi_{i}:\Omega\to{\Bbb {R}}\); \(i=1,2\) is a measurable function, \(g_{i}:[1,T]\times{\Bbb {R}}\times{\Bbb {R}}\times\Omega\to{\Bbb {R}}\) is a given function, \({}^{H}I_{1}^{1\gamma_{i}}\) is the leftsided mixed Hadamard integral of order \(1\gamma_{i}\), and \({}^{H}D_{1}^{\alpha_{i},\beta_{i}}\) is the Hilfer–Hadamard fractional derivative of order \(\alpha_{i}\) and type \(\beta_{i}\).
The present paper initiates the study of coupled systems for Hilfer fractional differential equations with random effects.
Preliminaries
Let C be the Banach space of all continuous functions v from I into \({\Bbb {R}}\) with the supremum (uniform) norm
As usual, \(AC(I)\) denotes the space of absolutely continuous functions from I into \({\Bbb {R}}\). We denote by \(AC^{1}(I)\) the space defined by
By \(L^{1}(I)\), we denote the space of Lebesgueintegrable functions \(v:I\rightarrow{\Bbb {R}}\) with the norm
Let \(L^{\infty}(I)\) be the Banach space of measurable functions \(u:I\to {\Bbb {R}}\) which are essentially bounded, equipped with the norm
By \(C_{\gamma}(I)\) and \(C^{1}_{\gamma}(I)\), we denote the weighted spaces of continuous functions defined by
with the norm
and with
with the norm
Also, by \({\mathcal {C}}:=C_{\gamma_{1}}\times C_{\gamma_{2}}\) we denote the product weighted space with the norm
Definition 2.1
A function \(T:\Omega\times{\Bbb {R}}\to {\Bbb {R}}\) is called jointly measurable if \(T(\cdot,u)\) is measurable for all \(u\in{\Bbb {R}}\) and \(T(w,\cdot)\) is continuous for all \(w\in\Omega\).
Definition 2.2
A function \(f:I\times{\Bbb {R}}\times\Omega\to {\Bbb {R}}\) is called random Carathéodory if the following conditions are satisfied:

(i)
The map \((t,w)\to f(t,u,w)\) is jointly measurable for all \(u\in{\Bbb {R}}\), and

(ii)
the map \(u\to f(t,u,w)\) is continuous for all \(t\in I\) and \(w\in\Omega\).
Let E be a Banach space and \(T:\Omega\times E\to E\) be a mapping. Then T is called a random operator if \(T(w,u)\) is measurable in w for all \(u\in E\) and it expressed as \(T(w)u=T(w,u)\). In this case we also say that \(T(w)\) is a random operator on E. A random operator \(T(w)\) on E is called continuous (resp. compact, totally bounded and completely continuous) if \(T(w,u)\) is continuous (resp. compact, totally bounded and completely continuous) in u for all \(w\in\Omega\). The details of completely continuous random operators in Banach spaces and their properties appear in Itoh [14].
Definition 2.3
Let \({\mathcal {P}}(Y)\) be the family of all nonempty subsets of Y and C be a mapping from ω into \({\mathcal {P}}(Y)\). A mapping \(T:\{(w,y):w\in\Omega, y\in C(w)\}\to Y\) is called random operator with stochastic domain C if C is measurable (i.e., for all closed \(A\subset Y\), \(\{w\in\Omega, C(w)\cap A\neq\emptyset\}\) is measurable) and for all open \(D \subset Y\) and all \(y\in Y\), \(\{w\in\Omega: y\in C(w), T(w,y)\in D\}\) is measurable. T will be called continuous if every \(T(w)\) is continuous. For a random operator T, a mapping \(y:\Omega\to Y\) is called random (stochastic) fixed point of T if for Palmost all \(w\in\Omega\), \(y(w)\in C(w)\) and \(T(w)y(w)=y(w)\) and for all open \(D\subset Y\), \(\{w\in\Omega: y(w)\in D\}\) is measurable.
Now, we give some results and properties of fractional calculus.
Definition 2.4
The leftsided mixed Riemann–Liouville integral of order \(r>0\) of a function \(w\in L^{1}(I)\) is defined by
where \(\Gamma(\cdot)\) is the (Euler’s) Gamma function defined by
Notice that, for all \(r,r_{1},r_{2}>0\) and each \(w\in C\), we have \(I_{0}^{r}w\in C\), and
Definition 2.5
The Riemann–Liouville fractional derivative of order \(r\in(0,1]\) of a function \(w\in L^{1}(I)\) is defined by
Let \(r\in(0,1]\), \(\gamma\in[0,1)\) and \(w\in C_{1\gamma}(I)\). Then the following expression leads to the leftinverse operator.
Moreover, if \(I_{0}^{1r}w\in C^{1}_{1\gamma}(I)\), then the following composition is proved in [22]:
Definition 2.6
The Caputo fractional derivative of order \(r\in(0,1]\) of a function \(w\in L^{1}(I)\) is defined by
In [13], Hilfer studied applications of a generalized fractional operator having the Riemann–Liouville and the Caputo derivatives as specific cases (see also [16, 24]).
Definition 2.7
(Hilfer derivative)
Let \(\alpha\in(0,1)\), \(\beta\in[0,1]\), \(w\in L^{1}(I)\), \(I_{0}^{(1\alpha )(1\beta)}\in AC^{1}(I)\). The Hilfer fractional derivative of order α and type β of w is defined as
Properties
Let \(\alpha\in(0,1)\), \(\beta\in[0,1]\), \(\gamma =\alpha +\beta\alpha\beta\), and \(w\in L^{1}(I)\).

1.
The operator \((D_{0}^{\alpha,\beta}w)(t)\) can be written as
$$\bigl(D_{0}^{\alpha,\beta}w \bigr) (t)= \biggl(I_{0}^{\beta(1\alpha)} \frac{d}{dt} I_{0}^{1\gamma}w \biggr) (t) = \bigl(I_{0}^{\beta(1\alpha)} D_{0}^{\gamma}w \bigr) (t); \quad \mbox{for a.e. } t\in I. $$Moreover, the parameter γ satisfies
$$\gamma\in(0,1],\qquad \gamma\geq\alpha,\qquad \gamma>\beta,\qquad 1\gamma < 1 \beta (1\alpha). $$ 
2.
The generalization (3) for \(\beta=0\), coincides with the Riemann–Liouville derivative and for \(\beta=1\) with the Caputo derivative.
$$D_{0}^{\alpha,0}=D_{0}^{\alpha}\quad \mbox{and} \quad D_{0}^{\alpha,1}= {}^{c}D_{0}^{\alpha}. $$ 
3.
If \(D_{0}^{\beta(1\alpha)}w\) exists and is in \(L^{1}(I)\), then
$$\bigl(D_{0}^{\alpha,\beta}I_{0}^{\alpha}w \bigr) (t)= \bigl(I_{0}^{\beta(1\alpha )}D_{0}^{\beta (1\alpha)}w \bigr) (t); \quad \mbox{for a.e. }t\in I. $$Furthermore, if \(w\in C_{\gamma}(I)\) and \(I_{0}^{1\beta(1\alpha )}w\in C^{1}_{\gamma}(I)\), then
$$\bigl(D_{0}^{\alpha,\beta}I_{0}^{\alpha}w \bigr) (t)=w(t); \quad \mbox{for a.e. }t\in I. $$ 
4.
If \(D_{0}^{\gamma}w\) exists and is in \(L^{1}(I)\), then
$$\bigl(I_{0}^{\alpha}D_{0}^{\alpha,\beta}w \bigr) (t)= \bigl(I_{0}^{\gamma}D_{0}^{\gamma}w \bigr) (t) =w(t)\frac{I_{0}^{1\gamma}(0^{+})}{\Gamma(\gamma)}t^{\gamma1};\quad \mbox{for a.e. }t\in I. $$
Corollary 2.8
Let \(h\in C_{\gamma}(I)\). Then the Cauchy problem
has a unique solution given by
From the above corollary, we conclude with the following lemma.
Lemma 2.9
Let \(f_{i}:I\times{\Bbb {R}}\times{\Bbb {R}}\times\Omega\rightarrow{\Bbb {R}}\); \(i=1,2\) be such that \(f(\cdot,u_{1}(\cdot,w),u_{2}(\cdot,w),w)\in C_{\gamma_{i}}\) for all \(w\in \Omega\), and any \(u_{i}(w)\in C_{\gamma_{i}}\). Then the coupled system (1) is equivalent to the problem of solutions of the following system of fractional integral equations:
We need the following Itoh random fixed point theorem; see [14].
Theorem 2.10
Let X be a nonempty, closed convex bounded subset of the separable Banach space E and let \(N:\Omega\times X\to X\) be a compact and continuous random operator. Then the random equation \(N(w)u=u\) has a random solution.
Coupled systems of Hilfer fractional random differential equations
In this section, we are concerned with the existence of solutions for the system (1). Let us start by defining what we mean by a random solution of the system (1).
Definition 3.1
By a random solution of the problem (1) we mean a coupled measurable function \((u_{1},u_{2}):\Omega\to C_{\gamma_{1}}\times C_{\gamma_{2}}\) that satisfies the conditions \((I_{0}^{1\gamma_{i}}u_{i})(0^{+},w)=\phi_{i}(w)\); \(i=1,2\), and the equations \((D_{0}^{\alpha_{i},\beta_{i}}u_{i})(t,w)=f_{i}(t,u_{1}(t,w),u_{2}(t,w),w)\); \(i=1,2\) on \(I\times\Omega\).
The following hypotheses will be used in the sequel.
 (H_{1}):

The functions \(f_{i}\); \(i=1,2\) are random Carathéodory on \(I\times{\Bbb {R}}\times{\Bbb {R}}\times\Omega\),
 (H_{2}):

there exist measurable and bounded functions \(p_{i},q_{i}:\Omega\to L^{\infty}(I,[0,\infty))\); \(i=1,2\), such that
$$\begin{aligned}& \bigl\vert f_{i}(t,u_{1},u_{2},w) \bigr\vert \leq\frac {p_{i}(t,w)u_{1}+q_{i}(t,w)u_{2}}{1+u_{1}+u_{2}}; \\& \quad \textit{for a.e. }t\in I, \textit{and each } u_{i}\in{\Bbb {R}}, w\in\Omega. \end{aligned}$$
Now, we shall prove the following theorem concerning the existence of random solutions of the system (1).
Theorem 3.2
Assume that the hypotheses (H_{1}) and (H_{2}) hold. Then the system (1) has at least one random solution defined on \(I\times\Omega\).
Proof
Define the following operators \(N_{i}:\Omega\times C_{\gamma _{i}}\rightarrow C_{\gamma_{i}}\); \(i=1,2\):
and consider the continuous operator \(N:\Omega\times{\mathcal {C}}\to {\mathcal {C}}\) defined by
Set
For each \(i=1,2\), the map \(\phi_{i}\) is measurable for all \(w\in\Omega\). Again, as the indefinite integral is continuous on I, then \(N_{i}(w)\) defines a mapping \(N_{i}:\Omega\times C_{\gamma_{i}}\to C_{\gamma_{i}}\). Thus \((u_{1},u_{2})\) is a random solution for the system (1) if and only if \((u_{1},u_{2})=N(w)(u_{1},u_{2})\).
Next, for any \(u_{i}\in C_{\gamma_{i}}\); \(i=1,2\), and each \(t\in I\) and \(w\in \omega\), we have
Thus
This proves that \(N(w)\) transforms the ball
into itself. We shall show that the operator \(N:\Omega\times B_{R}\to B_{R}\) satisfies all the assumptions of Theorem 2.10. The proof will be given in several steps.
Step 1. \(N(w)\) is a random operator on \(\Omega\times B_{R}\) into \(B_{R}\).
Since for each \(i=1,2\), \(f_{i}(t,u_{1},u_{2},w)\) is random Carathéodory, the map \(w\to f_{i}(t,u_{1},u_{2},w)\) is measurable in view of Definition 2.1. Similarly, the product \((ts)^{\alpha_{i}1}f_{i}(s,u_{1}(s,w), u_{2}(s,w),w)\) of a continuous and a measurable function is again measurable. Further, the integral is a limit of a finite sum of measurable functions, therefore, the map
is measurable. As a result, \(N(w)\) is a random operator on \(\Omega\times B_{R}\) into \(B_{R}\).
Step 2. \(N(w)\) is continuous.
Let \(\{(u_{1n},u_{2n})\}_{n\in {\mathbb {N}}}\) be a sequence such that \((u_{1n},u_{2n})\rightarrow(u_{1},u_{2})\) in \(B_{R}\). Then, for each \(i=1,2\), \(t\in I\), and \(w\in\Omega\), we have
Since \((u_{1n},u_{2n})\rightarrow(u_{1},u_{2})\) as \(n\rightarrow \infty \) and \(f_{i}\) is random Carathéodory, then, by the Lebesgue dominated convergence theorem, Eq. (7) implies
Step 3. \(N(w)B_{R}\) is uniformly bounded.
This is clear since \(N(w)B_{R}\subset B_{R}\) and \(B_{R}\) is bounded.
Step 4. \(N(w)B_{R}\) is equicontinuous.
Let \(t_{1},t_{2}\in I\), \(t_{1}< t_{2}\) and let \((u_{1},u_{2})\in B_{R}\). Then, for each \(i=1,2\), and \(w\in\Omega\), we have
Thus, we get
As \(t_{1}\longrightarrow t_{2}\), the righthand side of the above inequality tends to zero.
As a consequence of steps 1 to 4 together with the Arzelá–Ascoli theorem, we can conclude that \(N:\Omega\times B_{R}\to B_{R}\) is continuous and compact. Theorem 2.10 implies that the operator equation \(N(w)(u_{1},u_{2})=(u_{1},u_{2})\) has a random solution. This shows that the random system (1) has a random solution. □
Hilfer–Hadamard fractional random differential equations
Now, we are concerned with some existence results for the coupled system (2).
Set \(C:=C([1,T])\). Denote the weighted space of continuous functions defined by
with the norm
Let us recall some definitions and properties of Hadamard fractional integration and differentiation. We refer to [18] for a more detailed analysis.
Definition 4.1
(Hadamard fractional integral [18])
The Hadamard fractional integral of order \(q>0\) for a function \(g\in L^{1}([1,T])\), is defined as
provided the integral exists.
Example 4.2
Let \(0< q<1\). Then
Set
and
Analogous to the Riemann–Liouville fractional calculus, the Hadamard fractional derivative is defined in terms of the Hadamard fractional integral in the following way.
Definition 4.3
(Hadamard fractional derivative [18])
The Hadamard fractional derivative of order \(q>0\) applied to the function \(w\in AC_{\delta}^{n}\) is defined as
In particular, if \(q\in(0,1]\), then
Example 4.4
Let \(0< q<1\). Then
It has been proved (see e.g. Kilbas [17, Theorem 4.8]) that in the space \(L^{1}(I,{\Bbb {R}})\), the Hadamard fractional derivative is the leftinverse operator to the Hadamard fractional integral, i.e.
From Theorem 2.3 of [18], we have
Analogous to the Hadamard fractional calculus, the Caputo–Hadamard fractional derivative is defined in the following way.
Definition 4.5
(Caputo–Hadamard fractional derivative)
The Caputo–Hadamard fractional derivative of order \(q>0\) applied to the function \(w\in AC_{\delta}^{n}\) is defined as
In particular, if \(q\in(0,1]\), then
From the Hadamard fractional integral, the Hilfer–Hadamard fractional derivative (introduced for the first time in [20]) is defined in the following way.
Definition 4.6
(Hilfer–Hadamard fractional derivative)
Let \(\alpha\in(0,1)\), \(\beta\in[0,1]\), \(\gamma=\alpha+\beta\alpha \beta\), \(w\in L^{1}(I)\), and \({}^{H}I_{1}^{(1\alpha)(1\beta)}w\in AC^{1}(I)\). The Hilfer–Hadamard fractional derivative of order α and type β applied to the function w is defined as
This new fractional derivative (4.6) may be viewed as interpolating the Hadamard fractional derivative and the Caputo–Hadamard fractional derivative. Indeed for \(\beta=0\) this derivative reduces to the Hadamard fractional derivative and when \(\beta=1\), we recover the Caputo–Hadamard fractional derivative.
From Theorem 21 in [21], we concluded the following lemma.
Lemma 4.7
Let \(g_{i}:I\times{\Bbb {R}}\times{\Bbb {R}}\times\Omega\rightarrow{\Bbb {R}}\); \(i=1,2\) be such that \(g_{i}(\cdot,u_{1}(\cdot,w),u_{2}(\cdot,w),w)\in C_{\gamma_{i},\ln }([1,T])\) for any \(u_{i}(\cdot,w)\in C_{\gamma_{i},\ln}([1,T])\). Then problem (2) is equivalent to the following system of fractional integral equations:
Now we give (without proof) an existence result for the system (2). Let us introduce the following hypotheses.
 (\(\mathrm{H}'_{1}\)):

The functions \(g_{i}\); \(i=1,2\) are random Carathéodory on \([1,T]\times{\Bbb {R}}\times{\Bbb {R}}\times\Omega\),
 (\(\mathrm{H}'_{2}\)):

there exist measurable and bounded functions \(p_{i},q_{i}:\Omega\to L^{\infty}([1,T],[0,\infty))\), such that
$$\begin{aligned}& \bigl\vert g_{i}(t,u_{1},u_{2},w) \bigr\vert \leq\frac{p_{i}(t,w)u_{1}+q_{i}(t,w)u_{2}}{1+u_{1}+u_{2}}; \\& \quad \mbox{for a.e. }t\in[1,T], \mbox{and each }u_{i}\in{\Bbb {R}}, w \in\Omega. \end{aligned}$$
Theorem 4.8
Assume that the hypotheses (\(\mathrm{H}'_{1}\)) and (\(\mathrm{H}'_{2}\)) hold. Then the coupled system (2) has at least one random solution defined on \([1,T]\times\Omega\).
An example
Let \(\Omega=(\infty,0)\) be equipped with the usual σalgebra consisting of Lebesgue measurable subsets of \((\infty,0)\). As an application of our results we consider the following system of random Hilfer fractional differential equations:
where
and \(c=\frac{9\sqrt{\pi}}{16}\). Clearly, the functions f and g are random Carathéodory.
The hypothesis (H_{2}) is satisfied with
and \(q_{1}(t,w)=p_{2}(t,w)=0\), \(q_{2}(t,w)=\frac{ct^{\frac{1}{4}}}{64}\); \(t\in(0,1]\), \(w\in\Omega\). Hence, Theorem 3.2 implies that the coupled system (9) has at least one random solution defined on \([0,1]\times\Omega\).
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The work was supported by the National Natural Science Foundation of China (No. 11671339).
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Abbas, S., Benchohra, M. & Zhou, Y. Coupled Hilfer fractional differential systems with random effects. Adv Differ Equ 2018, 369 (2018). https://doi.org/10.1186/s1366201818323
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DOI: https://doi.org/10.1186/s1366201818323
MSC
 26A33
 45D05
 45G05
 45M10
Keywords
 Functional differential equation
 Leftsided mixed Riemann–Liouville integral of fractional order
 Leftsided mixed Hadamard integral of fractional order
 Hilfer fractional derivative
 Hilfer–Hadamard fractional derivative
 Coupled system
 Random solution
 Fixed point