Abstract
In this paper, we study the global convergence of successive approximations as well as the uniqueness of the random solution of a coupled random Hilfer fractional differential system. We prove a theorem on the global convergence of successive approximations to the unique solution of our problem. In the last section, we present an illustrative example.
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1 Introduction
Fractional calculus began in 1695, it is a theory of integrals and derivatives with a real order, several mathematicians interested in it as Leibniz, Liouville, Riemann, Abel, ...(see [17]). Several areas are interested in fractional calculus examples: biology, mechanics, electricity, control theory, biophysics, and other applied sciences [7, 9, 13, 17, 18, 20]. Recently, considerable attention has been given to the existence of solutions of fractional differential equations with Hilfer fractional derivative [10, 11, 13, 18, 19], and the references therein.
Functional differential equations with random effects play a fundamental role in the theory of random dynamical systems [2, 4, 12]. Random operator theory is usually the case that the mathematical models used to describe phenomena in the biological, physical, engineering and systems sciences contain certain parameters or coefficients that have specific interpretations, but whose values are unknown.
Recently, several articles on global convergence of successive approximations as well as the uniqueness of solutions for fractional differential equations were made, we refer [3, 5], and the references therein.
In [6], the authors studied the existence of random solutions for a random coupled Hilfer and Hadamard fractional differential systems in generalized Banach spaces. In this paper we study the uniformly convergence of successive approximations for the coupled random Hilfer fractional differential system:
with the initial conditions:
where \(T>0,\ \alpha _{i} \in (0,1), \beta _{i} \in [0,1],(\Omega , \mathcal {A})\) is a measurable space, \(\gamma _{i}=\alpha _{i}+\beta _{i}-\alpha _{i} \beta _{i},\) \(\phi _{i}: \Omega \rightarrow {\mathbb {R}}^{m} \), \(f_{i}: I \times {\mathbb {R}}^{m} \times {\mathbb {R}}^{m} \times \Omega \rightarrow {\mathbb {R}}^{m} ; i=1,2,\) are given functions, \(I_{0}^{1-\gamma _{i}}\) is the left-sided mixed Riemann–Liouville integral of order \(1-\gamma _{i},\) and \(D_{0}^{\alpha _{i}, \beta _{i}}\) is the generalized Riemann–Liouville derivative (Hilfer) operator of order \(\alpha _{i}\) and type \(\beta _{i}: i=1,2 .\)
This paper initiates the study of system (1)–(2) using successive approximations.
2 Preliminaries
We denote by C the Banach space of all continuous functions from I into \({\mathbb {R}}^{m}\) with the supremum (uniform) norm \(\Vert \cdot \Vert _{\infty } .\) As usual, AC(I) denotes the space of absolutely continuous functions from I into \({\mathbb {R}}^{m} .\) By \(L^{1}(I),\) we denote the space of Lebesgue-integrable functions \(v: I \rightarrow {\mathbb {R}}^{m}\) with the norm:
By \(C_{\gamma }(I)\) and \(C_{\gamma }^{1}(I),\) we denote the weighted spaces of continuous functions defined by
with the norm:
and
with the norm:
In addition, by \(\mathcal {C}\):\(=C_{\gamma _{1}} \times C_{\gamma _{2}},\) we denote the product weighted space with the norm:
Now, we will give some necessary definitions of fractional calculus.
Definition 2.1
[1, 15] The left-sided mixed Riemann–Liouville integral of order \(l>0\) of a function \(v \in L^{1}(I)\) is defined by
where \(\Gamma (\cdot )\) is the Gamma function. For all \( l, l_{1}, l_{2}>0\) and each \(v \in C,\) we have \(I_{0}^{l} v \in C,\) and
Definition 2.2
[1, 15] The Riemann–Liouville fractional derivative of order \(l \in (0,1]\) of a function \(v \in L^{1}(I)\) is defined by:
Lemma 2.3
[16] Let \(l \in (0,1], \gamma \in [0,1)\) and \(v \in C_{1-\gamma }(I).\) Then
In addition, if \(I_{0}^{1-l} v \in C_{1-\gamma }^{1}(I),\) then
Definition 2.4
[1, 15] The Caputo fractional derivative of order \(l \in (0,1]\) of a function \(v \in L^{1}(I)\) is defined by
Definition 2.5
[13] (Hilfer derivative). Let \(\alpha \in (0,1), \beta \in [0,1], v \in L^{1}(I),\) and \(I_{0}^{(1-\alpha )(1-\beta )} v \in A C(I).\) The Hilfer fractional derivative of order \(\alpha \) and type \(\beta \) of w is defined as
Property 2.6
Let \(\alpha \in (0,1), \beta \in [0,1], \gamma =\alpha +\beta -\alpha \beta ,\) and \(v \in L^{1}(I).\)
-
1.
The operator \(\left( D_{0}^{\alpha , \beta } v\right) (t)\) can be written as
$$\begin{aligned} \left( D_{0}^{\alpha , \beta } v\right) (t)=\left( I_{0}^{\beta (1-\alpha )} \frac{\mathrm{{d}}}{\mathrm{{d}} t} I_{0}^{1-\gamma } w\right) (t) =\left( I_{0}^{\beta (1-\alpha )} D_{0}^{\gamma } v\right) (t) ; \text{ for } \text{ a.e. } t \in I. \end{aligned}$$ -
2.
If \(D_{0}^{\gamma } v\) exists and is in \(L^{1}(I),\) then
$$\begin{aligned} \left( I_{0}^{\alpha } D_{0}^{\alpha , \beta } v\right) (t)=\left( I_{0}^{\gamma } D_{0}^{\gamma } v\right) (t)=v(t)-\frac{I_{0}^{1-\gamma }\left( 0^{+}\right) }{\Gamma (\gamma )} t^{\gamma -1} ; \text{ for } \text{ a.e. } t \in I. \end{aligned}$$
Corollary 2.7
Let \(\chi \in C_{\gamma }(I) .\) Then, the Cauchy problem
has the unique solution
Let \( \beta _{{\mathbb {R}}^{m}}\) be the Borel \(\sigma \)-algebra. A mapping \(\xi : \Omega \rightarrow {\mathbb {R}}^{m}\) is said to be measurable if for any \(B \in \beta _{{\mathbb {R}}^{m}} ; \text{ one } \text{ has } \)
Definition 2.8
Let \({\mathcal {A}} \times \beta _{{\mathbb {R}}^{m}}\) be the direct product of the \(\sigma \) -algebras \(\mathcal {A}\) and \(\beta _{{\mathbb {R}}^{m}}\) those defined in \(\Omega \) and \({\mathbb {R}}^{m}\), respectively. A mapping \(T: \Omega \times {\mathbb {R}}^{m} \rightarrow {\mathbb {R}}^{m}\) is called jointly measurable if for any \(D \in \beta _{{\mathbb {R}}^{m}},\) one has
Definition 2.9
A function \(T: \Omega \times {\mathbb {R}}^{m} \rightarrow {\mathbb {R}}^{m}\) is called jointly measurable if \(T(\cdot , v)\) is measurable for all \(v \in {\mathbb {R}}^{m}\) and \(T(w, \cdot )\) is continuous for all: \(w \in \Omega .\)
A random operator is a mapping \(T: \Omega \times {\mathbb {R}}^{m} \rightarrow {\mathbb {R}}^{m}\), such that T(w, v) is measurable in w for all \(u \in {\mathbb {R}}^{m},\) and it expressed as \(T(w) v=T(w, v) ;\) we also say that T(w) is a random operator on \({\mathbb {R}}^{m}.\) The random operator T(w) on E is called continuous (resp. compact, totally bounded, and completely continuous) if T(w, v) is continuous (resp. compact, totally bounded, and completely continuous) in v for all \(w \in \Omega \) (for more details, see [14]).
Definition 2.10
[8] Let \({\mathcal P}(X)\) be the family of all nonempty subsets of X and D be a mapping from \(\Omega \) into \({\mathcal P}(X).\) A mapping \(T:\{(w, x): w \in \Omega , y \in D(w)\} \rightarrow X\) is called a random operator with stochastic domain D if D is measurable (i.e., for all closed \(N \subset X,\{w \in \Omega , D(w) \cap N \ne \varnothing \}\) is measurable), and for all open \(G \subset X\) and all \(x \in X,\{w \in \Omega : x \in D(w), T(w, x) \in G\}\) is measurable. T will be called continuous if every T(w) is continuous.
Definition 2.11
. A function \(h: I \times {\mathbb {R}}^{m} \times {\mathbb {R}}^{m}\times \Omega \rightarrow {\mathbb {R}}^{m}\) is called random Carathéodory if the following conditions are satisfied:
-
(i)
The map \((t, w) \rightarrow h(t, y, v, w)\) is jointly measurable for all \(y,v \in {\mathbb {R}}^{m};\) and
-
(ii)
The map \((y,v) \rightarrow h(t, y, v, w)\) is continuous for a.e. \(t \in I\) and \(w \in \Omega .\)
3 Successive approximations and uniqueness results
In this section, we will give the main result of the global convergence of approximations of the problem (1) and (2).
Definition 3.1
By a generalized solution of the problem (1) and (2) we mean coupled measurable functions \((u,v)\in C_{\gamma _{i}}\times C_{\gamma _{2}}\) that satisfies the system (1) on I and the system (2).
Set \(I_{\eta }:=[0,\eta T];\) for any \(\eta \in [0,1].\) Let us introduce the following hypotheses.
- \((H_1)\):
-
The functions \(f_{i}: I \times {\mathbb {R}}^{m} \times {\mathbb {R}}^{m} \times \Omega \rightarrow {\mathbb {R}}^{m} ; i=1,2,\) are random Carathéodory,
- \((H_2)\):
-
There exist a constant \(\rho >0\) and continuous functions \(g_{i}: I \times [0, \rho ]^{m}\times [0, \rho ]^{m}\times \Omega \rightarrow \mathbb {R}_{+};\) \( i=1,2,\) such that \(g_{i}(t, \cdot ,\cdot ,w)\) is nondecreasing for any \(w \in \Omega \) and each \(t \in I,\) and
$$\begin{aligned} \bigg \Vert f_{i}(t,u,v,w)-f_{i}(t,\overline{u},\overline{v},w)\bigg \Vert \le g_{i}(t,\Vert u- \overline{u}\Vert _{C_{\gamma _{1}}},\Vert v- \overline{v}\Vert _{C_{\gamma _{2}}},w);\ i=1,2. \end{aligned}$$(3)for any \(w \in \Omega \) and each \(t \in I,\) \(u,\overline{u} \in C_{\gamma _{1}},\) and \(v,\overline{v}\in C_{\gamma _{2}},\) such that \(\Vert u-\overline{u}\Vert _{C_{\gamma _{1}}} \le \rho ,\) and \(\Vert v-\overline{v}\Vert _{C_{\gamma _{2}}} \le \rho ,\)
- \((H_3)\):
-
\((V,W)\equiv (0,0)\) is the only coupled functions in \(\Omega \times C_{\gamma _{i}}(I_{\lambda },[0, \rho ])\times C_{\gamma _{2}}(I_{\lambda },[0, \rho ])\), respectively, satisfying the integral inequalities:
$$\begin{aligned} V(t,w)\le \frac{1}{\Gamma \left( \alpha _{1}\right) } \int _{0}^{\lambda T} g_{1}(s,V(s,w),W(s,w),w)(t-s)^{\alpha _{1}-1} \mathrm{{d}} s, \end{aligned}$$(4)and
$$\begin{aligned} W(t,w)\le \frac{1}{\Gamma \left( \alpha _{2}\right) } \int _{0}^{\lambda T} g_{2}(s,V(s,w),W(s,w),w)(t-s)^{\alpha _{2}-1} \mathrm{{d}}s, \end{aligned}$$(5)with \(\eta \le \lambda \le 1.\)
Remark 3.2
From (3), for any \(w \in \Omega \) and each \(t \in I,\) \(u \in C_{\gamma _{1}},\ v\in C_{\gamma _{2}},\) and \(i=1,2,\) we get
where
and
Define the operators \(L_{1}: \mathcal {C} \times \) \(\Omega \) \(\rightarrow \) \( C_{\gamma _{1}},\) and \(L_{2}: \mathcal {C} \times \) \(\Omega \) \( \rightarrow \) \(C_{\gamma _{2}}\) by
and
Consider the operator \(L: \mathcal {C} \times \) \(\Omega \) \( \rightarrow \mathcal {C}:\)
For any \(w\in \Omega ,\) we define the successive approximations of the problem (1) and (2) as follows:
Theorem 3.3
Assume that the hypotheses \((H_1)-(H_3)\) hold. Then the successive approximations \(((u_n)_{n\in {\mathbb {N}}}, (v_n)_{n\in \mathbb {N}})\) are well defined and converge uniformly on I to the unique random solution of problem (1) and (2).
Proof
From \((H_1)\) the successive approximations are well defined. Thus, there exist \(\theta _{1}, \theta _{2}>0\), such that \(\Vert u\Vert _{C_{\gamma _{1}}} \le \theta _{1},\) \(\Vert v\Vert _{C_{\gamma _{1}}} \le \theta _{2}.\) Next, for any \(w\in \Omega ,\) and each \(t_1,t_2\in I\) with \(t_{1}<t_{2},\) we have
Then, from Remark 3.2, we get
Thus,
In addition, we obtain that
Hence
So, the sequence \(\left\{ (u_{n}, v_{n}) ; n \in \mathbb {N}\right\} \) is equi-continuous on I, for any \(w\in \Omega .\)
Let
If \(\tau =1,\) then we have the global convergence of successive approximations. Suppose that \(\tau <1,\) then the sequence \(\left\{ (u_{n}, v_{n})\right\} \) converges uniformly on \(I_{\tau }.\) Since this sequence is equi-continuous, then it converges uniformly to a continuous function \((\tilde{u}(t),\tilde{v}(t)).\) If we prove that there exists \(\lambda \in (\tau ,1]\), such that \(\left\{ (u_{n}, v_{n})\right\} \) converges uniformly on \(I_{\lambda },\) for any \(w\in \Omega .\) This will yield a contradiction.
Put \(u(t,w)=\tilde{u}(t,w) \) and \(v(t,w)=\tilde{v}(t,w);\) for each \(t \in I_{\tau }\) and any \(w\in \Omega .\)
From \(\left( H_{3}\right) ,\) there exists a constant \(\rho >0\) and a function \(g_{i}: I \times [0, \rho ]^{m}\times [0, \rho ]^{m}\times \Omega \rightarrow \mathbb {R}_{+}\) satisfying inequality (3). In addition, there exist \(\lambda \in [\tau , 1]\) and \(n_{0} \in \mathbb {N},\) such that for all \(t \in I_{\lambda }\) and any \(w\in \Omega ,\) and \(n, m>n_{0},\) we have
For each \(t\in I_\lambda ,\) and any \(w\in \Omega ,\) we put
Since the sequence \((V_k(t,w),W_k(t,w))\) is non-increasing, it is convergent to a function (V(t, w), W(t, w)) for each \(t\in I_\lambda ,\) and any \(w\in \Omega .\) From the equi-continuity of \(\{(V_k(t,w),W_k(t,w))\}\) it follows that \(\displaystyle \lim \nolimits _{k\rightarrow \infty }V_k(t,w)=V(t,w)\) and \(\displaystyle \lim \nolimits _{k\rightarrow \infty }W_k(t,w)=W(t,w)\) uniformly on \(I_\lambda .\) Furthermore, for each \(t\in I_\lambda ,\) and any \(w\in \Omega ,\) and for \(n,m\ge k,\) we have
Thus, from (3), we get
Hence
By the Lebesgue dominated convergence theorem, we get
In addition, we find that
Then, from \((H_1)\) and \((H_3)\), we get \(V\equiv 0\) and \(W\equiv 0\) on \(I_\lambda \times \Omega ,\) which yields that \(\displaystyle \lim \nolimits _{k\rightarrow \infty }(V_k(t,w),W_k(t,w))=(0,0)\) uniformly on \(I_\lambda \times \Omega .\) Thus \(\{(u_k(t,w),v_k(t,w))\}_{k=1}^{\infty }\) is a Cauchy sequence on \(I_\lambda \times \Omega .\) Consequently \(\{(u_k(t,w),v_k(t,w))\}_{k=1}^{\infty }\) is uniformly convergent on \(I_\lambda \) which yields the contradiction.
Thus, \(\{(u_k(t,w),v_k(t,w))\}_{k=1}^{\infty }\) converges uniformly on I for any \(w\in \Omega \) to a continuous function \((u_{*}(t,w),v_{*}(t,w)).\) By the Lebesgue dominated convergence theorem, we get
and
for each \(t\in I.\) This yields that \((u_{*},v_{*})\) is a solution of the problem (1) and (2). \(\square \)
Finally, we show the uniqueness of solutions of the problem (1) and (2). Let \((u_1,v_1)\) and \((u_2,v_2)\) be two solutions. As above, put
and suppose that \(\tau <1.\) There exist a constant \(\rho >0\) and a comparison function \(g_{i}: I \times [0, \rho ]^{m}\times [0, \rho ]^{m}\times \Omega \rightarrow \mathbb {R}_{+}\) ; \( i=1,2,\) satisfying inequality (3). We choose \(\lambda \in (\sigma ,1)\), such that
and
Again, by \((H_1)\) and \((H_3)\), we get \(u_1-u_2\equiv 0\) and \(v_1-v_2\equiv 0\) on \(I_\lambda \times \Omega .\) This gives \(u_1=u_2\) and \(v_1=v_2\) on \(I_\lambda \times \Omega ,\) which yields a contradiction. Consequently, \(\tau =1\) and the solution of the problem (1) and (2) is unique.
4 An example
We equip the space \(\mathbb {R}_{-}^{*}\):\(=(-\infty , 0)\) with the usual \(\sigma \) -algebra consisting of Lebesgue measurable subsets of \(\mathbb {R}_{-}^{*}\). Consider the following random coupled Hilfer fractional differential system:
where
For each \(u,\ v,\ \overline{u},\ \overline{v} \in {\mathbb {R}},\ p\in \mathbb {N}^{*}\) and \(t\in [0,1]\) we have
In addition, we obtain
This means that condition (3) holds for \(t\in [0,1],\ \rho >0\) and the comparison functions \(g_i:[0,1]\times [0,\rho ]\rightarrow [0,\infty );\ i=1,2\) given by
Consequently, Theorem 3.3 implies that the successive approximations \((u_n,v_n);\ n\in \mathbb {N},\) defined by
where
and
converge uniformly on [0, 1] to the unique solution of the problem (6).
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Si Bachir, F., Abbas, S., Benbachir, M. et al. Successive approximations for random coupled Hilfer fractional differential systems. Arab. J. Math. 10, 301–310 (2021). https://doi.org/10.1007/s40065-021-00326-0
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DOI: https://doi.org/10.1007/s40065-021-00326-0