Nonlocal coupled hybrid fractional system of mixed fractional derivatives via an extension of Darbo's theorem

1.   Introduction

In the past years, fractional differential equations and coupled systems of those equations have attracted a lot of regard from many researches as they have played a key role in many basic sciences such as chemistry, control theory, biology and other arenas ^[1,2,3]. In addition, boundary conditions of differential models are strongest tools to extend applications of those equations. In fact, differential equations of fractional models can be extended by creating different types of boundary conditions. Newly, many authors have studied various types of boundary conditions to obtain new results of differential models. The following hybrid differential equation was studied by Dhage and Lakshmikantham ^[4]:

in which $h$ and $\omega$ are continuous functions from $J\times \mathbb{R}$ into $\mathbb{R}\setminus\{0\}$ and $\mathbb{R}$, respectively. Based on the above work, the Caputo hybrid boundary value problem of the form:

was studied by Hilal and Kajouni ^[5] in which $0 < p < 1$, $h$ and $\omega$ are continuous functions from $J\times \mathbb{R}$ into $\mathbb{R}\setminus\{0\}$ and $\mathbb{R}$, respectively and $a_{1}, a_{2}, d$ are real constants with $a_{1}+a_{2}\neq 0$. For some more results on hybrid boundary value problems see ^[6,7,8,9] and references therein.

The fractional hybrid modeling is of great significance in different engineering fields, and it can be a unique idea for the future combined research between various applied sciences, for example see ^[10] in which a fractional hybrid modeling of a thermostat is simulated. For some recent results on hybrid fractional differential equations we refer to ^[11,12,13].

As in modern mathematics coupled fractional system have been applied to develop differential models of high complexity system, Ntouyas and Al-Sulami ^[14] have considered the following coupled system:

where $^{C}D^{\alpha_{1}}$ and $^{RL}D^{\alpha_{2}}$ indicate Caputo and Riemann-Liouville fractional derivitves of orders $\alpha_{1}$ and $\alpha_{2}$ respectively, $f_{1}, f_{2}: [0, L]\times {\mathbb R}\times {\mathbb R}\to {\mathbb R}$ are continuous functions, $I^q$ is the Riemann-Liouville fractional integral, $\lambda, \gamma\in {\mathbb R}$ and $\eta, \xi \in (0, L).$ They have applied Banach's fixed point theorem and Leray-Schauder alternative to obtain main results. Nonlocal boundary value problems involving mixed fractional derivatives have been considered in ^[15] and references cited therein.

Here, we combine mixed fractional derivatives and hybrid fractional differential equations. More precisely, in this paper the existence of solutions for the coupled hybrid system

is investigated supplemented with boundary conditions:

where $^{C}D^{\alpha_{1}},$ $^{RL}D^{\alpha_{2}}$ are the Caputo and Riemann-Liouville fractional derivatives of orders $\alpha_{1}\in (0, 1]$ and $\alpha_{2}\in (1, 2]$ respectively, $I^q$ is the Riemann-Liouville fractional integral, $f_{2}, g_{2}\in C(J\times {\mathbb R}\times {\mathbb R}, {\mathbb R}),$ $f_1, g_1\in C(J\times {\mathbb R}\times {\mathbb R}, {\mathbb R}\setminus\{0\}),$ $\theta, \gamma\in {\mathbb R}$ and $\eta, \xi \in (0, L).$ An existence result is obtained via a new extension of Darbo's theorem associated to measures of noncompactness.

Here we emphasize that the proposed coupled hybrid system includes:

● different orders of fractional derivatives ($\alpha_{1} \in (0, 1]$ and $\alpha_{2} \in (1, 2]$);

● two different kinds of fractional derivatives (Caputo and Riemann-Liouville);

● nonlocal type boundary conditions which contain both fractional derivatives and integrals.

We have organized the structure of the paper as follows. Section $2$ presents some main concepts which will be applied in the future. In the next section, we prove an existence result for the problem (1.1) and (1.2). Finally, we construct an example to illustrate the obtained result.

2.   Preliminaries

Now some basic notations are recalled from ^[2].

Definition 2.1. The Riemann-Liouville fractional integral of order $\alpha > 0$ of a continuous function $\xi:(0, \infty)\to {\mathbb R}$, is defined as

where $\Gamma(\alpha)$ is the Euler Gamma function.

Definition 2.2. For a continuous function $\xi:(0, \infty)\to {\mathbb R}$, the Reimann-Liouville fractional derivative of order $\alpha > 0$, $n-1 < \alpha < n$, $n\in \mathbb{N}$ is defined as:

Definition 2.3. Given a continuous function $\xi:[0, \infty)\longrightarrow {\mathbb R}$, the Caputo derivative of order $\alpha > 0$ is defined as

Now we present some basic facts about the notion measure of noncompactness.

Assume that $Z$ is the real Banach space with the norm $\|\cdot\|$ and zero element $\theta$. For a nonempty subset $X$ of $Z$, the closure and the closed convex hull of $X$ will be denoted by $\overline{X}$ and $Conv(X)$, respectively. Also, $M_{Z}$ and $N_{Z}$ denote the family of all nonempty and bounded subsets of $Z$ and its subfamily consisting of all relatively compact sets, respectively.

Definition 2.4. ^[16,19] We say that a mapping $h: M_{Z}\longrightarrow [0, \infty)$ is a measure of noncompactness, if the following conditions hold true:

(1) The family Ker$h = \{X\in M_{Z}: h(X) = 0\}$ is nonempty and Ker $h\subseteq N_{Z}$.

(2) $X_{1}\subseteq Y_{1}\Longrightarrow h(X_{1})\leq h(Y_{1}).$

(3) $h(\overline{X}) = h(X).$

(4) $h(Conv(X)) = h(X)$.

(5) $h(\alpha X + (1 - \alpha)Y)\leq \alpha h(X) + (1 - \alpha)h(Y)$ for $\alpha\in [0, 1]$.

(6) For the sequence $\{X_{n}\}$ of closed sets from $M_{Z}$ in which $X_{n+1}\subseteq X_{n}$ for $n = 1, 2, \dots$ and ${\lim_{n\rightarrow\infty}}h(X_{n}) = 0$, we have $\bigcap_{n = 1}^{\infty}X_{n}\neq \varnothing$.

In ^[17], some generalizations of Darbo's theorem have been proved by Samadi and Ghaemi. Also, in ^[18], Darbo's theorem was extended and the following result was presented which is basic for our main result.

Theorem 2.1. Let $T$ be a continuous self-mapping on the set $D$ where $D$ denotes a nonempty bounded, closed and convex subset of a Banach space $Z$. Assume that for all nonempty subset $X$ of $D$ we have

where $h$ is an arbitrary measure of noncompactness defined in $Z$ and $(\theta_{1}, \theta_{2})\in U.$ Then $T$ has a fixed point in $D$.

In Theorem 2.1, let $U$ indicate the set of all pairs $(\theta_{1}, \theta_{2})$ where the following conditions hold true:

($U_1$) $\theta_{1}(t_n)\nrightarrow 0$ for each strictly increasing sequence $\{t_n\};$

($U_2$) $\theta_{2}$ is strictly increasing function;

($U_3$) If $\{\alpha_n\}$ is a sequence of positive numbers, then $\lim_{n\to \infty}\alpha_{n} = 0$ $\longleftrightarrow$ $\lim_{n\to \infty}\theta_{2}(\alpha_n) = -\infty.$

($U_4$) Let $\{l_n\}$ be a decreasing sequence in which $l_n\to 0$ and $\theta_{1}(l_n) < \theta_{2}(l_n)-\theta_{2}(l_{n+1}),$ then we have $\sum_{n = 1}^{\infty}l_n < \infty.$

Next, the definition of a measure of noncompactness in the space $C([0, 1])$ is recalled which will be applied later. Fix $Y\in M_{C[0, 1]}$ and for $\varepsilon > 0$ and $y\in Y$ we define

Banas and Goebel ^[16] proved that $\varphi_{0}(Y)$ is a measure of noncompactness in the space $C([0, 1])$.

3.   Existence results

Now the coupled system (1.1) and (1.2) is investigated in the space $C([0, 1])$.

First following Lemma $2.5$ of ^[14], the following lemma is presented which will be applied later.

Lemma 3.1. Assume that the functions $\phi$ and $h$ are continuous real-valued functions on $[0, L]$. Then, the functions $u$ and $v$ satisfy the system

if and only if $u$ and $v$ satisfy the system

where $\Delta = L^{\alpha_{2}-1}+\theta\gamma\frac{\Gamma(\alpha_{2})\xi^p\eta^{\alpha_{2}-p-1}}{\Gamma(1+q)\Gamma(\alpha_{2}-p)}$. For convenience we set the notations:

Now we present the main result of this section as follows:

Theorem 3.1. Suppose that we have the following assumptions:

($\mathrm{D_{1})}$ The functions $f_{2}, g_{2}:I\times \mathbb{R}\times \mathbb{R}\longrightarrow \mathbb{R}$ are continuous provided that

where $l_{1}, l_{2}, k_{1}, k_{2}, \gamma_{1}, \gamma_{2}\geq 0$, $t\in I$ and $u, v, u_{1}, u_{2}, v_{1}, v_{2}\in \mathbb{R}$.

($\mathrm{D_{2})}$ The continuous functions $f_{1}, g_{1}$ have been defined from $I\times \mathbb{R}\times \mathbb{R}$ into $\mathbb{R}\setminus\{0\}$ provided that

where $d > 0$, $t, t_{1}, t_{2}\in [0, L]$ and $u, v, u_{1}, u_{2}, v_{1}, v_{2}\in \mathbb{R}$. Moreover, assume that $\overline{M} = \sup\big\{\vert f_{1}(t, 0, 0) \vert: t\in [0, L]\big\}$ and $\overline{N} = \sup\big\{\vert g_{1}(t, 0, 0) \vert:$ $t\in [0, L]\big\}.$

($\mathrm{D_{3})}$ There exists a positive solution $r_{0}$ of the following inequality:

exists. Moreover, assume that

Then the coupled hybrid fractional system (1.1) and (1.2) has a solution on $[0, L]$.

Proof. Define $G:C([0, L], \mathbb{R})\times C([0, L], \mathbb{R}) \longrightarrow C([0, L], \mathbb{R})\times C([0, L], \mathbb{R})$ by

where

with $\overline{f}(t) = f_{2}(t, u(t), v(t))$ and $\overline{g}(t) = g_{2}(t, u(t), v(t))$. Assume that the space $C([0, L], \mathbb{R})$ $\times C([0, L], \mathbb{R})$ has been equipped with the norm $\Vert (u, v) \Vert = \Vert u \Vert + \Vert v \Vert,$ where $\Vert u \Vert = \sup\{\vert u(t) \vert:\; t\in [0, L]\}.$ Define $D_{r_{0}} = \{u\in C([0, L], \mathbb{R}):\; \; \Vert u \Vert \leq r_{0}\}.$ First we show that $G(D_{r_{0}}\times D_{r_{0}})\subseteq D_{r_{0}}\times D_{r_{0}}$. Given $t\in [0, L]$ and $u, v\in D_{r_{0}}$, we earn

The above estimate yields that

Similarly we can prove that

Consequently, we have

Due to (3.9) and $\mathrm{(D_{3})}$ we derive that $G(D_{r_{0}}\times D_{r_{0}})\subseteq D_{r_{0}}\times D_{r_{0}}$. Now we verify the continuity property of $G$ on $D_{r_{0}}\times D_{r_{0}}$. Let $(x_{1}, y_{1}), (u_{1}, v_{1})\in D_{r_{0}}\times D_{r_{0}}$ and $\varepsilon > 0$ be arbitrarily such that $\Vert (x _{1}, y_{1})-(u_{1}, v_{1})\Vert < \frac{\varepsilon}{2}.$ Given $t\in [0, L]$ we get

Due to the above estimate we conclude that

Similarly, for $(x_{1}, y_{1}), (u_{1}, v_{1})\in D_{r_{0}}\times D_{r_{0}}$ and $t\in [0, L]$ we conclude that

Consequently, we get

In view of (3.10) and (3.11) we earn

where $E_{1}(\varepsilon), E_{2}(\varepsilon)\longrightarrow 0$. Hence $G$ is continuous on $D_{r_{0}} \times D_{r_{0}}$.

Next we show that the condition (2.1) of Theorem 2.1 is fulfilled. Let $X_{1}, X_{2}\subseteq D_{r_{0}}$ and $\overline{\varphi}(X) = \varphi_{0}(X_{1}) + \varphi_{0}(X_{2})$ where $X_{i}, i = 1, 2$ indicate the natural projection of $X$ into $C(I)$.

For convenience we put

Let $t_{1}, t_{2}\in [0, L]$ and $\varepsilon > 0$ be arbitrarily such that $\vert t_{2}-t_{1}\vert\leq \varepsilon$. Thus for $(x_{1}, y_{1})\in X_{1}\times X_{2}$ we get

Consequently, from assumption $(\mathrm{D_{3}})$ and the above estimate we conclude that

Hence we have

Similarly, we prove that

Combining (3.13) and (3.14) we conclude that

By taking logarithms, we earn

Hence we obtain all conditions of Theorem 2.1 with $\theta_{1}(t) = d$ and $\theta_{2}(t) = \ln(t)$. Consequently, from Theorem 2.1 a fixed point of the mapping $G$ is obtained which implies that the coupled system (1.1) and (1.2) has a solution on $[0, L]$ and the proof is completed.

Now an example is presented to show the applicability of the obtained result.

Example 3.1. We consider the following hybrid nonlocal system of mixed fractional derivatives:

Here, we have

The above system is a special case of the system (1.1) and (1.2). Now we show that the conditions of Theorem 3.1 are satisfied. Due to the definitions of $f_{2}$ and $g_{2}$, given $t\in [0, 1]$ and $u, v, u_{1}, u_{2}, v_{1}, v_{2}\in \mathbb{R}$ we have

Consequently, $f_{2}$ and $g_{2}$ satisfy condition $\mathrm({D_{1})}$ with $l_{1} = l_{2} = \frac{1}{100}$ and $k_{1} = k_{2} = \gamma_{1} = \gamma_{2} = \frac{1}{10000}.$ Besides, obviously the functions $f_{1}$ and $g_{1}$ satisfy the condition $\mathrm({D_{2})}$. Furthermore, $\overline{M} = \frac{e^{-d}}{6}$ and $\overline{N} = \frac{e^{-d}}{6}$. To verify condition $\mathrm({D_{3})}$, given that $M_{1}\approx 1.52$, $M_{2}\approx 0.58$, $M_{3}\approx 0.25$ and $M_{4}\approx 1.26$, so the existent inequality in $\mathrm({D_{3})}$ has the form

Obviously, the above inequality has a positive solution $r_{0}$, for example $r_{0} = e^{-d}$. Moreover, we have

Therefore all conditions of Theorem 3.1 are satisfied. Hence, by Theorem 3.1 the system (3.17) has a solution on $[0, 1]$.

4.   Conclusions

We have studied a nonlocal coupled hybrid fractional system consisting from one Caputo and one Riemann-Liouville fractional derivatives and nonlocal hybrid boundary conditions. An existence result is established via a new generalization of Darbo's fixed point theorem associated with measures of noncompactness. The obtained result is well illustrated by a numerical example. The result obtained in this paper is new and significantly contributes to the existing literature on the topic.

Acknowledgments

This research was funded by King Mongkut's University of Technology North Bangkok. Contract no. KMUTNB-61-KNOW-021.

Conflict of interest

The authors declare that they have no competing interests.