Abstract
This paper is concerned with the solvability of coupled nonlinear fractional differential equations of different orders supplemented with nonlocal coupled boundary conditions on an arbitrary domain. The tools of the fixed point theory are applied to obtain the criteria ensuring the existence and uniqueness of solutions of the problem at hand. Examples illustrating the main results are presented.
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1 Introduction
We introduce and study a new class of coupled systems of mixed-order fractional differential equations equipped with nonlocal multi-point coupled boundary conditions. In precise terms, we consider the following fully coupled system:
where \({}^{C}{D}^{\chi }\) is the Caputo fractional derivative of order \(\chi \in \{\xi ,\zeta \}\), \(\varphi ,\psi :[a,b]\times \mathbb{R} \times \mathbb{R}\rightarrow \mathbb{R}\) are given functions, and \(p_{i}, \in \mathbb{R}\), \(i=1,2,3\).
The tools of fractional calculus are found to be of great help in modeling several real-world problems appearing in scientific and technical disciplines. For examples and details, see financial economics [1], ecology [2], immune systems [3], chaotic synchronization [4, 5], etc. The widespread interest in this branch of mathematical analysis motivated many researchers to explore it further. In particular, the area of fractional order boundary value problems has been extensively studied. For some recent works on nonlocal nonlinear and integral boundary value problems involving different types of fractional differential equations, for instance, see [6–17]. On the other hand, fractional differential systems equipped with a variety of boundary conditions also received great attention in view of the occurrence of such systems in the mathematical modeling of several physical and engineering processes [18–20]. Concerning the theoretical development of these systems, one can find the details in the articles [21–31].
Recently, in [32], the authors studied a new class of coupled systems of mixed-order fractional differential equations equipped with nonlocal multi-point coupled boundary conditions of the form:
where \(D^{\chi } \) is the Caputo fractional derivative of order \(\chi \in \{\xi , \zeta \}\), \(\varphi , \psi : [a,b]\times {\mathbb{R}}\times {\mathbb{R}}\to { \mathbb{R}}\) are given functions \(p, q, \delta _{i}\in {\mathbb{R}}\), \(i=1,2,\ldots , m\). In [33], the existence and uniqueness of solutions for the following system were investigated by using the Leray–Schauder alternative and the contraction mapping principle:
where \({}^{c} {D}^{\chi }\) is the Caputo fractional derivative of order \(\chi \in \{\xi ,\zeta \}\), \(\varphi ,\psi :[a,b]\times \mathbb{R} \times \mathbb{R}\rightarrow \mathbb{R}\) are given functions, \(p,q, \delta _{i}, x_{0}, y_{0} \in \mathbb{R}\), \(i=1,2,\ldots , m\).
In the present research, inspired by the published articles [32] and [33], we consider a coupled system (1.1) consisting of fractional differential equations of two different fractional-orders: \((1,2]\) and \((2,3]\) on an arbitrary domain supplemented with a new set of coupled nonlocal multi-point boundary conditions. We emphasize that the present study is novel and more general, and contributes significantly to the existing literature on the topic. Moreover, several new results follow as special cases of the results presented in this work (see Sect. 5).
The rest of the paper is organized as follows: In Sect. 2 we recall some definitions and prove a basic lemma helping us to transform system (1.1) into equivalent integral equations. The main results are established in Sect. 3. An existence result is proved via the Leray–Schauder alternative, and the existence of a unique solution is established by using Banach’s contraction mapping principle. Examples illustrating the obtained results are also constructed in Sect. 4.
2 Preliminaries
Let us begin this section with some definitions related to our study [34].
Definition 2.1
The Riemann–Liouville fractional integral of order \(\omega \in \mathbb{R}\) (\(\omega >0\)) for a locally integrable real-valued function h defined on \(-\infty \leq a< t< b\leq +\infty \), denoted by \(I_{a^{+}} ^{\omega }h\), is defined by
where Γ denotes the Euler gamma function.
Definition 2.2
Let \(h, h^{(m)} \in L^{1}[a,b]\) for \(-\infty \leq a< t< b\leq +\infty \). The Riemann–Liouville fractional derivative \(D_{a^{+}} ^{\omega }h\) of order \(\omega \in (m-1, m]\), \(m\in \mathbb{N}\), is defined as
while the Caputo fractional derivative \({{}^{C}{D}_{a^{+}}^{\omega }h}\) of order \(\omega \in (m-1, m]\), \(m\in \mathbb{N}\), is defined as
Remark 2.3
The Caputo fractional derivative of order \(\omega \in (m-1, m]\), \(m\in \mathbb{N}\) for a continuous function \(h: (0,\infty )\to {\mathbb{R}}\) such that \(h\in C^{m}[a,b]\), existing almost everywhere on \([a, b]\), is defined by
Now we present an important result to analyze problem (1.1).
Lemma 2.4
Let \(\Phi ,\Psi \in C([a,b],\mathbb{R})\) and \(\Lambda \ne 0\). Then the unique solution of the system
is given by a pair of integral equations
where
Proof
The solution of system (2.1) can be written as
where \(c_{i} \in \mathbb{R}\) (\(i=1,2,\ldots ,5\)) are unknown constants. Using the condition \(x(a)=0\) in (2.5), we get \(c_{1}=0 \), while making use of the conditions \(y(\theta _{1})=0\), \(y(\theta _{2})=0\) in (2.6) leads to the equations
Using the conditions \(x(b)=p_{1} y(\theta _{3})\) and \(y(b)=p_{2} x(\theta _{3})\) with \(c_{1}=0 \) yields
Subtracting (2.8) from (2.7), we get
where \(a_{1}=(\theta _{2} - a)^{2} - (\theta _{1} - a)^{2}\). Inserting the value of \(c_{4}\) in (2.7), we find that
Substituting the values of \(c_{3}\) and \(c_{4}\) in (2.9) and (2.10), we obtain
Solving the above system, we get
Now substituting the value of \(c_{5}\) in (2.11) and (2.12), we find that
Finally, inserting the values of the constants \(c_{i}\), \(i=1,2,\ldots ,5\), into (2.5) and (2.6) yields equations (2.2) and (2.3). This completes the proof. We can prove the converse by direct computation. The proof is finished. □
3 Main results
Let \(X=C([a,b], \mathbb{R})\) be a Banach space endowed with the norm \(\Vert x\Vert =\sup \vert x(t)\vert \), \(t \in [a,b]\).
In view of Lemma 2.4, we define an operator \(T:X \times X \rightarrow X\) by
where
Here \((X \times X, \Vert (x,y)\Vert )\) is a Banach space equipped with the norm \(\Vert (x,y)\Vert = \Vert x\Vert +\Vert y\Vert \), \(x,y\in X\).
In our first result, we establish the existence of a solution for system (1.1) by applying the Leray–Schauder alternative [35].
Lemma 3.1
(Leray–Schauder alternative)
: Let \(\mathfrak{J}:{\mathcal{U}}\longrightarrow \mathcal{U} \) be a completely continuous operator (i.e., a map restricted to any bounded set in \(\mathcal{U}\) is compact). Let \(\mathcal{Q}(\mathfrak{J})=\{x\in \mathcal{U}:x=\eta \mathfrak{J}(x)\textit{ for some }0<\eta <1\}\). Then either the set \(\mathcal{Q}(\mathfrak{J})\) is unbounded or \(\mathfrak{J}\) has at least one fixed point.
For computational convenience, we set
where
Theorem 3.2
Let \(\Lambda \ne 0\) (Λ is defined by (2.4)). In addition, we assume that:
- \((H_{1})\):
-
\(\varphi ,\psi :[a,b] \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) are continuous functions and there exist real constants \(k_{i},\gamma _{i} \geq 0\) (\(i=1,2\)) and \(k_{0}>0\), \(\gamma _{0}>0\) such that, for all \(t\in [a,b]\) and \(x,y \in \mathbb{R}\),
$$\begin{aligned}& \bigl\vert \varphi (t,x,y) \bigr\vert \leq k_{0}+ k_{1} \vert x \vert + k_{2} \vert y \vert , \\& \bigl\vert \psi (t,x,y) \bigr\vert \leq \gamma _{0}+ \gamma _{1} \vert x \vert + \gamma _{2} \vert y \vert . \end{aligned}$$
Then system (1.1) has at least one solution on \([a,b]\) provided that
where \(L_{1}\), \(M_{1}\), \(L_{2}\), \(M_{2}\) are given in (3.1).
Proof
Observe that the continuity of the operator \(T: X \times X \rightarrow X \times X\) follows that of the functions φ and ψ. Next, let \(\Omega \subset X \times X\) be bounded such that
for positive constants \(K_{1}\) and \(K_{2}\). Then, for any \((x,y)\in \Omega \), we have
which implies that
In a similar way, in view of notation (3.1), we have
which yields
From the above argument, we deduce that the operator T is uniformly bounded, as
Next, we show that T is equicontinuous. Let \(t_{1},t_{2}\in [a,b]\) with \(t_{1}< t_{2}\). Then we have
Analogously, we can obtain
Clearly the right-hand sides of inequalities (3.3) and (3.4) tend to zero independently of x and y as \(t_{1}\to t_{2}\). This shows that the operator \(T(x,y) \) is equicontinuous. In consequence, we deduce that the operator \(T(x,y) \) is completely continuous.
Finally, we consider the set \(\mathcal{P} = \lbrace (x,y) \in X \times X:(x,y)= \nu T(x,y), 0 \leq \nu \leq 1\rbrace \) and show that it is bounded.
Let \((x,y) \in \mathcal{P} \) with \((x,y)= \nu T(x,y)\). For any \(t \in [a,b]\), we have \(x(t)= \nu T_{1}(x,y)(t)\), \(y(t)= \nu T_{2}(x,y)(t)\). Then, by \((H_{1})\), we have
and
In consequence of the above arguments, we deduce that
and
which imply that
Thus
where \(M_{0}=\min \lbrace 1-[(L_{1}+L_{2})k_{1}+(M_{1}+M_{2})\gamma _{1}],1-[(L_{1}+L_{2})k_{2}+(M_{1}+M_{2}) \gamma _{2}] \rbrace \). Hence the set \(\mathcal{P}\) is bounded. Thus, by the Leray–Schauder alternative, we deduce that the operator T has at least one fixed point, which corresponds to the fact that problem (1.1) has at least one solution on \([a,b]\). The proof is completed. □
In the next theorem we prove the existence of a unique solution of system (1.1) by using the contraction mapping principle due to Banach.
Theorem 3.3
Let \(\Lambda \ne 0\) (Λ is defined by (2.4)). In addition, we assume that:
- \((H_{2})\):
-
\(\varphi ,\psi :[a,b]\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}\) are continuous functions and there exist positive constants \(l_{1}\) and \(l_{2}\) such that, for all \(t\in [a,b]\) and \(x_{i},y_{i}\in \mathbb{R}\), \(i=1,2\), we have
$$\begin{aligned}& \bigl\vert \varphi (t, x_{1}, x_{2}) - \varphi (t, y_{1} , y_{2}) \bigr\vert \leq l_{1} \bigl( \vert x_{1}- y_{1} \vert + \vert x_{2} -y_{2} \vert \bigr), \\& \bigl\vert \psi (t,x_{1},x_{2})- \psi (t,y_{1},y_{2}) \bigr\vert \leq l_{2}\bigl( \vert x_{1}- y_{1} \vert + \vert x_{2}- y_{2} \vert \bigr). \end{aligned}$$
If
where \(L_{1}\), \(M_{1}\) and \(L_{2}\), \(M_{2}\) are given by (3.1), then system (1.1) has a unique solution on \([a,b]\).
Proof
Define \(\sup_{t\in [a,b]}\varphi (t,0,0)=N_{1}<\infty \), \(\sup_{t\in [a,b]}\psi (t,0,0)=N_{2}<\infty \), and \(r>0\) such that
In the first step, we show that \(T B_{r} \subset B_{r} \), where \(B_{r} = \lbrace (x,y)\in X \times X : \Vert (x,y) \Vert \leq r \rbrace \). By assumption \((H_{2})\), for \((x,y) \in B_{r} \), \(t \in [a,b] \), we have
Similarly, we get
Then, we obtain
Taking the norm, we get
Likewise, we can find that
Consequently,
Now, for \((x_{1},y_{1}),(x_{2},y_{2}) \in X \times X \) and for any \(t \in [a,b] \), we get
which implies that
Similarly, we find that
It follows from (3.9) and (3.10) that
From the above inequality and (3.8), we deduce that T is a contraction. Hence it follows by Banach’s fixed point theorem that there exists a unique fixed point for the operator T, which corresponds to a unique solution of problem (1.1) on \([a,b]\). This completes the proof. □
4 Examples
Let us consider the following mixed-type coupled fractional differential systems:
Here \(\xi = 3/2\), \(\zeta = 5/2\), \(\theta _{1}=11/5\), \(\theta _{2}=11/4\), \(\theta _{3}=14/5\), \(p_{1}=1/100 \), \(p_{2}=1/50\). With the given data, it is found that \(L_{1}\simeq 1.5045\), \(L_{2}\simeq 0.23941\), \(M_{1}\simeq 1.2806 \times 10^{-3}\), \(M_{2} \simeq 5.3193\).
(1) In order to illustrate Theorem 3.2, we take
It is easy to check that condition \((H_{1})\) is satisfied with \(k_{0}= \sqrt{28}\), \(k_{1}= 1/40\), \(k_{2}= 1/(5e)\), \(\gamma _{0}=1/\sqrt{5}\), \(\gamma _{1}= 1/(10e)\), \(\gamma _{2}= 1/100\).
Furthermore, \((L_{1} + L_{2})k_{1} + (M_{1} + M_{2})\gamma _{1}\simeq 0.20009<1\) and \((L_{1} + L_{2})k_{2} + (M_{1} + M_{2}) \gamma _{2} \simeq 0.18152<1\). Clearly, the hypotheses of Theorem 3.2 are satisfied, and hence the conclusion of Theorem 3.2 applies to problem (4.1) with φ and ψ given by (4.2).
(2) In order to illustrate Theorem 3.3, we take
which clearly satisfy condition \((H_{2})\) with \(l_{1}= 1/ 6 \) and \(l_{2}= 1/(5e)\).
Moreover, \((L_{1} +L_{2})l_{1} + (M_{1} + M_{2})l_{2} \simeq 0.6811<1\). Thus the hypotheses of Theorem 3.3 hold true, and consequently there exists a unique solution of problem (4.1) with φ and ψ given by (4.3) on \([2,3]\).
5 Conclusions
In this paper, we have studied the existence of solution for a boundary value problem consisting of a coupled system of nonlinear fractional differential equations of different orders and five-point nonlocal coupled boundary conditions on an arbitrary domain. The given problem is transformed into an equivalent fixed point problem, which is solved by applying the standard tools of the modern functional analysis to obtain the existence and uniqueness results for the original problem. Our results are not only new in the given setting, but also reduce to some new results as special cases by fixing the parameters involved in the boundary conditions. For example, if we take \(p_{1}=0=p_{2}\) in the obtained results, we get the ones associated with four-point nonlocal boundary conditions: \(x(a)=0\), \(x(b)=0\), \(y(\theta _{1})=0\), \(y(\theta _{2})=0\), \(y(b)=0\).
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Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. (KEP-PhD-41-130-41). The authors, therefore, acknowledge with thanks DSR technical and financial support.
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This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. (KEP-PhD-41-130-41).
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Each of the authors, AA, SH, BA, and SKN, contributed equally to each part of this work. All authors read and approved the final manuscript.
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Alsaedi, A., Hamdan, S., Ahmad, B. et al. Existence results for coupled nonlinear fractional differential equations of different orders with nonlocal coupled boundary conditions. J Inequal Appl 2021, 95 (2021). https://doi.org/10.1186/s13660-021-02636-5
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DOI: https://doi.org/10.1186/s13660-021-02636-5