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Analysis of a coupled system of fractional differential equations with non-separated boundary conditions
Advances in Difference Equations volume 2020, Article number: 590 (2020)
Abstract
Solutions to fractional differential equations is an emerging part of current research, since such equations appear in different applied fields. A study of existence, uniqueness, and stability of solutions to a coupled system of fractional differential equations with non-separated boundary conditions is the main target of this paper. The existence and uniqueness results are obtained by employing the Leray–Schauder fixed point theorem and the Banach contraction principle. Additionally, we examine different types of stabilities in the sense of Ulam–Hyers such as Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. To prove the effectiveness of our main results, we study a few interesting examples.
1 Introduction
In the last few decades, the theory of fractional differential equations (FDEs) has performed a significant role in a new branch of applied mathematics. Many researchers addressed FDEs for various models because of the fact that FDEs are considered to be more applicable and realistic as compared to integer order or classical differential equations. Fractional order differential and integral equations, which constitute a coupled system, became an important field of research in view of their nonlocal nature and applications in many real-world problems like anomalous diffusion [30], disease models [8, 9, 22], synchronization of chaotic system, etc. [10, 45]. We refer the reader to a series of papers [1, 5, 12, 15, 16, 19, 28, 31, 33–35, 40] for the theoretical works on coupled systems of FDEs and classical differential equations.
In the area of mathematical analysis Ulam [32] stability of functional equations is one of the essential subjects. The verdict of this type of stability plays a key role concerning this subject. When Hyers [13] and Rassias [24] generalized this stability to Banach spaces, then a number of mathematicians spread the idea of stability to different classes of differential equations. Obloza [20, 21] proved for the first time Ulam–Hyers–Rassias stability of first order linear differential equations, after that many researchers furnished this idea with different results [7, 17, 25, 26, 37, 39, 41–44].
Nowadays, the investigation of initial and boundary value problems has attracted the attention of many mathematicians. Particularly, the existence, uniqueness, and stability properties of coupled systems supplemented with boundary conditions have grown to be one of the central interest areas in mathematical analysis, see [2–4, 36, 38].
Li et al. [18] investigated the existence and uniqueness of solutions to the following FDEs system with non-separated boundary conditions:
where \(f \in ([0,T]\times \mathbb{R})\), \(^{c}D^{\alpha }\) represents the Caputo fractional derivative of order α and \(a_{i}\), \(b_{i}\), \(c_{i}\), for \(i=1,2\), are real constants with \(a_{1}+b_{1}\neq 0\) and \(b_{2}\neq 0\). Alsulami et al. [6] studied fractional order coupled systems with non-separated coupled boundary conditions:
where \(t \in [0,T]\), \(a,b \in (1,2]\), \(^{c}D^{a}\), \(^{c}D^{b}\) denote the Caputo fractional derivatives of order a and b, respectively, \(u,v:[0,T]\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}\) and \(\lambda _{j},\mu _{j},j=1,2\), are real constants.
Recently, Rao and Alesemi [23] investigated the existence and uniqueness of solutions for a coupled system of fractional differential equations with fractional non-separated coupled boundary conditions. As far as we know, the Ulam–Hyers stability analysis for the solutions of nonlinear coupled FDEs with non-separated coupled boundary conditions has been rarely investigated.
Motivated by the mentioned work, in this paper we study the existence, uniqueness, and stability results to the following nonlinear coupled FDEs:
with non-separated coupled boundary conditions
where the symbols \({}^{c}D^{m}_{0^{+}}\) and \({}^{c}D^{n}_{0^{+}}\) stand for Caputo fractional derivatives, \(m, n \in (2,3]\), \(\alpha _{i}\), \(\beta _{i}\), \(i=1,2,3 \), are real constants, and \(k,l \in C([0,1]\times \mathbb{R}_{+}\times \mathbb{R}_{+},\mathbb{R})\).
We can say that model (1.1)–(1.2) is of quite general and flexible nature, because the parameters involved in the problem cover a wide range of important cases. Some of them are explained here. If we choose \(\alpha _{1},\beta _{1}=0\) and \(t \in [0,T]\), then the results correspond to a problem with coupled flux type boundary conditions. Next, if we set \(t\in [0,T]\), \(\alpha _{1}=\alpha _{2}=\alpha _{3}=1\), and \(\beta _{1}=\beta _{2}=\beta _{3}=-1\) or vice versa, then the results correspond to nonlinear coupled fractional differential equations with coupled periodic and anti-periodic boundary conditions of the form: \(\varpi (0)=w(T)\), \(\varpi '(0)=w'(T)\), \(\varpi ''(0)=w''(T)\), \(w(0)=- \varpi (T)\), \(w'(0)=-\varpi '(T)\), \(w''(0)=-\varpi ''(T)\) or \(\varpi (0)=-w(T)\), \(\varpi '(0)=-w'(T)\), \(\varpi ''(0)=-w''(T)\), \(w(0) =\varpi (T)\), \(w'(0)=\varpi '(T)\), \(w''(0)=\varpi ''(T)\).
The remaining part of the paper is designed in the following way: In Section 2, we recall a few important definitions and lemmas from fractional calculus used throughout this article. In Section 3, we present the application of some standard fixed point approaches already mentioned in the abstract, through which the existence and uniqueness results for (1.1)–(1.2) are obtained. Ulam–Hyers stability results are established in Section 4. In Section 5, applications of the main results are provided, while in the final section, we present the conclusion of the paper.
2 Preliminaries
In this part we state some important lemmas and definitions about fractional derivatives and fractional integrals taken from [46].
Definition 2.1
(Riemann–Liouville fractional integral)
The Riemann–Liouville fractional integral of a function \(f:[0,\infty )\rightarrow \mathbb{R}\) of order \(\alpha >0\) is defined by
if the integral on the right-hand side exists, where Γ denotes the Euler gamma function.
Definition 2.2
(Riemann–Liouville fractional derivative)
The Riemann–Liouville fractional derivative of a function \(f:[0,\infty )\rightarrow \mathbb{R}\) of order \(\alpha >0\), \(n-1< \alpha <n\) is defined by
Definition 2.3
(Caputo fractional derivative)
The Caputo fractional derivative of a function \(f:[0,\infty )\rightarrow \mathbb{R}\) of order \(\alpha >0\), \(n-1< \alpha <n\) is given by
where \([\alpha ]\) is the integer part of α.
Theorem 2.4
([29], Banach contraction theorem)
Consider a Banach space \(Y\neq \emptyset \), and let a map \(\Omega :Y\rightarrow Y\) be a contraction on Y. Then Ω has precisely one fixed point.
Lemma 2.5
([14], Method of undetermined coefficients)
Let \(\alpha >0\), \(h(t) \in C([0,1],\mathbb{R})\), then the homogeneous fractional differential equation
has a solution of the form
where \(a_{i} \in \mathbb{R}\), \(i=0,1,\ldots , n-1\), and \(n = [\alpha ] + 1\).
Lemma 2.6
Let \(\theta , \tau \in C([0,1],\mathbb{R})\). Then the solution of the following boundary value problem
enriched with the boundary conditions (1.2) is given by
and
where
\(\varrho =(1-\alpha _{2}\beta _{2})(1-\alpha _{3}\beta _{3})\), \(\varrho ^{\ast }=\varrho (1-\alpha _{1}\beta _{1})\), \(z=(1-\alpha _{1}\beta _{1})(1-\alpha _{2}\beta _{2})\), \(\tilde{z}=(1-\alpha _{1}\beta _{1})(1-\alpha _{3}\beta _{3})\).
Proof
We know by the method of undetermined coefficients that the general solution of (2.1) can be drafted as follows:
where \(a_{i},b_{i}\in \mathbb{R}\), \(i=0,1,2\).
Using the boundary conditions \(\varpi (0)=\alpha _{1}w(1)\) and \(w(0)=\beta _{1}\varpi (1)\), we get
By applying the conditions \(\varpi '(0)=\alpha _{2}w'(\zeta )\), \(w'(0)=\beta _{2}\varpi '(\zeta )\) and using (2.4) and (2.5), we obtain
and
In view of \(\varpi ''(0)=\alpha _{3}w''(\eta )\) and \(w''(0)=\beta _{3}\varpi ''(\eta )\) along with (2.4) and (2.5), we have
By solving the last two equations, we get
Substituting the values of \(a_{2}\) and \(b_{2}\) in (2.7) and (2.8), we get
and
By substituting the values of \(b_{0}\), \(b_{1}\), and \(b_{2}\) in (2.6), we get
After some calculations, we get (2.2). By following the same steps, we obtain
With the help of \(b_{0}\), \(b_{1}\), and \(b_{2}\), we get (2.3). □
3 Main results
Let \(\mathbb{Y}=\{\varpi (t): \varpi \in C([0,1],\mathbb{R})\}\) denote the Banach space of all continuous functions from the interval \([0,1]\) into \(\mathbb{R}\) supplied by the norm \(\|\varpi \|=\sup \{|\varpi (t)| : t \in [0,1]\}\). Definitely, the product space \(\mathbb{X}=\mathbb{Y}\times \mathbb{Y}, \|(., .)\|\) is a Banach space equipped with the norm \(\|(\varpi ,w)\|=\|\varpi \|+\|w\|\).
Next, in view of Lemma 3.2, we define an operator \(\Upsilon :\mathbb{X}\rightarrow \mathbb{X}\) by
where
Here,
For convenience, we use the subsequent symbols:
and
Now we present our main results.
Theorem 3.1
Suppose that:
- \((\mathrm{E}_{1})\):
-
\(k,l:[0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}\) are continuous such that, for any \(y_{1},y_{2},z_{1},z_{2}\in \mathbb{R}\), there exist constants \(\wp _{1},\wp _{2}>0\) satisfying
$$\begin{aligned}& \bigl\vert k(t,y_{1},z_{1})-k(t,y_{2},z_{2}) \bigr\vert \leq \wp _{1}\bigl( \vert y_{1}-y_{2} \vert + \vert z_{1}-z_{2} \vert \bigr) \leq \wp _{1} \bigl\vert (y_{1},z_{1})-(y_{2},z_{2}) \bigr\vert , \\& \bigl\vert l(t,y_{1},z_{1})-l(t,y_{2},z_{2}) \bigr\vert \leq \wp _{2}\bigl( \vert y_{1}-y_{2} \vert + \vert z_{1}-z_{2} \vert \bigr) \leq \wp _{2} \bigl\vert (y_{1},z_{1})-(y_{2},z_{2}) \bigr\vert ; \end{aligned}$$ - \((\mathrm{E}_{2})\):
-
The linear operator \(D:\mathbb{R}\rightarrow \mathbb{R}\) is continuous such that, for each \(\varpi ,w ,u,v\in \mathbb{R}\), we can find constants \(0< l_{1},l_{2}<1\) satisfying
$$\begin{aligned}& \bigl\vert D \varpi (t)-D(u) \bigr\vert \leq l_{1} \bigl\vert \varpi (t)-u(t) \bigr\vert , \\& \bigl\vert D w(t)-D(v) \bigr\vert \leq l_{2} \bigl\vert w(t)-v(t) \bigr\vert . \end{aligned}$$
If \((\omega _{1}+\omega _{3})\wp _{1}+(\omega _{2}+\omega _{4})\wp _{2}<1\), then system (1.1)–(1.2) has a unique solution on the interval \([0,1]\).
Proof
Define \(\sup_{t \in [0,1]} k(t,0,0)=\delta _{1}<\infty \) and \(\sup_{t \in [0,1]} l(t,0,0)=\delta _{2}<\infty \) and \(q>0\), where
Define \(T_{q}=\{(\varpi ,w)\in \mathbb{X}: \|(\varpi ,w)\leq q\|\}\), we will show that \(\Upsilon (T_{q})\subset T_{q}\).
By assumptions \(\mathrm{E}_{1}\) and \(\mathrm{E}_{2}\) for each \((\varpi ,w) \in T_{q}\) and every \(t\in [0,1]\), we have
Following the same procedure, we obtain
which further yields
Thus,
Likewise,
Since
Therefore, combining inequalities (3.1) and (3.2) into (3.3), we obtain
Hence, ϒ maps a bounded subset of \(T_{q}\) into a bounded subset of \(T_{q}\).
Next, for any \((\varpi _{2},w_{2}), (\varpi _{1},w_{1}) \in \mathbb{X}\) and each \(t \in [0,1]\), we have
Employing assumptions \(\mathrm{E}_{1}\) and \(\mathrm{E}_{2}\), we obtain
The condition \(0< l_{1},l_{2}<1\) will lead us to
Similarly,
From inequalities (3.4) and (3.5), it follows that
As \([(\omega _{1}+\omega _{3})\wp _{1}+(\omega _{2}+\omega _{4})\wp _{2}]<1\). Therefore, ϒ is a contractive operator. By the Banach contraction principle, we deduce that the operator ϒ has a unique fixed point which is the unique solution of problem (1.1)–(1.2). □
The coming result is established on the basis of Leray–Schauder fixed point alternative.
Lemma 3.2
(Leray–Schauder alternative [11])
Let \(A:S\rightarrow S\) be a completely continuous operator (i.e., a map restricted to any bounded set in S is compact), and let
Then either the set \(Z(F)\) is unbounded or F has at least one fixed point.
Theorem 3.3
Assume that:
- \((\mathrm{E}_{3})\):
-
\(k,l: [0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}\) such that, for all x, x̄, y, ȳ, we can find positive real constants \(d_{0}\), \(d_{1}\), \(d_{2}\), \(e_{0}\), \(e_{1}\), \(e_{2}\) satisfying
$$\begin{aligned} &\bigl\vert k(t, x, \bar{x}) \bigr\vert \leq d_{0}+ d_{1} \vert x \vert +d_{2} \vert \bar{x} \vert , \\ &\bigl\vert l(t, y, \bar{y}) \bigr\vert \leq e_{0}+ e_{1} \vert y \vert + e_{2} \vert \bar{y} \vert . \end{aligned}$$
If \((\omega _{1}+\omega _{3})d_{2}+(\omega _{2}+\omega _{4})e_{1}<1\) and \((\omega _{1}+\omega _{3})d_{1}+(\omega _{2}+\omega _{4})e_{2}<1\), then the boundary value problem (1.1)–(1.2) has at least one solution on \([0,1]\).
Proof
The proof will be finished in the subsequent steps.
Step 1. First we show that the operator \(\Upsilon :\mathbb{X}\rightarrow \mathbb{X}\) is completely continuous. It is clear that ϒ is continuous due to the continuity of functions k and l.
Let Δ be any bounded subset of \(\mathbb{X}\). Then, for all \((\varpi ,w)\in \Delta \), there exist some positive constants \(A_{1}\) and \(A_{2}\) such that \(|k(t,w(t),D^{m}\varpi (t))|\leq A_{1}\), \(|l(t,\varpi (t),D^{n}w(t))|\leq A_{2}\). Therefore, for any \((\varpi ,w)\in \Delta \), we have
By the boundedness of k and l, we get
Consequently,
Similarly,
Thus, it follows from inequalities (3.6) and (3.7) that ϒ is uniformly bounded since \(\|\Upsilon (\varpi ,w)\|\leq (\omega _{1}+\omega _{3})A_{1}+(\omega _{2}+ \omega _{4})A_{2}\).
Step 2. Next, we show that ϒ is equicontinuous. Let \(t_{1},t_{2}\in [0,1]\) with \(t_{1}< t_{2}\). Then
or
By following the same procedure, we get
From inequalities (3.8) and (3.9), we conclude that \(|\Upsilon _{1}(\varpi (t_{2}),w(t_{2}))-\Upsilon _{1}(\varpi (t_{1},w(t_{1}))| \rightarrow 0\) as \(t_{1}\rightarrow t_{2}\) and \(|\Upsilon _{2}(\varpi (t_{2}),w(t_{2}))-\Upsilon _{2}(\varpi (t_{1},w(t_{1}))| \rightarrow 0 \) as \(t_{1}\rightarrow t_{2}\). Therefore, \(\Upsilon (\varpi ,w)\) is equicontinuous. Hence, by Arzela–Ascoli theorem the operator \(\Upsilon (\varpi ,w)\) is completely continuous.
Step 3. It remains to show that the set
is bounded. Let \((\varpi ,w) \in Z\) along with \((\varpi ,w)=\mu \Upsilon (\varpi ,w)\), then for any \(t \in [0,1]\), we have \(\varpi (t)=\mu \Upsilon _{1}(\varpi ,w)(t)\), \(w(t)=\mu \Upsilon _{2}(\varpi ,w)(t)\).
Also
which further gives
Consequently,
where
which implies that Z is bounded. Thus, the operator ϒ has at least one fixed point which is the solution of (1.1)–(1.2), thanks to Lemma 3.2. □
4 Ulam–Hyers stability
This section is dedicated to the investigation of Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability results corresponding to the solutions of (1.1)–(1.2). We will only establish the most general stability result, that is, generalized Ulam–Hyers–Rassias stability result. The following definitions are adopted from [27].
Let \(\varepsilon _{m},\varepsilon _{n}>0\) and \(\Theta _{m},\Theta _{n}:[0,1]\rightarrow \mathbb{R_{+}}\) be nondecreasing continuous functions. We will focus on the following inequalities for \(t\in [0,1]\):
Definition 4.1
Problem (1.1)–(1.2) is called Ulam–Hyers stable if we can find a constant \(\mathcal{P}_{m,n}>0\) (\(\mathcal{P}_{m,n}=\max \{\mathcal{P}_{m}, \mathcal{P}_{n}\}\)) such that, for each \(\varepsilon >0\) \((\varepsilon =\max \{\varepsilon _{m},\varepsilon _{n}\})\) and every solution \((\varpi ,w) \in \mathbb{X}\) of (4.1), there exists a solution \((s^{\ast },w^{\ast })\in \mathbb{X}\) of (1.1)–(1.2) with
Definition 4.2
Problem (1.1)–(1.2) is said to be generalized Ulam–Hyers stable if there is \(\Lambda \in C(\mathbb{R_{+}},\mathbb{R_{+}})\), \(\Lambda (0)=0\) such that, for every solution \((\varpi ,w) \in \mathbb{X}\) of (4.2), there is a solution \((s^{\ast },w^{\ast })\in \mathbb{X}\) of (1.1)–(1.2) with
Definition 4.3
Problem (1.1)–(1.2) is Ulam–Hyers–Rassias stable on the interval \([0,1]\) with respect to \(\Theta _{m,n}\in C([0,1],\mathbb{R})\), \((\Theta _{m,n}=\max \{\Theta _{m}, \Theta _{n}\})\) if there exists a positive real number C such that, for each solution \((\varpi ^{\ast },w^{\ast })\in \mathbb{X}\) of (4.3), there exists a solution \((\varpi ,w)\in \mathbb{X}\) of (1.1)–(1.2) with
Definition 4.4
Problem (1.1)–(1.2) is generalized Ulam–Hyers–Rassias stable on \([0,1]\) with respect to \(\Theta _{m,n}\in C([0,1],\mathbb{R})\) if there exists a real number \(C_{\Theta }>0\) such that, for each solution \((\varpi ,w) \in \mathbb{X}\) of (4.4), we have a solution \((s^{\ast },w^{\ast })\in \mathbb{X}\) of (1.1)–(1.2) with
Remark 4.5
We say that the functions \(\varpi ,w\in \mathbb{X}\) are the solutions of (4.4) if there exist functions \(\hbar _{1},\hbar _{1} \in \mathbb{X}\) which depend upon ϖ, w, respectively, such that
-
(I)
$$ \bigl\vert \hbar _{1}(t) \bigr\vert \leq \Theta _{m},\qquad \bigl\vert \hbar _{2}(t) \bigr\vert \leq \Theta _{n}$$
-
(II)
$$\begin{aligned}& ^{c}D^{m}\varpi (t)=k\bigl(t,w(t),^{c}D^{m} \varpi (t)\bigr)+\hbar _{1}(t),\quad 0< t< 1, \\& ^{c}D^{n}w(t)=l\bigl(t,\varpi (t),^{c}D^{n}w(t) \bigr)+\hbar _{2}(t), \quad 0< t< 1. \end{aligned}$$
Before going to the main result, we need the following assumption:
- \((\mathrm{E}_{4})\):
-
\(\hbar _{1},\hbar _{2}:J\rightarrow \mathbb{R}^{+}\) are nondecreasing functions such that
$$ \int _{0}^{t}\hbar _{1}(s)\,ds \leq \lambda _{m}\hbar _{1}(t), \qquad \int _{0}^{t} \hbar _{2}(s)\,ds\leq \lambda _{n}\hbar _{2}(t)$$for all \(t\in [0,1]\) and \(\lambda _{m},\lambda _{n} >0\).
Lemma 4.6
Let ϖ, w be the solutions of inequality (4.4), then
Proof
From Remark 4.5(II), we have
In view of Lemma 3.2, the solution of (4.5) will be equivalent to the subsequent integral equations:
It follows from (4.6) that
where
By using \((I)\) of Remark 4.5 and assumption \(\mathrm{E}_{4}\), (4.8) leads to
Also from (4.7) we have
where
Following the same procedures, we can write equation (4.9) as
□
Theorem 4.7
Under assumption \(\mathrm{E}_{4}\), system (1.1)–(1.2) has generalized Ulam–Hyers–Rassias stability provided that
Proof
Let \((\varpi ,w)\in \mathbb{X}\) be a solution of inequality (4.4) and \((\varpi ^{*},w^{*})\in \mathbb{X}\) be the unique solution of the following problem:
Thus, we write the solution of (4.10) as follows:
Now
which further gives
Similarly,
Inequality (4.11) can be written as
or
where
Also from (4.12) we gain
where
From (4.13) and (4.14), we have
Set
Simplification yields
Inequality (4.15) becomes
where
Thus, by Definition 4.4, the proposed system is generalized Ulam–Hyers–Rassias stable. □
Remark 4.8
From the last theorem, the Ulam–Hyers stability, generalized Ulam–Hyers stability, and Ulam–Hyers–Rassias stability of system (1.1)–(1.2) can be obtained as corollaries.
5 Example
ln this section, we present some examples to achieve the existence, uniqueness, and stability of the proposed system.
Example 5.1
Consider the following system of fractional differential equation with non-separated coupled boundary conditions:
Here, \(m=n=\frac{5}{2}\), \(\alpha _{1}=\frac{1}{2}\), \(\alpha _{2}=\frac{1}{3}\), \(\alpha _{3}= \frac{1}{4}\), \(\beta _{1}=\frac{3}{7}\), \(\beta _{2}=\frac{2}{3}\), \(\beta _{3}=\frac{1}{5}\), \(\zeta =\frac{1}{4}\), \(\eta =\frac{1}{6}\). We can easily find that \(\omega _{1}=0.648937084\), \(\omega _{2}=0.5431954534\), \(\omega _{3}=0.8409326373\), and \(\omega _{4}=0.5537849552\).
Also, we have
Also \((\omega _{1}+\omega _{3})\wp _{1}+(\omega _{2}+\omega _{4})\wp _{2} \approx 0.1616781 <1\). Since all the requirements of Theorem 3.1 are satisfied, hence problem (5.1) has a unique solution.
Example 5.2
Consider the following system of fractional differential equations:
Comparing (5.2) with system (1.1)–(1.2) yields \(m=n=\frac{5}{2}\), \(\alpha _{1}=\frac{1}{2}\), \(\alpha _{2}=\frac{1}{3}\), \(\alpha _{3}= \frac{1}{4}\), \(\beta _{1}=\frac{3}{7}\), \(\beta _{2}=\frac{2}{3}\), \(\beta _{3}=\frac{1}{5}\), \(\zeta =\frac{1}{4}\), \(\eta =\frac{1}{6}\). Furthermore,
As \(d_{0}=1\), \(d_{1}=\frac{1}{9}\), \(d_{2}=\frac{1}{9}\), \(e_{0}=\frac{1}{3}\), \(e_{1}= \frac{1}{9}\), \(e_{2}=\frac{1}{9}\), we can easily find that \(\omega _{1}=0.648937084\), \(\omega _{2}=0.5431954534\), \(\omega _{3}=0.8409326373\), and \(\omega _{4}=0.5537849552\). Note that \((\omega _{1}+\omega _{3})d_{2}+(\omega _{2}+\omega _{4})e_{1} \approx 0.2874278 <1\) and \((\omega _{1}+\omega _{3})d_{1}+(\omega _{2}+\omega _{4})e_{2}<1\). As a conclusion, we can say that a coupled system has at least one solution. Furthermore,
and condition \(\mathrm{E}_{4}\) is also satisfied. Consequently, problem (5.2) is generalized Ulam–Hyers–Rassias stable.
6 Conclusion
In this manuscript, we have successfully derived the existence and uniqueness results for nonlinear coupled FODEs with non-separated boundary conditions. To guarantee the existence and uniqueness of solutions, some sufficient criteria have been established by the Banach contraction principle and the Leray–Schauder alternative. Moreover, we presented the generalized Ulam–Hyers–Rassias stability for model (1.1)–(1.2) by classical functional analysis. At the end, for the justification of our results, we stated some examples. The results obtained in this article are of quite general nature, because by changing the parameters and the interval from \((0,1)\) to \([0,T]\) in the proposed system, one can get different types of boundary conditions like coupled flux type conditions, periodic and anti periodic boundary conditions, etc.
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This work was jointly supported by the National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province [(2016)4006], Major Research Project of Innovative Group in Guizhou Education Department [(2018)012].
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Luo, D., Zada, A., Shaleena, S. et al. Analysis of a coupled system of fractional differential equations with non-separated boundary conditions. Adv Differ Equ 2020, 590 (2020). https://doi.org/10.1186/s13662-020-03045-6
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DOI: https://doi.org/10.1186/s13662-020-03045-6