Abstract
In this work, we study the coupled system of fractional integro-differential equations, which includes the fractional derivatives of the Riemann–Liouville type and the fractional q-integral of the Riemann–Liouville type. We focus on the utilization of two significant fixed-point theorems, namely the Schauder fixed theorem and the Banach contraction principle. These mathematical tools play a crucial role in investigating the existence and uniqueness of a solution for a coupled system of fractional q-integro-differential equations. Our analysis specifically incorporates the fractional derivative and integral of the Riemann–Liouville type. To illustrate the implications of our findings, we present two examples that demonstrate the practical applications of our results. These examples serve as tangible scenarios where the aforementioned theorems can effectively address real-world problems and elucidate the underlying mathematical principles. By leveraging the power of the Schauder fixed theorem and the Banach contraction principle, our work contributes to a deeper understanding of the solutions to coupled systems of fractional q-integro-differential equations. Furthermore, it highlights the potential practical significance of these mathematical tools in various fields where such equations arise, offering a valuable framework for addressing complex problems.
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1 Introduction
The topics of q-calculus and fractional calculus have received a lot of interest in view of their role in describing some real-world problems in numerous fields. It is worth noting that both fractional calculus and q-calculus are generalizations of classical calculus for any order. Note that the field of fractional calculus has a large number of fractional operators, but in our paper, we are interested in studying the equations that contain the fractional derivative of the Riemann–Liouville type. Mathematics modeling is used to convert a range of applied problems into a set of fractional differential and integral equation [1,2,3]. The study of theoretical aspects of q-calculus and fractional calculus has been the focus of many studies, and it is now considered a significant area of research. There are numerous papers on the solvability of nonlinear fractional differential equations. At the same time, the study of coupled systems of nonlinear fractional differential equations is also important due to their numerous applications. In Ahmad and Nieto [4] studied the existence of result for the following coupled system:
where \(\alpha , \lambda , \beta , \zeta , \gamma , and \varsigma \) satisfy certain conditions. In [5], Zhang et al. applied a variety of fixed-point theorems to the following coupled system of nonlinear fractional differential equations to investigate the existence and uniqueness of solutions:
where \({\mathcal {D}}^{\rho _i}\) and \({\mathcal {D}}^{\alpha _i}\) represent the standard Riemann–Liouville fractional derivative, \(2<\rho _i\le 3,\) \(0<\alpha _i\le 1,\) \(0<\mu _i\le 1,\) \(\gamma _i,\beta _i>0, i=1,2\). For more details, see [6,7,8,9,10,11,12,13,14,15,16,17,18,19].
In contrast, several studies have been written about the existence and uniqueness of solutions for fractional q-integro-differential equations; for further information, see [20,21,22,23,24]. In [25, 26], the authors discussed the numerical and analytical solutions of the Fredholm and Fredholm–Volterra integro-differential equations of the first and second orders, respectively. In addition, they study the numerical solution using a merge of finite difference with Simpson’s and finite difference with trapezoidal methods. In [27], they discussed the existence and uniqueness of a solution for the nonlocal fractional q-integro-differential equation:
under the q-nonlocal condition:
where \({^{CF}{\mathcal {D}}^{\beta }}{\mathfrak {U}}({{\L } })\) is the fractional derivative of Caputo–Fabrizio type, \(I_q^{\alpha }\) is the Riemann–Liouville fractional q-integral, and \(\rho _0,\alpha _0\) are constants, \(q,\beta \in (0,1)\). In addition, they solved it numerically by using two methods: the finite-trapezoidal method and the cubic-trapezoidal method.
Inspired by the aforementioned results, we examine in this work the following coupled system of a nonlocal fractional q-integro-differential equation:
considering the q-nonlocal conditions:
where \({\mathcal {D}}^{\beta _i}\) and \({\mathcal {D}}^{\alpha _i}\) represent the standard fractional derivative of the Riemann–Liouville type, \(I_{q_i}^{\delta _i}\) represent the Riemann–Liouville fractional q-integrals of the order \(\delta _i>0,\) \(2<\beta _i\le 3,\) \(0<\alpha _i\le 1,\) \(\rho _i\) are constants, and \(q_i,\alpha _i \in (0,1), i=1,2\).
This is how the essay is organized: We list several lemmas and definitions that are indeed in this work in Sect. 2. The existence and uniqueness of the solution to the nonlocal coupled system of a fractional q-integro-differential equation (1)–(2) were examined in Sect. 3. Section 4 contains applications. In Sect. 5, the conclusion is presented.
2 Preliminaries
We now go through some fundamental ideas in q-calculus and fractional calculus, as well as some lemmas that will be applied in this article.
Definition 2.1
[20, 21] Let z be a function that is defined on the interval [0, 1]. The Riemann–Liouville fractional q-integral of order \(\varrho > 0\) can be defined as
where
Lemma 2.2
[20] As a result of q-integration by parts, we obtain
Definition 2.3
[28] The Riemann–Liouville fractional derivative of order \(\beta >0\) of a function \({\mathcal {F}}:(0,\infty )\rightarrow {\mathbb {R}} \) can be defined as
where \( \nu = [\beta ]+1, [\beta ]\) represents the integer part of number \(\beta \), provided that the right-hand side is pointwise defined on \((0,\infty )\).
Definition 2.4
[28] The Riemann–Liouville fractional integral of order \(\beta >0\) of a function \({\mathcal {F}}:(0,\infty )\rightarrow {\mathbb {R}} \) is given by
Lemma 2.5
[28] Suppose that \(\beta >0\), \(\nu -1<\beta <\nu , \nu \in N.\) Then
-
1.
For any \(z \in L^1(c,d), D^{\beta }(I^{\beta }z)=z.\)
-
2.
If \(I^{\nu -\beta }z\in AC^\nu [c,d]\), then
$$\begin{aligned} I^{\beta } {\mathcal {D}}^{\beta } z({{\L } })=z({{\L } })+k_1 {{\L } }^{\beta -1}+k_2 {{\L } }^{\beta -2}+\cdots +k_\nu {{\L } }^{\beta -\nu }, \end{aligned}$$
where \( k_i \in {\mathbb {R}} (i=1,2,\ldots , \nu ), \nu \) is the lowest integer smaller than or equal to \(\beta .\)
Lemma 2.6
[28]
-
1.
If \( {\mathcal {F}} \in L^1 (c,d), \sigma>\varphi >0,\) then
$$\begin{aligned} I^\sigma I^\varphi {\mathcal {F}}({{\L } }) = I^{\sigma +\varphi } {\mathcal {F}}({{\L } }),\quad {\mathcal {D}}^\varphi I^\sigma {\mathcal {F}}({{\L } })=I^{\sigma -\varphi } {\mathcal {F}}({{\L } }),\quad {\mathcal {D}}^\sigma I^\sigma {\mathcal {F}}({{\L } })= {\mathcal {F}}({{\L } }). \end{aligned}$$ -
2.
If \(\sigma> \varphi >0,\) then
$$\begin{aligned} I^{\sigma } {{\L } }^{\varphi }=\frac{\Gamma (\varphi +1)}{\Gamma (\varphi +1+\sigma )} {{\L } }^{\sigma +\varphi },\quad {\mathcal {D}}^{\varphi } {{\L } }^{\sigma }=\frac{\Gamma (\sigma +1)}{\Gamma (\sigma +1-\varphi )} {{\L } }^{\sigma -\varphi }, \quad {\mathcal {D}}^{\sigma } {{\L } }^{\varphi }=0. \end{aligned}$$
Lemma 2.7
[29] Suppose that \(\sigma >0, {\mathfrak {U}}\in L^1([c,d],{\mathbb {R}}).\) Then, we have
For the convenience, we set
Lemma 2.8
Suppose that \({\mathfrak {U}} \in AC^3[0,1]\), and \(2<\beta _1\le 3.\) Therefore, we can get a unique solution to the following nonlocal problem:
as
Proof
Considering the lemma (), we can get the solution of (5) as follows:
Using the condition \({\mathfrak {U}}(0)={\mathfrak {U}}'(0)=0\), we get \(k_2=k_3=0\). Therefore,
Using the q-nonlocal condition \((1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1} q_1^\kappa {\mathfrak {U}}(q_1^\kappa \chi _1)=\rho _1\), we obtain
Therefore,
Substituting \(k_1\) into (7), we obtain
The proof is finished. \(\square \)
In the same way, the solution of
is
Denote that \(L^1([0, 1],{\mathbb {R}} )\) is the Banach space of Lebesgue integrable functions from \([0, 1]\rightarrow {\mathbb {R}} \) with the norm \( \Vert {\mathfrak {U}}\Vert = \int _{0}^{1} |{\mathfrak {U}}({{\L } })| d {{\L } }.\)
3 Main results
The spaces \({\mathcal {X}}\) and \({\mathcal {Y}}\) should now be introduced as follows: \({\mathcal {X}}=\{{\mathfrak {U}} | {\mathfrak {U}}\in {\mathcal {C}}[0,1], {\mathcal {D}}^{\beta _1}{\mathfrak {U}}\in {\mathcal {C}}[0,1] \) and \( I_{q_1}^{\delta _1}{\mathfrak {U}} \in {\mathcal {C}}[0,1]\}\) equipped with the norm \(\Vert {\mathfrak {U}}\Vert =\max _{{{\L } }\in [0,1]}|{\mathfrak {U}}({{\L } })|+\max _{{{\L } }\in [0,1]}|{\mathcal {D}}^{\beta _1} {\mathfrak {U}}({{\L } })|+\max _{{{\L } }\in [0,1]}|I_{q_1}^{\delta _1}{\mathfrak {U}}({{\L } })|,\) and also \({\mathcal {Y}}=\{w | w\in {\mathcal {C}}[0,1],{\mathcal {D}}^{\beta _2}w\in {\mathcal {C}}[0,1] \) and \( I_{q_2}^{\delta _2}w \in {\mathcal {C}}[0,1]\}\) equipped with the norm \(\Vert w\Vert =\max _{{{\L } }\in [0,1]}|w({{\L } })|+\max _{{{\L } }\in [0,1]}|{\mathcal {D}}^{\beta _2} w({{\L } })|+\max _{{{\L } }\in [0,1]}|I_{q_2}^{\delta _2}w({{\L } })|.\) Evidently, \(({\mathcal {X}},\Vert .\Vert )\) and \(({\mathcal {Y}},\Vert .\Vert )\) are Banach spaces. Therefore, the product space \(({\mathcal {X}} \times {\mathcal {Y}}, \Vert ({\mathfrak {U}},w)\Vert _{{\mathcal {X}}\times {\mathcal {Y}}}) \) is also a Banach space equipped with the norm \( \Vert ({\mathfrak {U}},w)\Vert _{{\mathcal {X}}\times {\mathcal {Y}}}=\Vert {\mathfrak {U}}\Vert _{\mathcal {X}}+\Vert w\Vert _{\mathcal {Y}}.\) Now, we define the operator \( G: {\mathcal {X}} \times {\mathcal {Y}}\rightarrow {\mathcal {X}} \times {\mathcal {Y}}\) by
where
and
Conveniently, we set
-
1.
$$\begin{aligned} {\mathcal {A}}_{i j}= \bigg (1+|\Lambda _i|(1-q_i) \chi _i \sum _{\kappa =0}^{\nu _i}q_i^\kappa \bigg ) \Vert I^{\beta _i-1}b_{ij}\Vert _{L^1}, i = 1,2,\quad j = 1, \ldots ,7. \end{aligned}$$
-
2.
$$\begin{aligned} {\mathcal {B}}_{i j}=\Vert I^{\beta _i-\alpha _i-1}b_{ij}\Vert _{L^1}+\frac{|\Lambda _i|\Gamma (\beta _i)}{\Gamma (\beta _i-\alpha _i)} (1-q_i) \chi _i \sum _{\kappa =0}^{\nu _i}q_i^\kappa \Vert I^{\beta _i-1}b_{ij}\Vert _{L^1}, \end{aligned}$$
\(i = 1,2,\quad j = 1, \ldots ,7.\)
-
3.
$$\begin{aligned} C_{ij}=\bigg (1+\frac{1}{\Gamma _{q_i}(\delta _i+1)}\bigg ) {\mathcal {A}}_{i j} +{\mathcal {B}}_{i j}, i = 1,2,\quad j = 1, \ldots ,7. \end{aligned}$$
-
4.
$$\begin{aligned} E_i=\bigg (1+\frac{1}{\Gamma _{q_i}(\delta _i+1)} +\frac{\Gamma (\beta _i)}{\Gamma (\beta _i-\alpha _i)} \bigg )|\rho _i\Lambda _i|, \qquad i = 1,2. \end{aligned}$$
-
5.
$$\begin{aligned} D_i=\max \{C_{i1},C_{i2},C_{i3},C_{i4},C_{i5},C_{i6}\},\qquad i = 1,2. \end{aligned}$$
Theorem 3.1
Suppose that \({\mathcal {F}}_i\) are continuous for almost every \( {{\L } } \in (0,1)\) and measurable in \( {{\L } } \) for any \( w_1, w_2,w_3,w_4,w_5,w_6 \in {\mathbb {R}}\). There exist nonnegative functions \(b_{i j}({{\L } }) \in L^1([0,1],\mathbb {R_+}),i = 1, 2, j=1,2,\ldots ,7,\) such that
-
(a)
$$\begin{aligned} |{\mathcal {F}}_i({{\L } },w_1,w_2,w_3,w_4,w_5,w_6)|\le \sum _{j=1}^{6} b_{i j}({{\L } })|w_j|^{\tau _{i j}}+b_{i 7}({{\L } }),\quad \tau _{i j} \in (0,1). \end{aligned}$$
Consequently, there is at least one solution for the nonlocal coupled system (1)-(2).
Proof
To demonstrate that the operator G defined by (8) has a fixed point, we shall employ the Schauder fixed-point theorem.
First, let \( Q_{{\mathfrak {r}}}\subset {\mathcal {X}}\times {\mathcal {Y}} = \lbrace ({\mathfrak {U}},w)({{\L } }) \in {\mathbb {R}}^2: \Vert ({\mathfrak {U}},w)\Vert _{{\mathcal {X}}\times {\mathcal {Y}}} \le {\mathfrak {r}} \rbrace \), where
It is obvious that \(Q_{\mathfrak {r}}\) is nonempty, bounded, closed and convex subset of \({\mathcal {C}}[0, 1]\). We demonstrate \(G: Q_{\mathfrak {r}}\rightarrow Q_{\mathfrak {r}}.\) For any \(({\mathfrak {U}},w)({{\L } })\in Q_{\mathfrak {r}},\) we use Lemma () and condition (a), to obtain
Similarly, by using Lemma (), we get
Analogously, we get
Using lemma (), we get
Therefore,
Similarly, we have
Thus,
Then, \( G: Q_{{\mathfrak {r}}}\rightarrow Q_{{\mathfrak {r}}}\), and also, the class of functions \( \lbrace G({\mathfrak {U}},w)({{\L } }) \rbrace \) is uniformly bounded in \( Q_{{\mathfrak {r}}}\). Observe that \(G_1({\mathfrak {U}},w)({{\L } }),G_2({\mathfrak {U}},w)({{\L } }),{\mathcal {D}}^{\alpha _1}G_1({\mathfrak {U}},w)({{\L } }),{\mathcal {D}}^{\alpha _2}G_2({\mathfrak {U}},w)({{\L } }),I_{q_1}^{\delta _1}G_1({\mathfrak {U}},w)({{\L } }),I_{q_2}^{\delta _2}G_2({\mathfrak {U}},w)({{\L } })\) are continuous on [0, 1]. Clearly, G is also continuous. Next, we demonstrate that G is equicontinuous. Let
For \( {{\L } }_1, {{\L } }_2\in [0,1], {{\L } }_1< {{\L } }_2\), we obtain
In contrast, we obtain
and also,
Similar to that, we can demonstrate
Letting \({{\L } }_1\rightarrow {{\L } }_2\), then
Therefore,
That is, as \({{\L } }_1 \rightarrow {{\L } }_2\),
As a result, we establish that the operator G is equicontinuous. Then, using the Arzela–Ascoli theorem, we conclude that G is a completely continuous operator. So, it follows from the Schauder fixed theorem that (1)–(2) possesses at least one solution \(({\mathfrak {U}}, w) \in Q_{\mathfrak {r}}\). This proof has been established. \(\square \)
Now, we demonstrate the uniqueness of solutions using the Banach contraction principle.
Theorem 3.2
Suppose that \({\mathcal {F}}_i\) are continuous for almost all \( {{\L } } \in (0,1)\) and measurable in t for any \( w_1, w_2,w_3,w_4,w_5,w_6 \in {\mathbb {R}}\). There exist nonnegative functions \(b_{i j}({{\L } }) \in L^1([0,1],\mathbb {R_+}),i = 1, 2, j=1,2,\ldots ,7,\) such that the following requirements are fulfilled:
- \((H_1)\):
-
\(|{\mathcal {F}}_i({{\L } },w_1,w_2,w_3,w_4,w_5,w_6)-{\mathcal {F}}_i({{\L } },z_1,z_2,z_3,z_4,z_5,z_6)|\le \sum _{j=1}^{6}b_{i j}({{\L } })|w_j-z_j|,i=1,2.\)
- \((H_1)\):
-
\(3 D_1+3 D_2<1.\)
Following that, the coupled system (1)–(2) possesses a unique solution.
Proof
Let \(\sup _{{{\L } }\in [0,1]} {\mathcal {F}}_i({{\L } },0,0,0,0,0,0)=\mu _i<\infty ,i=1,2\) and take
where \(\mu _i'=|\rho _i\Lambda _i |+\frac{ |\rho _i\Lambda _i| \Gamma (\beta _i)}{\Gamma (\beta _i-\alpha _i)}+\frac{ |\rho _i\Lambda _i|}{\Gamma _{q_i}(\delta _i+1)}\) \(+\bigg [\frac{1}{\Gamma (\beta _i-\alpha _i+1)}+\frac{(1+|\Lambda _i|(1-q_i)\chi _i \sum _{\kappa =0}^{\nu _i} q_i^\kappa )}{\Gamma (\beta _i+1)}(1+\frac{1}{\Gamma _{q_i}(\delta _i+1)})\) \(+\frac{|\Lambda _i|(1-q_i) \chi _i \sum _{\kappa =0}^{\nu _i}q_i^\kappa }{\Gamma (\beta _i-\alpha _i) \Gamma (\beta _i+1)}\bigg ]\mu _i.\) First, we prove that \(G(Q_R)\subset Q_R\), where \( Q_R=\{({\mathfrak {U}},w)|({\mathfrak {U}},w)\in {\mathcal {X}} \times {\mathcal {Y}}:\Vert ({\mathfrak {U}},w)\Vert _{{\mathcal {X}}\times {\mathcal {Y}}}\le R\}\). For any \(({\mathfrak {U}},w)({{\L } }) \in Q_R\), we can obtain
As an alternative, using (9), we may get
Also,
Therefore,
Similarly, we can get
Thus,
Second, for any \(({\mathfrak {U}}_1,w_1)({{\L } })\), \((\mathfrak { U}_2,w_2)({{\L } })\in Q_R,\) we have
and
also,
Therefore,
Similarly, we can get
Thus,
Since \(3 D_1 + 3 D_2 < 1\), G is a contraction operator. This implies that G has a unique fixed point, and thus, (1)–(2) has a unique solution. \(\square \)
4 Applications
We now give the two examples below to demonstrate our findings.
Test problem 1: Consider the following coupled system:
where \( {\beta _1=\frac{9}{4},} \quad {\beta _2=\frac{5}{2},} \quad \alpha _1=\frac{1}{2},\quad \alpha _2=\frac{1}{5},\quad q_1=\frac{1}{7},\quad q_2=\frac{1}{3},\quad \delta _1=\frac{1}{4},\quad \delta _2=\frac{1}{2},\quad \chi _1=0.5, \quad \chi _2=0.4,\quad \nu _1=1,\quad \nu _2= 2, \quad 0<\tau _{i j}<1(j=1,2,\ldots ,6,\quad i=1,2)\) and \(b_{i j}({{\L } }) ( j=1,2,\ldots ,7,\quad i=1,2)\) are nonnegative functions. The test problem 1 must have at least one solution, according to Theorem ().
Test problem 2: Consider the following coupled system:
where \( {\beta _1=\frac{5}{2},} \quad {\beta _2=\frac{7}{3},} \quad \alpha _1=\frac{1}{2},\quad \alpha _2=\frac{1}{5},\quad q_1=0.2,\quad q_2=0.5,\quad \delta _1=\frac{1}{4},\quad \delta _2=\frac{1}{3},\) \( \chi _1=0.5, \quad \chi _2=0.4,\quad \nu _1=\nu _2=4. \)
Then, we have
and
where
\( b_{11}=\frac{{{\L } }^5}{30},\quad b_{12}=\frac{{{\L } }^4}{60},\quad b_{13}=\frac{{{\L } }^4}{40}, \quad b_{14}=\frac{{{\L } }^6}{25}, \quad b_{15}=\frac{{{\L } }^7}{45},\quad b_{16}=\frac{{{\L } }^5}{50},\quad b_{17}=\frac{(1-{{\L } })^5}{50},\quad b_{21}=\frac{(1-{{\L } })^3}{150},\quad b_{22}=\frac{(1-{{\L } })^5}{100},\quad b_{23}=\frac{{{\L } }^7}{100},\quad b_{24}=\frac{{{\L } }^5}{80},\quad b_{25}=\frac{{{\L } }^8}{50},\quad b_{26}=\frac{{{\L } }^8}{50},\quad b_{27}=\frac{(1-{{\L } })^8}{100}.\) By direct calculation, we get
\( {\mathcal {A}}_{11}=0.0095576,\quad {\mathcal {A}}_{12}=0.0062124,\quad {\mathcal {A}}_{13}=0.0093187,\quad {\mathcal {A}}_{14}=0.0091753, \quad {\mathcal {A}}_{15}=0.00419785, \) \( {\mathcal {A}}_{16}=0.0057346,\quad {\mathcal {A}}_{17}=0.0155236,\quad {\mathcal {A}}_{21}=0.0108042,\quad {\mathcal {A}}_{22}=0.0110885,\quad {\mathcal {A}}_{23}=0.00381566,\) \( {\mathcal {A}}_{24}=0.0069399,\quad {\mathcal {A}}_{25}=0.0065411,\quad {\mathcal {A}}_{26}=0.0032706,\quad {\mathcal {A}}_{27}=0.0075244,\quad {\mathcal {B}}_{11}=0.0154193,\) \( {\mathcal {B}}_{12}=0.00974474,\quad {\mathcal {B}}_{13}=0.0146171,\quad {\mathcal {B}}_{14}=0.0151834,\quad B_{15}=0.00711007,\quad {\mathcal {B}}_{16}=0.00925156, \) \({\mathcal {B}}_{17}=0.01935415,\quad {\mathcal {B}}_{21}=0.0118813,\quad {\mathcal {B}}_{22}=0.0121674,\quad {\mathcal {B}}_{23}=0.00452789,\quad {\mathcal {B}}_{24}=0.00814884,\) \( {\mathcal {B}}_{25}=0.00779734,\quad {\mathcal {B}}_{26}=0.0038987,\quad {\mathcal {B}}_{27}=0.008244,\quad C_{11}=0.0195134,\quad C_{12}=0.0126837,\) \( C_{13}=0.0190256,\quad C_{14}=0.0339163,\quad C_{15}=0.0156807,\quad C_{16}=0.0209596,\quad C_{17}=0.0510482,\) \( C_{21}=0.0343762,\)
\( C_{22}=0.0352543,\quad C_{23}=0.0124723,\quad C_{24}=0.022598,\quad C_{25}=0.0214163,\quad \) \(C_{26}=0.0107081,\quad C_{27}=0.0239103.\) Thus, \(D_1=0.051048,\quad D_2=0.03525\). Therefore, \(3D_1+3D_2=0.258907<1.\) The coupled system (1)–(2) must have at least one solution, according to Theorem ().
5 Conclusion
In the present work, we have discussed the existence of solutions for a coupled system of Riemann–Liouville fractional q-integro-differential equations. We have established the conditions under which these solutions exist. Furthermore, we have demonstrated the uniqueness of the solution by utilizing the contraction principle. By employing the powerful tools of fixed-point theorems, specifically the Schauder fixed theorem and the Banach contraction principle, we have provided a rigorous analysis of the coupled system. These theorems have allowed us to establish the existence and uniqueness of solutions, showcasing their effectiveness in addressing complex mathematical problems. Additionally, we have presented two illustrative examples that highlight the practical applications of our results. These examples serve to demonstrate how the findings of our work can be applied to real-world scenarios, further emphasizing the relevance and significance of our research. In conclusion, this work contributes to the understanding of coupled systems of Riemann–Liouville fractional q-integro-differential equations by discussing the existence and uniqueness of solutions. Our findings provide a solid foundation for future research in this area and offer practical insights for solving similar problems in various fields of study.
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References
Aslam M, Murtaza R, Abdeljawad T, ur Rahman G, Khan A, Khan H, Gulzar H (2021) A fractional order HIV/AIDS epidemic model with Mittag–Leffler kernel. Adv Differ Equ 2021:15
Khan A, Khan ZA, Abdeljawad T, Khan H (2022) Analytical analysis of fractional-order sequential hybrid system with numerical application. Adv Cont Discret Mod 2022:19
Khan H, Ahmed S, Alzabut J, Azar AT (2023) A generalized coupled system of fractional differential equations with application to finite time sliding mode control for Leukemia therapy. Chaos, Solitons Fract 174:113901
Ahmad B, Nieto JJ (2009) Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput Math Appl 58:1838–1843
Zhang H, Li Y, Wei L (2016) Existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations with fractional integral boundary conditions. J Nonlinear Sci Appl 9:2434–2447
Su X (2009) Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl Math Lett 22:64–69
Wang J, Xiang H, Liu Z (2010) Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Int J Differ Equ 2010:12
Zhao Y, Sun S, Han Z, Feng W (2011) Positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders. Adv Differ Equ 2011:13
Zhang Y, Bai Z, Feng T (2011) Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance. Comput Math Appl 61:1032–1047
Chen Y, Chen D, Lv Z (2012) The existence results for a coupled system of nonlinear fractional differential equations with multi-point boundary conditions. Bull Iran Math Soc 38:607–624
Zhang K, Xu J, O’Regan D (2015) Positive solutions for a coupled systems of nonlinear fractional differential equations. Math Methods Appl Sci 38:1662–1672
He J, Zhang X, Yonghong W (2016) Existence of positive solution for a fractional order nonlinear differential system involving a changing sign perturbation. J Nonlinear Sci Appl 9:2076–2085
Zhang Y (2018) Existence results for a coupled system of nonlinear fractional multi-point boundary value problems at resonance. J Inequal Appl 2018:17
Agarwal RP, Luca R (2019) Positive solutions for a semipositone singular Riemann–Liouville fractional differential problem. Int J Nonlinear Sci Numer Simul 20:823–831
Bragdi A, Assia F, Guezane LA (2020) Existence of solutions for nonlinear fractional integro-differential equations. Adv Differ Equ 2020:9
Ntouyas SK, Broom A, Alsaedi A, Saeed T, Ahmad B (2020) Existence results for a nonlocal coupled system of differential equations involving mixed right and left fractional derivatives and integrals. Symmetry 578:17
Tudorache A, Luca R (2021) On a singular Riemann–Liouville fractional boundary value problem with parameters. Nonlinear Anal Model Control 26:151–168
Ahmad B, Alghamdi B, Alsaedi A, Ntouyas SK (2021) Existence results for Riemann–Liouville fractional integro-differential inclusions with fractional nonlocal integral boundary conditions. AIMS Math 6:7093–7110
Ahmad B, Alghamdi B, Agarwal RP, Alsaedi A (2022) Riemann–Liouville fractional integro-differential equations with fractional nonlocal multi-point boundary conditions. Fractals 30:11
Ahmad B, Nieto JJ, Alsaedi A, Al-Hutami H (2014) Existence of solutions for nonlinear fractional q-difference integral equations with two fractional orders and nonlocal four-point boundary conditions. J Frank Inst 351:2890–2909
Ntouyas SK, Tariboon J, Asawasamrit S, Tariboon J (2015) A coupled system of fractional q-integro-difference equations with nonlocal fractional q-integral boundary conditions. Adv Differ Equ 2015:2015
Ntouyas SK, Samei ME (2019) Existence and uniqueness of solutions for multi-term fractional q-integro-differential equations via quantum calculus. Adv Differ Equ 2019:20
Liang S, Samei ME (2020) New approach to solutions of a class of singular fractional q-differential problem via quantum calculus. Adv Differ Equ 2020:22
Alsaedi A, Al-Hutami H, Ahmad B, Agarwal RP (2022) Existence results for a coupled system of nonlinear fractional q-integro-difference equations with q-integral coupled boundary conditions. Fractals 30:19
Ibrahim AA, Zaghrout AAS, Raslan KR, Ali KK (2020) On the analytical and numerical study for nonlinear Fredholm integro-differential equations. Appl Math Inf Sci 14:921–929
Raslan KR, Ali KK, Ahmed RG, Al-Jeaid HK, Ibrahim AA (2022) Study of nonlocal boundary value problem for the Fredholm–Volterra integro-differential equation. Hindawi J Funct Spaces 2022:16
Ibrahim AA, Zaghrout AAS, Raslan KR, Ali KK (2022) On study nonlocal integro differetial equation involving the Caputo–Fabrizio fractional derivative and q-integral of the Riemann Liouville Type. Appl Math Inform Sci 16:983–993
Podlubny I (1999) Fractional differential equations. Academic Press, San Diego
Guezane-Lakoud A, Khaldi R (2012) Solvability of a fractional boundary value problem with fractional integral condition. Nonlinear Anal 75:2692–2700
Acknowledgements
The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through the Large Research Project under grant number RGP2/222/45.
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Alaofi, Z.M., Raslan, K.R., Ibrahim, A.AE. et al. Comprehensive analysis on the existence and uniqueness of solutions for fractional q-integro-differential equations. J Supercomput 80, 23848–23866 (2024). https://doi.org/10.1007/s11227-024-06305-4
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DOI: https://doi.org/10.1007/s11227-024-06305-4
Keywords
- Fractional derivative
- q-Integro-differential equation
- Existence and uniqueness of solution
- Applications