Abstract
The main objective of this paper is to investigate the existence, uniqueness, and Ulam–Hyers stability of positive solutions for fractional integro-differential boundary values problem. Uniqueness result is obtained by using the Banach principle. For obtaining two positive solutions, we apply another fixed point criterion due to Avery–Anderson–Henderson on cones by establishing some inequalities. An illustrative example is presented to indicate the validity of the obtained results. The results are new and provide a generalization to some known results in the literature.
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1 Introduction
The topic of fractional boundary values problems gained a largest share of interest of researchers and scientists due to its great and important role in many fields such as engineering, physics, chemistry, and many other applications, see [1–3] and the references therein.
The subject of analysis of differential systems such as existence, uniqueness, and stability of solution for various boundary values problems has received the attention of many researchers, since the shape of the solution of differential models is obtained by its boundary [4–15]. One form of active research is the hybrid system that has been used as a model of several physical systems and has an unusual differential form, see [16–23].
The fixed point theorems of many versions are the main core of obtaining the necessary and sufficient criteria implying the existence and uniqueness of solution for fractional boundary values problems [24–42]. In particular, Banach fixed point theorem is the most popular one to find the unique solution of the problem. The existence of more than one solution has been obtained by many fixed point theorems such as Schauder’s and Krasnosel’skii’s fixed point theorems according to the stated given conditions on nonlinear terms. The existence of at least two solutions of the nonlinear boundary value problem (BVP) are given by Avery–Anderson–Henderson fixed point principle [43].
The in-depth qualitative behavior of the solution for fractional BVPs is the positivity of such solutions. The study of existence and stability of positive solution in boundary value problems is characterized by more investigation in all components of the fractional models along with the involved boundary conditions [44, 45]. Most researchers avoid the multi nonzero components in initial or boundary conditions such as constants, functions, integrals, or even derivatives of functions. The using of the zero-valued-conditions fasciate these investigations and avoid any conflicts of the components.
Sun et al. [46] investigated the required conditions for confirming the existence and uniqueness of the solution to a nonlinear fractional differential equation (FDE) whose nonlinearity involves an explicit fractional derivative using Avery–Anderson–Henderson fixed point theorem. Devi et al. [44] studied the existence and uniqueness along with the Ulam–Hyers (UH) stability of positive solution of general nonlinear FDEs containing p-Laplacian operator. The authors of [47] turned to the existence and multiplicity of positive solutions for a system consisting of Riemann–Liouville FDEs equipped with the p-Laplacian operators and singular nonnegative nonlinearities, and also furnished with nonlocal boundary conditions which possess the integrals of Riemann–Stieltjes type. The existence criterion and its stability of a hybrid fractional differential equation with fractional integral, fractional derivative in the Caputo sense, and p-Laplacian operator are also investigated in the research article by Al-Sadi et al. [48].
In this article, we focus on the fractional integro-differential boundary value problem of a hybrid system given as
where \(m_{1},n_{0}\neq 0\), \(D^{\alpha }\) denotes the Caputo fractional derivative and \(f,g,h_{k}:[t_{0},T]\times \mathbb{R} \rightarrow \mathbb{R}\), \(k=0,1,\ldots,n-1\), are given continuous functions, and \(\delta _{0}, \dots , \delta _{n-1}>0\). Note that the novelty of the paper in the above system is that we shall investigate the qualitative criteria for two positive solutions to a new hybrid system with the finite number of integro-differential boundary conditions in terminal points with the help of a complicated case of fixed point techniques due to Avery–Anderson–Henderson. By taking different values for existing parameters and functions, one can get some well-known FDEs studied in the previous research works.
The other five sections of the manuscript are summarized as follows: In Sect. 2, we offer basic preliminaries of results in fractional calculus and fixed point theories. In Sect. 3, the solution of the fractional linear model of (1) is obtained. Therefore, an application of Banach fixed point theorem on the integral solution for system (1) implies the existence of one and only one solution of the system. In Sect. 4, to apply Avery–Anderson–Henderson fixed point theorem, we obtain sufficient criteria and conditions for the positivity and existence of at least two of them for fractional hybrid system (1) by showing the complete continuity of the operator that represents the integral solution of the system. In Sect. 5, the UH stability of the solution is investigated and sufficient conditions for this kind of stability are obtained. Finally, in Sect. 6, we design an example to ensure the consistency of the results. The conclusion section closes this paper.
2 Preliminaries and notations
We again introduce several specifications and facts about fractional calculus and topics of fixed point theorems.
Definition 2.1
([49])
The Riemann–Liouville (left-sided) fractional integral of a real-valued function \(\phi \in C[t_{0},T]\) is introduced as
if it exists.
Definition 2.2
([49])
The Caputo fractional derivative of order \(q \in (n-1,n]\) for \(\phi \in C^{n}[t_{0},T]\) is defined as
if it exists.
Lemma 2.3
([49])
Let \(n-1< q < n\), then
for \(a_{k}\in \mathbb{R} \), \(k=0,1,\ldots ,n-1\).
For example, the γth Caputo derivative of \(\phi (t)=(t-t_{0})^{\zeta }\) is given by
Theorem 2.4
(Banach principle [24])
Let \((E, \Vert \cdot \Vert )\) be a Banach space and Ω be a closed and bounded subset of E. If \(\Psi :\Omega \rightarrow \Omega \) is a contraction operator, then Ψ has a unique fixed point in Ω. We mean by a contraction that it is an operator Ψ that satisfies
Theorem 2.5
(Avery–Anderson–Henderson theorem [43])
Consider \((E, \Vert \cdot \Vert )\) as a Banach space, \(P\in E\) as a cone, and μ and ϕ as two increasing nonnegative continuous functionals on P, and let ω be a nonnegative continuous functional on P with \(\omega (0)=0\) provided that, for some \(r_{3}>0\) and \(M>0\), the inequalities \(\phi (x)\leq \omega (x)\leq \mu (x)\) and \(\Vert x\Vert \leq M\phi (x)\) fulfill \(\forall x\in \overline{P(\phi ,r_{3})}\), in which \(P(\phi ,r_{3})=\{x\in P:\phi (x)< r_{3}\}\). Let positive numbers \(r_{1}< r_{2}< r_{3}\) exist so that \(\omega (lx)\leq l\omega (x)\) for \(0\leq l\leq 1\), and \(x\in \partial P(\omega ,r_{2})\). If \(\Psi :\overline{P(\phi ,r_{3})}\rightarrow P\) is an operator with the complete continuity property satisfying:
-
(C1)
\(\phi (\Psi x)>r_{3}\), \(\forall x\in \partial P(\phi ,r_{3})\);
-
(C2)
\(\omega (\Psi x)< r_{2}\), \(\forall x\in \partial P(\omega ,r_{2})\);
-
(C3)
\(P(\mu ,r_{1})\neq \emptyset \), and \(\mu (\Psi x)>r_{1}\), \(\forall x\in \partial P(\mu ,r_{1})\),
then Ψ admits at least two fixed points \(x_{1}\) and \(x_{2}\) provided that \(r_{1}<\mu (x_{1})\) with \(\omega (x_{1})< r_{2}\) and \(r_{2}<\omega (x_{2})\) with \(\phi (x_{2})< r_{3}\).
3 Results regarding unique solution
We obtain firstly a solution of the corresponding linear system of (1).
Theorem 3.1
Let \(m_{1},n_{0}\neq 0\) and \(\delta _{0}, \dots , \delta _{n-1}>0\). Then the linear hybrid fractional boundary value problem (FBVP)
has an integral solution in the following form:
Proof
Taking the fractional integral to both sides of differential equation (3) and using Lemma 2.3, we get
By definition of the Caputo derivative of the fractional order \(\rho _{i}<\alpha \), and using equation (2), it becomes
Particularly, we find
and
Then
and
Also, we have
and
The boundary condition \(m_{0}x(t_{0})+n_{0}D^{\rho _{0}}x(T)=I^{\delta _{0}}h_{0}(T)\) gives
and the condition \(m_{1}x(t_{0})+n_{1}D^{\rho _{1}}x(T)=I^{\delta _{1}}h_{1}(T)\) implies
The boundary condition \(x^{(k)}(t_{0})=I^{\delta _{k}}h_{k}(T)\), \(k=2,3,\ldots ,n-1\), implies that
Substituting (8) into (6) and (7), we obtain
and
Then
Substituting the constants \(c_{k}\), \(k=0,1,2,\ldots,n-1\), into (5), we obtain solution (4), and the proof is finished. □
If \(\rho _{0},\rho _{1}\in (0,1)\), then the integral solution (4) will be different. This case is explained in the next result.
Theorem 3.2
Let \(\frac{m_{0}n_{1}}{n_{0}m_{1}}\neq \frac{\Gamma (2-\rho _{1})}{\Gamma (2-\rho _{0})}(T-t_{0})^{\rho _{1}- \rho _{0}}\) and \(\delta _{0}, \dots , \delta _{n-1}>0\). Then the linear hybrid FBVP
has an integral solution in the following form:
Proof
We sketch the proof. Equation (7) will become
Solving equations (6) and (11) and taking into account (8), we deduce that
and
Substituting the constants \(c_{k}\), \(k=0,1,2,\ldots,n-1\), into (5), we obtain solution (10), and this completes our proof. □
Now, by Theorem 2.4, we prove the existence of a unique solution to system (1). For this, define an operator \(\Psi :{C([t_{0},T],\mathbb{R} )}\rightarrow {\mathbb{R} }\) provided
The required criterion for finding a unique solution of the nonlinear hybrid FBVP (1) is given in the next result.
Theorem 3.3
Assume
- [H1]:
-
f, \(g^{(k)}\), \(h_{k}:[t_{0},T]\times {C([t_{0},T], \mathbb{R})} \rightarrow \mathbb{R} \), \(k=0,1,\ldots ,n-1\), are continuous functions such that
$$ \textstyle\begin{cases} \vert f(t,\mathrm{u}_{1})-f(s,\mathrm{v}_{1}) \vert \leq {C}_{f} \vert \mathrm{u}_{1}-\mathrm{v}_{1} \vert , \\ \vert g^{(k)}(t,\mathrm{u}_{1})-g^{(k)}(s,\mathrm{v}_{1}) \vert \leq {C}_{g^{(k)}} \vert \mathrm{u}_{1}-\mathrm{v}_{1} \vert , \\ \vert h_{k}(t,\mathrm{u}_{1})-h_{k}(s,\mathrm{v}_{1}) \vert \leq {C}_{h_{k}} \vert \mathrm{u}_{1}-\mathrm{v}_{1} \vert ,\end{cases}$$where \(t,s\in {}[ t_{0},T]\), \(\mathrm{u}_{1},\mathrm{v}_{1}\in {C([t_{0},T], \mathbb{R})}\), \({C}_{f}\), \({C}_{g^{(k)}}\), \({C}_{h_{k}}\) are nonnegative constants.
Then the hybrid FBVP system (1) admits one and only one solution provided that \(\Delta <1 \), where
Proof
Let Ω be any closed bounded subset of E. Then the continuity of Ψ is followed by that of constitutive functions and Lebesgue dominated convergence theorem. By enlarging the set Ω, one can deduce that Ψ maps Ω into itself. We need to show the contraction property of the operator Ψ. For this, let \(x,y\in \Omega \), then
where Δ is introduced in (12). Thus, the operator Ψ satisfies the contraction property of Banach fixed point Theorem 2.4 with constant \(\Delta <1\). Hence Ψ admits a fixed point in Ω, which is the same unique solution of hybrid FBVP (1). □
4 Results regarding two positive solutions
We establish sufficient conditions of the existence of two positive solutions of system (1) with the help of the existing hypotheses on cones presented in Theorem 2.5 due to Avery–Anderson–Henderson.
Define the cone \(P= \{ {x\in C([t_{0},T], \mathbb{R} ):x(t)\geq 0} \} \). We want to obtain firstly sufficient conditions to make \(\Psi x\in P\), whenever \(x\in P\).
The next assumptions are essential for the coming results.
- [H2]:
-
Assume that
$$ \textstyle\begin{cases} m_{0},n_{1}\leq 0,\quad m_{1},n_{0}>0, \\ \frac{\Gamma (3-\rho _{1})}{\Gamma (3-\rho _{0})}< \frac{m_{0}n_{1}}{n_{0}m_{1}}(T-t_{0})^{\rho _{0}-\rho _{1}}, \\ f(t,x(t))\geq 0, \\ g(t,x(t))\geq g(t_{0},x(t_{0}))\geq 0,\quad t\in {}[ t_{0},T], \\ \frac{n_{1}m_{0}}{m_{1}n_{0}}I^{\alpha -\rho _{1}}f(T,x(T))\geq I^{ \alpha -\rho _{0}}f(T,x(T)), \\ Kg(\kappa ,x(\kappa ))\geq \frac{n_{1}m_{0}}{m_{1}n_{0}}D^{\rho _{1}}g(T,x(T)) \geq D^{\rho _{0}}g(T,x(T))\geq 0,\quad K>0,\kappa \in {}[ t_{0},T], \\ 0\leq {g^{(k)}(t_{0},x(t_{0}))\leq I^{\delta _{k}}{h_{k}}(T,x(T))},\quad k=2,3,\ldots ,n-1.\end{cases}$$ - [H3]:
-
f, \(g^{(k)}\), \(h_{k}:[t_{0},T]\times P\rightarrow \mathbb{R} \), \(k=0,1,\ldots ,n-1\), are all bounded and continuous functions. Moreover, for \(r>0\), there exist a positive real number \({L}_{r}\) and a continuous nonnegative function \(\mathfrak{g}\) so that \(\forall (t,x),(s,y)\in {}[ t_{0},T]\times {}[ 0,r]\),
$$ \bigl\vert g(t,x)-g(s,y) \bigr\vert \leq {L}_{r} \bigl\vert \mathfrak{g}(x)-\mathfrak{g}(y) \bigr\vert . $$
Lemma 4.1
If [H2] holds and \(x\in P\), then \(\Psi x\in P\).
Proof
For any \(t\in {}[ t_{0},T]\), let
where
and
We first need to show that \(\xi _{k}(t)\leq 0\), that is,
Therefore, it suffices to show that \(C_{1}+(C_{3}-C_{1}C_{2})(t-t_{0})\leq 0 \). Notice that \(\xi _{k}(t_{0})=C_{1}<0\), then we need to show that \(C_{3}\leq C_{1}C_{2}\), that is,
By induction on k, it is obvious, by assumption, that it is true for \(k=2\). We assume it is true for the case k and show it for the case \(k+1\). We have
We deduce now
The remainder of the proof is obvious by the given assumptions. Hence, the result follows. □
Lemma 4.2
If [H2] and [H3] are fulfilled, then \(\Psi :P\rightarrow P\) admits the complete continuity property.
Proof
Define a bounded subset \(B_{r}=\{x\in P:x(t)\leq r\}\) of P, and let
for positive constants \(L_{f}\), \(L_{g^{(k)}}\), and \(L_{h_{k}}\), \(k=0,1,2,\ldots ,n-1\). The proof consists of three steps.
(Step 1) Ψ is a continuous operator.
Assuming \(x\in P\), then by Lemma 4.1, \(\Psi x\in P\) which implies that \(\Psi :P\rightarrow P\). Let \(\{x_{m}\}\) be a sequence in the cone P such that \(\lim_{m\rightarrow \infty }x_{m}=x\) in P. The continuity of f, \(g^{(k)}\), and \(h_{k}\) implies that \(\lim_{m\rightarrow \infty }f(t,x_{m}(t))=f(t,x(t))\), \(\lim {}_{m\rightarrow \infty }g^{(k)}(t,x_{m}(t))=g^{(k)}(t,x(t))\), and \(\lim_{m\rightarrow \infty }h_{k}(t,x_{m}(t))=h_{k}(t,x(t))\). In this case, by the dominated convergence theorem,
Thus, Ψ is a continuous operator.
(Step 2) The operator Ψ is uniformly bounded. \(\forall t\in {}[ t_{0},T]\), we get
Hence, Ψ maps a bounded set \(B_{r}\) into a uniformly bounded subset of P.
(Step 3) \(\Psi B_{r}\) is an equicontinuous set in P. Let \(x\in B_{r}\) and \(t_{2},t_{1}\in {}[ t_{0},T]\) such that \(t_{1}< t_{2}\), then
If \(t_{2}-t_{1}\rightarrow 0\), then \(|\Psi x(t_{2})-\Psi x(t_{1})|\rightarrow 0\) independently of the values of x. Hence, \(\Psi B_{r} \) is equicontinuous. By the means of the Arzela–Ascoli theorem, we follow that \(\Psi :P\rightarrow P\) is completely continuous. □
We now show the existence of at least two solutions for the hybrid FBVP (1).
For simplifications, we use the following notations in the coming results:
and
where the involved constants exist and are positive.
Theorem 4.3
Let [H2] and [H3] hold. If there exist \(0< r_{1}< r_{2}< r_{3}\) satisfying
-
(i)
\(f(t,x)>n_{f}\frac{r_{3}}{R_{3}}\), \(h_{i}(t,x)>n_{h_{i}}\frac{r_{3}}{R_{3}}\), \(i=0\), \(1,t\in {}[ \tau ,T]\), \(x\geq r_{3}\),
-
(ii)
\(f(t,x)\leq M_{f}\frac{r_{2}}{R_{2}}\), \(h_{i}(t,x)\leq M_{h_{k}}\frac{r_{2}}{R_{2}}\), \(i=0,1\), \((t,x)\in {}[ t_{0},T]\times {}[ 0,r_{2}]\), and
-
(iii)
\(f(t,x)>m_{f}\frac{r_{1}}{R_{1}}\), \(h_{i}(t,x)>m_{h_{i}}\frac{r_{1}}{R_{1}}\), \(i=0\), \(1,(t,x)\in {}[ t_{0},T]\times {}[ 0,r_{1}]\),
then hybrid system (1) possesses at least two positive solutions \(x_{1}\) and \(x_{2}\) provided that \(r_{1}<\Vert x_{1}\Vert \) with \(\Vert x_{1}\Vert < r_{2}\) along with \(r_{2}<\Vert x_{2}\Vert \) with \(\min_{t\in {}[ \tau ,T]}x(t)< r_{3}\).
Proof
Let \(\tau \in {}[ t_{0},T)\) and define the functionals ϕ, ω, and μ on the cone P such that
It is obvious that \(\omega (0)=0\), \(\omega (lx)= \vert l \vert \omega (x)\), and \(\phi (x)\leq \omega (x)\leq \mu (x)\). Using Lemmas 4.1and 4.2, we have \(\Psi :\overline{P(\phi ,r_{3})}\rightarrow P\) is completely continuous. We start with the first condition of Theorem 2.5, namely \(\phi (\Psi x)>r_{3}\) for all \(x\in \partial P(\phi ,r_{3})\). Let \(\min_{t\in {}[ \tau ,T]}x(t)=r_{3}\), we get
Next, we check the second condition. Let \(x\in \partial P(\omega ,r_{2})\), then
This shows that \(\omega (\Psi x)=\Vert \Psi x\Vert < r_{2}\). Since \(0\in P\) and \(r_{1}>0\), hence \(P(\mu ,r_{1})\neq \emptyset \). By assuming \(x\in \partial P(\mu ,r_{1})\), we have \(0\leq x(t)\leq r_{1}\), \(\forall t\in {}[ t_{0},T]\). By assumption (iii), we have
Hence, \(\mu (\Psi x)>r_{1}\). All the conditions of Theorem 2.5 are established, and the desired result follows. □
5 Hyers–Ulam stability
The notion of the stability of functional differential equations was first introduced by Ulam [50], and then it was extended by Hyers [51]. Later on, this type of stability and its generalization were called of Hyers–Ulam (HU) and Hyers–Ulam–Rassias (HUR) type, respectively. Investigation of the UH and GUH stability has been given a special attention in studying all fractional differential equations. Here, we discuss the Hyers–Ulam (HU) and Hyers–Ulam-Rassias (HUR) stability results about the hybrid FBVP (1) on the interval \([t_{0},T]\).
Definition 5.1
System (1) is Hyers–Ulam stable whenever for every \(\epsilon >0\) and \(y\in {C([t_{0},T],\mathbb{R})}\) satisfying
there exists \(x(t)\) as a solution of (1) such that
where C is independent of both y and x.
Definition 5.2
System (1) is Hyers–Ulam–Rassias stable if \(\forall y\in {C([t_{0},T],\mathbb{R} )}\) satisfying
where \(\varphi :[t_{0},T]\rightarrow \mathbb{R}\) is continuous, there is \(x(t)\) as a solution of (1), provided
where C is independent of both y and x.
For simplification, set
Remark 5.1
\(y\in {C([t_{0},T],\mathbb{R} )}\) is a solution of (13) iff we can find \(\hbar \in {C([t_{0},T],\mathbb{R} )}\) so that
-
(1)
\(\vert \hbar (t) \vert \leq \epsilon \), \(t\in {}[ t_{0},T]\);
-
(2)
y satisfies the equation
$$ y(t)=\Theta \bigl(y,\hbar (t)+f \bigl(t,y(t) \bigr) \bigr)=\Theta \bigl(y,f \bigl(t,y(t) \bigr) \bigr)+I^{\alpha } \hbar (t). $$(15)
A similar remark can be obtained on considering inequality (14).
Lemma 5.3
A function \(y\in {C([t_{0},T],\mathbb{R} )}\) satisfying (13) also satisfies the following integral inequality:
Proof
According to Remark 5.1, y satisfies equation (15). As a result,
□
Theorem 5.4
If [H1] is fulfilled, hybrid system (1) is Hyers–Ulam stable, provided that \(\Delta <1\).
Proof
Take \(\epsilon >0\) and \(y\in {C([t_{0},T],\mathbb{R} )}\) satisfying (13), and let \(x\in {C([t_{0},T],\mathbb{R} )}\) be the unique solution of (1). Thus,
Hence,
where
Hence, system (1) is Hyers–Ulam stable. □
Remark 5.2
The Hyers–Ulam-Rassias stability can be established in a similar manner.
6 Example
In this portion, we give an example to defend our pivot results of the theory attained above.
Example 6.1
Due to (1), regard the following fractional hybrid differential system:
where \(\alpha =2.5\), \(\rho _{0}=0.5\), \(\rho _{1}=1.5\), and \(\delta _{0}=\delta _{1}=\delta _{2}=2\). Since the functions f, \(h_{k}\), \(g^{(k)}\), \(k=0,1,2\), are Lipschitz with \({C}_{f}={C}_{h_{k}}={C}_{g^{(k)}}=0.01\). Therefore, we find that
Hence, problem (16) has a unique solution by Theorem 3.3. Moreover, using Theorem 5.4, system (16) is Hyers–Ulam (–Rassias) stable with \(C=2.39\).
Let us, in particular, assume that
One can find figures of the functions \(f(t,x)\) and \(g(t,x)\) in [52]. The calculations of the basic conditions give the following:
Moreover, it is obvious that condition [H3] is also valid. To continue our investigation, we need to justify the conditions of Theorem 4.3. Let \(r_{1}=1\), \(r_{2}=5\), and \(r_{3}=6\). Also assume that
Then we find that
Therefore, hypotheses (i)–(iii) are satisfied, since if \(t\in {}[ 0.5,1]\) and \(x\geq 6\), we have
For \((t,x)\in {}[ 0,1]\times {}[ 0,5]\), we obtain
Finally, \(\forall (t,x)\in {}[ 0,1]\times {}[ 0,1]\), we get
Therefore, using Theorem 4.3, for system (16), the existence of at least two positive solutions \(x_{1}\) and \(x_{2}\) is guaranteed provided \(1<\Vert x_{1}\Vert \) with \(\Vert x_{1}\Vert <5\) and \(5<\Vert x_{2}\Vert \) with \(\min_{t\in {}[ 0.5,1]}x(t)<6\).
7 Conclusion
The fractional integro-differential boundary problem of a hybrid system is a generalization of many existing problems. Many basic expressions are gathered in this model such as hybrid model, fractional derivatives of any order, fractional intro-differential boundary conditions, etc. Based on some well-known fixed point theorems of operator theory and the technique of fractional nonlinear differential systems, the existence and uniqueness criteria for the considered system (1) have been obtained. To do this, we used some notions on cones and verified some inequalities. Likewise, under specific assumptions and conditions, we have found the Hyers–Ulam stability result regarding solutions of hybrid system (1). The future research may continue to develop many qualitative properties of a modified system with the very recent fractional derivatives containing nonsingular kernels.
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References
Herrmann, R.: Fractional Calculus. An Introduction for Physicists. World Scientific, Singapore (2011)
Lakshmikantham, V., Leela, S., Vasundhara, D.J.: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, London (2009)
Sun, H., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018). https://doi.org/10.1016/j.cnsns.2018.04.019
Ahmad, M., Zada, A., Alzabut, J.: Hyers–Ulam stability of a coupled system of fractional differential equations of Hilfer–Hadamard type. Demonstr. Math. 52(1), 283–295 (2019). https://doi.org/10.1515/dema-2019-0024
Ahmad, M., Zada, A., Alzabut, J.: Stability analysis of a nonlinear coupled implicit switched singular fractional differential system with p-Laplacian. Adv. Differ. Equ. 2019, 436 (2019). https://doi.org/10.1186/s13662-019-2367-y
Riaz, U., Zada, A., Ali, Z., Ahmad, M., Xu, J., Fu, Z.: Analysis of nonlinear coupled systems of impulsive fractional differential equations with Hadamard derivatives. Math. Probl. Eng. 2019, Article ID 5093572 (2019). https://doi.org/10.1155/2019/5093572
Baleanu, D., Aydogan, S.M., Mohammadi, H., Rezapour, S.: On modelling of epidemic childhood diseases with the Caputo-Fabrizio derivative by using the Laplace Adomian decomposition method. Alex. Eng. J. 59(5), 3029–3039 (2020). https://doi.org/10.1016/j.aej.2020.05.007
Bachir, F.S., Abbas, S., Benbachir, M., Benchora, M.: Hilfer–Hadamard fractional differential equations; existence and attractivity. Adv. Theory Nonlinear Anal. Appl. 5(1), 49–57 (2021). https://doi.org/10.31197/atnaa.848928
Baitiche, Z., Derbazi, C., Benchora, M.: ψ-Caputo fractional differential equations with multi-point boundary conditions by topological degree theory. Res. Nonlinear Anal. 3(4), 167–178 (2020)
Tuan, H.T., Mohammadi, H., Rezapour, S.: A mathematical model for COVID-19 transmission by using the Caputo fractional derivative. Chaos Solitons Fractals 140, 110107 (2020). https://doi.org/10.1016/j.chaos.2020.110107
Thabet, S.T.M., Etemad, S., Rezapour, S.: On a coupled Caputo conformable system of pantograph problems. Turk. J. Math. 45(1), 496–519 (2021). https://doi.org/10.3906/mat-2010-70
Mohammadi, H., Kumar, S., Rezapour, S., Etemad, S.: A theoretical study of the Caputo–Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control. Chaos Solitons Fractals 144, 110668 (2021). https://doi.org/10.1016/j.chaos.2021.110668
Baleanu, D., Jajarmi, A., Mohammadi, H., Rezapour, S.: A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative. Chaos Solitons Fractals 134, 109705 (2020). https://doi.org/10.1016/j.chaos.2020.109705
Rezapour, S., Ntouyas, S.K., Iqbal, M.Q., Hussain, A., Etemad, S., Tariboon, J.: An analytical survey on the solutions of the generalized double- order ϕ-integrodifferential equation. J. Funct. Spaces 2021, Article ID 6667757 (2021). https://doi.org/10.1155/2021/6667757
Shah, K., Khan, R.A.: Iterative scheme for a coupled system of fractional-order differential equations with three-point boundary conditions. Math. Methods Appl. Sci. 41(3), 1047–1053 (2018). https://doi.org/10.1002/mma.4122
Matar, M.M.: Existence of solution for fractional neutral hybrid differential equations with finite delay. Rocky Mt. J. Math. 50(6), 2141–2148 (2020). https://doi.org/10.1216/rmj.2020.50.2141
Matar, M.M.: Qualitative properties of solution for hybrid nonlinear fractional differential equations. Afr. Math. 30, 1169–1179 (2019). https://doi.org/10.1007/s13370-019-00710-2
Baleanu, D., Etemad, S., Rezapour, S.: On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators. Alex. Eng. J. 59(5), 3019–3027 (2020). https://doi.org/10.1016/j.aej.2020.04.053
Baleanu, D., Mohammadi, H., Rezapour, S.: Analysis of the model of HIV-1 infection of CD4+ T-cell with a new approach of fractional derivative. Adv. Differ. Equ. 2020, 71 (2020). https://doi.org/10.1186/s13662-020-02544-w
Amara, A.: Existence results for hybrid fractional differential equations with three-point boundary conditions. AIMS Math. 5(2), 1074–1088 (2020). https://doi.org/10.3934/math.2020075
Tuan, H.T., Ganji, R.M., Jafari, H.: A numerical study of fractional rheological models and fractional Newell-Whitehead-Segel equation with non-local and non-singular kernel. Chinese J. Phys. 68, 308–320 (2020). https://doi.org/10.1016/j.cjph.2020.08.019
Matar, M.M.: Approximate controllability of fractional nonlinear hybrid differential systems via resolvent operators. J. Math. 2019, Article ID 8603878 (2019). https://doi.org/10.1155/2019/8603878
Baleanu, D., Etemad, S., Rezapour, S.: A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions. Bound. Value Probl. 2020, 64 (2020). https://doi.org/10.1186/s13661-020-01361-0
Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)
Nikan, O., Jafari, H., Golbabai, A.: Numerical analysis of the fractional evolution model for heat flow in materials with memory. Alex. Eng. J. 59(4), 2627–2637 (2020). https://doi.org/10.1016/j.aej.2020.04.026
Ganji, R.M., Jafari, H., Baleanu, D.: A new approach for solving multi variable orders differential equations with Mittag-Leffler kernel. Chaos Solitons Fractals 130, 109405 (2020). https://doi.org/10.1016/j.chaos.2019.109405
Abdeljawad, T., Karapinar, E., Pada, S.K., Mlaiki, N.: Solutions of boundary value problems on extended-Branciari b-distance. J. Inequal. Appl. 2020, 103 (2020). https://doi.org/10.1186/s13660-020-02373-1
Jarad, F., Alqudah, M.A., Abdeljawad, T.: On more general forms of proportional fractional operators. Open Math. 18(1), 167–176 (2020). https://doi.org/10.1515/math-2020-0014
Aksuy, U., Karapinar, E., Erhan, I.M.: Fixed point theorems in complete modular metric spaces and an application to anti-periodic boundary value problems. Filomat 31(17), 5475–5488 (2017). https://doi.org/10.2298/FIL1717475A
Lazreg, J.E., Abbas, S., Benchohra, M., Karapinar, E.: Impulsive Caputo–Fabrizio fractional differential equations in b-metric spaces. Open Math. 19(1), 167–176 (2021). https://doi.org/10.1515/math-2021-0040
Karapinar, E., Fulga, A., Rashid, M., Shahid, L., Aydi, H.: Large contractions on quasi-metric spaces with an application to nonlinear fractional differential equations. Mathematics 7(5), 444 (2019). https://doi.org/10.3390/math7050444
Alqahtani, B., Aydi, H., Karapinar, E., Rakocevic, V.: A solution for Volterra fractional integral equations by hybrid contractions. Mathematics 7(8), 694 (2019). https://doi.org/10.3390/math7080694
Abdeljawad, T., Agarwal, R.P., Karapinar, E., Kumari, P.S.: Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space. Symmetry 11(5), 686 (2019). https://doi.org/10.3390/sym11050686
Adiguzel, R.S., Aksoy, U., Karapinar, E., Erhan, I.M.: On the solutions of fractional differential equations via Geraghty type hybrid contractions. Appl. Comput. Math. Int. J. 20(2), 313–333 (2021)
Baleanu, D., Sajjadi, S.S., Jajarmi, A., Defterli, O., Asad, J.H.: The fractional dynamics of a linear triatomic molecule. Rom. Rep. Phys. 73(1), 105 (2021)
Jajarmi, A., Baleanu, D.: On the fractional optimal control problems with a general derivative operator. Asian J. Control 23(2), 1062–1071 (2021). https://doi.org/10.1002/asjc.2282
Sajjadi, S.S., Baleanu, D., Jajarmi, A., Pirouz, H.M.: A new adaptive synchronization and hyperchaos control of a biological snap oscillator. Chaos Solitons Fractals 138, 109919 (2020). https://doi.org/10.1016/j.chaos.2020.109919
Baleanu, D., Ghanbari, B., Asad, J.H., Jajarmi, A., Pirouz, H.M.: Planar system-masses in an equilateral triangle: numerical study within fractional calculus. Comput. Model. Eng. Sci. 124(3), 953–968 (2020). https://doi.org/10.32604/cmes.2020.010236
Jajarmi, A., Baleanu, D.: A new iterative method for the numerical solution of high-order nonlinear fractional boundary value problems. Front. Phys. 8, 220 (2020). https://doi.org/10.3389/fphy.2020.00220
Aydogan, S.M., Baleanu, D., Mousalou, A., Rezapour, S.: On high order fractional integro-differential equations including the Caputo–Fabrizio derivative. Bound. Value Probl. 2018, 90 (2018). https://doi.org/10.1186/s13661-018-1008-9
Baleanu, D., Mousalou, A., Rezapour, S.: On the existence of solutions for some infinite coefficient-symmetric Caputo–Fabrizio fractional integro-differential equations. Bound. Value Probl. 2017, 145 (2017). https://doi.org/10.1186/s13661-017-0867-9
Baleanu, D., Rezapour, S., Saberpour, Z.: On fractional integro-differential inclusions via the extended fractional Caputo–Fabrizio derivation. Bound. Value Probl. 2019, 79 (2019). https://doi.org/10.1186/s13661-019-1194-0
Avery, R., Anderson, D., Henderson, J.: Some fixed point theorems of Leggett–Williams type. Rocky Mt. J. Math. 41(2), 371–386 (2011)
Devi, A., Kumar, A., Baleanu, D., Khan, A.: On stability analysis and existence of positive solutions for a general non-linear fractional differential equations. Adv. Differ. Equ. 2020, 300 (2020). https://doi.org/10.1186/s13662-020-02729-3
Dong, X., Bai, Z., Zhang, W.: Positive solutions for nonlinear eigenvalue problems with conformable fractional differential derivatives. J. Shandong Univ. Sci. Technol. Nat. Sci. 2016(3), 85–91 (2016)
Sun, A., Su, Y., Yuan, Q., Li, T.: Existence of solutions to fractional differential equations with fractional-order derivative terms. J. Appl. Anal. Comput. 11(1), 486–520 (2021). https://doi.org/10.11948/20200072
Alsaedi, A., Luca, R., Ahmad, B.: Existence of positive solutions for a system of singular fractional boundary value problems with p-Laplacian operators. Mathematics 8(11), 1890 (2020). https://doi.org/10.3390/math8111890
Al-Sadi, W., Zhenyou, H., Alkhazzan, A.: Existence and stability of a positive solution for nonlinear hybrid fractional differential equations with singularity. J. Taibah Univ. Sci. 13(1), 951–960 (2019). https://doi.org/10.1080/16583655.2019.1663783
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Philadelphia (1993)
Ulam, S.M.: A Collection of Mathematical Problems. Interscience, New York (1960)
Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27(4), 222–224 (1941). https://doi.org/10.1073/pnas.27.4.222
Boutiara, A., Etemad, S., Alzabut, J., Hussain, A., Subramanian, M., Rezapour, S.: On a nonlinear sequential four-point fractional q-difference equation involving q-integral operators in boundary conditions along with stability criteria. Adv. Differ. Equ. 2021, 367 (2021). https://doi.org/10.1186/s13662-021-03525-3
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The fifth and sixth authors would like to thank Azarbaijan Shahid Madani University. The authors would like to thank dear referees for their constructive and valuable comments.
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Matar, M.M., abu Jarad, M., Ahmad, M. et al. On the existence and stability of two positive solutions of a hybrid differential system of arbitrary fractional order via Avery–Anderson–Henderson criterion on cones. Adv Differ Equ 2021, 423 (2021). https://doi.org/10.1186/s13662-021-03576-6
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DOI: https://doi.org/10.1186/s13662-021-03576-6