A collaborative prediction approach to defend against amplified reflection and exploitation attacks

Arvind Prasad; Shalini Chandra; Ibrahim Atoum; Naved Ahmad; Yazeed Alqahhas; Arvind Prasad; Shalini Chandra; Ibrahim Atoum; Naved Ahmad; Yazeed Alqahhas

doi:10.3934/era.2023308

Electronic Research Archive

2023, Volume 31, Issue 10: 6045-6070. doi: 10.3934/era.2023308

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Research article Special Issues

A collaborative prediction approach to defend against amplified reflection and exploitation attacks

1.
Department of Computer Science, BBA University, Lucknow, India
2.
Department of Computer Science and Information Systems, AlMaarefa University, Riyadh, Saudi Arabia

Received: 10 April 2023 Revised: 17 August 2023 Accepted: 23 August 2023 Published: 07 September 2023

An amplified reflection and exploitation-based distributed denial of service (DDoS) attack allows an attacker to launch a volumetric attack on the target server or network. These attacks exploit network protocols to generate amplified service responses through spoofed requests. Spoofing the source addresses allows attackers to redirect all of the service responses to the victim's device, overwhelming it and rendering it unresponsive to legitimate users. Mitigating amplified reflection and exploitation attacks requires robust defense mechanisms that are capable of promptly identifying and countering the attack traffic while maintaining the availability and integrity of the targeted systems. This paper presents a collaborative prediction approach based on machine learning to mitigate amplified reflection and exploitation attacks. The proposed approach introduces a novel feature selection technique called closeness index of features (CIF) calculation, which filters out less important features and ranks them to identify reduced feature sets. Further, by combining different machine learning classifiers, a voting-based collaborative prediction approach is employed to predict network traffic accurately. To evaluate the proposed technique's effectiveness, experiments were conducted on CICDDoS2019 datasets. The results showed impressive performance, achieving an average accuracy, precision, recall and F1 score of 99.99%, 99.65%, 99.28% and 99.46%, respectively. Furthermore, evaluations were conducted by using AUC-ROC curve analysis and the Matthews correlation coefficient (MCC) statistical rate to analyze the approach's effectiveness on class imbalance datasets. The findings demonstrated that the proposed approach outperforms recent approaches in terms of performance. Overall, the proposed approach presents a robust machine learning-based solution to defend against amplified reflection and exploitation attacks, showcasing significant improvements in prediction accuracy and effectiveness compared to existing approaches.

Keywords:

Citation: Arvind Prasad, Shalini Chandra, Ibrahim Atoum, Naved Ahmad, Yazeed Alqahhas. A collaborative prediction approach to defend against amplified reflection and exploitation attacks[J]. Electronic Research Archive, 2023, 31(10): 6045-6070. doi: 10.3934/era.2023308

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Abstract

1. Introduction

The models of the differential and integral equations have been appeared in different applications (see ^{[1,2,3,4,6,7,9,12,13,14,15,16,17,18,19,20]}).

Boundary value problems involving fractional differential equations arise in physical sciences and applied mathematics. In some of these problems, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed. It is sometimes better to impose nonlocal conditions since the measurements needed by a nonlocal condition may be more precise than the measurement given by a local condition. Consequently, a variety of excellent results on fractional boundary value problems (abbreviated BVPs) with resonant conditions have been achieved. For instance, Bai ^[4] studied a type of fractional differential equations with m-points boundary conditions. The existence of nontrivial solutions was established by using coincidence degree theory. Applying the same method, Kosmatov ^[17] investigated the fractional order three points BVP with resonant case.

Although the study of fractional BVPs at resonance has acquired fruitful achievements, it should be noted that such problems with Riemann-Stieltjes integrals are very scarce, so it is worthy of further explorations. Riemann-Stieltjes integral has been considered as both multipoint and integral in a single framework, which is more common, see the relevant works due to Ahmad et al. ^[1].

The boundary value problems with nonlocal, integral and infinite points boundary conditions have been studied by some authors (see, for example ^[8,10,11,12]).

Here, we discuss the boundary value problem of the nonlinear differential inclusions of arbitrary (fractional) orders

$\begin{equation} \frac{dx}{dt}\in F_{1}(t,x(t),I^{\gamma}f_{2}(t,D^{\alpha}x(t))),\; \; \alpha,\gamma\in(0,1] \; \; \; \; t\in(0,1) \end{equation}$

(1.1)

with the nonlocal boundary condition

$\begin{equation} {{\sum\limits^{m}_{k = 1} a_{k}}} x(\tau_{k} ) = x_{0},\; a_{k} > 0\; \; \tau_{k}\in[0,1], \end{equation}$

(1.2)

the integral condition

$\begin{equation} \int_{0}^{1}x(s)dg(s) = x_{0} \end{equation}$

(1.3)

and the infinite point boundary condition

$\begin{equation} {{\sum\limits^{\infty}_{k = 1} a_{k}}} x(\tau_{k} ) = x_{0},\; a_{k} > 0\; and\; \tau_{k}\in[0,1]. \end{equation}$

(1.4)

We study the existence of solutions $x\in C[0, 1]$ of the problems (1.1) and (1.2), and deduce the existence of solutions of the problem of (1.1) with the conditions (1.3) and (1.4). Then the existence of the maximal and minimal solutions will be proved. The sufficient condition for the uniqueness and continuous dependence of the solution will be studied.

This paper is organised as: In Section 2, we prove the existence of continuous solutions of the problems (1.1) and (1.2), and deduce the existence of solutions of the problem of (1.1) with the conditions (1.3) and (1.4). In Section 3, the existence of the maximal and minimal solutions is proved. In Section 4, the sufficient condition for the uniqueness and continuous dependence of the solution are studied. Next, in Section 5, we extend our results to the nonlocal problems (1.3) and (2.1). Finally, some existence results is proved for the nonlocal problems (1.4) and (2.1) in Section 6.

2. Main results

Consider the following assumptions:

(I) (i) The set $F_{1}(t, x, y)$ is nonempty, closed and convex for all $(t, x, y)\in [0, 1]\times R\times R$ .

(ii) $F_{1}(t, x, y)$ is measurable in $t\in [0, 1]$ for every $x, y\in R$ .

(iii) $F_{1}(t, x, y)$ is upper semicontinuous in $x$ and $y$ for every $t\in [0, 1]$ .

(iv) There exist a bounded measurable function $a_{1}: [0, 1]\longrightarrow R$ and a positive constant $K_{1}$ , such that

$\begin{eqnarray*} \|F_{1}(t,x,y)\|& = &\sup\{|f_{1}|:f_{1}\in F_{1}(t,x,y)\}\\ &\leq& |a_{1}(t)|+K_{1}(|x|+|y|). \end{eqnarray*}$

Remark 2.1. From the assumptions (i)–(iv) we can deduce that (see ^[3,6,7,13]) there exists $f_{1}\in F_{1}(t, x, y)$ , such that

(v) $f_{1}:[0, 1]\times R\times R\longrightarrow R$ is measurable in $t$ for every $x, y\in R$ and continuous in $x, y$ for $t\in [0, 1]$ , and there exist a bounded measurable function $a_{1}:[0, 1]\rightarrow R$ and a positive constant $K_{1} > 0$ such that

$|f_{1}(t,x,y)|\leq |a_{1}(t)|+K_{1}(|x|+|y|),$

and the functional $f_{1}$ satisfies the differential equation

$\begin{equation} \frac{dx}{dt} = f_{1}(t,x(t),I^{\gamma}f_{2}(t,D^{\alpha}x(t))),\; \; \alpha,\gamma\in(0,1]\; and\; t\in(0,1]. \end{equation}$

(2.1)

(II) $f_{2}:[0, 1]\times R\longrightarrow R$ is measurable in $t$ for any $x\in R$ and continuous in $x$ for $t\in [0, 1]$ , and there exist a bounded measurable function $a_{2}:[0, 1]\rightarrow R$ and a positive constant $K_{2} > 0$ such that

$|f_{2}(t,x)|\leq |a_{2}(t)|+K_{2}|x|,\; \forall t\in [0,1]\; \; \mbox {and}\; x\in R$

and

$\sup\limits_{t\in [0,1]}|a_{i}(t)|\leq a_{i}, i = 1,2.$

(III) $2K_{1}\gamma+K_{1}K_{2}\alpha < \alpha\gamma\Gamma(2-\alpha), \; \alpha, \gamma \in (0, 1].$

Remark 2.2. From (I) and (v) we can deduce that every solution of (1.1) is a solution of (2.1). Now, we shall prove the following lemma.

Lemma 2.1. If the solution of the problems (1.2)–(2.1) exists then it can be expressed by the integral equation

$\begin{eqnarray} y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\limits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}} {\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}} {\Gamma(\gamma)}f_{2}(\phi,y(\theta))d\theta)ds. \end{eqnarray}$

(2.2)

Proof. Consider the boundary value problems (1.2)–(2.1) be satisfied. Operating by $I^{1-\alpha}$ on both sides of (2.1) we can obtain

$\begin{equation} D^{\alpha}x(t) = I^{1-\alpha}\frac{dx}{dt} = I^{1-\alpha} f_{1}(t,x(t),I^{\gamma}f_{2}(t,D^{\alpha}x(t))). \end{equation}$

(2.3)

Taking

$\begin{equation} D^{\alpha}x(t) = y(t), \end{equation}$

(2.4)

then we obtain

$\begin{equation} x(t) = x(0)+I^{\alpha}y(t). \end{equation}$

(2.5)

Putting $t = \tau$ and multiplying (2.5) by $\Sigma_{k = 1}^{m}a_{k}$ , then we get

$\begin{equation} \Sigma_{k = 1}^{m}a_{k}x(\tau_{k}) = \Sigma_{k = 1}^{m}a_{k}x(0)+\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k}), \end{equation}$

(2.6)

$\begin{equation} x_{0} = \Sigma_{k = 1}^{m}a_{k}x(0)+\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k}) \end{equation}$

(2.7)

and

$\begin{equation} x(0) = \frac{1}{\Sigma_{k = 1}^{m}a_{k}}[x_{0}-\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k})]. \end{equation}$

(2.8)

Then

$\begin{equation} x(t) = \frac{1}{\Sigma_{k = 1}^{m}a_{k}}[x_{0}-\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k})]+I^{\alpha}y(t). \end{equation}$

(2.9)

Substituting (2.8) and (2.9) in (2.5), which completes the proof.

Theorem 2.1. Let assumptions (I)–(III) be satisfied. Then the integral equation (2.2) has at least one continuous solution.

Proof. Define a set $Q_{r}$ as

$Q_{r} = \{y\in C[0,1]:\|y\|\leq r\},\; \;$

$\begin{eqnarray*} r = \frac{\Gamma(\gamma+1)[a_{1}+K_{1}A|x_{0}|]+k_{1}a_{2}}{\alpha\gamma\Gamma(2-\alpha)-[2K_{1}\alpha+K_{1}K_{2}\gamma]},\; \; \; (\sum\limits^{m}_{k = 1}a_{k})^{-1} = A, \end{eqnarray*}$

and the operator $F$ by

$\begin{eqnarray*} Fy(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\limits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y(\theta))d\theta)ds. \end{eqnarray*}$

For $y\in Q_{r}$ , then

$\begin{eqnarray*} |Fy(t)|& = &|\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\limits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}} {\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}} {\Gamma(\gamma)}f_{2}(\theta,y(\theta))d\theta)ds|\\ &\leq& K_{1}\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)}[\frac{|x_{0}|}{\sum\limits^{m}_{k = 1}a_{k}}+\frac{\sum\limits^{m}_{k = 1} a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta}{\sum\limits^{m}_{k = 1}a_{k}}\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta+\int_{0}^{s}\frac{(s-\phi)^{\gamma-1}} {\Gamma(\gamma)}(K_{2}|y(\theta)|+|a_{2}(t)|)d\theta]+|a_{1}(t)|)ds\\ &\leq& \frac{1}{\Gamma(2-\alpha)}[K_{1}A|x_{0}|+\frac{K_{1}\|y\|}{\Gamma(\alpha+1)}+K_{1}\frac{\|y\|}{\Gamma(\alpha+1)} +\frac{K_{1}K_{2}\|y\|}{\Gamma(\gamma+1)}+\frac{K_{1}a_{2}}{\Gamma(\gamma+1)}+a_{1}]\\ &\leq& \frac{1}{\Gamma(2-\alpha)}[K_{1}A|x_{0}|+\frac{K_{1}r}{\Gamma(\alpha+1)}+K_{1}\frac{r}{\Gamma(\alpha+1)}+ \frac{K_{1}K_{2}r}{\Gamma(\gamma+1)}+\frac{K_{1}a_{2}}{\Gamma(\gamma+1)}+a_{1}]\\ &\leq& \frac{1}{\Gamma(2-\alpha)}[K_{1}A|x_{0}|+\frac{2K_{1}r}{\Gamma(\alpha+1)}+\frac{K_{1}K_{2}r}{\Gamma(\gamma+1)}+ \frac{K_{1}a_{2}}{\Gamma(\gamma+1)}+a_{1}]\; \leq r. \end{eqnarray*}$

Thus, the class of functions $\{Fy\}$ is uniformly bounded on $Q_{r}$ and $F:Q_{r}\rightarrow Q_{r}$ . Let $y\in Q_{r}$ and $t_{1}, t_{2}\in [0, 1]$ such that $|t_{2}-t_{1}| < \delta$ , then

$\begin{eqnarray*} &&|Fy(t_{2})-Fy(t_{1})|\\ & = &|\int_{0}^{t_{2}}\frac{(t_{2}-s)^{-\alpha}}{\Gamma(1-\alpha)} (f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds-\int_{0}^{t_{1}}\frac{(t_{1}-s)^{-\alpha}}{\Gamma(1-\alpha)} (f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds|\\ &\leq&|\int_{0}^{t_{2}}\frac{(t_{2}-s)^{-\alpha}}{\Gamma(1-\alpha)} (f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds-\int_{0}^{t_{1}}\frac{(t_{2}-s)^{-\alpha}}{\Gamma(1-\alpha)} (f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds\\ &&+\int_{0}^{t_{1}}\frac{(t_{2}-s)^{-\alpha}}{\Gamma(1-\alpha)} (f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds-\int_{0}^{t_{1}}\frac{(t_{1}-s)^{-\alpha}}{\Gamma(1-\alpha)} (f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds|\\ &\leq&\int_{t_{1}}^{t_{2}}\frac{(t_{2}-s)^{-\alpha}}{\Gamma(1-\alpha)} |(f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))|ds\\ &&+\int_{0}^{t_{1}}\frac{(t_{2}-s)^{-\alpha}-(t_{1}-s)^{-\alpha}}{\Gamma(1-\alpha)} |(f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds|\\ &\leq&\int_{t_{1}}^{t_{2}}\frac{(t_{2}-s)^{-\alpha}}{\Gamma(1-\alpha)} |(f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))|ds\\ &&+\int_{0}^{t_{1}}\frac{(t_{2}-s)^{\alpha}-(t_{1}-s)^{\alpha}}{\Gamma(1-\alpha)(t_{2}-s)^{\alpha}(t_{1}-s)^{\alpha}}. |(f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))|ds. \end{eqnarray*}$

Thus, the class of functions $\{Fy\}$ is equicontinuous on $Q_{r}$ and $\{Fy\}$ is compact operator by the Arzela-Ascoli Theorem ^[5].

Now we prove that $F$ is continuous operator. Let ${y_{n}}\subset Q_{r}$ be convergent sequence such that $y_{n}\rightarrow y$ , then

$\begin{eqnarray*} Fy_{n}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{n}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{n}(\theta)d\theta, \int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y_{n}(\theta)))ds. \end{eqnarray*}$

Using Lebesgue dominated convergence Theorem ^[5] and assumptions (iv)–(II) we have

$\begin{eqnarray*} \lim\limits_{n\to \infty} Fy_{n}(t)& = &\lim\limits_{n\to \infty}\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{n}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{n}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y_{n}(\theta)))ds\\ & = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}\lim\limits_{n\to \infty}y_{n}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}\lim\limits_{n\to \infty}y_{n}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,\lim\limits_{n\to \infty}y_{n}(\theta)))ds\\ & = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2} (\theta,y(\theta)))ds\; = \; Fy(t). \end{eqnarray*}$

Then $F:Q_{r}\rightarrow Q_{r}$ is continuous, and by Schauder Fixed Point Theorem^[5] there exists at least one solution $y\in C[0, 1]$ of (2.2). Now

$\begin{eqnarray*} \frac{dx}{dt}& = &\frac{d}{dt}[x(0)+I^{\alpha}y(t)]\\ & = &\frac{d}{dt}I^{\alpha}I^{1-\alpha}f_{1}(t,x(t),I^{\gamma}f_{2}(t,y(t)))\\ & = &\frac{d}{dt}If_{1}(t,x(t),I^{\gamma}f_{2}(t,y(t)))\\ & = &f_{1}(t,x(t),I^{\gamma}f_{2}(t,y(t))). \end{eqnarray*}$

Putting $t = \tau$ and using (2.9), we obtain

$\begin{eqnarray*} &&x(t) = \frac{1}{\Sigma_{k = 1}^{m}a_{k}}[x_{0}-\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k})]+I^{\alpha}y(t),\\ &&\Sigma_{k = 1}^{m}a_{k}x(\tau_{k}) = \Sigma_{k = 1}^{m}a_{k}\frac{1}{\Sigma_{k = 1}^{m}a_{k}}[x_{0}-\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k})]+\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k}), \end{eqnarray*}$

then

$\begin{eqnarray*} &&\Sigma_{k = 1}^{m}a_{k}x(\tau_{k}) = x_{0}, \; a_{k} > 0\; and\; \tau_{k}\in[0,1]. \end{eqnarray*}$

This proves the equivalence between the problems (1.2)–(2.1) and the integral equation (2.2). Then there exists at least one solution $y\in C[0, 1]$ of the problems (1.2)–(2.1).

3. Maximal and minimal solutions

Here, we shall study the maximal and minimal solutions for the problems (1.2) and (2.1). Let $y(t)$ be any solution of (2.2), let $u(t)$ be a solution of (2.2), then $u(t)$ is said to be a maximal solution of (2.2) if it satisfies the inequality

$y(t)\leq u(t),\; \; \; t\in[0,1].$

A minimal solution can be defined by similar way by reversing the above inequality.

Lemma 3.1. Let the assumptions of Theorem 2.1 be satisfied. Assume that $x(t)$ and $y(t)$ are two continuous functions on $[0, 1]$ satisfying

$\begin{eqnarray*} y(t)&\leq&\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y(\theta)))ds\; \; t\in[0,1],\\ x(t)&\geq&\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}x(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}x(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2} (\theta,x(\theta)))ds\; \; t\in[0,1], \end{eqnarray*}$

where one of them is strict.

Let functions $f_{1}$ and $f_{2}$ be monotonic nondecreasing in $y$ , then

$\begin{equation} y(t) < x(t),\; \; \; t > 0. \end{equation}$

(3.1)

Proof. Let the conclusion (3.1) be not true, then there exists $t_{1}$ with

$y(t_{1}) < x(t_{1}),\; \; \; t_{1} > 0\; \; \; \mbox{and}\; \; \; y(t) < x(t),\; \; \; 0 < t < t_{1}.$

Since $f_{1}$ and $f_{2}$ are monotonic functions in $y,$ then we have

$\begin{eqnarray*} y(t_{1})&\leq&\int_{0}^{t_{1}}\frac{(t_{1}-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y(\theta)))ds\\ & < &\int_{0}^{t_{1}}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}x(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}x(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2} (\theta,x(\theta)))ds\\ & < &x(t_{1}),\; t_{1}\in[0,1]. \end{eqnarray*}$

This contradicts the fact that $y(t_{1}) = x(t_{1})$ , then $y(t) < x(t)$ . This completes the proof.

For the existence of the continuous maximal and minimal solutions for (2.1), we have the following theorem.

Theorem 3.1. Let the assumptions of Theorem 2.1 be hold. Moreover, if $f_{1}$ and $f_{2}$ are monotonic nondecreasing functions in $y$ for each $t\in [0, 1]$ , then Eq (2.1) has maximal and minimal solutions.

Proof. First, we should demonstrate the existence of the maximal solution of (2.1). Let $\epsilon > 0$ be given. Now consider the integral equation

$\begin{eqnarray} y_{\epsilon}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1,\epsilon}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}} {\Gamma(\alpha)}y_{\epsilon}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_\epsilon(\theta)d\theta,\int_{0}^{s} \frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2,\epsilon}(\theta,y_{\epsilon}(\theta))d\theta \bigg)ds,\; \; t\in[0,1], \end{eqnarray}$

where

$f_{1,\epsilon}(s,x_{\epsilon}(s),y_{\epsilon}(s) ) = f_{1}(s,x_{\epsilon}(s),y_{\epsilon}(s))+\epsilon,$

$f_{2,\epsilon}(s,x_{\epsilon}(s)) = f_{2}(s,x_{\epsilon}(s))+\epsilon.$

For $\epsilon_{2} > \epsilon_{1},$ we have

$\begin{eqnarray} y_{\epsilon_{2}}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1,\epsilon_2}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{2}}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{2}}(\theta)d\theta,\int_{0}^{s} \frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2,\epsilon_{2}}(\theta,y_{\epsilon_{2}}(\theta))d\theta\bigg )ds,\; \; t\in[0,1], \end{eqnarray}$

$\begin{eqnarray} y_{\epsilon_{2}}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} \bigg[f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{2}}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{2}}(\theta)d\theta,\int_{0}^{s} \frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}(f_{2}(\theta,y_{\epsilon_{2}}(\theta))+\epsilon_{2})d\theta \bigg)+\epsilon_2\bigg]ds,\; t\in[0,1]. \end{eqnarray}$

Also

$\begin{eqnarray*} y_{\epsilon_{1}}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1,\epsilon_1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{1}}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{1}}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^ {\gamma-1}}{\Gamma(\gamma)}f_{2,\epsilon_{1}}(\theta,y_{\epsilon_{1}}(\theta))d\theta\bigg)ds,\\ y_{\epsilon_{1}}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} \bigg[f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\phi)^{\alpha-1}} {\Gamma(\alpha)}y_{\epsilon_{1}}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{1}}(\theta)d\theta,\int_{0}^{s} \frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}(f_{2}(\theta,y_{\epsilon_{1}}(\theta))+\epsilon_{1})d\theta \bigg)+\epsilon_1\bigg] ds \\ & > &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} \bigg[f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}} {\Gamma(\alpha)}y_{\epsilon_{2}}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{2}}(\theta)d\theta,\int_{0}^{s} \frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}(f_{2}(\theta,y_{\epsilon_{2}}(\tau))+\epsilon_{2})d\theta \bigg)+\epsilon_2\bigg]ds. \end{eqnarray*}$

Applying Lemma 3.1, we obtain

$y_{\epsilon_{2}} < y_{\epsilon_{1}},\; t\in[0,1].$

As shown before, the family of function $y_{\epsilon}(t)$ is equi-continuous and uniformly bounded, then by Arzela Theorem, there exist decreasing sequence $\epsilon_{n}$ , such that $\epsilon_{n}\rightarrow 0$ as $n\rightarrow \infty$ , and $u(t) = {\lim_{n\rightarrow \infty}}y_{\epsilon_{n}}(t)$ exists uniformly in $[0, 1]$ and denote this limit by $u(t)$ . From the continuity of the functions, $f_{2, \epsilon_n}(t, y_{\epsilon_n}(t))$ , we get $f_{2, \epsilon_n}(t, y_{\epsilon_n}(t))\longrightarrow f_{2}(t, y(t))$ as $n\rightarrow \infty$ and

$\begin{eqnarray*} &&u(t) = \lim\limits_{n\rightarrow \infty}y_{\epsilon_{n}}(t) = \int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} \bigg[f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{n}}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{n}}(\theta)d\theta, \int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}(f_{2}(\theta,y_{\epsilon_{n}}(\theta))+\epsilon_{n})\bigg)+\epsilon_n \bigg]ds,\; \; t\in[0,1]. \end{eqnarray*}$

Now we prove that $u(t)$ is the maximal solution of (2.1). To do this, let $y(t)$ be any solution of (2.1), then

$\begin{eqnarray*} y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y(\theta))d\theta \bigg)ds,\\ y_{\epsilon}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} \bigg[f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)} y_{\epsilon}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}(f_{2} (\theta,y_{\epsilon}(\theta))+\epsilon)d\theta \bigg)+\epsilon \bigg]ds \end{eqnarray*}$

and

$\begin{eqnarray*} y_{\epsilon}(t)& > &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)} f_{2}(\theta,y_{\epsilon}(\theta))\bigg)ds. \end{eqnarray*}$

Applying Lemma 3.1, we obtain

$y(t) < y_{\epsilon}(t), t\in [0,1].$

From the uniqueness of the maximal solution it clear that $y_{\epsilon}(t)$ tends to $u(t)$ uniformly in $[0, 1]$ as $\epsilon\rightarrow 0.$

By a similar way as done above, we can prove the existence of the minimal solution.

4. Uniqueness of the solution

Here, we study the sufficient condition for the uniqueness of the solution $y\in C[0, 1]$ of problems (1.2) and (2.1). Consider the following assumptions:

( $I^{*}$ ) (i) The set $F_{1}(t, x, y)$ is nonempty, closed and convex for all $(t, x, y)\in [0, 1]\times R\times R$ .

(ii) $F_{1}(t, x, y)$ is measurable in $t\in [0, 1]$ for every $x, y\in R$ .

(iii) $F_{1}$ satisfies the Lipschitz condition with a positive constant $K_{1}$ such that

$H(F_{1}(t,x_{1},y_{1}),F_{1}(t,x_{2},y_{2})|\leq K_{1}(|x_{1}-x_{_{2}}|+|y_{1}-y_{2}|),$

where $H(A, B)$ is the Hausdorff metric between the two subsets $A, B\in [0, 1]\times E.\;$

Remark 4.1. From this assumptions we can deduce that there exists a function $f_{1}\in F_{1}(t, x, y)$ , such that

(iv) $f_{1}:[0, 1]\times R\times R \rightarrow R$ is measurable in $t\in [0, 1]$ for every $x, y\in R$ and satisfies Lipschitz condition with a positive constant $K_{1}$ such that (see ^[3,7])

$|f_{1}(t,x_{1},y_{1})-f_{1}(t,x_{2},y_{2})|\leq K_{1}(|x_{1}-x_{2}|+|y_{1}-y_{2}|).$

$(II^{*}) \; f_{2}:[0, 1]\times R\longrightarrow R$ is measurable in $t \in [0, T]$ and satisfies Lipschitz condition with positive constant $K_{2}$ , such that

$|f_{2}(t,x)-f_{2}(t,y)|\leq K_{2}|x-y|.$

From the assumption ( $I^{*}$ ), we have

$|f_{1}(t,x,y)|-|f_{1}(t,0,0)|\leq |f_{1}(t,x,y)-f_{1}(t,0,0)|\leq K_{1}(|x|+|y|).$

Then

$\begin{eqnarray*} |f_{1}(t,x,y)|&\leq& K_{1}(|x|+|y|)+|f_{1}(t,0,0)|\\ &\leq& K_{1}(|x|+|y|)+\mid a_{1}(t)\mid, \end{eqnarray*}$

where $|a_{1}(t)| = \sup\limits_{t\in I}|f_{1}(t, 0, 0)|.$

From the assumption ( $II^{*}$ ), we have

$|f_{2}(t,y)|-|f_{2}(t,0)|\leq|f_{2}(t,y)-f_{2}(t,0)|\leq K_{2}|y|.$

Then

$\begin{eqnarray*} |f_{2}(t,x)|&\leq& K_{2}|x|+|f_{2}(t,0)|\\ &\leq& K_{2}|x|+\mid a_{2}(t)\mid, \end{eqnarray*}$

where $|a_{2}(t)| = \sup\limits_{t\in I}|f_{1}(t, 0)|$ .

Theorem 4.1. Let the assumptions $(I^{*})$ and $(II^{*})$ be satisfied. Then the solution of the problems (1.2) and (2.1) is unique.

Proof. Let $y_{1}(t)$ and $y_{2}(t)$ be solutions of the problems (1.2) and (2.1), then

Then

$\|y_{1}-y_{2}\|[\alpha\gamma\Gamma(2-\alpha)-(2K_{1}\alpha+K_{1}K_{2}\gamma)] < 0.$

Since $(\alpha\gamma\Gamma(2-\alpha)-(2K_{1}\alpha+K_{1}K_{2}\gamma)) < 1$ , then $y_{1}(t) = y_{2}(t)$ and the solution of (1.2) and (2.1) is unique.

4.1. Continuous dependence of the solution

Definition 4.1. The unique solution of the problems (1.2) and (2.1) depends continuously on initial data $x_{0}$ , if $\epsilon > 0$ , $\exists \delta > 0$ , such that

$|x_{0}-x^{*}_{0}|\leq \delta\; \; \; \; \Rightarrow \|y-y^{*}\|\leq \epsilon,$

where $y^{*}$ is the unique solution of the integral equation

$\begin{eqnarray*} y^{*}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x^{*}_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y^{*}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y^{*}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}} {\Gamma(\gamma)}f_{2}(\theta,y^{*}(\theta))d\theta)ds|. \end{eqnarray*}$

Theorem 4.2. Let the assumptions $(I^{*})$ and $(II^{*})$ be satisfied, then the unique solution of (1.2) and (2.1) depends continuously on $x_{0}.$

Proof. Let $y(t)$ and $y^{*}(t)$ be the solutions of problems (1.2) and (2.1), then

then we obtain

$\begin{eqnarray*} &&|y(t)-y^{*}(t)|\leq(AK_{1}\delta)(\alpha\gamma\Gamma(2-\alpha)-(2K_{1}\alpha+K_{1}K_{2}\gamma))^{-1}\leq\epsilon \end{eqnarray*}$

and

$\begin{eqnarray*} &&\|y(t)-y^{*}(t)\|\leq\epsilon. \end{eqnarray*}$

Definition 4.2. The unique solution of the problems (1.2) and (2.1) depends continuously on initial data $a_{k}$ , if $\epsilon > 0$ , $\exists \delta > 0$ , such that

$\sum\limits^{m}_{k = 1}|a_{k}-a_{k}^{*}|\leq\delta\; \; \; \; \Rightarrow \|y-y^{*}\|\leq \epsilon,$

where $y^{*}$ is the unique solution of the integral equation

$\begin{eqnarray*} y^{*}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}^{*}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y^{*}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y^{*}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)} f_{2}(\theta,y^{*}(\theta))d \theta)ds|. \end{eqnarray*}$

Theorem 4.3. Let the assumptions $(I^{*})$ and $(II^{*})$ be satisfied, then the unique solution of problems (1.2) and (2.1) depends continuously on $a_{k}$ .

Proof. Let $y(t)$ and $y^{*}(t)$ be the solutions of problems (1.2) and (2.1) and $(\sum\nolimits^{m}_{k = 1}a^{*}_{k})^{-1} = A^{*}$ ,

$\begin{eqnarray*} |y(t)-y^{*}(t)|& = &|\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma (\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^ {\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y(\theta))d \theta)ds \\ &&-\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a^{*}_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a^{*}_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y^{*}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y^{*}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}} {\Gamma(\gamma)}f_{2}(\theta,y^{*}(\theta))d \theta)ds|\\ &\leq&\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)}[\frac{|x_{0}|(\sum\nolimits^{m}_{k = 1}a^{*}_{k}-\sum\nolimits^{m}_{k = 1}a_{k})}{\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+\frac{\sum\nolimits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta}{\sum\nolimits^{m}_{k = 1}a_{k}} -\frac{\sum\nolimits^{m}_{k = 1}a^{*}_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y^{*}(\theta)|d\theta}{\sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+K_{1}\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)-y^{*}(\theta)|d\theta+K_{1}K_{2} \int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}|y(\phi)-y^{*}(\theta)|d\theta]ds\\ &\leq&\frac{1}{\Gamma(2-\alpha)}[\frac{K_{1}|x_{0}|\sum\nolimits^{m}_{k = 1}|a^{*}_{k}-a_{k}|}{\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+\frac{K_{1}\sum\nolimits^{m}_{k = 1}a_{k}(\sum\nolimits^{m}_{k = 1}a^{*}_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta- \sum\nolimits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta)}{\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+\frac{K_{1}\sum\nolimits^{m}_{k = 1}a^{*}_{k}(\sum\nolimits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta- \sum\nolimits^{m}_{k = 1}a^{*}_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta)}{\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+K_{1}\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)-y^{*}(\theta)|d\theta+K_{1}K_{2} \int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}|y(\phi)-y^{*}(\theta)|d\theta]ds\\ &\leq&\frac{1}{\Gamma(2-\alpha)}[\frac{K_{1}|x_{0}|\delta}{\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}+ \frac{K_{1}(\sum\nolimits^{m}_{k = 1}a_{k}\delta\frac{r}{\Gamma(\alpha+1)}+\sum\nolimits^{m}_{k = 1}a^{*}_{k}\delta\frac{r}{\Gamma(\alpha+1)})}{\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+\frac{K_{1}\|y-y^{*}\|}{\Gamma(\alpha+1)}+\frac{K_{1}K_{2}\|y-y^{*}\|}{\Gamma(\gamma+1)}\\ &\leq&\frac{K_{1}\delta|x_{0}|}{\Gamma(2-\alpha)\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}+ \frac{K_{1}(\sum\nolimits^{m}_{k = 1}a_{k}\delta\frac{r}{\Gamma(\alpha+1)}+\sum\nolimits^{m}_{k = 1}a^{*}_{k}\delta\frac{r}{\Gamma(\alpha+1)})}{\Gamma(2-\alpha)\Gamma(\alpha+1)\sum\nolimits^{m}_{k = 1}a_{k} \sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+\frac{K_{1}\|y-y^{*}\|}{\Gamma(2-\alpha)\Gamma(\alpha+1)}+\frac{K_{1}K_{2}\|y-y^{*}\|}{\Gamma(\gamma+1)\Gamma(2-\alpha)}, \end{eqnarray*}$

then we obtain

$\begin{eqnarray*} |y(t)-y^{*}(t)|\leq(AA^{*}K_{1}\delta [|x_{0}|+ r(\sum\limits^{m}_{k = 1}a_{k}+\sum\limits^{m}_{k = 1}a^{*}_{k})]).\; (\alpha\gamma\Gamma(2-\alpha)-(2K_{1}\alpha+K_{1}K_{2}\gamma))^{-1}\leq\epsilon \end{eqnarray*}$

and

$\begin{eqnarray*} &&\|y(t)-y^{*}(t)\|\leq\epsilon. \end{eqnarray*}$

5. Riemann-Stieltjes integral condition

Let $y\in C[0, 1]$ be the solution of the nonlocal boundary value problems (1.2) and (2.1). Let $a_{k} = (g(t_{k})-g(t_{k-1})$ , $g$ is increasing function, $\tau_{k}\in(t_{k-1}-t_{k})$ , $0 = t_{0} < t_{1} < t_{2}, ... < t_{m} = 1$ , then, as $m\longrightarrow \infty$ the nonlocal condition (1.2) will be

$\sum\limits^{m}_{k = 1}(g(t_{k})-g(t_{k-1})y(\tau_{k}) = x_{0}.$

As the limit $m\longrightarrow \infty$ , we obtain

$\lim\limits_{m\rightarrow \infty}\sum\limits^{m}_{k = 1}(g(t_{k})-g(t_{k-1})y(\tau_{k}) = \int_{0}^{1}y(s)dg(s) = x_{0}.$

Theorem 5.1. Let the assumptions (I)–(III) be satisfied. If $\sum\nolimits^{m}_{k = 1}a_{k}$ be convergent, then the nonlocal boundary value problems of (1.3) and (2.1) have at least one solution given by

$\begin{eqnarray*} y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{g(1)-g(0)}[x_{0}-\int_{0}^{1}\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta dg(s)]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2} (\phi,y(\theta))d\theta)ds. \end{eqnarray*}$

Proof. As $m\rightarrow \infty$ , the solution of the nonlocal boundary value problem (2.1) will be

$\begin{eqnarray*} \lim\limits_{m\rightarrow \infty}y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,(g(1)-g(0))^{-1}x_{0}\\ &&-(g(1)-g(0))^{-1}.\lim\limits_{m\rightarrow \infty}\sum\limits^{m}_{k = 1}[g(t_{k})-g(t_{k-1}]\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)} f_{2}(\phi,y(\theta))d\theta)ds\\ & = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{g(1)-g(0)}[x_{0}-\int_{0}^{1}\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta dg(s)]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)} f_{2}(\phi,y(\theta))d\theta)ds. \end{eqnarray*}$

6. Infinite-point boundary condition

Theorem 6.1. Let the assumptions (I)–(III) be satisfied, then the nonlocal boundary value problems of (1.4) and (2.1) have at least one solution given by

$\begin{eqnarray} y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{\infty}_{k = 1}a_{k}}[x_{0}-\sum\limits^{\infty}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}} {\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)} f_{2}(\phi,y(\theta))d\theta)ds. \end{eqnarray}$

Proof. Let the assumptions of Theorem 2.1 be satisfied. Let $\sum\nolimits^{m}_{k = 1} a_{k}$ be convergent, then take the limit to (1.4), we have

$\begin{eqnarray} \lim\limits_{m\rightarrow \infty}y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\lim\limits_{m\rightarrow m}\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\lim\limits_{m\rightarrow \infty}\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\phi,y(\theta))d\theta)ds. \end{eqnarray}$

Now

$\begin{eqnarray*} |a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta| \leq|a_{k}|\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta\leq\frac{|a_{k}|\|y\|}{\Gamma(\alpha+1)} \leq\frac{|a_{k}|r}{\Gamma(\alpha+1)} \end{eqnarray*}$

and by the comparison test $(\sum\nolimits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta)$ is convergent,

$\begin{eqnarray*} y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{\infty}_{k = 1}a_{k}}[x_{0}-\sum\limits^{\infty}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\nonumber\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\phi,y(\theta))d\theta)ds.\nonumber \end{eqnarray*}$

Furthermore, from (2.9) we have

$\begin{eqnarray*} \sum\limits^{\infty}_{k = 1}a_{k}x(\tau_{k})& = &\sum\limits^{\infty}_{k = 1}a_{k}[\frac{1}{\sum\nolimits^{\infty}_{k = 1}a_{k}}(x_{0}-\sum\limits^{\infty}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-s)^{\alpha-1}}{\Gamma(\alpha)}y(s)ds)]\\ &&+\int_{0}^{\tau_{k}}\frac{(\tau_{k}-s)^{\alpha-1}}{\Gamma(\alpha)}y(s)ds)]\\ & = &x_{0}. \end{eqnarray*}$

Example 6.1. Consider the following nonlinear integro-differential equation

$\begin{equation} \frac{dx}{dt} = t^{4}e^{-t}+\frac{x(t)}{\sqrt{t+2}}+\frac{1}{3}I^{\gamma}(\cos(5t+1)+\frac{1}{9}[t^{5}\sin D^{\frac{1}{3}}x(t)+e^{-3t}x(t)]) \end{equation}$

(6.1)

with boundary condition

$\begin{equation} {\sum\limits^{m}_{k = 1}\bigg[\frac{1}{k}-\frac{1}{k+1}\bigg]} x(\tau_k ) = x_{0},\; a_{k} > 0\; \; \tau_{k}\in[0,1]. \end{equation}$

(6.2)

Let

$\begin{eqnarray*} f_{1}(t,x(t),I^{\gamma}f_{2}(t,D^{\alpha}x(t))) = t^{4}e^{-t}+\frac{x(t)}{\sqrt{t+2}}+\frac{1}{3}I^{\gamma}(\cos(5t+1)+\frac{1}{9}[t^{5}\sin D^{\frac{1}{3}}x(t)+e^{-3t}x(t)]), \end{eqnarray*}$

then

$\begin{eqnarray*} |f_{1}(t,x(t),I^{\gamma}f_{2}(t,D^{\alpha}x(t))) = |t^{4}e^{-t}+\frac{x(t)}{\sqrt{2t+4}}+\\ \frac{1}{4}(\frac{4}{3}I^{\gamma}(\cos(5t+1)+\frac{1}{9}[t^{5}\sin D^{\frac{1}{3}}x(t)+e^{-3t}x(t)]))| \end{eqnarray*}$

and also

$\begin{eqnarray*} &&|I^{\gamma}f_{2}(t,D^{\alpha}x(t))|\leq \frac{1}{4}(\frac{4}{3}I^{\gamma}|\cos(5t+1)+\frac{1}{3}[t^{5}\sin D^{\frac{1}{3}}x(t)+e^{-3t}x(t)]|. \end{eqnarray*}$

It is clear that the assumptions (I) and (II) of Theorem 2.1 are satisfied with $a_{1}(t) = t^{4}e^{-t}\in L^{1}[0, 1]$ , $a_{2}(t) = \frac{4}{3}I^{\gamma}|(\cos(5t+1)|\in L^{1}[0, 1]$ , and let $\alpha = \frac{1}{3}$ , $\gamma = \frac{2}{3}$ , then $2K_{1}\gamma+K_{1}K_{2}\alpha = \frac{1}{2} < \alpha\gamma\Gamma(2-\alpha) = \frac{1}{2}\Gamma(2-\alpha)$ . Therefore, by applying Theorem 2.1, the nonlocal problems (6.1) and (6.2) has a continuous solution.

7. Conclusions

In this paper, we have studied a boundary value problem of fractional order differential inclusion with nonlocal, integral and infinite points boundary conditions. We have prove some existence results for that a single nonlocal boundary value problem, in of proving some existence results for a boundary value problem of fractional order differential inclusion with nonlocal, integral and infinite points boundary conditions. Next, we have proved the existence of maximal and minimal solutions. Then we have established the sufficient conditions for the uniqueness of solutions and continuous dependence of solution on some initial data and on the coefficients $a_k$ are studied. Finally, we have proved the existence of a nonlocal boundary value problem with Riemann-Stieltjes integral condition and with infinite-point boundary condition. An example is given to illustrate our results.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	Y. Jia, F. Zhong, A. Alrawais, B. Gong, X. Cheng, FlowGuard: An intelligent edge defense mechanism against IoT DDoS attacks, IEEE Internet Things J., 7 (2020), 9552–9562. https://doi.org/10.1109/JIOT.2020.2993782 doi: 10.1109/JIOT.2020.2993782
[2]	A. Prasad, S. Chandra, Machine learning to combat cyberattack: a survey of datasets and challenges, J. Def. Model. Simul. Appl. Methodol. Technol., 2022 (2022). https://doi.org/10.1177/15485129221094881 doi: 10.1177/15485129221094881
[3]	H. Wang, H. He, W. Zhang, W. Liu, P. Liu, A. Javadpour, Using honeypots to model botnet attacks on the internet of medical things, Comput. Electr. Eng., 102 (2022), 108212. https://doi.org/10.1016/j.compeleceng.2022.108212 doi: 10.1016/j.compeleceng.2022.108212
[4]	Y. Lee, H. Chae, K. Lee, Countermeasures against large-scale reflection DDoS attacks using exploit IoT devices, Automatika, 62 (2021), 127–136. https://doi.org/10.1080/00051144.2021.1885587 doi: 10.1080/00051144.2021.1885587
[5]	M. Anagnostopoulos, S. Lagos, G. Kambourakis, Large-scale empirical evaluation of DNS and SSDP amplification attacks, J. Inf. Secur. Appl., 66 (2022), 103168. https://doi.org/10.1016/j.jisa.2022.103168 doi: 10.1016/j.jisa.2022.103168
[6]	K. B. Dasari, N. Devarakonda, Detection of different DDoS attacks using machine learning classification algorithms, Ing. Des Syst. d Inf., 26 (2021), 461–468. https://doi.org/10.18280/isi.260505 doi: 10.18280/isi.260505
[7]	C. Rossow, Amplification hell: Revisiting network protocols for DDoS abuse, inNDSS, (2021), 1–15.
[8]	J. D. Case, M. Fedor, M. L. Schoffstall, J. Davin, Simple network management protocol (SNMP), 1989.
[9]	D. Kshirsagar, S. Sawant, A. Rathod, S. Wathore, CPU load analysis & minimization for TCP SYN flood detection, Procedia Comput. Sci., 85 (2016), 626–633. https://doi.org/10.1016/j.procs.2016.05.230 doi: 10.1016/j.procs.2016.05.230
[10]	S. Muthurajkumar, A. Geetha, S. Aravind, H. Barakath Meharajnisa, UDP flooding attack detection using entropy in software-defined networking, in Proceedings of International Conference on Communication and Computational Technologies, Springer, (2023), 549–560. https://doi.org/10.1007/978-981-19-3951-8_42
[11]	N. N. Mohamed, Y. Mohd Yussoff, M. A. Mat Isa, H. Hashim, Extending hybrid approach to secure Trivial File Transfer Protocol in M2M communication: a comparative analysis, Telecommun. Syst., 70 (2019), 511–523. https://doi.org/10.1007/s11235-018-0522-5 doi: 10.1007/s11235-018-0522-5
[12]	H. Aydın, Z. Orman, M. A. Aydın, A long short-term memory (LSTM)-based distributed denial of service (DDoS) detection and defense system design in public cloud network environment, Comput. Secur., 118 (2022), 102725. https://doi.org/10.1016/j.cose.2022.102725 doi: 10.1016/j.cose.2022.102725
[13]	S. Pundir, M. S. Obaidat, M. Wazid, A. K. Das, D. P. Singh, J. Rodrigues, MADP-IIME: malware attack detection protocol in IoT-enabled industrial multimedia environment using machine learning approach, Multimedia Syst., 29 (2023), 1785–1797. https://doi.org/10.1007/s00530-020-00743-9 doi: 10.1007/s00530-020-00743-9
[14]	M. Gallagher, N. Pitropakis, C. Chrysoulas, P. Papadopoulos, A. Mylonas, S. Katsikas, Investigating machine learning attacks on financial time series models, Comput. Secur., 123 (2022), 102933. https://doi.org/10.1016/j.cose.2022.102933 doi: 10.1016/j.cose.2022.102933
[15]	A. Prasad, S. Chandra, VMFCVD: An optimized framework to combat volumetric DDoS attacks using machine learning, Arabian J. Sci. Eng., 47 (2022), 9965–9983. https://doi.org/10.1007/s13369-021-06484-9 doi: 10.1007/s13369-021-06484-9
[16]	C. S. Kalutharage, X. Liu, C. Chrysoulas, N. Pitropakis, P. Papadopoulos, Explainable AI-based DDOS attack identification method for IoT networks, Computers, 12 (2023), 32. https://doi.org/10.3390/computers12020032 doi: 10.3390/computers12020032
[17]	A. Prasad, S. Chandra, BotDefender: A collaborative defense framework against botnet attacks using network traffic analysis and machine learning, Arabian J. Sci. Eng., (2023). https://doi.org/10.1007/s13369-023-08016-z doi: 10.1007/s13369-023-08016-z
[18]	M. Bhattacharya, S. Roy, A. K. Das, S. Chattopadhyay, S. Banerjee, A. Mitra, DDoS attack resisting authentication protocol for mobile based online social network applications, J. Inf. Secur. Appl., 65 (2022), 103115. https://doi.org/10.1016/j.jisa.2022.103115 doi: 10.1016/j.jisa.2022.103115
[19]	O. Thorat, N. Parekh, R. Mangrulkar, TaxoDaCmachine learning: Taxonomy based Divide and Conquer using machine learning approach for DDoS attack classification, Int. J. Inf. Manage. Data Insights, 1 (2021), 100048. https://doi.org/10.1016/j.jjimei.2021.100048 doi: 10.1016/j.jjimei.2021.100048
[20]	M. E. Ahmed, H. Kim, M. Park, Mitigating DNS query-based DDoS attacks with machine learning on software-defined networking, in IEEE Military Communications Conference (MILCOM), (2017), 11–16. https://doi.org/10.1109/MILCOM.2017.8170802
[21]	I. Sreeram, V. P. K. Vuppala, HTTP flood attack detection in application layer using machine learning metrics and bio inspired bat algorithm, Appl. Comput. Inf., 15 (2019), 59–66. https://doi.org/10.1016/j.aci.2017.10.003 doi: 10.1016/j.aci.2017.10.003
[22]	O. Salman, I. H. Elhajj, A. Chehab, A. Kayssi, A machine learning based framework for IoT device identification and abnormal traffic detection, Trans. Emerging Telecommun. Technol., 33 (2022). https://doi.org/10.1002/ett.3743 doi: 10.1002/ett.3743
[23]	X. Liu, L. Zheng, S. Helal, W. Zhang, C. Jia, J. Zhou, A broad learning-based comprehensive defence against SSDP reflection attacks in IoTs, Digital Commun. Networks, 2022 (2022). https://doi.org/10.1016/j.dcan.2022.02.008 doi: 10.1016/j.dcan.2022.02.008
[24]	S. Ismail, Z. El Mrabet, H. Reza, An ensemble-based machine learning approach for cyber-attacks detection in wireless sensor networks, Appl. Sci., 13 (2022), 30. https://doi.org/10.3390/app13010030 doi: 10.3390/app13010030
[25]	D. Kshirsagar, S. Kumar, A feature reduction based reflected and exploited DDoS attacks detection system, J. Ambient Intell. Hum. Comput., 13 (2022), 393–405. https://doi.org/10.1007/s12652-021-02907-5 doi: 10.1007/s12652-021-02907-5
[26]	A. Mishra, N. Gupta, B. B. Gupta, Defensive mechanism against DDoS attack based on feature selection and multi-classifier algorithms, Telecommun. Syst., 82 (2023), 229–244. https://doi.org/10.1007/s11235-022-00981-4 doi: 10.1007/s11235-022-00981-4
[27]	I. Sharafaldin, A. H. Lashkari, S. Hakak, A. A. Ghorbani, Developing realistic Distributed Denial of Service (DDoS) attack dataset and taxonomy, in International Carnahan Conference on Security Technology (ICCST), (2019), 1–8. https://doi.org/10.1109/CCST.2019.8888419
[28]	A. Prasad, S. Chandra, Defending ARP spoofing-based MitM attack using machine learning and device profiling, in 2019 International Carnahan Conference on Security Technology (ICCST), (2022), 978–982. https://doi.org/10.1109/ICCCIS56430.2022.10037723
[29]	D. Tang, L. Tang, R. Dai, J. Chen, X. Li, J. Rodrigues, MF-Adaboost: LDoS attack detection based on multi-features and improved Adaboost, Future Gener. Comput. Syst., 106 (2020), 347–359. https://doi.org/10.1016/j.future.2019.12.034 doi: 10.1016/j.future.2019.12.034
[30]	B. Sabir, M. A. Babar, R. Gaire, A. Abuadbba, Reliability and robustness analysis of machine learning based phishing URL detectors, arXiv preprint, (2022), arXiv: 2005.08454. https://doi.org/10.48550/arXiv.2005.08454
[31]	S. A. Khanday, H. Fatima, N. Rakesh, Implementation of intrusion detection model for DDoS attacks in Lightweight IoT Networks, Expert Syst. Appl., 215 (2023), 119330. https://doi.org/10.1016/j.eswa.2022.119330 doi: 10.1016/j.eswa.2022.119330
[32]	M. M. Alani, E. Damiani, XRecon: An explainbale IoT reconnaissance attack detection system based on ensemble learning, Sensors, 23 (2023), 5298. https://doi.org/10.3390/s23115298 doi: 10.3390/s23115298
[33]	R. Verma, S. Chandra, RepuTE: A soft voting ensemble learning framework for reputation-based attack detection in fog-IoT milieu, Eng. Appl. Artif. Intell., 118 (2023), 105670. https://doi.org/10.1016/j.engappai.2022.105670 doi: 10.1016/j.engappai.2022.105670
[34]	S. Pokhrel, R. Abbas, B. Aryal, IoT security: botnet detection in IoT using machine learning, arXiv preprint, (2021), arXiv: 2104.02231. https://doi.org/10.48550/arXiv.2104.02231
[35]	A. P. Bradley, The use of the area under the ROC curve in the evaluation of machine learning algorithms, Pattern Recognit., 30 (1997), 1145–1159. https://doi.org/10.1016/S0031-3203(96)00142-2 doi: 10.1016/S0031-3203(96)00142-2
[36]	D. Chicco, G. Jurman, The advantages of the Matthews correlation coefficient (MCC) over F1 score and accuracy in binary classification evaluation, BMC Genomics, 21 (2020), 6. https://doi.org/10.1186/s12864-019-6413-7 doi: 10.1186/s12864-019-6413-7
[37]	Md. M. Rashid, J. Kamruzzaman, M. Ahmed, N. Islam, S. Wibowo, S. Gordon, Performance enhancement of intrusion detection system using bagging ensemble technique with feature selection, in 2020 IEEE Asia-Pacific Conference on Computer Science and Data Engineering (CSDE), (2020), 1–5. https://doi.org/10.1109/CSDE50874.2020.9411608
[38]	Md. Raihan-Al-Masud, H. A. Mustafa, Network intrusion detection system using voting ensemble machine learning, in 2019 IEEE International Conference on Telecommunications and Photonics (ICTP), (2019), 1–4. https://doi.org/10.1109/ICTP48844.2019.9041736
[39]	S. V. J. Rani, I. Ioannou, P. Nagaradjane, C. Christophorou, V. Vassiliou, S. Charan, et al., Detection of DDoS attacks in D2D communications using machine learning approach, Comput. Commun., 198 (2023), 32–51. https://doi.org/10.1016/j.comcom.2022.11.013 doi: 10.1016/j.comcom.2022.11.013
[40]	S. ur Rehman, M. Khaliq, S. I. Imtiaz, A. Rasool, M. Shafiq, A. R. Javed, et al., DIDDOS: An approach for detection and identification of Distributed Denial of Service (DDoS) cyberattacks using Gated Recurrent Units (GRU), Future Gener. Comput. Syst., 118 (2021), 453–466. https://doi.org/10.1016/j.future.2021.01.022 doi: 10.1016/j.future.2021.01.022
[41]	R. J. Alzahrani, A. Alzahrani, Security analysis of DDoS attacks using machine learning algorithms in networks traffic, Electronics, 10 (2021), 2919. https://doi.org/10.3390/electronics10232919 doi: 10.3390/electronics10232919
[42]	S. Sindian, S. Sindian, An enhanced deep autoencoder-based approach for DDoS attack detection, Wseas Trans. Syst. Control, 15 (2020), 716–724. https://doi.org/10.37394/23203.2020.15.72 doi: 10.37394/23203.2020.15.72
[43]	I. Ortet Lopes, D. Zou, F. A. Ruambo, S. Akbar, B. Yuan, Towards effective detection of recent DDoS attacks: A deep learning approach, Secur. Commun. Netw., 2021 (2021), 1–14. https://doi.org/10.1155/2021/5710028 doi: 10.1155/2021/5710028
[44]	D. Javeed, T. Gao, M. T. Khan, SDN-enabled hybrid DL-driven framework for the detection of emerging cyber threats in IoT, Electronics, 10 (2021), 918. https://doi.org/10.3390/electronics10080918 doi: 10.3390/electronics10080918

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