## 2 Statement of the Problem  
Consider the scalar nonlinear Riemann-Liouville fractional differential equation with supremum (FrDES)  
\[ \begin{array}{ll}\,_{0}^{RL}D_{t}^{q}\,y(t)=F(t,y(t),\sup_{s\in[t-r,t]}y(s))& \mbox{for}\,\,\,t\in(0,T]\end{array}  \tag{1}\]  
with the initial condition  
\[ \begin{array}{ll}y(s)=\phi(s)&\mbox{for}\,\,\,s\in[-r,0]\\ t^{1-q}y(t)|_{t=0}=\lim_{t\to 0+}t^{1-q}y(t)=\phi(0),\end{array}  \tag{2}\]  
where \(q\in(0,1)\), \(F:[0,T]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}\), \(\phi:[-r,0]\to\mathbb{R}:\,\,\phi(0)<\infty\), \(r,T>0\) are given numbers and the Riemann-Liouville (RL) fractional derivative of order \(q\in(0,1)\) is defined by (see, for example, [2])  
\[ \begin{array}{ll}\,_{0}^{RL}D_{t}^{q}m(t)=\frac{1}{\Gamma\,(1-q)}\frac{d}{dt }\left(\int\limits_{0}^{t}\,(t-s)^{-q}\,m(s)ds\right),&t\geq 0.\end{array} \]  
The main characteristics of the initial value problem (1), (2) are the RL fractional derivative, the impulsive conditions and the presence of the special type of the delay, which is defined by the supremum of the unknown function on a finite past time interval. All these characteristics lead to the impossibilities of finding the exact solution of the IVP. As a result we require well grounded algorithms for approximate solutions. In this paper we will present an algorithm for obtaining two monotone sequences of functions given in explicit forms, which will approach monotonically (increasing and decreasing) to the exact solution of the IVP (1), (2). In connection with this, we will initially give some definitions. Consider the set of functions  
\[C_{1-q}([a,b])=\{u(t):\,\,[a,b]\to\mathbb{R}:\,\,(t-a)^{1-q}u(t)\in C([a,b], \mathbb{R})\},\]  
where \(a,b,\,\,a<b\) are real numbers and the norm \(||u||_{C_{1-q}[a,b]}=\max\limits_{t\in[a,b]}|(t-a)^{1-q}u(t)|\). Consider the scalar RL fractional equation with supremum and a linear part of the type  
\[ \begin{array}{ll}\,_{0}^{RL}D_{t}^{q}y(t)=\mu y(t)+f(t,y(t),\sup_{s\in[t-r,t ]}y(s))&\mbox{for}\,\,t\in(0,T],\end{array}  \tag{3}\]  
with initial condition (2), where \(\mu\) is a real constant and \(f\in C([0,T]\times\mathbb{R}^{2},\mathbb{R})\). Similar to [23], we consider the integral form for the solution of (3) given by:\[y(t)= \begin{cases}\phi(t)&\text{for}\ \ t\in[-r,0],\\ \phi(0)\Gamma(q)E_{q,q}(\mu t^{q})t^{q-1}\\ &\ \ \ \ \ \ +\int_{0}^{t}(t-s)^{q-1}E_{q,q}(\mu(t-s)^{q})f(s,y(s),\sup_{ \sigma\in[s-r,s]}y(\sigma))ds,\ t\in(0,T]\end{cases}  \tag{4}\]  
where \(E_{\lambda,\tilde{\theta}}(z)=\sum\limits_{k=0}^{\infty}\frac{z^{k}}{\Gamma( k+\tilde{\beta})}\) is the Mittag-Leffler function with two parameters. Similar to [7], taking into account that \(f(t,x,y)=\mu x+(f(t,x,y)-\mu x)\) with an arbitrary constant \(\mu\) and the integral presentation (4) of (3) we define mild solutions of (1), (2):  
Definition 1: The function \(y\in PC([-r,T],\mathbb{R})\) is a mild solution of the IVP for FrDES (1), (2), if it satisfies  
\[y(t)= \begin{cases}\phi(t)&\text{for}\ \ t\in[-r,0],\\ \phi(0)\Gamma(q)E_{q,q}(\mu t^{q})t^{q-1}\\ &\ \ \ \ +\int_{0}^{t}(t-s)^{q-1}E_{q,q}(\mu(t-s)^{q})\Big{(}f(s,y(s), \sup_{\sigma\in[s-r,s]}y(\sigma))-\mu y(s)\Big{)}ds,\ t\in(0,T],\end{cases}  \tag{5}\]  
where \(\mu\) is a real constant. Definition 2: The function \(y\in PC([-r,T],\mathbb{R})\) is a mild maximal solution (a mild minimal solution) of the IVP for FrDES (1), (2), if it is a mild solution of (1), (2) and for any mild solution \(u(t)\in PC([-r,T],\mathbb{R})\) of (1), (2) the inequality \(y(t)\leq u(t)\) (\(y(t)\geq u(t)\)) holds on 1 and \(t^{1-q}y(t)|_{t=0}\leq(\geq)t^{1-q}u(t)|_{t=0}\). Definition 3: The function \(\alpha(t)\in PC([-r,T],\mathbb{R})\) is a mild lower (a mild upper) solution of the IVP for FrDES (1) with a constant \(\mu\in\mathbb{R}\), if it satisfies the inequalities  
\[\alpha(t)<(>)\phi(t),\ \ \text{for}\ \ t\in[-r,0],\ \ \ \ t^{1-q}\alpha(t)|_{t=0}<(>)t^{1-q}\phi(t)|_{t=0}\] (6) \[\alpha(t)\leq(\geq)\begin{cases}\alpha(0)\Gamma(q)E_{q,q}(\mu t ^{q})t^{q-1}+\\ \ \ \  
Step 4.