## 2 Statement of the Problem
Obtain the lower successive approximation  
\[\alpha^{(n)}(t)= \begin{cases}\phi(t)-\frac{h}{n},\ \ \ \ \ t\in[-r,0]\\ (\phi(0)-\frac{h}{n})\Gamma(q)E_{q,q}(Mt^{q})t^{q-1}\\ \ \ \ \ \ +\int_{0}^{t}(t-s)^{q-1}E_{q,q}(M(t-s)^{q})\Gamma(s,\alpha^{(n-1)}(s), \sup_{\sigma\in[s-r,s]}\alpha^{(n-1)}(\sigma))ds\\ \ \ \ \ \ -\int_{0}^{t}(t-s)^{q-1}E_{q,q}(M(t-s)^{q})\Big{(}M\alpha^{(n-1)}(s)\\ \ \ \ \ \ \ \ \ +L(\sup_{\sigma\in[s-r,s]}\alpha^{(n-1)}(\sigma)-\sup_{ \sigma\in[s-r,s]}\alpha^{(n)}(\sigma))\Big{)}ds,\ \ t\in(0,T].\end{cases} \]  
Step 5. Obtain the upper successive approximation  
\[\beta^{(n)}(t)= \begin{cases}\phi(t)-\frac{g}{n},\ \ \ \ \ t\in[-r,0]\\ (\phi(0)-\frac{g}{n})\Gamma(q)E_{q,q}(Mt^{q})t^{q-1}\\ \ \ \ \ \ +\int_{0}^{t}(t-s)^{q-1}E_{q,q}(M(t-s)^{q})F(s,\beta^{(n-1)}(s), \sup_{\sigma\in[s-r,s]}\beta^{(n-1)}(\sigma))ds\\ \ \ \ \ \ \ -\int_{0}^{t}(t-s)^{q-1}E_{q,q}(M(t-s)^{q})\Big{(}M\beta^{(n-1)}(s)\\ \ \ \ \ \ \ \ \ \ \ +L(\sup_{\sigma\in[s-r,s]}\beta^{(n-1)}(\sigma)-\sup_{ \sigma\in[s-r,s]}\beta^{(n)}(\sigma))\Big{)}ds,\ \ t\in(0,T].\end{cases} \]  
Step 6. Obtain \(A=||\alpha^{(n)}-\beta^{(n)}||_{C_{1-q}[0,T]}=\max_{t\in[0,T]}|t^{1-q}\Big{(} \alpha^{(n)}(t)-\beta^{(n)}(t)\Big{)}|\). Step 7. If \(A<\varepsilon\), then the approximate mild solution \(x(t)=\frac{\alpha^{(n)}(t)+\beta^{(n)}(t)}{2}\) for \(t\in(0,T]\). If not, then \(n:=n+1\) and go to step 4. **II. Special case of IVP (1), (2)**  
In the special case when the right hand side part of the Equation (1) does not depend on the current state of the unknown function we obtain the following easier iterative scheme without an application of the Mittag-Leffler function. This scheme is similar to the above one in which Steps 4 and 5 are replaced by  
Step 4. Obtain the lower successive approximation  
\[\alpha^{(n)}(t)= \begin{cases}\phi(t)-\frac{h}{n},\ \ \ \ \ t\in[-r,0]\\ (\phi(0)-\frac{h}{n})t^{q-1}\\ \ \ \ \ \ \ +\frac{1}{\Gamma(q)}\int_{0}^{t}(t-s)^{q-1}F(s,\alpha^{(n-1 )}(s),\sup_{\sigma\in[s-r,s]}\alpha^{(n-1)}(\sigma))ds\\ \ \ \ \ \ \ -\frac{1}{\Gamma(q)}\int_{0}^{t}(t-s)^{q-1}\Big{(}M\alpha^{(n-1 )}(s)\\ \ \ \ \ \ \ \ \ \ +L(\sup_{\sigma\in[s-r,s]}\alpha^{(n-1)}(\sigma)-\sup_{ \sigma\in[s-r,s]}\alpha^{(n)}(\sigma))\Big{)}ds,\ \ t\in(0,T].\end{cases} \]  
Figure 1: Graphs of the Mittag–Leffer function and its bounds. Figure 2: Graph of the function \(\beta(t)+0.2t^{-0.5}+0.25\). _In this case we have_ \(g=\inf_{t\in[-0.5,0)}(0.1t-0.05+0.15)=0.1(-0.5)-0.05+0.15=0.05\)_._  
_The function_  
\[\alpha(t)= \begin{cases}0.1(t+0.2)^{2}-0.2,&t\in[-0.5,0]\\ -0.2(t^{-0.5}+1),&t\in(0,1]\end{cases} \]  
_is a mild lower solution of IVP (7) because \(\alpha(t)=0.1(t+0.2)^{2}-0.2\leq 0.1(-0.5+0.2)^{2}-0.2=-0.191<\phi(t)=-0.15\) for \(t\in[-0.5,0]\) and \(t^{0.5}(-0.2t^{-0.5}-0.2)|_{t=0}=-0.2<-0.15\)._  
_Also, for \(t\in(0,1]\) applying the inequalities \(\alpha(t)+0.2t^{-0.5}+0.25=0.05>0\), \(t\in[0,1]\), \(\alpha(t)-0.05=-0.2t^{-0.5}-0.25<0\), and \(0>\sup_{\sigma\in[s-0.5,s]}\alpha(\sigma)\geq\alpha(0.5)=-0.2(0.5)^{-0.5}-0.2>-0.49\) we get_  
\[\alpha(t) \leq-0.196\Gamma(0.5)E_{0.5,0.5}(-0.49t^{0.5})t^{-0.5} \tag{9}\] \[\quad+0.56(0.49)\int_{0}^{t}(t-s)^{-0.5}\bigg{(}-0.2s^{-0.5}-0.25 \bigg{)}ds\] \[\leq\alpha(0)\Gamma(0.5)E_{0.5,0.5}(-0.49t^{0.5})t^{-0.5}\] \[\quad+\int_{0}^{t}(t-s)^{-0.5}E_{0.5,0.5}(-0.49(t-s)^{0.5})\times\] \[\quad\times\bigg{(}\frac{\alpha(s)+0.2t^{-0.5}+0.25}{s+1}\sup_{ \sigma\in[s-0.5,s]}\alpha(\sigma)+0.49\alpha(s)\bigg{)}ds.\]  
_From Figure 5 it can be seen that inequalities (9) hold for \(t\in(0,1]\). In this case \(h=\inf_{t\in[-0.5,0]}(-0.15-0.1(t+0.2)^{2}+0.2)=0.041\)._  
_As it can be seen on Figure 6 the inequality \(\alpha(t)\leq\beta(t)\) is valid on \([-0.5,1]\), i.e., the considered mild lower and mild upper solutions of IVP (7) are properly defined._  
Figure 4: Graphs of the function \(\beta(t)\) and its bound in (8). To determine the constants \(M\) and \(L\) in Algorithm 1 we need to check the conditions for the application of the considered algorithm given in the theoretical Section 5 and Theorem 1.