## 2 Statement of the Problem
Then \(\sup_{\sigma\in[s-r,s]}\alpha(\sigma)\leq\sup_{\sigma\in[s-r,s]}\beta(\sigma)\) for \(s\in[0,t]\) and  
\[\alpha(t^{*})-\beta(t^{*}) =\Omega(u,\alpha,\varepsilon)(t^{*})-\Omega(v,\beta, \varepsilon+\delta)(t^{*})\] \[=-\delta\,\Gamma(q)E_{q,q}(M(t^{*})^{q})(t^{*})^{q-1}\] \[\quad+L\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q}) \Big{(}\sup_{\sigma\in[s-r,s]}\alpha(\sigma)-\sup_{\sigma\in[s-r,s]}\beta( \sigma)\Big{)}ds\] \[\quad+\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})\times\] \[\quad\times\Big{(}F(s,u(s),\sup_{\sigma\in[s-r,s]}u(\sigma))-F(s, \alpha(s),\sup_{\sigma\in[s-r,s]}\alpha(\sigma))\Big{)}ds\] Figure 8:0 Figure 8:1 Figure 8:2 Figure 8:3 Figure 8:4 Figure 8:5 Figure 8:6 The obtained contradiction proves the claim. **Lemma 2**. : _Let_  
1. _The functions_ \(u,\alpha\in PC([-r,T],\mathbb{R})\) _be such that_ \(u(t)\geq\alpha(t),\ t\in[-r,T]\) _and_ \(\alpha(t)=\Omega(u,\alpha,\varepsilon+\delta)(t)\) _with_ \(\varepsilon\in\mathbb{R}\)_,_ \(\delta>0\)_For any \(t\in(0,T]\) the inequality_  
Figure 8:7 Figure 8:8 _holds with_ \(M\in\mathbb{R}\)_,_ \(L>0\)_._
3. _The function_ \(\beta\in PC([-r,T],\mathbb{R})\) _be such that_ \(\beta(t)=\Omega(v,\beta,\varepsilon)(t)\)_,_ \(t\in[-r,T]\)_._  
_Then the inequality_ \(\alpha(t)>\beta(t)\) _holds for all_ \(t\in[-r,T]\)_._  
Proof. : The proof is similar to the one in Theorem 1. Assume the contrary. Since \(\alpha(t)>\beta(t)\) for \(t\in[-r,0]\) and \(t^{1-q}\alpha(t)|_{t=0}=\phi(0)+\varepsilon+\delta>\phi(0)+\varepsilon=t^{1-q} \beta(t)|_{t=0}\) there exists a point \(t^{*}\in(0,T]\) such that \(\alpha(t)>\beta(t)\) for \(t\in[-r,t^{*})\) and \(\alpha(t^{*})=\beta(t^{*})\). Then \(\sup_{\sigma\in[s-r,s]}\alpha(\sigma)\geq\sup_{\sigma\in[s-r,s]}\beta(\sigma)\) for \(s\in[0,t]\) and  
Figure 8:9 Figure 9:0 Figure 9:1 Figure 9:2 Figure 9:3 Figure 9:4 Figure 9:5 Figure 9:6 The obtained contradiction proves the claim. Now we will prove the convergence. **Theorem 1**. : _Let the following conditions be fulfilled:_  
1. _Let the functions_ \(\alpha,\beta\in PC([-r,T],\mathbb{R})\cup C_{1-q}([0,T])\) _be a mild lower solution and a mild upper solution of the IVP for FrDES (_1_), respectively, defined by Definition_ 3 _with_ \(\mu=M\) _such that_ \(\alpha(t)\leq\beta(t)\) _for_ \(t\in[0,T]\) _and_ Figure 9:7 Figure 9:8 (13)
2. _The function_ \(F\in C([0,T]\times\mathbb{R}\times\mathbb{R},\mathbb{R})\) _and there exist constants_ \(M\in\mathbb{R}\) _and_ \(L>0\) _such that for any_ \(t\in[0,T]\)_,_ \(x,y,u,v\in\mathbb{R}:\;\alpha(t)\leq x\leq y\leq\beta(t)\)_,_ \(\sup_{\sigma\in[t-r,t]}\alpha(\sigma)\leq u\leq v\leq\sup_{\sigma\in[t-r,t]}\beta (\sigma)\) _the inequality_ \(F(t,x,u)-F(t,y,v)\leq M(x-y)+L(u-v)\) _holds.__Then there exists two sequences of functions \(\{\alpha^{(n)}(t)\}_{0}^{\infty}\) and \(\{\beta^{(n)}(t)\}_{0}^{\infty}\), \(t\in[-r,T]\), such that:_  
1. _The sequences_ \(\{\alpha^{(n)}(t)\}\) _and_ \(\{\beta^{(n)}(t)\}\) _are defined by_ \(\alpha^{(0)}(t)=\alpha(t),\ \ \beta^{(0)}(t)=\beta(t)\) _and_ Figure 9:9 _where the constants_ \(M,L\) _are defined in condition_ 2_._
2. _The sequence_ \(\{\alpha^{(j)}(t)\}_{j=0}^{\infty}\) _is increasing, i.e.,_ \(\alpha^{(j-1)}(t)\leq\alpha^{(j)}(t)\) _for_ \(t\in[-r,T]\) _and_ \(t^{1-q}\alpha^{(j-1)}(t)|_{t=0}\leq t^{1-q}\alpha^{(j)}(t)|_{t=0}\)_,_ \(j=1,2,\ldots\) _._
3. _The sequence_ \(\{\beta^{(j)}(t)\}_{j=0}^{\infty}\) _is decreasing, i.e.,_ \(\beta^{(j-1)}(t)\geq\beta^{(j)}(t)\) _for_ \(t\in(0,T]\) _and_ \(t^{1-q}\beta^{(j)}(t)|_{t=0}\leq t^{1-q}\beta^{(j-1)}(t)|_{t=0}\)_,_ \(j=1,2,\ldots\)_._
4. _The inequalities_ \[F\left(t,u(t),\sup_{s\in[t-r,t]}u(s)\right)-F\left(t,\alpha(t),\sup_ {s\in[t-r,t]}\alpha(s)\right)\]0 (14) _hold._
5. \(\lim_{k\to\infty}\alpha^{(n)}(t)=\lim_{k\to\infty}\beta^{(n)}(t)=\phi(t)\) _uniformly on_ \([0,T]\)_._
6.