## 2 Statement of the Problem
_The sequences_ \(\{t^{1-q}\alpha^{(n)}(t)\}_{0}^{\infty}\) _and_ \(\{t^{1-q}\beta^{(n)}(t)\}_{0}^{\infty}\) _converge uniformly on the interval_ \([0,T]\) _and let_ \(\tilde{V}(t)=\lim_{k\to\infty}t^{1-q}\alpha^{(n)}(t)\)_,_ \(\tilde{W}(t)=\lim_{k\to\infty}t^{1-q}\beta^{(n)}(t)\) _on_ \([0,T]\)_._
7. _The inequalities_ \(\alpha^{(n)}(t)\leq V(t)\leq W(t)\leq\beta^{(n)}(t)\) _hold on_ \([0,T]\) _for any_ \(n=0,1,2,\ldots\) _where_ \[V(t)= \begin{cases}\phi(t),&t\in[-r,0]\\ t^{q-1}\tilde{V}(t)&t\in(0,T],\end{cases} \qquad W(t)= \begin{cases}\phi(t),&t\in [-r,0]\\ t^{q-1}\tilde{W}(t)&t\in(0,T].\end{cases} \]  
_The functions_ \(V(t)\) _and_ \(W(t)\) _are mild solutions of the IVP for FrDES (1), (2) on_ \([-r,T]\)_._  
Proof. : Let \(\alpha^{(0)}(t)=\alpha(t)\), \(\beta^{(0)}(t)=\beta(t)\) for \(t\in[-r,T]\) and \(\alpha^{(n)}(t)=\Omega\left(\alpha^{(n-1)},\alpha^{(n)},-\frac{h}{n}\right)(t)\), \(\beta^{(n)}(t)=\Omega\left(\beta^{(n-1)},\beta^{(n)},\frac{g}{n}\right)(t)\), \(n=1,2,3,\ldots\). We will prove the inequality  
\[\alpha^{(0)}(t)\leq\alpha^{(1)}(t),\ t\in[-r,T]. \tag{15}\]  
For \(t\in[-r,0]\) from Definition 3 and the definition of the constant \(h\) we get \(\alpha^{(1)}(t)=\phi(t)-h\geq\phi(t)-(\phi(t)-\alpha^{(0)}(t))=\alpha^{(0)}(t),\ t\in[-r,0]\). Also, from Condition 1\(\alpha^{(0)}(0)<\phi(0)-h=\alpha^{(1)}(0)\). Assume there exists a point \(t^{*}\in(0,T]\) such that \(\alpha^{(0)}(t)<\alpha^{(1)}(t)\), \(t\in[0,t^{*})\) and \(\alpha^{(0)}(t^{*})=\alpha^{(1)}(t^{*})\). Then from Condition 1, Definition 3, and the inequality \(\alpha(0)=\phi(0)-(\phi(0)-\alpha(0))<\phi(0)-h\) the inequality \[\alpha^{(0)}(t^{*}) \leq \alpha(0)\Gamma(q)E_{q,q}(M(t^{*})^{q})(t^{*})^{q-1}\] \[+\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})F(s,\alpha^{ (0)}(s),\sup_{\sigma\in[s-r,s]}\alpha^{(0)}(\sigma)ds\] \[-\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})M\alpha^{(0 )}(t)ds\] \[< (\phi(0)-h)\Gamma(q)E_{q,q}(M(t^{*})^{q})(t^{*})^{q-1}\] \[+\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})F(s,\alpha ^{(0)}(s),\sup_{\sigma\in[s-r,s]}\alpha^{(0)}(\sigma))ds\] \[-\int_{0}^{t^{*}}(t-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})\times\] \[\times\Big{(}M\alpha^{(0)}(t)+L\Big{(}\sup_{\sigma\in[s-r,s]} \alpha^{(0)}(\sigma)-\sup_{\sigma\in[s-r,s]}\alpha^{(1)}(\sigma)\Big{)}\Big{)}ds\] \[= \Omega(\alpha^{(0)},\alpha^{(1)},-h)(t^{*})=\alpha^{(1)}(t^{*})\]  
holds. The obtained contradiction proves the inequality (15) on \((0,T]\). We will prove the inequality  
\[\beta^{(0)}(t)\geq\beta^{(1)}(t),\ t\in[-r,T]. \tag{16}\]  
For \(t\in[-r,0]\) from Definition 3 and the definition of the constant \(g\) we get \(\beta^{(1)}(t)=\phi(t)+g\leq\phi(t)+(\beta(t)-\phi(t))=\beta^{(0)}(t),\ t\in[-r,0]\). Also, from Condition 1\(\beta^{(0)}(0)>\phi(0)+g=\beta^{(1)}(0)\). Assume there exists a point \(t^{*}\in(0,T]\) such that \(\beta^{(0)}(t)>\beta^{(1)}(t)\), \(t\in[0,t^{*})\) and \(\beta^{(0)}(t^{*})=\beta^{(1)}(t^{*})\). Then from Condition 1, Definition 3, and the inequalities \(\beta(0)=\phi(0)+(\beta(0)-\phi(0))>\phi(0)+g\) and \(\sup_{\sigma\in[s-r,s]}\beta^{(0)}(\sigma)-\sup_{\sigma\in[s-r,s]}\beta^{(1)} (\sigma)\geq 0\), \(s\in[0,t^{*}]\) we obtain  
\[\alpha^{(0)}(t^{*}) \leq \alpha(0)\Gamma(q)E_{q,q}(M(t^{*})^{q})(t^{*})^{q-1}\]0 \[\alpha^{(0)}(t^{*}) \leq \alpha(0)\Gamma(q)E_{q,q}(M(t^{*})^{q})(t^{*})^{q-1}\]1 \[\alpha^{(0)}(t^{*}) \leq \alpha(0)\Gamma(q)E_{q,q}(M(t^{*})^{q})(t^{*})^{q-1}\]2 \[\alpha^{(0)}(t^{*}) \leq \alpha(0)\Gamma(q)E_{q,q}(M(t^{*})^{q})(t^{*})^{q-1}\]3 \[\alpha^{(0)}(t^{*}) \leq \alpha(0)\Gamma(q)E_{q,q}(M(t^{*})^{q})(t^{*})^{q-1}\]4 \[\alpha^{(0)}(t^{*}) \leq \alpha(0)\Gamma(q)E_{q,q}(M(t^{*})^{q})(t^{*})^{q-1}\]5 \[\alpha^{(0)}(t^{*}) \leq \alpha(0)\Gamma(q)E_{q,q}(M(t^{*})^{q})(t^{*})^{q-1}\]6  
holds. The contradiction proves the inequality (16). We use induction to prove properties of the sequences of successive approximations. Assume that for a natural number \(n\geq 2\) the inequality \(\alpha^{(n-2)}(t)\leq\alpha^{(n-1)}(t)\), \(t\in[-r,T]\) holds.