## 2 Statement of the Problem
We will prove that \(\alpha^{(n-1)}(t)\leq\alpha^{(n)}(t)\), \(t\in[-r,T]\). For \(t\in[-r,0]\) we get \(\alpha^{(n)}(t)=\phi(t)-\frac{h}{n}=\phi(t)-\frac{h}{n-1}-\frac{h}{n}+\frac{h} {n-1}=\alpha^{(n-1)}(t)+\frac{h}{n(n-1)}\). Therefore, for the functions \(u\equiv\alpha^{(n-2)},\alpha\equiv\alpha^{(n-1)},\beta\equiv\alpha^{(n)}\) and the constants \(\delta=\frac{h}{n(n-1)}>0\) and \(\varepsilon=-\frac{h}{n-1}\) according to Lemma 1 we have from the assumption \(u(t)\leq\alpha(t)\), \(t\in[-r,T]\) and \(\alpha(t)=\Omega(u,\alpha,\varepsilon)(t)\) and \(\beta(t)=\Omega(\alpha,\beta,\varepsilon+\delta)\) the inequality \(\alpha(t)=\alpha^{(n-1)}(t)<\beta(t)=\alpha^{(n)}(t)\) holds for all \(t\in[-r,T]\). Similarly, applying Lemma 2 instead of Lemma 1 we can prove  
\[\alpha^{(0)}(t^{*}) \leq \alpha(0)\Gamma(q)E_{q,q}(M(t^{*})^{q})(t^{*})^{q-1}\]7  
Using induction we will prove that for any natural number \(n\) the inequality  
\[\alpha^{(0)}(t^{*}) \leq \alpha(0)\Gamma(q)E_{q,q}(M(t^{*})^{q})(t^{*})^{q-1}\]8  
holds. Assume the inequality \(\alpha^{(n-1)}(t)\leq\beta^{(n-1)}(t),\ \ t\in[-r,T]\), holds. For \(t\in[-r,0]\) we have \(\alpha^{(n)}(t)=\phi(t)-\frac{h}{n}<\phi(t)<\phi(t)+\frac{g}{n}=\beta^{(n)}(t)\), i.e., (17) holds. Assume there exists a point \(t^{*}\in(0,T]\) such that \(\alpha^{(n)}(t)<\beta^{(n)}(t),\ t\in[0,t^{*})\) and \(\alpha^{(n)}(t^{*})=\beta^{(n)}(t^{*})\). Then we get the inequality  
\[\alpha^{(0)}(t^{*}) \leq \alpha(0)\Gamma(q)E_{q,q}(M(t^{*})^{q})(t^{*})^{q-1}\]9 \[+\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})F(s,\alpha^{ (0)}(s),\sup_{\sigma\in[s-r,s]}\alpha^{(0)}(\sigma)ds\]0 \[+\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})F(s,\alpha^{ (0)}(s),\sup_{\sigma\in[s-r,s]}\alpha^{(0)}(\sigma)ds\]1 \[+\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})F(s,\alpha^{ (0)}(s),\sup_{\sigma\in[s-r,s]}\alpha^{(0)}(\sigma)ds\]2 \[+\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})F(s,\alpha^{ (0)}(s),\sup_{\sigma\in[s-r,s]}\alpha^{(0)}(\sigma)ds\]3 \[+\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})F(s,\alpha^{ (0)}(s),\sup_{\sigma\in[s-r,s]}\alpha^{(0)}(\sigma)ds\]4 \[+\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})F(s,\alpha^{ (0)}(s),\sup_{\sigma\in[s-r,s]}\alpha^{(0)}(\sigma)ds\]5 \[+\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})F(s,\alpha^{ (0)}(s),\sup_{\sigma\in[s-r,s]}\alpha^{(0)}(\sigma)ds\]6  
The obtained contradiction proves the inequality (17) and the claim d). The claim e) follows from the definition of the functions \(\alpha^{(n)}(t)\) and \(\beta^{(n)}(t)\) for \(t\in[-r,0]\). and  
\[+\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})F(s,\alpha^{ (0)}(s),\sup_{\sigma\in[s-r,s]}\alpha^{(0)}(\sigma)ds\]7 \[+\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})F(s,\alpha^{ (0)}(s),\sup_{\sigma\in[s-r,s]}\alpha^{(0)}(\sigma)ds\]8 \[+\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})F(s,\alpha^{ (0)}(s),\sup_{\sigma\in[s-r,s]}\alpha^{(0)}(\sigma)ds\]9 \[-\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})M\alpha^{(0 )}(t)ds\]0  
Both sequences are increasing and decreasing, respectively, and bounded. Therefore, they are uniformly convergent on \([0,T]\). Denote, \(\tilde{V}(t)=\lim_{n\to\infty}\tilde{\alpha}^{(n)}(t)\) and \(\tilde{W}(t)=\lim_{n\to\infty}\tilde{\beta}^{(n)}(t),\;\;t\in[0,T]\). According to the above proved claims b), c) and d) the inequalities  
\[-\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})M\alpha^{(0 )}(t)ds\]1 \[-\int_{0}^{t^{*}}(t^{*}-s)^{q-1}E_{q,q}(M(t^{*}-s)^{q})M\alpha^{(0 )}(t)ds\]2  
hold. Therefore, the claim f) is proved. For any \(t\in[0,T]\) we have \(\lim_{n\to\infty}t^{q-1}\tilde{\alpha}^{(n)}(t)=V(t)\) and \(\lim_{n\to\infty}t^{q-1}\tilde{\alpha}^{(n)}(t)=\phi(t)=V(t)\) for \(t\in[-r,0]\). Therefore, \(\lim_{n\to\infty}\sup_{s\in[t-r,t]}s^{q-1}\tilde{\alpha}^{(n)}(s)=\sup_{s\in[t -r,t]}V(t)\).