It suffices to derive (20), (22), and (23) from (5), (7), and (8) by letting \(\Phi _{t,\theta }=F\delta _{\theta +t-\lambda _{t}}\) because (4) and (19) as well as (6) and (21) are the same, while having the entries *ψ*, *Q*, and *R* coming from different equations. We start from the derivation of (22) from (7).

We look to the integral form (10) of (7) and substitute \(\Phi _{t,\theta }=F\delta _{\theta +t-\lambda _{t}}\). Then, \(\Phi _{t-\theta ,s-t+\theta }=F\delta _{s-\lambda _{t-\theta }}\) and (10) yields

$$\begin{aligned} Q_{t,\theta } ={}& \int _{\max (0,t-\theta -\varepsilon )}^{t}\bigl(Q_{s,s-t+ \theta }A^{*} \\ & {}+R_{s,s-t+\theta ,0}-Q_{s,s-t+\theta }C^{*}CP_{s}-FCP_{s} \delta _{s-\lambda _{t-\theta }}\bigr)\,ds. \end{aligned}$$

(27)

Taking into account that the inequality \(t-\theta -\varepsilon <\lambda _{t-\theta }\) holds by \(\mathrm{(C)}\), we conclude that the integral of the term with the delta function is zero if either \(t<\lambda _{t-\theta }\) or \(t-\theta -\varepsilon <\lambda _{t-\theta }\le 0\). Solving them in *θ* and taking into account the definitions of the sets *G*, \(G'\), and \(G''\), we conclude that (27) can be written as

$$\begin{aligned} Q_{t,\theta } ={}& \int _{\max (0,t-\theta -\varepsilon )}^{t}\bigl(Q_{s,s-t+ \theta }A^{*}+R_{s,s-t+\theta ,0}-Q_{s,s-t+\theta }C^{*}CP_{s} \bigr)\,ds \\ & {}- \textstyle\begin{cases} FCP_{\lambda _{t-\theta }} & \text{if } (t,\theta )\in G, \\ 0 & \text{if } (t,\theta )\in G'\cup G''. \end{cases}\displaystyle \end{aligned}$$

(28)

We claim that \(Q_{t,\theta }=0\) for \((t,\theta )\in G'\cup G''\).

First, let \((t,\theta )\in G''\). In this case, \(\max (0,t-\theta -\varepsilon )< s\le t \) implies \(-\varepsilon < s-t+\theta \le \theta < t-\lambda ^{-1}_{t}\) and, therefore,

$$ (t,\theta )\in G'' \quad \text{and}\quad \max (0,t-\theta -\varepsilon )< s\le t \quad \Rightarrow \quad (s,s-t+\theta )\in G''. $$

(29)

This will be used below. Letting \(\Phi _{t,\theta }=F\delta _{\theta +t-\lambda _{t}}\) in (10), we obtain

$$ Q_{t,\theta }= \int _{\max (0,t-\theta -\varepsilon )}^{t}\bigl(Q_{s,s-t+ \theta }A^{*}+R_{s,s-t+\theta ,0}-Q_{s,s-t+\theta }C^{*}CP_{s}-FCP_{s} \delta _{s-\lambda _{t-\theta }}\bigr)\,ds. $$

For \((t,\theta )\in G''\), \(t<\lambda ^{-1}_{t-\theta }\). Therefore, the integral of the term with a delta function vanishes and we obtain

$$ Q_{t,\theta }= \int _{\max (0,t-\theta -\varepsilon )}^{t}\bigl(Q_{s,s-t+ \theta }A^{*}+R_{s,s-t+\theta ,0}-Q_{s,s-t+\theta }C^{*}CP_{s} \bigr)\,ds. $$

(30)

To substitute *R* in (30) in terms of *Q*, we let \(\Phi _{t,\theta }=F\delta _{\theta +t-\lambda _{t}}\) in (11) and obtain

$$\begin{aligned} R_{t,\theta ,\tau } ={}&- \int _{\max (0,t-\theta -\varepsilon )}^{t}\bigl(Q_{s,s-t+ \theta }C^{*}CQ^{*}_{s,s-t+\tau } \\ &{} +Q_{s,s-t+\theta }C^{*}F^{*}\delta _{s-\lambda _{t-\tau }}+FCQ^{*}_{s,s-t+ \tau } \delta _{s-\lambda _{t-\theta }}\bigr)\,ds. \end{aligned}$$

(31)

The case \((t,\theta )\in G''\) assumes \(t<\lambda _{t-\theta }\). Therefore,

$$\begin{aligned} R_{t,\theta ,0} ={}&- \int _{\max (0,t-\theta -\varepsilon )}^{t}\bigl(Q_{s,s-t+ \theta }C^{*}CQ^{*}_{s,s-t} \\ &{} +Q_{s,s-t+\theta }C^{*}F^{*}\delta _{s-\lambda _{t}}+FCQ^{*}_{s,s-t} \delta _{s-\lambda _{t-\theta }}\bigr)\,ds \\ ={}&- \int _{\max (0,t-\theta -\varepsilon )}^{t}Q_{s,s-t+\theta }\bigl(C^{*}CQ^{*}_{s,s-t}+C^{*}F^{*} \delta _{s-\lambda _{t}}\bigr)\,ds. \end{aligned}$$

(32)

Then,

$$ R_{s,s-t+\theta ,0}=- \int _{\max (0,t-\theta -\varepsilon )}^{s}Q_{r,r-t+ \theta }\bigl(C^{*}CQ^{*}_{r,r-s}+C^{*}F^{*} \delta _{r-\lambda _{s}}\bigr)\,dr. $$

Using this in (30), we obtain

$$\begin{aligned} Q_{t,\theta } ={}& \int _{\max (0,t-\theta -\varepsilon )}^{t}Q_{s,s-t+ \theta }\bigl(A^{*}-C^{*}CP_{s} \bigr)\,ds \\ &{} - \int _{\max (0,t-\theta -\varepsilon )}^{t} \int _{\max (0,t- \theta -\varepsilon )}^{s}Q_{r,r-t+\theta }\bigl(C^{*}CQ^{*}_{r,r-s}+C^{*}F^{*} \delta _{r-\lambda _{s}}\bigr)\,dr\,ds \\ ={}& \int _{\max (0,t-\theta -\varepsilon )}^{t}Q_{s,s-t+\theta }K_{t,s}\,ds, \end{aligned}$$

where

$$ K_{t,s}=A^{*}-C^{*}CP_{s}- \int _{s}^{t}C^{*}C\bigl(Q^{*}_{s,s-r}+C^{*}F^{*} \delta _{s-\lambda _{r}}\bigr)\,dr. $$

By (29), *Q* on \(G''\) is expressed linearly by the values of *Q* on \(G''\). This implies that

$$ Q_{t,\theta }=0 \quad \text{if } (t,\theta )\in G''. $$

Now, we assume \((t,\theta )\in G'\). In this case, \(0\le s\le t\) implies \(0\le s\le \lambda ^{-1}_{0}\) and \(s-\lambda ^{-1}_{0}\le s-t+\theta \le 0\). Therefore,

$$ 0\le s\le t \quad \Rightarrow\quad (s,s-t+\theta )\in G'. $$

(33)

This will be used below. For \((t,\theta )\in G'\), we have \(t-\theta -\varepsilon \le \lambda ^{-1}_{0}-\varepsilon <0\). Therefore, letting \(\Phi _{t,\theta }=F\delta _{\theta +t-\lambda _{t}}\) in (10), we obtain

$$ Q_{t,\theta }= \int _{0}^{t}\bigl(Q_{s,s-t+\theta }A^{*}+R_{s,s-t+\theta ,0}-Q_{s,s-t+ \theta }C^{*}CP_{s}-FCP_{s} \delta _{s-\lambda _{t-\theta }}\bigr)\,ds. $$

Additionally, in this case, \(\lambda _{t-\theta }\le 0\) implying

$$ Q_{t,\theta }= \int _{0}^{t}\bigl(Q_{s,s-t+\theta }A^{*}+R_{s,s-t+\theta ,0}-Q_{s,s-t+ \theta }C^{*}CP_{s} \bigr)\,ds. $$

(34)

Performing the same operations with (11) yields

$$\begin{aligned} R_{t,\theta ,0} ={}&- \int _{0}^{t}\bigl(Q_{s,s-t+\theta }C^{*}CQ^{*}_{s,s-t} \\ &{} +Q_{s,s-t+\theta }C^{*}F^{*}\delta _{s-\lambda _{t}}+FCQ^{*}_{s,s-t} \delta _{s-\lambda _{t-\theta }}\bigr)\,ds \\ ={}&- \int _{0}^{t}Q_{s,s-t+\theta }\bigl(C^{*}CQ^{*}_{s,s-t}+C^{*}F^{*} \delta _{s-\lambda _{t}}\bigr)\,ds. \end{aligned}$$

(35)

Then,

$$ R_{s,s-t+\theta ,0}=- \int _{0}^{s}Q_{r,r-t+\theta }\bigl(C^{*}CQ^{*}_{r,r-s}+C^{*}F^{*} \delta _{r-\lambda _{s}}\bigr)\,dr. $$

Substituting this into (34) yields

$$\begin{aligned} Q_{t,\theta } ={}& \int _{0}^{t}Q_{s,s-t+\theta }\bigl(A^{*}-C^{*}CP_{s} \bigr)\,ds \\ & {}- \int _{0}^{t} \int _{0}^{s}Q_{r,r-t+\theta }\bigl(C^{*}CQ^{*}_{r,r-s}+C^{*}F^{*} \delta _{r-\lambda _{s}}\bigr)\,dr\,ds \\ ={}& \int _{0}^{t}Q_{s,s-t+\theta } \biggl( A^{*}-C^{*}CP_{s}- \int _{s}^{t}C^{*}C\bigl(Q^{*}_{s,s-r}+C^{*}F^{*} \delta _{s-\lambda _{r}}\bigr)\,dr \biggr)\,ds. \end{aligned}$$

By (33), *Q* on \(G'\) is expressed linearly by the values of *Q* on \(G'\). This implies that

$$ Q_{t,\theta }=0 \quad \text{if } (t,\theta )\in G'. $$

Resuming, we can update equation (28) for \((t,\theta )\in G\) by removing from the interval \((\max (0,t-\theta -\varepsilon ),t]\) of integration those values of *s* for which \((s,s-t+\theta )\notin G\). According to the definitions of \(G'\) and \(G''\), these values of *s* are specified by the inequalities

$$ s-t+\theta < s-\lambda ^{-1}_{s} \quad \text{and}\quad s-\lambda ^{-1}_{0}\le s-t+ \theta . $$

Solving these inequalities, we obtain \(s<\lambda _{t-\theta }\le 0\). Therefore, the interval of integration in (28) must be \((\lambda _{t-\theta },t]\). Therefore, (28) in the updated form becomes the same as (25) if \((t,\theta )\in G\) and \(Q_{t,\theta }=0\) if \((t,\theta )\in G'\cup G''\).

Next, we derive (23) from (8) or (11). Letting \(\Phi _{t,\theta }=F\delta _{\theta +t-\lambda _{t}}\) in (11), we obtain

$$\begin{aligned} R_{t,\theta ,\tau } ={}&- \int _{\max (0,t-\theta -\varepsilon )}^{t}\bigl(Q_{s,s-t+ \theta }C^{*}CQ^{*}_{s,s-t+\tau } \\ & {}+Q_{s,s-t+\theta }C^{*}F^{*}\delta _{s-\lambda _{t-\tau }}+FCQ^{*}_{s,s-t+ \tau } \delta _{s-\lambda _{t-\theta }}\bigr)\,ds. \end{aligned}$$

(36)

Implementing zeros of *Q*, one can see that \(R_{t,\theta ,\tau }=0\) on the sets \(H'\) and \(H''\). Therefore, the integral of the first two terms in (36) can be updated to the interval \((\lambda _{t-\theta },t]\). Then, the integral of the second term vanishes since \(\theta \le \tau \) implies \(\lambda _{t-\tau }\le \lambda _{t-\theta }\) and, therefore, \(\lambda _{t-\tau }\) remains out of the interval \((\lambda _{t-\theta },t]\) of integration. Finally, the integral of the last term produces \(-FCQ^{*}_{\lambda _{t-\theta },\lambda _{t-\theta }-t+\tau }\). Thus, we obtain that (36) in the updated form becomes the same as (26) if \((t,\theta ,\tau )\in H\) and \(R_{t,\theta ,\tau }=0\) if \((t,\theta ,\tau )\in H'\cup H''\).

Finally, we turn to (5). Its solution can be written as (9). Letting \(\Phi _{t,\theta }=F\delta _{\theta +t-\lambda _{t}}\) and implementing zeros of *Q*, we find that (9) in the updated form becomes the same as (24) if \((t,\theta )\in G\) and \(\psi _{t,\theta }=0\) if \((t,\theta )\in G'\cup G''\). This completes the proof.