Appendix 1: Proof of Lemma 3
Set \(P_{j} = P_{j}^{R} + {\rm{i}}P_{j}^{I} ,X_{j} = X_{j}^{R} + {\rm{i}}X_{j}^{I} ,Y_{j} = Y_{j}^{R} + {\rm{i}}Y_{j}^{I} ,\) then \(P_{j}^{H} = P_{j} \Leftrightarrow (P_{j}^{R} )^{{ \top }} = P_{j}^{R} ,(P_{j}^{I} )^{{ \top }} = - P_{j}^{I} ,j = 1,2.\) Applying Lemma 1 to (2) yields
$$\left[ {\begin{array}{*{20}c} {P_{1}^{R} - X_{1}^{R} } & {Y_{1}^{R} } & { - P_{1}^{I} + X_{1}^{I} } & { - Y_{1}^{I} } \\ * & {P_{2}^{R} } & {(Y_{1}^{I} )^{{ \top }} } & { - P_{2}^{I} } \\ * & * & {P_{1}^{R} - X_{1}^{R} } & {Y_{1}^{R} } \\ * & * & * & {P_{2}^{R} } \\ \end{array} } \right] \ge 0,$$
and
$$\left[ {\begin{array}{*{20}c} {P_{1}^{R} } & {Y_{2}^{R} } & { - P_{1}^{I} } & { - Y_{2}^{I} } \\ * & {P_{2}^{R} - X_{2}^{R} } & {(Y_{2}^{I} )^{{ \top }} } & { - P_{2}^{I} + X_{2}^{I} } \\ * & * & {P_{1}^{R} } & {Y_{2}^{R} } \\ * & * & * & {P_{2}^{R} - X_{2}^{R} } \\ \end{array} } \right] \ge 0.$$
That is
$$\left[ {\begin{array}{*{20}c} {P_{1}^{R} - X_{1}^{R} } & { - P_{1}^{I} + X_{1}^{I} } & {Y_{1}^{R} } & { - Y_{1}^{I} } \\ * & {P_{1}^{R} - X_{1}^{R} } & {Y_{1}^{I} } & {Y_{1}^{R} } \\ * & * & {P_{2}^{R} } & { - P_{2}^{I} } \\ * & * & * & {P_{2}^{R} } \\ \end{array} } \right] \ge 0,$$
(16)
and
$$\left[ {\begin{array}{*{20}c} {P_{1}^{R} } & { - P_{1}^{I} } & {Y_{2}^{R} } & { - Y_{2}^{I} } \\ * & {P_{1}^{R} } & {Y_{2}^{I} } & {Y_{2}^{R} } \\ * & * & {P_{2}^{R} - X_{2}^{R} } & { - P_{2}^{I} + X_{2}^{I} } \\ * & * & * & {P_{2}^{R} - X_{2}^{R} } \\ \end{array} } \right] \ge 0.$$
(17)
Based on the definition of \(\begin{array}{*{20}l} {\begin{array}{*{20}l} {\begin{array}{*{20}l} {\begin{array}{*{20}l} {\chi ( \cdot )} \hfill \\ \end{array} } \hfill \\ \end{array} } \hfill \\ \end{array} } \hfill \\ \end{array}\) in Lemma 1, (12) and (13) can be rewritten as
$$\left[ {\begin{array}{*{20}c} {\chi (P_{1} ) - \chi (X_{1} )} & {\chi (Y_{1} )} \\ * & {\chi (P_{2} )} \\ \end{array} } \right] \ge 0,\quad {\rm{and}}\quad \left[ {\begin{array}{*{20}c} {\chi (P_{1} )} & {\chi (Y_{2} )} \\ * & {\chi (P_{2} ) - \chi (X_{2} )} \\ \end{array} } \right] \ge 0.$$
Letting \(z_{j} = z_{j}^{R} + {\rm{i}}z_{j}^{I} \in {\mathbb{C}}^{n} ,\) it follows from the improved RCI [36] that
$$\begin{aligned} & \frac{1}{\alpha }z_{1}^{H} P_{1} z_{1} + \frac{1}{1 - \alpha }z_{2}^{H} P_{2} z_{2} \\ & \quad = \frac{1}{\alpha }\left[ {\begin{array}{*{20}c} {z_{1}^{R} } \\ {z_{1}^{I} } \\ \end{array} } \right]^{{ \top }} \chi (P_{1} )\left[ {\begin{array}{*{20}c} {z_{1}^{R} } \\ {z_{1}^{I} } \\ \end{array} } \right] + \frac{1}{1 - \alpha }\left[ {\begin{array}{*{20}c} {z_{2}^{R} } \\ {z_{2}^{I} } \\ \end{array} } \right]^{{ \top }} \chi (P_{2} )\left[ {\begin{array}{*{20}c} {z_{2}^{R} } \\ {z_{2}^{I} } \\ \end{array} } \right] \\ \end{aligned}$$
$$\ge \left[ {\begin{array}{*{20}c} {z_{1}^{R} } \\ {z_{1}^{I} } \\ {z_{2}^{R} } \\ {z_{2}^{I} } \\ \end{array} } \right]^{{ \top }} \left[ {\begin{array}{*{20}c} {\chi (P_{1} ) + (1 - \alpha )\chi (X_{1} )} & {\alpha \chi (Y_{1} ) + (1 - \alpha )\chi (Y_{2} )} \\ * & {\chi (P_{2} ) + \alpha \chi (X_{2} )} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {z_{1}^{R} } \\ {z_{1}^{I} } \\ {z_{2}^{R} } \\ {z_{2}^{I} } \\ \end{array} } \right]$$
$$\begin{aligned} & \quad = \left[ {\begin{array}{*{20}c} {z_{1}^{R} } \\ {z_{2}^{R} } \\ {z_{1}^{I} } \\ {z_{2}^{I} } \\ \end{array} } \right]^{{ \top }} \left[ {\begin{array}{*{20}c} {P_{1}^{R} + (1 - \alpha )X_{1}^{R} } & {\alpha Y_{1}^{R} + (1 - \alpha )Y_{2}^{R} } \\ * & {P_{2}^{R} + \alpha X_{2}^{R} } \\ * & * \\ * & * \\ \end{array} } \right.\left. {\;\begin{array}{*{20}c} { - P_{1}^{I} - (1 - \alpha )X_{1}^{I} } & { - \alpha Y_{1}^{I} - (1 - \alpha )Y_{2}^{I} } \\ {\alpha (Y_{1}^{I} )^{{ \top }} + (1 - \alpha )(Y_{2}^{I} )^{{ \top }} } & { - P_{2}^{I} - \alpha X_{2}^{I} } \\ {P_{1}^{R} + (1 - \alpha )X_{1}^{R} } & {\alpha Y_{1}^{R} + (1 - \alpha )Y_{2}^{R} } \\ * & {P_{2}^{R} + \alpha X_{2}^{R} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {z_{1}^{R} } \\ {z_{2}^{R} } \\ {z_{1}^{I} } \\ {z_{2}^{I} } \\ \end{array} } \right] \\ & \quad = \left[ {\begin{array}{*{20}c} {z_{1} } \\ {z_{2} } \\ \end{array} } \right]^{H} \left[ {\begin{array}{*{20}c} {P_{1} + (1 - \alpha )X_{1} } & {\alpha Y_{1} + (1 - \alpha )Y_{2} } \\ * & {P_{2} + \alpha X_{2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {z_{1} } \\ {z_{2} } \\ \end{array} } \right]. \\ \end{aligned}$$
This completes the proof.
Appendix 2: Proof of Lemma 5
We only prove inequalities (7) and (8), inequality (6) can be proved similarly.
Setting \(z(s) = z^{R} (s) + {\rm{i}}z^{I} (s),U = U^{R} + {\rm{i}}U^{I} ,{\varrho }_{j} = {\varrho }_{j}^{R} + {\rm{i}}{\varrho }_{j}^{R} ,\) \(j = 5,6,...,10,\) from Lemma 5 of [7] we have
$$\begin{aligned} & \int_{a}^{b} {\int_{\theta }^{b} z } (s)^{H} Uz(s){\rm{d}}s{\rm{d}}\theta \\ & \quad = \int_{a}^{b} {\int_{\theta }^{b} {\left[ {\begin{array}{*{20}c} {z^{R} (s)} \\ {z^{I} (s)} \\ \end{array} } \right]^{{ \top }} } } \chi (U)\left[ {\begin{array}{*{20}c} {z^{R} (s)} \\ {z^{I} (s)} \\ \end{array} } \right]{\rm{d}}s{\rm{d}}\theta \\ & \quad \ge \frac{1}{2}\left[ {\begin{array}{*{20}c} {{\tilde{\varrho }}_{5} } \\ {{\tilde{\varrho }}_{6} } \\ {{\tilde{\varrho }}_{7} } \\ \end{array} } \right]^{{ \top }} \left[ {\begin{array}{*{20}c} {\chi (U)} & 0 & 0 \\ 0 & {2\chi (U)} & 0 \\ 0 & 0 & {3\chi (U)} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\tilde{\varrho }}_{5} } \\ {{\tilde{\varrho }}_{6} } \\ {{\tilde{\varrho }}_{7} } \\ \end{array} } \right] \\ \end{aligned}$$
$$= \frac{1}{2}\left[ {\begin{array}{*{20}c} {{\varrho }_{5}^{R} } \\ {{\varrho }_{6}^{R} } \\ {{\varrho }_{7}^{R} } \\ {{\varrho }_{5}^{I} } \\ {{\varrho }_{6}^{I} } \\ {{\varrho }_{7}^{I} } \\ \end{array} } \right]^{{ \top }} \left[ {\begin{array}{*{20}c} {U^{R} } & 0 & 0 & { - U^{I} } & 0 & 0 \\ 0 & {2U^{R} } & 0 & 0 & { - 2U^{I} } & 0 \\ 0 & 0 & {3U^{R} } & 0 & 0 & { - 3U^{I} } \\ { - U^{I} } & 0 & 0 & {U^{R} } & 0 & 0 \\ 0 & { - 2U^{I} } & 0 & 0 & {2U^{R} } & 0 \\ 0 & 0 & { - 3U^{I} } & 0 & 0 & {3U^{R} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\varrho }_{5}^{R} } \\ {{\varrho }_{6}^{R} } \\ {{\varrho }_{7}^{R} } \\ {{\varrho }_{5}^{I} } \\ {{\varrho }_{6}^{I} } \\ {{\varrho }_{7}^{I} } \\ \end{array} } \right]$$
$$\begin{aligned} & \quad = \frac{1}{2}\left[ {\begin{array}{*{20}c} {{\varrho }_{5} } \\ {{\varrho }_{6} } \\ {{\varrho }_{7} } \\ \end{array} } \right]^{H} \left[ {\begin{array}{*{20}c} U & 0 & 0 \\ 0 & {2U} & 0 \\ 0 & 0 & {3U} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\varrho }_{5} } \\ {{\varrho }_{6} } \\ {{\varrho }_{7} } \\ \end{array} } \right] \\ & \quad = \frac{1}{2}\sum\limits_{j = 5}^{7} {(j - 4)} {\varrho }_{j}^{H} U{\varrho }_{j} , \\ \end{aligned}$$
$$\begin{aligned} & \int_{a}^{b} {\int_{a}^{\theta } z } (s)^{H} Uz(s){\rm{d}}s{\rm{d}}\theta \\ & \quad = \int_{a}^{b} {\int_{a}^{\theta } {\left[ {\begin{array}{*{20}c} {z^{R} (s)} \\ {z^{I} (s)} \\ \end{array} } \right]^{{ \top }} } } \chi (U)\left[ {\begin{array}{*{20}c} {z^{R} (s)} \\ {z^{I} (s)} \\ \end{array} } \right]{\rm{d}}s{\rm{d}}\theta \\ & \quad \ge \frac{1}{2}\left[ {\begin{array}{*{20}c} {{\tilde{\varrho }}_{8} } \\ {{\tilde{\varrho }}_{9} } \\ {{\tilde{\varrho }}_{10} } \\ \end{array} } \right]^{{ \top }} \left[ {\begin{array}{*{20}c} {\chi (U)} & 0 & 0 \\ 0 & {2\chi (U)} & 0 \\ 0 & 0 & {3\chi (U)} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\tilde{\varrho }}_{8} } \\ {{\tilde{\varrho }}_{9} } \\ {{\tilde{\varrho }}_{10} } \\ \end{array} } \right] \\ \end{aligned}$$
$$= \frac{1}{2}\left[ {\begin{array}{*{20}c} {{\varrho }_{8}^{R} } \\ {{\varrho }_{9}^{R} } \\ {{\varrho }_{10}^{R} } \\ {{\varrho }_{8}^{I} } \\ {{\varrho }_{9}^{I} } \\ {{\varrho }_{10}^{I} } \\ \end{array} } \right]^{{ \top }} \left[ {\begin{array}{*{20}c} {U^{R} } & 0 & 0 & { - U^{I} } & 0 & 0 \\ 0 & {2U^{R} } & 0 & 0 & { - 2U^{I} } & 0 \\ 0 & 0 & {3U^{R} } & 0 & 0 & { - 3U^{I} } \\ { - U^{I} } & 0 & 0 & {U^{R} } & 0 & 0 \\ 0 & { - 2U^{I} } & 0 & 0 & {2U^{R} } & 0 \\ 0 & 0 & { - 3U^{I} } & 0 & 0 & {3U^{R} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\varrho }_{8}^{R} } \\ {{\varrho }_{9}^{R} } \\ {{\varrho }_{10}^{R} } \\ {{\varrho }_{8}^{I} } \\ {{\varrho }_{9}^{I} } \\ {{\varrho }_{10}^{I} } \\ \end{array} } \right]$$
$$\begin{aligned} & \quad = \frac{1}{2}\left[ {\begin{array}{*{20}c} {{\varrho }_{8} } \\ {{\varrho }_{9} } \\ {{\varrho }_{10} } \\ \end{array} } \right]^{H} \left[ {\begin{array}{*{20}c} U & 0 & 0 \\ 0 & {2U} & 0 \\ 0 & 0 & {3U} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\varrho }_{8} } \\ {{\varrho }_{9} } \\ {{\varrho }_{10} } \\ \end{array} } \right] \\ & \quad = \frac{1}{2}\sum\limits_{j = 8}^{10} {(j - 7)} {\varrho }_{j}^{H} U{\varrho }_{j} , \\ \end{aligned}$$
where \({\tilde{\varrho }}_{j} = {\rm{col}}\{ {\varrho }_{j}^{R} ,{\varrho }_{j}^{I} \} ,\) \(j = 5,6,...,10.\) This ends the proof.
Appendix 3: Proof of Theorem 1
Firstly, we verify the existence and uniqueness of the EP. Let
$$H(w) = - C(t)w + [A(t) + B(t)]h(w) + E(t)H(w) + J(t).$$
(18)
Now, we verify that \(H(w)\) is injective on \({\mathbb{C}}^{n} .\) For \(\aleph, {^{-}}\!\!\!{\lambda} \in {\mathbb{C}}^{n}\) with \(\aleph \ne {^{-}}\!\!\!{\lambda},\) we need to prove \(H(\aleph {) \ne }H{(}{^{-}}\!\!\!{\lambda} ).\) From (18), one has
$$H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} ) = - C(t)(\aleph - {^{-}}\!\!\!{\lambda} ) + [A(t) + B(t)][h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} )] + E(t)[H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )].$$
Thus
$$\begin{aligned} (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} [H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )] & = - (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} C(t)(\aleph - {^{-}}\!\!\!{\lambda} ) \\ & \quad + (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} [A(t) + B(t)][h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} )] + (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} E(t)[H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )]. \\ \end{aligned}$$
(19)
and
$$\begin{aligned} [H(\aleph ) - H({^{-}}\!\!\!{\lambda} )]^{H} P_{1} (\aleph - {^{-}}\!\!\!{\lambda} ) & = - (\aleph - {^{-}}\!\!\!{\lambda} )^{H} C(t)P_{1} (\aleph - {^{-}}\!\!\!{\lambda} ) \\ & \quad + [h(\aleph ) - h({^{-}}\!\!\!{\lambda} )]^{H} [A(t) + B(t)]^{H} P_{1} (\aleph - {^{-}}\!\!\!{\lambda} ) + [H(\aleph ) - H({^{-}}\!\!\!{\lambda} )]^{H} E(t)^{H} P_{1} (\aleph - {^{-}}\!\!\!{\lambda} ). \\ \end{aligned}$$
(20)
In addition, it follows from (18) that
$$\begin{aligned} & [H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )]^{H} Q[H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )] \\ & \quad = \{ - C(t)(\aleph - {^{-}}\!\!\!{\lambda} ) + [A(t) + B(t)][h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} )] + E(t)[H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )]\}^{H} \\ & \quad { \times }Q\{ - C(t)(\aleph - {^{-}}\!\!\!{\lambda} ) + [A(t) + B(t)][h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} )] + E(t)[H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )]\} . \\ \end{aligned}$$
(21)
Combining (19)–(21) yields
$$\begin{aligned} & {\rm{sym}}\{ (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} [H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )]\} \\ & \quad = (\aleph - {^{-}}\!\!\!{\lambda} )^{H} ( - P_{1} C(t) - C(t)P_{1} )(\aleph - {^{-}}\!\!\!{\lambda} ) + {\rm{sym}}\{ (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} [A(t) + B(t)][h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} )]\} \\ & \quad \quad + {\rm{sym}}\{ (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} E(t)[H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )]\} - [H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )]^{H} Q[H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )] \\ & \quad \quad + ( * )^{H} Q\{ - C(t)(\aleph - {^{-}}\!\!\!{\lambda} ) + [A(t) + B(t)][h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} )] + E(t)[H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )]\} . \\ \end{aligned}$$
(22)
Applying Lemma 3 in [9] and Assumption 1 yields
$$\begin{aligned} & (\aleph - {^{-}}\!\!\!{\lambda} )^{H} ( - P_{1} C(t) - C(t)P_{1} )(\aleph - {^{-}}\!\!\!{\lambda} ) \\ & \quad \le (\aleph - {^{-}}\!\!\!{\lambda} )^{H} ( - P_{1} C - CP_{1} + \nu_{1} N_{C}^{H} N_{C} + \nu_{1}^{ - 1} P_{1} M_{C}^{H} M_{C} P_{1} )(\aleph - {^{-}}\!\!\!{\lambda} ), \\ \end{aligned}$$
(23)
$$\begin{aligned} & {\rm{sym}}\{ (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} A(t)[h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} )]\} \le \nu_{2}^{ - 1} ( * )^{H} M_{A}^{H} P_{1} (\aleph - {^{-}}\!\!\!{\lambda} ) \\ & \quad + \nu_{2} ( * )^{H} N_{A} [h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} )] + {\rm{sym}}\{ (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} A[h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} )]\} , \\ \end{aligned}$$
(24)
$$\begin{aligned} & {\rm{sym}}\{ (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} B(t)[h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} )]\} \le \nu_{3}^{ - 1} ( * )^{H} M_{B}^{H} P_{1} (\aleph - {^{-}}\!\!\!{\lambda} ) \\ & \quad + \nu_{3} ( * )^{H} N_{B} [h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} )] + {\rm{sym}}\{ (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} B[h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} )]\} , \\ \end{aligned}$$
(25)
$$\begin{aligned} & {\rm{sym}}\{ (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} E(t)[H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )]\} \le \nu_{4}^{ - 1} ( * )^{H} M_{E}^{H} P_{1} (\aleph - {^{-}}\!\!\!{\lambda} ) \\ & \quad + \nu_{4} ( * )^{H} N_{E} [H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )] + {\rm{sym}}\{ (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} E[H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )]\} . \\ \end{aligned}$$
(26)
From Assumption 2, one gets
$$0 \le ( * )^{H} KR_{j} K(\aleph - {^{-}}\!\!\!{\lambda} ) - ( * )^{H} R_{j} [h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} )],j = 1,2.$$
(27)
Adding (22)–(27) yields
$${\rm{sym}}\{ (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} [H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )]\} \le \xi^{H} \Pi (t)\xi ,$$
(28)
where \(\xi = {\rm{col}}\{ \aleph - {^{-}}\!\!\!{\lambda} ,h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} ),h(\aleph {) - }h{(}{^{-}}\!\!\!{\lambda} ),H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )\} ,\) and
$$\Pi (t) = \left[ {\begin{array}{*{20}c} {\Pi_{11} (t)} & {P_{1} A - C(t)QA(t)} & {P_{1} B - C(t)QB(t)} & {P_{1} E - C(t)QE(t)} \\ * & {\Pi_{22} (t)} & {A(t)^{H} QB(t)} & 0 \\ * & * & {\Pi_{33} (t)} & 0 \\ * & 0 & 0 & {\Pi_{44} (t)} \\ \end{array} } \right],$$
with \(\Pi_{11} (t) = \Psi_{11}^{0} + \Psi_{11}^{1} + C(t)QC(t),\;\Pi_{22} (t) = \nu_{2} N_{A}^{H} N_{A} + A(t)^{H} QA(t) - R_{1} ,\;\Pi_{33} (t) = \nu_{3} N_{B}^{H} N_{B} + B(t)^{H}\)\(QB(t) - R_{2} ,\;\Pi_{44} (t) = \nu_{4} N_{E}^{H} N_{E} + E(t)^{H} QE(t) - Q,\) and \(\Psi_{11}^{1} = \nu_{1}^{ - 1} P_{1} M_{C} M_{C} P_{1} + \nu_{2}^{ - 1} P_{1} M_{A} M_{A}^{H} P_{1} + \nu_{3}^{ - 1} P_{1} M_{B}\)\(M_{B}^{H} P_{1} + \nu_{4}^{ - 1} P_{1} M_{E} M_{E}^{H} P_{1} .\)
Applying the SCL and (9) gives
$$\left[ {\begin{array}{*{20}c} {\tilde{\Psi }_{11} } & {\Psi_{12} } & 0 & {\tau N^{H} } \\ * & { - Q} & {QM} & 0 \\ 0 & * & { - \tau I} & 0 \\ * & 0 & 0 & { - \tau I} \\ \end{array} } \right] < 0,$$
(29)
where
$$\tilde{\Psi }_{11} = \left[ {\begin{array}{*{20}c} {\Psi_{11}^{0} + \Psi_{11}^{1} } & {P_{1} A} & {P_{1} B} & {P_{1} E} \\ * & {\nu_{2} N_{A}^{H} N_{A} - R_{1} } & 0 & 0 \\ * & 0 & {\nu_{3} N_{B}^{H} N_{B} - R_{2} } & 0 \\ * & 0 & 0 & {\nu_{4} N_{E}^{H} N_{E} - Q} \\ \end{array} } \right].$$
Again applying the SCL and (29) yields
$$\left[ {\begin{array}{*{20}c} {\tilde{\Psi }_{11} } & {\Psi_{12} } \\ * & { - Q} \\ \end{array} } \right] + \tau^{ - 1} \left[ {\begin{array}{*{20}c} 0 \\ {QM} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} 0 \\ {QM} \\ \end{array} } \right]^{H} + \tau \left[ {\begin{array}{*{20}c} {N^{H} } \\ 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {N^{H} } \\ 0 \\ \end{array} } \right]^{H} < 0.$$
(30)
From Assumption 1, Lemma 3 in [9] and (30), one has
$$\begin{gathered} \left[ {\begin{array}{*{20}c} {\tilde{\Psi }_{11} } & {\Psi_{12} } \\ * & { - Q} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} 0 \\ {QM} \\ \end{array} } \right]F(t)\left[ {\begin{array}{*{20}c} 0 \\ {QM} \\ \end{array} } \right]^{H} + \left[ {\begin{array}{*{20}c} {N^{H} } \\ 0 \\ \end{array} } \right]F(t)^{H} \left[ {\begin{array}{*{20}c} {N^{H} } \\ 0 \\ \end{array} } \right]^{H} \hfill \\ \le \left[ {\begin{array}{*{20}c} {\tilde{\Psi }_{11} } & {\Psi_{12} } \\ * & { - Q} \\ \end{array} } \right] + \tau^{ - 1} \left[ {\begin{array}{*{20}c} 0 \\ {QM} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} 0 \\ {QM} \\ \end{array} } \right]^{H} + \tau \left[ {\begin{array}{*{20}c} {N^{H} } \\ 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {N^{H} } \\ 0 \\ \end{array} } \right]^{H} < 0, \hfill \\ \end{gathered}$$
that is
$$\left[ {\begin{array}{*{20}c} {\tilde{\Psi }_{11} } & {\Psi_{12} + N^{H} F(t)^{H} M^{H} Q} \\ * & { - Q} \\ \end{array} } \right] < 0,$$
where \(F(t) = {\rm{diag}}\{ F_{C} (t),F_{A} (t),F_{B} (t),F_{E} (t)\} .\)
Again using the SCL gives
$$\Pi (t) = \tilde{\Psi }_{11} + ( * )^{H} Q[\Psi_{12}^{H} + QMF(t)N] < 0.$$
(31)
As \(\xi \ne 0\) and \(\aleph \ne {^{-}}\!\!\!{\lambda} ,\) from (28) and (31) one has
$${\rm{sym}}\{ (\aleph - {^{-}}\!\!\!{\lambda} )^{H} P_{1} [H(\aleph {) - }H{(}{^{-}}\!\!\!{\lambda} )]\} < 0,$$
which means \(H(\aleph {)} \ne H{(}{^{-}}\!\!\!{\lambda} ).\) Thus \(H(w)\) is injective on \({\mathbb{C}}^{n} .\)
Next, we assure \(||H(w)|| \mapsto \infty\) as \(||w|| \mapsto \infty .\) Similar to the proof of (28), it is easy to get
$${\rm{sym}}\{ w^{H} P_{1} [H(w{) - }H{(}0)]\} \le \ell^{H} \Pi (t)\ell ,$$
(32)
where \(\ell = {\rm{col}}\{ w,h(w{) - }h{(}0),h(w{) - }h{(}0),H(w{) - }H{(}0)\} .\)
Applying (32) and Lemma 2 in [9] gives
$$\ell^{H} \Pi (t)\ell \le - \lambda_{\min } ( - \Pi (t))||\ell ||^{2} \le - \lambda_{\min } ( - \Pi (t))||w||^{2} ,$$
thus one has
$$\begin{aligned} & \lambda_{\min } ( - \Pi (t))||w||^{2} \le - {\rm{sym}}\{ w^{H} P_{1} [H(w{) - }H{(}0)]\} \\ & \quad = - 2{\rm{Re}} \{ w^{H} P_{1} [H(w{) - }H{(}0)]\} \\ & \quad \le 2|w^{H} P_{1} [H(w{) - }H{(}0)] \\ & \quad \le 2||w|| \cdot ||P_{1} || \cdot ||H(w{) - }H{(}0) \\ & \quad \le 2||w|| \cdot ||P_{1} || \cdot (||H(w{)}|| + ||H{(}0). \\ \end{aligned}$$
That is
$$||H(w{)}|| \ge \frac{1}{{2||P_{1} ||}}\lambda_{\min } ( - \Pi (t))||w||^{2} - ||H{(}0)||.$$
Therefore, \(||H(w)|| \mapsto \infty\) as \(||w|| \mapsto \infty .\) From Lemma 5 in [9], one ensures that \(H(w)\) is homeomorphic on \({\mathbb{C}}^{n} ,\) which concludes that system (1) has unique EP.
Finally, we prove the GRS of EP. Let \(\overline{w} = [\overline{w}_{1} ,\overline{w}_{2} ,...,\overline{w}_{n} ]^{{ \top }}\) be the EP for system (1) and \(z(t) = w(t) - \overline{w},\) then (1) can be rewritten as
$$D^{\alpha } z(t) = - C(t)z_{t} + A(t)f(z_{t} ) + B(t)f(z_{\sigma } ) + E(t)D^{\alpha } z(t - \upsilon ),$$
(33)
where \(\sigma = \sigma (t),z_{t} = z(t),z_{\sigma } = z(t - \sigma (t)),z_{{\overline{\sigma }}} = z(t - \overline{\sigma }),f(z_{t} ) = h(z_{t} + \overline{w}) - h(\overline{w}).\)
From Assumption 2, one gets
$$0 \le z_{t}^{H} KR_{1} Kz_{t} - f(z_{t} )^{H} R_{1} f(z_{t} ),$$
(34)
$$0 \le z_{\sigma }^{H} KR_{2} Kz_{\sigma } - f(z_{\sigma } )^{H} R_{2} f(z_{\sigma } ).$$
(35)
Consider the following LKF
$$\overline{V}(z_{t} ,t) = \sum\limits_{j = 1}^{2} {V_{j} } (z_{t} ,t) + \frac{1}{{1 - \sigma_{d} }}\int_{t - \sigma }^{t} {( * )^{H} KR_{2} Kz(s){\rm{d}}s} ,$$
with \(V_{1} (z_{t} ,t) = \int_{0}^{\infty } {\varepsilon (\gamma )Z(\gamma ,t)^{H} P_{1} Z(\gamma ,t){\rm{d}}\gamma } ,V_{2} (z_{t} ,t) = \int_{t - \upsilon }^{t} {( * )^{H} Q(D^{\alpha } z(s)){\rm{d}}s} .\)
Because \(\varepsilon (\gamma ) > 0\) for any \(\gamma > 0\), \(V_{1} (z_{t} ,t) \ge 0.\) That is \(\overline{V}(z_{t} ,t) \ge 0\) for any \(\gamma > 0.\)
Calculation yields
$$\begin{gathered} \dot{\overline{V}}(z_{t} ,t) = \sum\limits_{j = 1}^{2} {\dot{V}_{j} } (z_{t} ,t) + \frac{1}{{1 - \sigma_{d} }}z_{t}^{H} KR_{2} Kz_{t} - \frac{{1 - \dot{\sigma }(t)}}{{1 - \sigma_{d} }}z_{\sigma }^{H} KR_{2} Kz_{\sigma } \hfill \\ \le \sum\limits_{j = 1}^{2} {\dot{V}_{j} } (z_{t} ,t) + \frac{1}{{1 - \sigma_{d} }}z_{t}^{H} KR_{2} Kz_{t} - z_{\sigma }^{H} KR_{2} Kz_{\sigma } . \hfill \\ \end{gathered}$$
(36)
By Lemma 2 it follows that
$$\begin{gathered} \dot{V}_{1} (z_{t} ,t) = 2\int_{0}^{\infty } {\varepsilon (\gamma )Z(\gamma ,t)^{H} P_{1} \frac{\partial Z(\gamma ,t)}{{\partial t}}{\rm{d}}\gamma } \hfill \\ \quad \quad \quad = 2\int_{0}^{\infty } {\varepsilon (\gamma )Z(\gamma ,t)^{H} P_{1} [ - \gamma Z(\gamma ,t) + D^{\alpha } z(t)]{\rm{d}}\gamma } \hfill \\ \quad \quad \quad = - 2\int_{0}^{\infty } {\gamma \varepsilon (\gamma )Z(\gamma ,t)^{H} P_{1} Z(\gamma ,t){\rm{d}}\gamma } + 2z(t)^{H} P_{1} (D^{\alpha } z(t)). \hfill \\ \end{gathered}$$
As \(P_{1} > 0,\int_{0}^{\infty } {\gamma \varepsilon (\gamma )Z(\gamma ,t)^{H} P_{1} Z(\gamma ,t){\rm{d}}\gamma } \ge 0.\) Thus one has
$$\begin{aligned} \dot{V}_{1} (z_{t} ,t) & \le 2z(t)^{H} P_{1} (D^{\alpha } z(t)) \\ & = 2z(t)^{H} P_{1} [ - C(t)z_{t} + A(t)f(z_{t} ) + B(t)f(z_{\sigma } ) + E(t)D^{\alpha } z(t - \upsilon )], \\ \end{aligned}$$
(37)
$$\begin{aligned} \dot{V}_{2} (z_{t} ,t) & = ( * )^{H} Q(D^{\alpha } z(t)) - ( * )^{H} Q(D^{\alpha } z(t - \upsilon )) \\ & = ( * )^{H} Q[ - C(t)z_{t} + A(t)f(z_{t} ) + B(t)f(z_{\sigma } ) + E(t)D^{\alpha } z(t - \upsilon )] - ( * )^{H} Q(D^{\alpha } z(t - \upsilon )). \\ \end{aligned}$$
(38)
Similar to (20)-(23), combining (31)-(35) gives
$$\dot{\overline{V}}(z_{t} ,t) \le v(t)^{H} \Pi (t)v(t)\quad {\rm{and}}\quad v(t) = {\rm{col}}\{ z_{t} ,f(z_{t} ),f(z_{\sigma } ),D^{\alpha } z(t - \upsilon )\} .$$
Applying (31) gives that the unique EP of system (1) is GRS.
Appendix 4: Some parameters of Theorem 2
$$\begin{aligned} \Omega & = \sum\limits_{j = 1}^{6} {\Omega_{j} } , \\ \Omega_{1} & = \mu_{13}^{{ \top }} Q\mu_{13} - \mu_{14}^{{ \top }} Q\mu_{14} + {\rm{sym}}\{ \mu_{13}^{{ \top }} G^{H} ( - \mu_{13} - C\mu_{1} + A\mu_{4} + B\mu_{5} + E\mu_{14} )\} \\ & \quad + \nu_{1} \mu_{1}^{{ \top }} N_{C}^{H} N_{C} \mu_{1} + \nu_{2} \mu_{4}^{{ \top }} N_{A}^{H} N_{A} \mu_{4} + \nu_{3} \mu_{5}^{{ \top }} N_{B}^{H} N_{B} \mu_{5} + \nu_{4} \mu_{14}^{{ \top }} N_{E}^{H} N_{E} \mu_{14} , \\ \Omega_{2} & = ( * )^{H} ({\mathbf{\mathbb{Q}}}_{1} + {\mathbf{\mathbb{Q}}}_{2} ){\rm{col}}\{ \mu_{1} ,\mu_{4} \} - {(}1 - \sigma_{d} {)}( * )^{H} {\mathbf{\mathbb{Q}}}_{1} {\rm{col}}\{ \mu_{2} ,\mu_{5} \} - ( * )^{H} {\mathbf{\mathbb{Q}}}_{2} {\rm{col}}\{ \mu_{3} ,\mu_{6} \} {,} \\ \Omega_{3} & = {\rm{sym}}\{ (\mu_{1} - \mu_{3} )^{{ \top }} P_{2} (\mu_{7} - \mu_{9} )\} + \mu_{1}^{{ \top }} (P_{3} + \overline{\sigma }^{2} P_{4} )\mu_{1} - z_{{\overline{\sigma }}}^{H} P_{3} z_{{\overline{\sigma }}} - \eta_{1}^{{ \top }} {\mathbf{\mathbb{P}}}_{4} \eta_{1} - \eta_{2}^{{ \top }} {\mathbf{\mathbb{P}}}_{4} \mu_{2} , \\ \Omega_{4} & = \tfrac{1}{2}\overline{\sigma }^{2} \mu_{1}^{{ \top }} (P_{5} + P_{6} )\mu_{1} - \tfrac{1}{2}\mu_{9}^{{ \top }} P_{5} \mu_{9} - 4( * )^{{ \top }} P_{5} (\mu_{9} - \mu_{11} ) - \tfrac{1}{2}\mu_{10}^{{ \top }} P_{5} \mu_{10} - 4( * )^{{ \top }} P_{5} (\mu_{10} - \mu_{12} ) \\ & \quad - \tfrac{1}{2}( * )^{{ \top }} P_{6} (2\mu_{7} - \mu_{9} ) - 4( * )^{{ \top }} P_{6} (\mu_{7} - 2\mu_{9} + \mu_{11} ) - \tfrac{1}{2}( * )^{{ \top }} P_{6} (2\mu_{8} - \mu_{10} ) - 4( * )^{{ \top }} P_{6} (\mu_{8} - 2\mu_{10} + \mu_{12} ), \\ \Omega_{5} & = \mu_{1}^{{ \top }} KR_{1} K\mu_{1} - \mu_{4}^{{ \top }} R_{1} \mu_{4} + \mu_{2}^{{ \top }} KR_{2} K\mu_{2} - \mu_{5}^{{ \top }} R_{2} \mu_{5} + \mu_{3}^{{ \top }} KR_{3} K\mu_{3} - \mu_{6}^{{ \top }} R_{3} \mu_{6} , \\ \end{aligned}$$
$$\begin{aligned} \Omega_{6} & = - ( * )^{{ \top }} KR_{4} K(\mu_{1} - \mu_{2} ) - ( * )^{{ \top }} R_{4} (\mu_{4} - \mu_{5} ) \\ & \quad - ( * )^{{ \top }} KR_{5} K(\mu_{1} - \mu_{3} ) - ( * )^{{ \top }} R_{5} (\mu_{4} - \mu_{6} ) - ( * )^{{ \top }} KR_{6} K(\mu_{2} - \mu_{3} ) - ( * )^{{ \top }} R_{6} (\mu_{5} - \mu_{6} ). \\ \end{aligned}$$
Appendix 5: Proof of Theorem 2
Consider the following LKF:
$$V(z_{t} ,t) = \sum\limits_{j = 1}^{5} {V_{j} } (z_{t} ,t),$$
with
$$\begin{aligned} V_{3} (z_{t} ,t) & = \int_{t - \sigma }^{t} {( * )^{H} {\mathbf{\mathbb{Q}}}_{1} \varpi {(}s{\rm{)d}}} s + \int_{{t - \overline{\sigma }}}^{t} {( * )^{H} {\mathbf{\mathbb{Q}}}_{2} \varpi {(}s{\rm{)d}}} s, \\ V_{4} (z_{t} ,t) & = ( * )^{H} P_{2} \int_{{t - \overline{\sigma }}}^{t} {z_{s} {\rm{d}}} s + \int_{{t - \overline{\sigma }}}^{t} {z_{s}^{H} P_{3} z_{s} {\rm{d}}} s + \overline{\sigma }\int_{{t - \overline{\sigma }}}^{t} {\int_{\theta }^{t} {z_{s}^{H} P_{4} z_{s} {\rm{d}}} s{\rm{d}}} \theta , \\ V_{5} (z_{t} ,t) & = \int_{{t - \overline{\sigma }}}^{t} {\int_{u}^{t} {\int_{\theta }^{t} {z_{s}^{H} P_{5} z_{s} {\rm{d}}} s} {\rm{d}}} \theta {\rm{d}}u + \int_{{t - \overline{\sigma }}}^{t} {\int_{{t - \overline{\sigma }}}^{u} {\int_{\theta }^{t} {z_{s}^{H} P_{6} z_{s} {\rm{d}}} s} {\rm{d}}} \theta {\rm{d}}u, \\ \end{aligned}$$
where \(\varpi (s) = {\rm{col}}\{ z_{s} ,f(z_{s} )\}\) and \(V_{j} (z_{t} ,t)(j = 1,2)\) are defined in Theorem 1.
Calculation gives
$$\dot{V}(z_{t} ,t) = \sum\limits_{j = 1}^{5} {\dot{V}_{j} } (z_{t} ,t).$$
(39)
where
$$\dot{V}_{3} (z_{t} ,t) = ( * )^{H} ({\mathbf{\mathbb{Q}}}_{1} + {\mathbf{\mathbb{Q}}}_{2} )\varpi {(}t{)} - [1 - \dot{\sigma }{(}t{)}]( * )^{H} {\mathbf{\mathbb{Q}}}_{1} \varpi {(}t - \sigma {(}t{))} - ( * )^{H} {\mathbf{\mathbb{Q}}}_{2} \varpi {(}t - \overline{\sigma }{)},$$
(40)
$$\dot{V}_{4} (z_{t} ,t) = 2(z_{t} - z_{{\overline{\sigma }}} )^{H} P_{2} {(}\beta_{1t} + \beta_{2t} {)} + z_{t}^{H} (P_{3} + \overline{\sigma }^{2} P_{4} )z_{t} - z_{{\overline{\sigma }}}^{H} P_{3} z_{{\overline{\sigma }}} - \overline{\sigma }\int_{{t - \overline{\sigma }}}^{t} {z_{s}^{H} P_{4} z_{s} {\rm{d}}s} ,$$
(41)
$$\dot{V}_{5} (z_{t} ,t) = \frac{{\overline{\sigma }^{2} }}{2}z_{t}^{H} (P_{5} + P_{6} )z_{t} - \int_{{t - \overline{\sigma }}}^{t} {\int_{\theta }^{t} {z_{s}^{H} P_{5} z_{s} {\rm{d}}s} {\rm{d}}\theta } - \int_{{t - \overline{\sigma }}}^{t} {\int_{{t - \overline{\sigma }}}^{\theta } {z_{s}^{H} P_{6} z_{s} {\rm{d}}s} {\rm{d}}\theta } .$$
(42)
Set \(\beta_{1t} = \int_{t - \sigma }^{t} {z_{\theta } {\rm{d}}\theta } ,\;\beta_{2t} = \int_{{t - \overline{\sigma }}}^{t - \sigma } {z_{\theta } {\rm{d}}\theta } ,\;\beta_{3t} = \frac{2}{\sigma }\int_{t - \sigma }^{t} {\int_{s}^{t} {z_{\theta } {\rm{d}}\theta } } {\rm{d}}s,\;\beta_{4t} = \frac{2}{{\overline{\sigma } - \sigma }}\int_{{t - \overline{\sigma }}}^{t - \sigma } {\int_{s}^{t - \sigma } {z_{\theta } {\rm{d}}\theta } } {\rm{d}}s,\)\(\beta_{5t} = \frac{6}{{\sigma^{2} }}\int_{t - \sigma }^{t} {\int_{s}^{t} {\int_{u}^{t} {z_{\theta } } {\rm{d}}\theta {\rm{d}}} } u{\rm{d}}s,\;\beta_{6t} = \frac{6}{{(\overline{\sigma } - \sigma )^{2} }}\int_{{t - \overline{\sigma }}}^{t - \sigma } {\int_{s}^{t - \sigma } {\int_{u}^{t - \sigma } {z_{\theta } } {\rm{d}}\theta {\rm{d}}} } u{\rm{d}}s.\)
Utilizing the mean-value Theorem for integral gives
$$\mathop {\lim }\limits_{{\sigma \to 0^{ + } }} \beta_{{(2{\jmath } - 1)t}} = \;\mathop {\lim }\limits_{{\sigma \to \overline{\sigma }^{ - } }} \beta_{{(2{\jmath })t}} = 0,\;{\jmath } = 2,3.$$
Therefore \(\beta_{3t} ,\;\beta_{4t} ,\beta_{5t} ,\;\beta_{6t}\) are well defined if we set
$$\begin{array}{*{20}l} {\beta_{{(2{\jmath } - 1)t}} |_{\sigma = 0} = \beta_{{(2{\jmath })t}} |_{{\sigma = \overline{\sigma }}} = 0,\;{\jmath } = 2,3.} \hfill \\ \end{array}$$
Denoting \(\iota (t) = \sigma/{\overline{\sigma }} ,\) when \(0 < \sigma < \overline{\sigma },\) applying Lemma 5 yields
$$\begin{aligned} \overline{\sigma }\int_{t - \sigma }^{t} {z_{s}^{H} P_{4} z_{s} {\rm{d}}s} & \ge \frac{{\overline{\sigma }}}{\sigma }[( * )^{H} P_{4} \beta_{1t} + 3( * )^{H} P_{4} (\beta_{1t} - \beta_{3t} ) + 5( * )^{H} P_{4} (\beta_{1t} - 3\beta_{3t} + 2\beta_{5t} )] \\ & = \frac{1}{\iota (t)}\varphi (t)^{H} (\eta_{1}^{{ \top }} {\mathbf{\mathbb{P}}}_{4} \eta_{1} )\varphi (t), \\ \end{aligned}$$
(43)
$$\begin{aligned} \overline{\sigma }\int_{{t - \overline{\sigma }}}^{t - \sigma } {z_{s}^{H} P_{4} z_{s} {\rm{d}}s} & \ge \frac{{\overline{\sigma }}}{{\overline{\sigma } - \sigma }}[( * )^{H} P_{4} \beta_{2t} + 3( * )^{H} P_{4} (\beta_{2t} - \beta_{4t} ) + 5( * )^{H} P_{4} (\beta_{2t} - 3\beta_{4t} + 2\beta_{6t} )] \\ & = \frac{1}{1 - \iota (t)}\varphi (t)^{H} (\eta_{2}^{{ \top }} {\mathbf{\mathbb{P}}}_{4} \eta_{2} )\varphi (t). \\ \end{aligned}$$
(44)
It is obviously that
$$\int_{{t - \overline{\sigma }}}^{t} {\int_{\theta }^{t} {z_{s}^{H} P_{5} z_{s} {\rm{d}}s} {\rm{d}}\theta } = \int_{t - \sigma }^{t} {\int_{\theta }^{t} {z_{s}^{H} P_{5} z_{s} {\rm{d}}s} {\rm{d}}\theta + (\overline{\sigma } - \sigma )} \int_{t - \sigma }^{t} {z_{s}^{H} P_{5} z_{s} {\rm{d}}s} + \int_{{t - \overline{\sigma }}}^{t - \sigma } {\int_{\theta }^{t - \sigma } {z_{s}^{H} P_{5} z_{s} {\rm{d}}s} {\rm{d}}\theta } ,$$
(45)
$$\int_{{t - \overline{\sigma }}}^{t} {\int_{{t - \overline{\sigma }}}^{\theta } {z_{s}^{H} P_{6} z_{s} {\rm{d}}s} {\rm{d}}\theta } = \int_{t - \sigma }^{t} {\int_{t - \sigma }^{\theta } {z_{s}^{H} P_{6} z_{s} {\rm{d}}s} {\rm{d}}\theta } + \sigma \int_{{t - \overline{\sigma }}}^{t - \sigma } {z_{s}^{H} P_{6} z_{s} {\rm{d}}s} + \int_{{t - \overline{\sigma }}}^{t - \sigma } {\int_{{t - \overline{\sigma }}}^{\theta } {z_{s}^{H} P_{6} z_{s} {\rm{d}}s} {\rm{d}}\theta } .$$
(46)
When \(0 < \sigma < \overline{\sigma },\) applying Lemma 5 yields
$$\int_{t - \sigma }^{t} {\int_{\theta }^{t} {z_{s}^{H} P_{5} z_{s} {\rm{d}}s} {\rm{d}}\theta } \ge \frac{1}{2}\beta_{3t}^{H} P_{5} \beta_{3t} + 4( * )^{H} P_{5} (\beta_{3t} - \beta_{5t} ),$$
(47)
$$(\overline{\sigma } - \sigma )\int_{t - \sigma }^{t} {z_{s}^{H} P_{5} z_{s} {\rm{d}}s} \ge [1/ \iota (t)-1 ] \varphi (t)^{H} (\eta_{1}^{{ \top }} {\mathbf{\mathbb{P}}}_{5} \eta_{1} )\varphi (t),$$
(48)
$$\int_{{t - \overline{\sigma }}}^{t - \sigma } {\int_{\theta }^{t - \sigma } {z_{s}^{H} P_{5} z_{s} {\rm{d}}s} {\rm{d}}\theta } \ge \frac{1}{2}\beta_{4t}^{H} P_{5} \beta_{4t} + 4( * )^{H} P_{5} (\beta_{4t} - \beta_{6t} ),$$
(49)
$$\int_{t - \sigma }^{t} {\int_{t - \sigma }^{\theta } {z_{s}^{H} P_{6} z_{s} {\rm{d}}s} {\rm{d}}\theta } \ge \frac{1}{2}( * )^{H} P_{6} (2\beta_{1t} - \beta_{3t} ) + 4( * )^{H} P_{6} (\beta_{1t} - 2\beta_{3t} + \beta_{5t} ),$$
(50)
$$\sigma \int_{{t - \overline{\sigma }}}^{t - \sigma } {z_{s}^{H} P_{6} z_{s} {\rm{d}}s} \ge \left( {\frac{1}{1 - \iota (t)} - 1} \right)\varphi (t)^{H} (\eta_{2}^{{ \top }} {\mathbf{\mathbb{P}}}_{6} \eta_{2} )\varphi (t),$$
(51)
$$\int_{{t - \overline{\sigma }}}^{t - \sigma } {\int_{{t - \overline{\sigma }}}^{\theta } {z_{s}^{H} P_{6} z_{s} {\rm{d}}s} {\rm{d}}\theta \ge \frac{1}{2}( * )^{H} P_{6} (2\beta_{2t} - \beta_{4t} ) + 4( * )^{H} P_{6} (\beta_{2t} - 2\beta_{4t} + \beta_{6t} )} .$$
(52)
By Lemma 4 one gets
$$\begin{aligned} & \frac{1}{{\iota (t)}}( * )^{H} ({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{5} )\eta _{1} \varphi (t) + \frac{1}{{1 - \iota (t)}}( * )^{H} ({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{6} )\eta _{2} \varphi (t) \\ & \quad = ( * )^{H} \left( {\begin{array}{*{20}c} {({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{5} )/\iota (t)} & 0 \\ 0 & {({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{6} )/[1 - \iota (t)]} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\eta _{1} } \\ {\eta _{2} } \\ \end{array} } \right)\varphi (t) \\ & \quad \ge ( * )^{{ \top }} \left( {\begin{array}{*{20}c} {{\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{5} + [1 - \iota (t)]{\mathbf{\mathbb{W}}}_{1} } & {\iota (t)Y_{1} + [1 - \iota (t)]Y_{2} } \\ * & {{\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{6} + \iota (t){\mathbf{\mathbb{W}}}_{2} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\eta _{1} } \\ {\eta _{2} } \\ \end{array} } \right)\varphi (t), \\ \end{aligned}$$
(53)
where \({\mathbf{\mathbb{W}}}_{1} = {\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{5} - Y_{1} ({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{6} )^{ - 1} Y_{1}^{H} ,\,{\mathbf{\mathbb{W}}}_{2} = {\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{6} - Y_{2}^{H} ({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{5} )^{ - 1} Y_{2} .\)
From Assumption 2, the following inequalities are true
$$\begin{aligned} & \vartheta_{q} (w) \triangleq w^{H} KR_{q} Kw - ( * )^{H} R_{q} f(w) \ge 0, \\ & \zeta_{r} (w,\ell ) \triangleq ( * )^{H} KR_{3 + r} K(w - \ell ) - ( * )^{H} R_{3 + r} [f(w) - f(\ell )] \ge 0, \\ \end{aligned}$$
where \(q,r = 1,2,3.\)
Hence, the next inequalities are true:
$$\vartheta_{1} (z_{t} ) + \vartheta_{2} (z_{\sigma } ) + \vartheta_{3} (z_{{\overline{\sigma }}} ) \ge 0,$$
(54)
$$\zeta_{1} (z_{t} ,z_{\sigma } ) + \zeta_{2} (z_{t} ,z_{{\overline{\sigma }}} ) + \zeta_{3} (z_{\sigma } ,z_{{\overline{\sigma }}} ) \ge 0.$$
(55)
For any matrix \(G\) with appropriate dimension, from (33) one gets
$$2(D^{\alpha } z(t))^{H} G^{H} [ - D^{\alpha } z(t) - C(t)z_{t} + A(t)f(z_{t} ) + B(t)f(z_{\sigma } ) + E(t)D^{\alpha } z(t - \upsilon )] = 0.$$
For any positive scalars \(\nu_{j} (j = 1,2,3,4),\) from Assumption 1 and Lemma 3 in [9] one has
$$\begin{aligned} 0 & = 2(D^{\alpha } z(t))^{H} G^{H} \{ - D^{\alpha } z(t) - [C + M_{C} F_{C} (t)N_{C} ]z_{t} + [A + M_{A} F_{A} (t)N_{A} ]f(z_{t} ) \\ & \quad + [B + M_{B} F_{B} (t)N_{B} ]f(z_{\sigma } ) + [E + M_{E} F_{E} (t)N_{E} ]D^{\alpha } z(t - \upsilon )\} \\ & \le 2(D^{\alpha } z(t))^{H} G^{H} [ - D^{\alpha } z(t) - Cz_{t} + Af(z_{t} ) + Bf(z_{\sigma } ) + ED^{\alpha } z(t - \upsilon )] + \nu_{1} ( * )^{H} N_{C} z_{t} \\ & \quad + \nu_{1}^{ - 1} ( * )^{H} M_{C}^{H} G(D^{\alpha } z(t)) + \nu_{2} ( * )^{H} N_{A} f(z_{t} ) + \nu_{2}^{ - 1} ( * )^{H} M_{A}^{H} G(D^{\alpha } z(t)) + \nu_{3} ( * )^{H} N_{B} f(z_{\sigma } ) \\ & \quad + \nu_{3}^{ - 1} ( * )^{H} M_{B}^{H} G(D^{\alpha } z(t)) + \nu_{4} ( * )^{H} N_{E} (D^{\alpha } z(t - \upsilon )) + \nu_{4}^{ - 1} ( * )^{H} M_{E}^{H} G(D^{\alpha } z(t)). \\ \end{aligned}$$
(56)
Substituting (37), (38) and (40)–(56) into (39) gives
$$\dot{V}(z_{t} ,t) \le ( * )^{H} \Omega (\iota (t))\varphi (t),$$
(57)
where \(\varphi (t) = {\rm{col}}\{ z_{t} ,\;z_{\sigma } ,\;z_{{\overline{\sigma }}} ,\;f(z_{t} ),\;f(z_{\sigma } ),\;f(z_{{\overline{\sigma }}} ),\;\beta_{1t} ,\;\beta_{2t} ,\;\beta_{3t} ,\;\beta_{4t} ,\beta_{5t} ,\;\beta_{6t} ,D^{\alpha } z(t),D^{\alpha } z(t - \upsilon )\} ,\)
$$\Omega (\iota (t)) = \Omega + \Omega_{0} - ( * )^{{ \top }} \left( {\begin{array}{*{20}c} {[1 - \iota (t)][{\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{5} - Y_{1} ({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{6} )^{ - 1} Y_{1}^{H} ]} & {\iota (t)Y_{1} + [1 - \iota (t)]Y_{2} } \\ * & {\iota (t)[{\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{6} - Y_{2}^{H} ({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{5} )^{ - 1} Y_{2} ]} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\eta_{1} } \\ {\eta_{2} } \\ \end{array} } \right),$$
with \(\Omega_{0} = ( * )^{H} X^{ - 1} M^{H} G\mu_{13} .\)
It is easy to verify that inequality (57) is still true for \(\sigma (t) = 0\) or \(\sigma (t) = \overline{\sigma }.\)
Note that \(\Omega (\iota (t))\) is linear about \(\iota (t),\) inequality \(\Omega (\iota (t)) < 0\) is equal to the following two conditions:
$$\Omega (\iota (t) = 0) = \Omega + \Omega_{0} - ( * )^{{ \top }} \left( {\begin{array}{*{20}c} {{\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{5} - Y_{1} ({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{6} )^{ - 1} Y_{1}^{H} } & {Y_{2} } \\ * & 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\eta_{1} } \\ {\eta_{2} } \\ \end{array} } \right) < 0,$$
(58)
$$\Omega (\iota (t) = 1) = \Omega + \Omega_{0} - ( * )^{{ \top }} \left( {\begin{array}{*{20}c} 0 & {Y_{1} } \\ * & {{\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{6} - Y_{2}^{H} ({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{5} )^{ - 1} Y_{2} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\eta_{1} } \\ {\eta_{2} } \\ \end{array} } \right) < 0.$$
(59)
By the SCL, inequalities (58), (59) are equal to the following ones respectively:
$$\left( {\begin{array}{*{20}c} {\Omega + \Omega_{0} - \eta_{1}^{{ \top }} ({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{5} )\eta_{1} - {\rm{sym}}\{ \eta_{1}^{{ \top }} Y_{2} \eta_{2} \} } & {\eta_{1}^{{ \top }} Y_{1} } \\ * & { - ({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{6} )} \\ \end{array} } \right) < 0,$$
(60)
$$\left( {\begin{array}{*{20}c} {\Omega + \Omega_{0} - {\rm{sym}}\{ \eta_{1}^{{ \top }} Y_{1} \eta_{2} \} - \eta_{2}^{{ \top }} ({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{6} )\eta_{2} } & {\eta_{2}^{{ \top }} Y_{2}^{H} } \\ * & { - ({\mathbf{\mathbb{P}}}_{4} + {\mathbf{\mathbb{P}}}_{5} )} \\ \end{array} } \right) < 0.$$
(61)
By the SCL again, inequalities (60), (61) are equal to (10) and (11) respectively. This ends the proof.
