Appendix
Proof of Theorem 3.1
Step 1. Choose \(x^0(t) \in L^\textsf{p}(\Omega , \mathcal {H})\) be a fixed initial approximation for (6), then by using Remark 2.3 for every \(n \ge 1\), we have
$$\begin{aligned} \left\| x^{n}(t)\right\| ^\textsf{p}&\le 4^{\textsf{p}-1}\Bigg \{\left\| S_q(t) \big [\varphi (0)-\mathcal {N}^1(0,0,\varphi )\big ]\right| ^\textsf{p}+\left\| \mathcal {N}^1(t,x_t^n,y_t^n)\right\| ^\textsf{p}\\&\quad +\,\left\| \int _0^tT_q(t-\varsigma )\big [\mathfrak {f}^1\left( \varsigma ,x_\varsigma ^{n-1},y_\varsigma ^{n-1}\right) -\mathfrak {f}^1(\varsigma ,0,0)+\mathfrak {f}^1(\varsigma ,0,0)\big ]d\varsigma \right\| ^\textsf{p}\\&\quad +\!\left\| \int _0^tT_q(t\!-\!\varsigma )\big [\Lambda ^1(\varsigma ,x_\varsigma ^{n-1},y_\varsigma ^{n-1})\!-\!\Lambda ^1(\varsigma ,0,0)\!+\!\Lambda ^1(\varsigma ,0,0)\big ]d\tilde{Z}_Q^{\mathbb {H}}(\varsigma )\right\| ^\textsf{p}\Bigg \}\\&\le 4^{\textsf{p}-1}\Bigg \{\tilde{M}_R^\textsf{p} \left\| \big [\varphi (0)-\mathcal {N}(0,0,\varphi )\big ]\right\| ^\textsf{p}+\left\| \mathcal {N}^1(t,x_t^n,y_t^n)\right\| ^\textsf{p}\\&\quad +2^{\textsf{p}-1}\tilde{M}_T^\textsf{p}\Bigg [\Big [\int _0^t(t-\varsigma )^{\frac{\textsf{p} (q-1)}{\textsf{p}-1}}d\varsigma \Big ]^{\textsf{p}-1}\int _0^t\Big (\left\| \mathfrak {f}^1(\varsigma ,x_\varsigma ^{n-1},y_\varsigma ^{n-1})-\mathfrak {f}(\varsigma ,0,0)\right\| ^\textsf{p}\\&\quad +\left\| \mathfrak {f}^1(\varsigma ,0,0)\right\| ^\textsf{p}\Big )d\varsigma +\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\\&\quad \times \,\int _0^t\Big (\left\| \Lambda ^1(\varsigma ,x_\varsigma ^{n-1},y_\varsigma ^{n-1})-\Lambda ^1(\varsigma ,0,0)\right\| ^\textsf{p}+\left\| \Lambda ^1(\varsigma ,0,0)\right\| ^\textsf{p}\Big )d\varsigma \Bigg ]\Bigg \}. \end{aligned}$$
Further,
$$\begin{aligned} E\left\| x^n(t)\right\| ^\textsf{p}&\le 4^{\textsf{p}-1}\Bigg \{2^{\textsf{p}-1}\tilde{M}_R^\textsf{p}\big [\big ( E\left\| \varphi \right\| ^\textsf{p}-M_{\mathcal {L}}^*E\left\| \varphi \right\| |^\textsf{p}\big )+\big (E\left\| \hat{\varphi }\right\| ^\textsf{p}-M_{\mathcal {L}}^*E\left\| \hat{\varphi }\right\| ^\textsf{p}\big )\big ]\\ {}&\quad +M_{\mathcal {L}}^*\big (E\left\| x_t^n\right\| ^\textsf{p}+E\left\| y_t\right\| ^\textsf{p}\big )+2^{\textsf{p}-1}\Bigg [\tilde{M}_T^\textsf{p}t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\\&\quad +\,\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Bigg ]\\&\quad \times \,\int _0^t\vartheta \big (E\left\| x_\varsigma ^{n-1}\right\| ^\textsf{p}\big )+\hat{\vartheta } \big (E\left\| y_\varsigma ^{n-1}\right\| ^\textsf{p}+\vartheta _0+\hat{\vartheta }_0\big )d\varsigma \Bigg ]\Bigg \}. \end{aligned}$$
Hence
$$\begin{aligned} E\left\| x^n(t)\right\| ^\textsf{p}&\le \eta _1+4^{\textsf{p}-1}\Bigg \{M_{\mathcal {L}}^*\left( E\left\| x_t^n\right\| |^\textsf{p}+E\left\| y_t^n\right\| |^\textsf{p}\right) \\&\quad +2^{\textsf{p}-1}\Bigg [\tilde{M}_T^\textsf{p}t^{\textsf{p}q-1}\left( \frac{{\textsf{p}-1}}{\textsf{p}q-1}\right) ^{\textsf{p}-1}+\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}}t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}}\\&\quad \times \, \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\left( \frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\right) ^{\frac{\textsf{p}-2}{2}}\int _0^t\Big (\vartheta \left( E\left\| x_\varsigma ^{n-1}\right\| ^\textsf{p}\right) \\&\quad +\hat{\vartheta } \left( E\left\| y_\varsigma ^{n-1}\right\| ^\textsf{p}\right) \Big )d\varsigma \Bigg ]\Bigg \}, \end{aligned}$$
where
$$\begin{aligned} \eta _1&=4^{\textsf{p}-1}\Bigg [\tilde{M}_R^\textsf{p}\left( E\left\| (\varphi _0\right\| ^\textsf{p}+M_{\mathcal {L}}^*E\left\| \varphi _0\right\| ^\textsf{p}\right) +2^{\textsf{p}-1}\Bigg [t^{\textsf{p}q-1}\left( \frac{{\textsf{p}-1}}{\textsf{p}q-1}\right) ^{\textsf{p}-1}\tilde{M}_T^\textsf{p}\\&\quad +\,\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Bigg ](\vartheta _0+\hat{\vartheta }_0)\Bigg ]. \end{aligned}$$
Since \(\varrho (\cdot )\) is concave and \(\varrho (0)=0\), \(\exists \) positive constants \(\alpha , \beta \) such that \(\varrho (\tilde{u}) \le \alpha +\beta \tilde{u}, \ \forall \tilde{u} \ge 0, t \in J.\) It is to be noted that
$$\begin{aligned}&\Big (\sup \limits _{0\le \varsigma \le t}E\left\| x^n(\varsigma )\right\| ^\textsf{p}\Big )\\&\quad \le \eta _1\!+\!\alpha \Big (t^{\textsf{p}q\!-\!1}\Big (\frac{\textsf{p}\!-\!1}{\textsf{p}q\!-\!1}\Big )^{\textsf{p}\!-\!1}\tilde{M}_T^\textsf{p}\!+\!\tilde{K}C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} \!-\! \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big )\tilde{T}\\&\qquad +\,4^{\textsf{p}-1}M_{\mathcal {L}}^*E\left\| x_{\varsigma }\right\| ^\textsf{p}+4^{\textsf{p}-1}\beta \Bigg \{2^{\textsf{p}-1}\bigg [\tilde{M}_T^\textsf{p}t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\\&\qquad +\,\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big )\\&\quad \le \eta _2+4^{\textsf{p}-1}M_{\mathcal {L}}^*E\left\| x_{\varsigma }\right\| ^\textsf{p}+4^{\textsf{p}-1}\beta \Bigg \{2^{\textsf{p}-1}\bigg [\tilde{M}_T^\textsf{p}\beta t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\\&\qquad +\,\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big )\\&\qquad \times \,\int _0^t\Big (\vartheta (E\left\| x_\varsigma ^{n-1}\right\| ^\textsf{p})+\hat{\vartheta } (E\left\| y_\varsigma ^{n-1}\right\| ^\textsf{p})\Big )d\varsigma \Bigg ]\Bigg \},\ n = 1, 2, \ldots , \end{aligned}$$
$$\begin{aligned} \text {here} \ \ \eta _2&=\eta _1+4^{\textsf{p}-1}\textsf{p}\Bigg \{2^{\textsf{p}-1}\Big (\tilde{M}_T^\textsf{p} t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\\&\quad +\,\tilde{K} C_\mathbb {H}^{\frac{p}{2}} t^{p\mathbb {H} - \frac{p}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Bigg \}\tilde{T} \end{aligned}$$
and seeing that
$$\begin{aligned} E\left\| x_t\right\| ^\textsf{p} \!&=\!\sup \limits _{-s\le \hat{\theta }\le 0}E\left\| x(t+\hat{\theta })\right\| ^\textsf{p}\!\le \!2^{\textsf{p}-1}\!\!\sup \limits _{-s\le \hat{\pi }\le 0}E\left\| x(\hat{\pi })\right\| ^\textsf{p}\!+\!2^{\textsf{p}-1}\!\sup \limits _{0\le \hat{\pi }\le \tilde{T}}E\left\| x(\hat{\pi })\right\| ^\textsf{p}\\ \text {and}\\E\left\| x_\varsigma ^{n}\right\| ^\textsf{p}&\le 2^{\textsf{p}-1}E\left\| \varphi \right\| ^\textsf{p}+2^{\textsf{p}-1}\sup \limits _{0\le \varsigma \le t}E\left\| x^{n-1}(\varsigma )\right\| ^\textsf{p}.\\E\left\| y_t\right\| ^\textsf{p} \!&=\!\sup \limits _{-s\le \hat{\theta }\le 0}E\left\| y(t+\hat{\theta })\right\| ^\textsf{p}\!\le \!2^{\textsf{p}-1}\!\!\sup \limits _{-s\le \hat{\pi }\le 0}E\left\| y(\hat{\pi })\right\| ^\textsf{p}\!+\!2^{\textsf{p}-1}\!\sup \limits _{0\le \hat{\pi }\le \tilde{T}}E\left\| y(\hat{\pi })\right\| ^\textsf{p}\\\text {and}\\E\left\| y_\varsigma ^{n}\right\| ^\textsf{p}&\le 2^{\textsf{p}-1}E\left\| \hat{\varphi }\right\| ^\textsf{p}+2^{\textsf{p}-1}\sup \limits _{0\le \hat{\varsigma }\le t}E\left\| y^{n-1}(\varsigma )\right\| ^\textsf{p}.\\ \text {Now}, \\&\Big (\sup \limits _{0\le \varsigma \le t}E\left\| x^n(\varsigma )\right\| ^\textsf{p}\Big )+\Big (\sup \limits _{0\le \varsigma \le t}E\left\| y^n(\varsigma )\right\| ^\textsf{p}\Big )\\&\quad \le \eta _2+10^{\textsf{p}-1}M_{\mathcal {L}}^*(\left\| \varphi \right\| ^\textsf{p}+ \left\| \hat{\varphi }\right\| ^\textsf{p})+10^{\textsf{p}-1}M_{\mathcal {L}}^*\sup \limits _{0\le \varsigma \le t}E\left\| x^n(\varsigma )\right\| ^\textsf{p}\\&\qquad +\sup \limits _{0\le \varsigma \le t}E\left\| y^n(\varsigma )\right\| ^\textsf{p} +\,4^{\textsf{p}-1}\beta \Bigg \{2^{\textsf{p}-1}\bigg [\tilde{M}_T^\textsf{p}\beta t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\\&\qquad +\,\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big )\Bigg ]\tilde{T}\Bigg \}\\&\qquad +\,4^{\textsf{p}-1}\beta \Bigg \{2^{\textsf{p}-1}\bigg [\tilde{M}_T^\textsf{p}\beta t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\\&\qquad +\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\bigg ]\\&\qquad \times \,\int _0^t(E\left\| x^{n-1}(\varsigma )\right\| ^\textsf{p}+E\left\| y^{n-1}(\varsigma )\right\| ^\textsf{p})d\varsigma \Bigg \},\ n = 1, 2, \ldots \end{aligned}$$
$$\begin{aligned}&\sup \limits _{0\le \varsigma \le t}E\left\| x^n(\varsigma )\right\| ^\textsf{p}+\Big (\sup \limits _{0\le \varsigma \le t}E\left\| y^n(\varsigma )\right\| ^\textsf{p}\Big )\Big (1-10^{\textsf{p}-1}M_{\mathcal {L}}^*\Big )\\&\quad \le \eta _3+4^{\textsf{p}-1}\beta \Bigg \{2^{\textsf{p}-1}\bigg [\tilde{M}_T^\textsf{p}\beta t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\\&\qquad +\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\bigg ]\\&\qquad \times \,\int _0^t(E\left\| x^{n-1}(\varsigma )\right\| ^\textsf{p}+E\left\| y^{n-1}(\varsigma )\right\| ^\textsf{p})d\varsigma \Bigg \},\ n = 1, 2, \ldots \\&\Big (\sup \limits _{0\le \varsigma \le t}E\left\| x^n(\varsigma )\right\| ^\textsf{p}\Big )+\Big (\sup \limits _{0\le \varsigma \le t}E\left\| y^n(\varsigma )\right\| ^\textsf{p}\Big )\\&\quad \le \frac{1}{\Big (1-10^{\textsf{p}-1}M_{\mathcal {L}}^*\Big )} \Bigg \{\eta _3+10^{\textsf{p}-1}\beta \bigg [\tilde{M}_T^\textsf{p}\beta t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\\&\qquad +\,\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big )\bigg ]\\&\qquad \times \int _0^t(E\left\| x^{n-1}(\varsigma )\right\| ^\textsf{p}+E\left\| y^{n-1}(\varsigma )\right\| ^\textsf{p}) d\varsigma \Bigg \}, \end{aligned}$$
where, \(\eta _3=\eta _2+\frac{4^{\textsf{p}-1}}{1-8^{\textsf{p}-1}}\Bigg \{2^{\textsf{p}-1}\beta \Big (\tilde{M}_T^\textsf{p}t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}+\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big )\tilde{T}\Bigg \}\). Also, for every \(\hat{K} \ge 1\)
$$\begin{aligned}{} & {} \max \limits _{1\le n\le \hat{K}}\sup \limits _{0 \le \varsigma \le t}E\left\| x^{n-1}(\varsigma )\right\| ^\textsf{p} \le E\left\| x^0(\varsigma )\right\| ^\textsf{p}+\max \limits _{1\le n\le \hat{K}}\sup \limits _{0 \le \varsigma \le t}E\left\| x^n(\varsigma )\right\| ^\textsf{p}\text {and}\\{} & {} \max \limits _{1\le n\le \hat{K}}\sup \limits _{0 \le \varsigma \le t}E\left\| y^{n-1}(\varsigma )\right\| ^\textsf{p} \le E\left\| y^0(\varsigma )\right\| ^\textsf{p}+\max \limits _{1\le n\le \hat{K}}\sup \limits _{0 \le \varsigma \le t}E\left\| y^n(\varsigma )\right\| ^\textsf{p} \end{aligned}$$
$$\begin{aligned}&\max \limits _{1\le n\le \hat{K}}\sup \limits _{0 \le \varsigma \le t}E\left\| x^ {n-1}(\varsigma )\right\| ^\textsf{p}+\max \limits _{1\le n\le \hat{K}}E\sup \limits _{0 \le \varsigma \le t}\left\| y^ {n-1}(\varsigma )\right\| ^\textsf{p} \\&\quad \le \frac{1}{\Big (1-10^{\textsf{p}-1}M_{\mathcal {L}}^*\Big )} \Bigg \{\eta _3+10^{\textsf{p}-1}\beta \bigg [\tilde{M}_T^\textsf{p}\beta t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\\&\qquad +\tilde{K}C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} \!-\! \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q\!-\!\textsf{p}\!-\!1}{2}}\Big (\frac{{\textsf{p}\!-\!2}}{2\textsf{p}q\!-\!\textsf{p}\!-\!2}\Big )^{\frac{\textsf{p}\!-\!2}{2}}\Big )\bigg ]\int _0^t(E\left\| x^{0}(\varsigma )\right\| ^\textsf{p}\!+\!E\left\| y^{0}(\varsigma )\right\| ^\textsf{p})d\varsigma \\&\qquad +10^{\textsf{p}\!-\!1}\beta \bigg [\tilde{M}_T^\textsf{p}\beta t^{\textsf{p}q\!-\!1}\Big (\frac{{\textsf{p}\!-\!1}}{\textsf{p}q\!-\!1}\Big )^{\textsf{p}\!-\!1}\!+\!\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} \!-\! \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q\!-\!\textsf{p}\!-\!1}{2}}\Big (\frac{{\textsf{p}\!-\!2}}{2\textsf{p}q\!-\!\textsf{p}\!-\!2}\Big )^{\frac{\textsf{p}\!-\!2}{2}}\bigg ]\\&\qquad \times \, \int _0^t(\max \limits _{1\le n\le \hat{K}}\sup \limits _{0 \le \tilde{r} \le \varsigma }E\left\| x^{n}(\tilde{r})\right\| ^\textsf{p}+\max \limits _{1\le n\le \hat{K}}\sup \limits _{0 \le \tilde{r} \le \varsigma }E\left\| y^{n}(\tilde{r})\right\| ^\textsf{p})d\varsigma \Bigg \}\\ \end{aligned}$$
$$\begin{aligned}&\quad \le \frac{1}{\Big (1-10^{\textsf{p}-1}M_{\mathcal {L}}^*\Big )} \Bigg \{\eta _3+10^{\textsf{p}-1}\beta \bigg [\tilde{M}_T^\textsf{p}\beta t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\\&\qquad +\,\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\bigg ](E\left\| \varphi \right\| ^\textsf{p}+E\left\| \hat{\varphi }\right\| ^\textsf{p})\tilde{T}\\&\qquad +\,8^{\textsf{p}-1}\beta \bigg [\tilde{M}_T^\textsf{p}\beta t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\\&\qquad +\,\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\bigg ]\int _0^t\max \limits _{1\le n\le \hat{K}}(\sup \limits _{0 \le \tilde{r} \le \varsigma }E\left\| x^{n}(\tilde{r})\right\| ^\textsf{p}\\ {}&\quad +E\left\| y^{n}(\tilde{r})\right\| ^\textsf{p})d\varsigma \Bigg \}\le \eta _4+\eta _5\int _0^t(E\left\| x^n(\varsigma )\right\| ^\textsf{p}+E\left\| y^n(\varsigma )\right\| ^\textsf{p})d\varsigma , \end{aligned}$$
where
$$\begin{aligned}\eta _4&= \frac{1}{\Big (1-10^{\textsf{p}-1}M_{\mathcal {L}}^*\Big )} \Bigg \{\eta _3+10^{\textsf{p}-1}\beta \bigg [\tilde{M}_T^\textsf{p}\beta t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\\&\quad +\,\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\bigg ](E\left\| \varphi \right\| ^\textsf{p}+E\left\| \hat{\varphi }\right\| ^\textsf{p})\tilde{T}\Bigg \},\\ \eta _5&=\frac{1}{\Big (1-10^{\textsf{p}-1}M_{\mathcal {L}}^*\Big )} \Bigg \{8^{\textsf{p}-1}\beta \bigg [\tilde{M}_T^\textsf{p}\beta t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\\&\quad +\,\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\bigg ]\Bigg \}, \end{aligned}$$
$$\begin{aligned} \max \limits _{1\le n\le \hat{K}}&\sup \limits _{0 \le \varsigma \le t}E\left\| x^{n-1}(\varsigma )\right\| ^\textsf{p}+\max \limits _{1\le n\le \hat{K}}\sup \limits _{0 \le \varsigma \le t}E\left\| y^{n-1}(\varsigma )\right\| ^\textsf{p}\\ {}&\le \eta _4+\eta _5\int _0^t\max \limits _{1\le n\le \hat{K}}\sup \limits _{0 \le \tilde{r} \le \varsigma }(E\left\| x^{n}(\tilde{r})\right\| ^\textsf{p}+E\left\| y^{n}(\tilde{r})\right\| ^\textsf{p})d\varsigma . \end{aligned}$$
Since \(\hat{K}\) is arbitrary, by the Gronwall inequality, one can have
$$\begin{aligned} \sup \limits _{0 \le \varsigma \le t}E\left\| x^{n-1}(\varsigma )\right\| ^\textsf{p}+\sup \limits _{0 \le \varsigma \le t}E\left\| y^{n-1}(\varsigma )\right\| ^\textsf{p}\le \eta _4 e^{\eta _5\tilde{T}}. \end{aligned}$$
Also,
$$\begin{aligned} E\left\| x^n\right\| ^\textsf{p}_{\hat{C}_{s}}= & {} E\left\| x_0^n\right\| ^\textsf{p}+E\left\| y_0^n\right\| ^\textsf{p}+\int _0^{\tilde{T}}E\left\| x_0^n\right\| ^\textsf{p}+E\left\| y_0^n\right\| ^\textsf{p}d\varsigma \le E\left\| \hat{\varphi }\right\| ^\textsf{p}\\{} & {} +E\left\| \varphi \right\| ^\textsf{p}+\tilde{T}\eta _4e^{\eta _5\tilde{T}} < \infty . \end{aligned}$$
This implies the boundedness.
Step 2. Choose the sequence \( x^n(t), \ n \ge 1, \) is a Cauchy sequence. For all \( m, n \ge 0 \) and \( t \in \hat{J} \) and \(M_{\mathcal {L}}^* < \frac{1}{3^{\textsf{p}-1}}\), one can have
$$\begin{aligned}&\sup \limits _{0 \le \varsigma \le t}E\left\| x^{n+1} (\varsigma ) -x^{m+1}(\varsigma )\right\| ^\textsf{p}+\sup \limits _{0 \le \varsigma \le t}E\left\| y^{n+1} (\varsigma ) -y^{m+1}(\varsigma )\right\| ^\textsf{p}\\&\quad \le 4^{\textsf{p}\!-\!1}\Bigg \{ M_{\mathcal {L}}^* \sup \limits _{0 \le \varsigma \le t}E\left\| x^{n+1}( \varsigma ) \!-\!x^{m+1}(\varsigma )\right\| ^\textsf{p} \!+\! M_{\mathcal {L}}^* \sup \limits _{0 \le \varsigma \le t}E\left\| y^{n\!+\!1}( \varsigma ) \!-\!y^{m\!+\!1}(\varsigma )\right\| ^\textsf{p}\\&\qquad +\,\left( \tilde{M}_T^\textsf{p}t^{\textsf{p}q-1}\left( \frac{{\textsf{p}-1}}{\textsf{p}q-1}\right) ^{\textsf{p}-1}+\tilde{K}C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\left( \frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\right) ^{\frac{\textsf{p}-2}{2}}\right) \\&\qquad \times \,\left( \int _0^t\vartheta \left( \sup \limits _{0 \le \varsigma \le t}E\left\| x_{\varsigma }^n-x_{\varsigma }^m\right\| ^\textsf{p}\right) d\varsigma +\int _0^t\hat{\vartheta }\left( \sup \limits _{0 \le \varsigma \le t}E\left\| y_{\varsigma }^n-y_{\varsigma }^m\right\| ^\textsf{p}\right) d\varsigma \right) \Bigg \}. \end{aligned}$$
$$\begin{aligned} \text {Thus,}&\left( \sup \limits _{0 \le \varsigma \le t}E\left\| x^{n+1}(t) \!-\!x^{m+1}(t)\right\| ^\textsf{p}\!+\!\sup \limits _{0 \le \varsigma \le t}E\left\| y^{n+1}(t) \!-\!y^{m+1}(t)\right\| ^\textsf{p}\right) (1\!-\!4^{\textsf{p}-1}M_{\mathcal {L}}^*)\\&\quad \le 4^{\textsf{p}-1}\Bigg \{ \left( \tilde{M}_T^\textsf{p}t^{\textsf{p}q-1}\left( \frac{{\textsf{p}\!-\!1}}{\textsf{p}q\!-\!1}\right) ^{\textsf{p}-1}\!+\!\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\left( \frac{{\textsf{p}\!-\!2}}{2\textsf{p}q\!-\!\textsf{p}-2}\right) ^{\frac{\textsf{p}-2}{2}}\right) \\&\qquad \times \,\left( \int _0^t\vartheta \left( \sup \limits _{0 \le \varsigma \le t}E\left\| x_{\varsigma }^n\!-\!x_{\varsigma }^m\right\| ^\textsf{p}\right) d\varsigma \!+\!\int _0^t\hat{\vartheta }\left( \sup \limits _{0 \le \varsigma \le t}E\left\| y_{\varsigma }^n\!-\!y_{\varsigma }^m\right\| ^\textsf{p}\right) d\varsigma \right) \Bigg \}. \end{aligned}$$
$$\begin{aligned} \text {That is}&\sup \limits _{0 \le \varsigma \le t}E\left\| x^{n+1}(t) -x^{m+1}(t)\right\| ^\textsf{p}+\sup \limits _{0 \le \varsigma \le t}E\left\| y^{n+1}(t) -y^{m+1}(t)\right\| ^\textsf{p}\\&\quad \le \frac{4^{\textsf{p}-1}}{1-4^{\textsf{p}-1}M_{\mathcal {L}}^*} \left\{ \Big (\tilde{M}_T^\textsf{p}t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\right. \\&\qquad +\,\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big )\\&\left. \qquad \times \,\left( \int _0^t\vartheta \left( \sup \limits _{0 \le \varsigma \le t}E\left\| x_{\varsigma }^n\!-\!x_{\varsigma }^m\right\| ^\textsf{p}\right) d\varsigma \!+\!\int _0^t\hat{\vartheta }\left( \sup \limits _{0 \le \varsigma \le t}E\left\| y_{\varsigma }^n\!-\!y_{\varsigma }^m\right\| ^\textsf{p}\right) d\varsigma \right) \right\} . \end{aligned}$$
Thus,
$$\begin{aligned}&\sup \limits _{0 \le \varsigma \le t}E\left\| x^{n+1}(t) -x^{m+1}(t)\right\| ^\textsf{p}+\sup \limits _{0 \le \varsigma \le t}E\left\| y^{n+1}(t) -y^{m+1}(t)\right\| ^\textsf{p}\nonumber \\&\quad \le \frac{\eta _64^{\textsf{p}-1}}{1-4^{\textsf{p}-1}M_{\mathcal {L}}^*}\left( \int _0^t\vartheta \left( \sup \limits _{0\le \varsigma \le t}E\left\| x_{\varsigma }^n-x_{\varsigma }^m\right\| ^\textsf{p}\right) d\varsigma \right. \nonumber \\&\qquad +\left. \int _0^t\hat{\vartheta }\left( \sup \limits _{0 \le \varsigma \le t}E\left\| y_{\varsigma }^n-y_{\varsigma }^m\right\| ^\textsf{p}\right) d\varsigma \right) , \end{aligned}$$
(24)
where \(\eta _6 = \tilde{M}_T^\textsf{p}t^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}+\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}.\) Through the integration and the application of Jesen’s inequality (21) becomes
$$\begin{aligned}&\int _0^t\sup \limits _{0\le l\le \varsigma }\left( E\left\| x^{n+1}(l)-x^{m+1}(l)\right\| ^\textsf{p}+E\left\| y^{n+1}(l) -y^{m+1}(l)\right\| ^\textsf{p}\right) d\varsigma \\&\quad = \frac{\eta _6}{1-4^{\textsf{p}-1}M_{\mathcal {L}}^*}\int _0^t\int _0^\varsigma \left[ \vartheta \left( \sup \limits _{0\le \tilde{r} \le l}E\left\| x^n(\tilde{r})-x^m(\tilde{r})\right\| ^\textsf{p}\right) \right. \\&\left. \qquad +\,\hat{\vartheta }\left( \sup \limits _{0\le \tilde{r} \le l}E\left\| y^n(\tilde{r})-y^m(\tilde{r})\right\| ^\textsf{p}\right) \right] dld\varsigma \\&\quad \le \frac{\eta _6}{1-4^{\textsf{p}-1}M_{\mathcal {L}}^*} \int _0^t\varsigma \int _0^\varsigma \left[ \vartheta \left( \sup \limits _{0\le \tilde{r} \le l}E\left\| x^n(\tilde{r})-x^m(\tilde{r})\right\| ^\textsf{p}\right) \right. \\&\qquad \left. +\,\hat{\vartheta }\left( \sup \limits _{0\le \tilde{r} \le l}E\left\| y^n(\tilde{r})-y^m(\tilde{r})\right\| ^\textsf{p}\right) \right] \frac{1}{\varsigma }dld\varsigma \\&\quad \le \frac{\eta _6}{1-4^{\textsf{p}-1}M_{\mathcal {L}}^*} t\left[ \int _0^t\vartheta \left( \int _0^\varsigma \sup \limits _{0\le \tilde{r} \le l}E\left\| x^n(\tilde{r})-x^m(\tilde{r})\right\| ^\textsf{p}\frac{1}{\varsigma }dl\right) \right. \\&\left. \qquad +\,\hat{\vartheta }\left( \int _0^\varsigma \sup \limits _{0\le \tilde{r} \le l}E\left\| x^n(\tilde{r})-x^m(\tilde{r})\right\| ^\textsf{p}\frac{1}{\varsigma }dl\right) \right] d\varsigma . \end{aligned}$$
Then
$$\begin{aligned} \Psi _{n+1, m+1}(t)\le \frac{\eta _6}{1-4^{\textsf{p}-1}M_{\mathcal {L}}^*} \int _0^t \big (\vartheta +\hat{\vartheta }\big )(\Psi _{n,m}(\varsigma ))d\varsigma , \end{aligned}$$
(25)
where
$$\begin{aligned} \Psi _{n,m}(t) = \dfrac{\int _0^t\big [\big (\sup \nolimits _{0\le \tilde{r} \le l}E\left\| x^n(\tilde{r})-x^m(\tilde{r})\right\| ^\textsf{p}\big )+\big (\sup \nolimits _{0\le \tilde{r} \le l}E\left\| y^n(\tilde{r})-y^m(\tilde{r})\right\| ^\textsf{p}\big )\big ]d\varsigma }{t}. \end{aligned}$$
From (20), one can see that
$$\begin{aligned} \sup \limits _{n,m}\Psi _{n,m}(t) < \infty . \end{aligned}$$
Using Fatou’s lemma and taking limits on both sides of the inequality (22), one can get
$$\begin{aligned} \Psi (t) = \frac{\eta _6}{1-4^{\textsf{p}-1}M_{\mathcal {L}}^*}\int _0^t\big (\vartheta +\hat{\vartheta }\big )(\Psi (\varsigma ))d\varsigma . \end{aligned}$$
Now, applying the Lemma 2.4., it says that \(\Psi (t) =0\) for any \(t \in \hat{J}\). Thus, \(\{x^n(t), n \in N\}\) is a Cauchy sequence in \(L^\textsf{p}(\Omega , \mathcal {H})\). So there is an \(x \in L^\textsf{p}(\Omega , \mathcal {H})\) such that
$$\begin{aligned} \lim \limits _{n \rightarrow \infty }\int _0^{\tilde{T}}\sup \limits _{0\le \varsigma \le t} E\left\| x_\varsigma ^n -x_\varsigma \right\| ^\textsf{p} dt =0. \end{aligned}$$
Then \(E\left\| x(t)\right\| ^\textsf{p} \le \eta _5\). Further, setting \(n \rightarrow \infty \), for \( t \in \hat{J}\), one can have
$$\begin{aligned} E\left\| T_q(t-\varsigma )\big [\int _0^t\mathfrak {f}(t,x_\varsigma ^{n-1},y_\varsigma ^{n-1})-\mathfrak {f}(t,x_\varsigma ,y_\varsigma )\big ]d\varsigma \right\| ^\textsf{p}&\rightarrow 0,\\ E\left\| \int _{0}^{t}T_q(t-\varsigma )\big [\Lambda (t,x_\varsigma ^{n-1},y_\varsigma ^{n-1})-\Lambda (t,x_\varsigma ,y_\varsigma )\big ]d\tilde{Z}_Q^{\mathbb {H}}\varsigma \right\| ^\textsf{p}&\rightarrow 0. \end{aligned}$$
Hence, taking limits on both sides of (11),
$$\begin{aligned} x(t)&=S_q(t)\big [\varphi _0-\mathcal {N}(0,0,\varphi _0)\big ]+\mathcal {N}(t,x_t,y_t) +\int _0^tT_q(t-\varsigma )\mathfrak {f}(\varsigma ,x_\varsigma ,y_\varsigma )d\varsigma \\ {}&\qquad +\int _0^tT_q(t-\varsigma )\Lambda (\varsigma ,x_\varsigma ,y_\varsigma )d\tilde{Z}_Q^{\mathbb {H}}(\varsigma ). \end{aligned}$$
Thus x(t) is a mild solution of (1) on the interval \(\hat{J}\) and similarly one can prove that y(t) is also a mild solution.
Step 3. Uniqueness of the solution
Set \(x,z \in L^\textsf{p}(\Omega ,\mathcal {H})\) are solutions of (1) on some interval \(\hat{J}\) and \(M_{\mathcal {L}}^* < \frac{1}{3^{\textsf{p}-1}}\). Thus uniqueness is obvious for \(t \in \hat{J}\), one can have
$$\begin{aligned} E\left\| x_t-z_t\right\| ^\textsf{p} \le \frac{\eta _6}{1-4^{\textsf{p}-1}M_{\mathcal {L}}^*}\int _0^t \vartheta (E\left\| x_\varsigma -z_\varsigma \right\| ^\textsf{p})d\varsigma . \end{aligned}$$
By utilizing Bihari’s inequality, one can obtain
$$\begin{aligned} \sup \limits _{t\in [0,\tilde{T}]}E\left\| x_t-z_t\right\| ^\textsf{p} =0, 0\le t \le \tilde{T}. \end{aligned}$$
Similarly uniqueness has also exist for the solution y(t). Hence, the proof. This shows the existence of stochastic disturbance in the visual perception trajectory of fish robot and robot driver.
Proof of second Theorem 3.2: From (12) and (14) one can compute the difference
$$\begin{aligned} x_{\epsilon }(t) - z_{\epsilon }(t)&=\big [\mathcal {N}^1(t,x_{\epsilon ,t},y_{\epsilon ,t})-\mathcal {N}^1(t,x_{\epsilon ,t},z_{\epsilon ,t})\big ]\\&\qquad +\epsilon \int _0^tT_q(t-\varsigma )\big [\mathfrak {f}^1(\varsigma ,x_{\epsilon ,\varsigma },y_{\epsilon ,\varsigma })-\tilde{\mathfrak {f}}(x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })]d\varsigma \\ {}&\qquad +\epsilon ^{\mathbb {H}}\int _0^tT_q(t-\varsigma )\big [\Lambda ^1(\varsigma ,x_{\epsilon ,\varsigma },y_{\epsilon ,\varsigma })-\tilde{\Lambda }(x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\big ]d\tilde{Z}_Q^{\mathbb {H}}(\varsigma ). \end{aligned}$$
Thus, for \(\tilde{u} \in \hat{J}\), we acquired
$$\begin{aligned}&\sup \limits _{0 \le t \le \tilde{u}}\left\| x_{\epsilon }(t) - z_{\epsilon }(t)\right\| ^\textsf{p} \\&\quad \le \ 3^{\textsf{p}-1}\Bigg \{ \left\| \mathcal {N}^1(t,x_{\epsilon ,t},y_{\epsilon ,t})-\mathcal {N}^1(t,x_{\epsilon ,t}, z_{\epsilon ,t})\right\| ^\textsf{p}\\&\qquad +\epsilon ^\textsf{p} \sup \limits _{0 \le t \le \tilde{u}} \left\| \int _0^tT_q(t-\varsigma )\big [\mathfrak {f}^1(\varsigma ,x_{\epsilon ,\varsigma },y_{\epsilon ,\varsigma }) - \tilde{\mathfrak {f}}^1(x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\big ]d\varsigma \right\| ^\textsf{p}\\&\qquad +\epsilon ^{\textsf{p}\mathbb {H}} \sup \limits _{0 \le t \le \tilde{u}} \left\| \int _0^t T_q(t-\varsigma )\big [\Lambda ^1(\varsigma ,x_{\epsilon ,\varsigma },y_{\epsilon ,\varsigma }) - \tilde{\Lambda }^1(x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\big ]d\tilde{Z}_Q^{\mathbb {H}}(\varsigma )\right\| ^\textsf{p}\Bigg \}\\&\quad \le \ 3^{\textsf{p}-1}\Bigg \{ \left\| \mathcal {N}^1(t,x_{\epsilon ,t},y_{\epsilon ,t})-\mathcal {N}^1(t,x_{\epsilon ,t},z_{\epsilon ,t})\right\| ^\textsf{p}+ \pi _1+\pi _2\Bigg \}, \end{aligned}$$
where \(\pi _1\) is given below and subsequent estimation follows.
$$\begin{aligned} \pi _1&= 2^{\textsf{p}-1}\epsilon ^\textsf{p} \sup \limits _{0 \le t \le \tilde{u}} \left\| \int _0^t T_q(t-\varsigma )\big [\mathfrak {f}^1(\varsigma ,x_{\epsilon ,\varsigma },y_{\epsilon ,\varsigma })-\mathfrak {f}^1(\varsigma ,x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\right. \\&\quad +\left. \mathfrak {f}^1(\varsigma ,x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma }) - \tilde{\mathfrak {f}}^1(x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\big ]d\varsigma \right\| ^\textsf{p}\\ {}&\le 4^{\textsf{p}-1} \epsilon ^\textsf{p} \sup \limits _{0 \le t \le \tilde{u}}\Big [\int _0^t(t-\varsigma )^{\frac{\textsf{p} (q-1)}{\textsf{p}-1}}d\varsigma \Big ]^{\textsf{p}-1}\tilde{M}_T^\textsf{p} \left\| \int _0^t\big [\mathfrak {f}^1(\varsigma ,x_{\epsilon ,\varsigma },y_{\epsilon ,\varsigma })-\mathfrak {f}^1(\varsigma ,x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\big ]\right\| ^\textsf{p}d\varsigma \\ {}&\quad +4^{\textsf{p}-1}\epsilon ^\textsf{p}\sup \limits _{0 \le t \le \tilde{u}} \Big [\int _0^t(t-\varsigma )^{\frac{\textsf{p} (q-1)}{\textsf{p}-1}}d\varsigma \Big ]^{\textsf{p}-1} \tilde{M}_T^\textsf{p} \left\| \int _0^t \big [\mathfrak {f}^1(\varsigma ,x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })-\tilde{\mathfrak {f}}^1(x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\big ]\right\| ^\textsf{p}d\varsigma . \end{aligned}$$
By Holder inequality and (\(A_1\)) & (\(A_4\)), one can have
$$\begin{aligned}&E(\pi _1)\\&\quad \le 4^{\textsf{p}-1} \epsilon ^\textsf{p}\! \sup \limits _{0 \le t \le \tilde{u}}\Big [\int _0^t(t\!-\!\varsigma )^{\frac{\textsf{p} (q-1)}{\textsf{p}-1}}d\varsigma \Big ]^{\textsf{p}-1}\tilde{M}_T^\textsf{p} E\left\| \int _0^t\big [\mathfrak {f}^1(\varsigma ,x_{\epsilon ,\varsigma },y_{\epsilon ,\varsigma })\!-\!\mathfrak {f}^1(\varsigma ,x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\big ]\right\| ^\textsf{p}d\varsigma \\&\qquad +4^{\textsf{p}-1}\epsilon ^\textsf{p}\!\sup \limits _{0 \le t \le \tilde{u}} \Big [\int _0^t(t\!-\!\varsigma )^{\frac{\textsf{p} (q-1)}{\textsf{p}-1}}d\varsigma \Big ]^{\textsf{p}-1} \tilde{M}_T^\textsf{p}E\left\| \int _0^t \big [\mathfrak {f}^1(\varsigma ,x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\!-\!\tilde{\mathfrak {f}}^1(x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\big ]\right\| ^\textsf{p}d\varsigma \\&\quad \le 4^{\textsf{p}-1} \epsilon ^\textsf{p} \tilde{u}^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1} \tilde{M}_T^\textsf{p} \int _0^{\tilde{u}}\big (\vartheta \big [E\left\| x_{\epsilon ,\varsigma }\!-\!y_{\epsilon ,\varsigma }\big ]\right\| ^\textsf{p}+\hat{\vartheta } \big [E\left\| x_{\epsilon ,\varsigma }\!-\!z_{\epsilon ,\varsigma }\big ]\right\| ^\textsf{p}\big )d\varsigma \\&\qquad +\,4^{\textsf{p}-1}\epsilon ^\textsf{p} \tilde{u}^{\textsf{p}q}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\tilde{M}_T^\textsf{p}E\sup \limits _{0 \le t \le \tilde{u}}\left\| \frac{1}{t}\int _0^t \big [\mathfrak {f}^1(\varsigma ,x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })-\tilde{\mathfrak {f}}^1(x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\big ]\right\| ^\textsf{p}d\varsigma \\&\quad \le 4^{\textsf{p}-1} \epsilon ^\textsf{p} \tilde{u}^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1} \tilde{M}_T^\textsf{p} \int _0^{\tilde{u}}\big (\vartheta \big [E\left\| x_{\epsilon ,\varsigma }-y_{\epsilon ,\varsigma }\big ]\right\| ^\textsf{p}+\hat{\vartheta } \big [E\left\| x_{\epsilon ,\varsigma }-y_{\epsilon ,\varsigma }\big ]\right\| ^\textsf{p}\big )d\varsigma \\&\qquad +\,4^{\textsf{p}-1}\epsilon ^\textsf{p} \tilde{u}^{\textsf{p}q}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\tilde{M}_T^\textsf{p}\sup \limits _{0 \le t \le \tilde{u}}\Big \{a_1(t)\big [1+ \sup \limits _{0 \le \varsigma \le t}\big (\left\| x_{\epsilon ,\varsigma }\right\| ^\textsf{p}+\left\| z_{\epsilon .\varsigma }\right\| ^\textsf{p}\big )\big ]\Big \}\\&\quad \le 4^{\textsf{p}-1} \epsilon ^\textsf{p}\tilde{u}^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\tilde{M}_T^\textsf{p} \int _0^{\tilde{u}}\big (\vartheta \big [E \left\| x_{\epsilon ,\varsigma }-y_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]+\hat{\vartheta }\big [E \left\| x_{\epsilon ,\varsigma }-z_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]\big )d\varsigma \\&\qquad +\,4^{\textsf{p}-1}\epsilon ^\textsf{p} \tilde{u}^{\textsf{p}q}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\tilde{M}_T^\textsf{p}a_1(\tilde{u})\big [1+ \sup \limits _{0 \le t \le \tilde{u}}\big (\left\| x_{\epsilon ,\varsigma }\right\| ^\textsf{p}+\left\| z_{\epsilon .\varsigma }\right\| ^\textsf{p}\big )\big ]. \end{aligned}$$
Through the properties of solutions, we know that \(\left\| x_0(t)\right\| ^\textsf{p} < \infty \), then for every \(t \ge 0\), \(\left\| x(t)\right\| ^\textsf{p} < \infty \). This is coupled with the reality that \(\lim \nolimits _{\tilde{T}_1 \rightarrow \infty }a_1(\tilde{T}_1) =0\).Hence,
$$\begin{aligned} E(\pi _1)&\le 4^{\textsf{p}-1} \epsilon ^\textsf{p}\tilde{u}^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\tilde{M}_T^\textsf{p} \int _0^{\tilde{u}}\big (\vartheta \big [E\left\| x_{\epsilon ,\varsigma }-y_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]\nonumber \\&\qquad +\hat{\vartheta }\big [E\left\| x_{\epsilon ,\varsigma }-z_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]\big )d\varsigma +4^{\textsf{p}-1}\epsilon ^\textsf{p}\tilde{M}_T^\textsf{p} \eta _7, \end{aligned}$$
(26)
where \(\eta _7 =\tilde{u}^{\textsf{p}q}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}a_1(\tilde{u})\big [1+ \sup \limits _{0 \le t \le \tilde{u}}\) \(\big [1+ \sup \limits _{0 \le t \le \tilde{u}}\big (\left\| x_{\epsilon ,\varsigma }\right\| ^\textsf{p}+\left\| z_{\epsilon .\varsigma }\right\| ^\textsf{p}\big )\big ]\). Following the above procedure, one can get
$$\begin{aligned} E(\pi _2)&\le 4^{\textsf{p}-1}\epsilon ^{\textsf{p}\mathbb {H}}\tilde{K}C_\mathbb {H}^{\frac{\textsf{p}}{2}} \tilde{u}^{\textsf{p}\mathbb {H}- \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}\tilde{u}^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\\&\qquad \times \, \sup \limits _{0 \le t \le \tilde{u} }E\left\| \int _0^{\tilde{u}}\big [\Lambda ^1(\varsigma ,x_{\epsilon ,\varsigma },y_{\epsilon ,\varsigma })-\Lambda ^1(\varsigma ,x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\big ]\right\| ^\textsf{p}d\varsigma \\&\qquad +\,4^{\textsf{p}-1}\epsilon ^{\textsf{p}\mathbb {H}}a_2(\tilde{u}) \tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}\tilde{u}^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\\&\qquad \times \, \sup \limits _{0 \le t \le \tilde{u}}E\left\| \int _0^{\tilde{u}}\big [\Lambda ^1(\varsigma ,x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })-\tilde{\Lambda }^1(x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\big ]\right\| ^\textsf{p}d\varsigma \\&= 4^{\textsf{p}-1} \epsilon ^{\textsf{p}\mathbb {H}}\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}\tilde{u}^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\\&\qquad \times \, \sup \limits _{0 \le t \le \tilde{u} }E\left\| \int _0^{\tilde{u}}\big [\Lambda ^1(\varsigma ,x_{\epsilon ,\varsigma },y_{\epsilon ,\varsigma })-\Lambda ^1(\varsigma ,x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\big ]\right\| ^\textsf{p}d\varsigma \\&\qquad +\,4^{\textsf{p}-1}\epsilon ^{\textsf{p}\mathbb {H}}a_2(\tilde{u}) \tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}\tilde{u}^{\frac{2\textsf{p}q-\textsf{p}+1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big )\\&\qquad \times \, \sup \limits _{0 \le t \le \tilde{u} }E\left\| \frac{1}{t}\int _0^{\tilde{u}} \big [\Lambda ^1(\varsigma ,x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })-\tilde{\Lambda }^1(x_{\epsilon ,\varsigma },z_{\epsilon ,\varsigma })\big ]\right\| ^\textsf{p}d\varsigma \\&\le 4^{\textsf{p}-1} \epsilon ^{\textsf{p}\mathbb {H}}\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}\\&\qquad \times \,\tilde{u}^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big )\int _0^{\tilde{u}}\big (\vartheta \big [ E\left\| x_{\epsilon ,\varsigma }-y_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]\\&\qquad +\,\hat{\vartheta }\big [ E\left\| x_{\epsilon ,\varsigma }-z_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]\big )d\varsigma \\&\qquad +\,4^{\textsf{p}-1}\epsilon ^{\textsf{p}\mathbb {H}} \tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}\tilde{u}^{\frac{2\textsf{p}q-\textsf{p}+1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big )\\&\qquad \times \, a_2(\tilde{u})\big [1+ \sup \limits _{0 \le t \le \tilde{u}}\big (E\left\| x_{\epsilon ,\varsigma }\right\| ^\textsf{p}+E\left\| z_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big )\big ]. \end{aligned}$$
Through the properties of solutions, we know that \(\left\| x_0(t)\right\| ^\textsf{p} < \infty \), then for any \(t \ge 0\), \(\left\| x(t)\right\| ^\textsf{p} < \infty \). This is coupled with the reality that \(\lim \limits _{\tilde{T}_1 \rightarrow \infty }a_2(\tilde{T}_1) =0\). Hence,
$$\begin{aligned} E(\pi _2)&\le 4^{\textsf{p}-1} \epsilon ^{\textsf{p}\mathbb {H}}\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}\tilde{u}^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big )\nonumber \\ {}&\qquad \times \int _0^{\tilde{u}}\big (\vartheta \big [E \left\| x_{\epsilon ,\varsigma }-y_{\epsilon ,\varsigma }\right\| ^\textsf{p}+\hat{\vartheta }\big [E \left\| x_{\epsilon ,\varsigma }-z_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]\big )d\varsigma \!+\!4^{\textsf{p}-1}\tilde{M}_T^\textsf{p}\epsilon ^{\textsf{p}\mathbb {H}} \eta _8, \end{aligned}$$
(27)
where \(\eta _8 = \tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{u}^{\frac{2\textsf{p}q-\textsf{p}+1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big ) a_2(\tilde{u})\big [1+ E\sup \limits _{0 \le t \le \tilde{u}} \big (\left\| x_{\epsilon ,\varsigma }\right\| ^\textsf{p}+\left\| z_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big )\big ]\). From (23) and (24), one can find that
$$\begin{aligned}&\sup \limits _{0\le t \le \tilde{u}}E\left\| x_{\epsilon }(t)-z_{\epsilon }(t)\right\| ^\textsf{p}\\&\quad \le 3^{\textsf{p}-1}\Bigg \{M_{\mathcal {L}}\sup \limits _{0\le t \le \tilde{u}}E\left\| x_{\epsilon }(t)-z_{\epsilon }(t)\right\| ^\textsf{p}+ \Big [\epsilon ^{\textsf{p}}\tilde{u}^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big ) ^{\textsf{p}-1}\tilde{M}_T^\textsf{p}\\&\qquad +\,\epsilon ^{\textsf{p}\mathbb {H}}\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}\tilde{u}^{\frac{2\textsf{p}q-\textsf{p}+1}{2}}\Big ]\\&\qquad \times \, \Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big ) \int _0^{\tilde{u}}\vartheta \big [E\left\| x_{\epsilon ,\varsigma }-y_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]\\&\qquad +\,\hat{\vartheta }\big [E\left\| x_{\epsilon ,\varsigma }-z_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]d\varsigma +\, 4^{\textsf{p}-1}\tilde{M}_T^\textsf{p}\Bigg (\epsilon ^{\textsf{p}}\eta _7+\epsilon ^{\textsf{p}\mathbb {H}}\eta _8\Bigg ) \Bigg \}. \end{aligned}$$
$$\begin{aligned}&\big (1-3^{\textsf{p}-1}M_{\mathcal {L}}\big )\sup \limits _{0\le t \le \tilde{u}}E\left\| x_{\epsilon }(t)-z_{\epsilon }(t)\right\| ^\textsf{p}\\&\quad \le 3^{\textsf{p}-1}\Bigg \{ \Big [\epsilon ^{\textsf{p}}\tilde{u}^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\tilde{M}_T^\textsf{p}+\epsilon ^{\textsf{p}\mathbb {H}}\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}\\&\qquad \times \,\tilde{u}^{\frac{2\textsf{p}q-\textsf{p}+1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big )\Big ]\int _0^{\tilde{u}}\big (\vartheta \big [E\left\| x_{\epsilon ,\varsigma }-y_{\epsilon ,\varsigma }\right\| ^\textsf{p}\\&\qquad +\hat{\vartheta }\big [E\left\| x_{\epsilon ,\varsigma }-z_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]\big )d\varsigma +\,4^{\textsf{p}-1}\tilde{M}_T^\textsf{p}\Bigg (\epsilon ^{\textsf{p}}\eta _7+\epsilon ^{\textsf{p}\mathbb {H}}\eta _8\Bigg ) \Bigg \}.\\&\sup \limits _{0\le t \le \tilde{u}}E\left\| x_{\epsilon }(t)-z_{\epsilon }(t)\right\| ^\textsf{p}\\&\quad \le \frac{3^{\textsf{p}-1}}{\big (1-3^{\textsf{p}-1}M_{\mathcal {L}}\big )}\Bigg \{ \Big [\epsilon ^{\textsf{p}}\tilde{u}^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\tilde{M}_T^\textsf{p}+\epsilon ^{\textsf{p}\mathbb {H}}\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}}\\&\qquad \times \, \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}\tilde{u}^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2} \Big )^{\frac{\textsf{p}-2}{2}}\Big )\Big ] \int _0^{\tilde{u}}\big (\vartheta \big [E\left\| x_{\epsilon ,\varsigma }-y_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ] \\&\qquad +\,\hat{\vartheta }\big [E\left\| x_{\epsilon ,\varsigma }-y_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]\big )d\varsigma +\,4^{\textsf{p}-1}\tilde{M}_T^\textsf{p}\Bigg (\epsilon ^{\textsf{p}}\eta _7+\epsilon ^{\textsf{p}\mathbb {H}}\eta _8\Bigg ) \Bigg \}. \end{aligned}$$
By the condition of concavity of \(\vartheta \), two constants \(\alpha \) and \(\beta \) such that \(\vartheta (x) \le \alpha x+\beta \), \(\forall x >0\) and \(\vartheta (0)=0\) are got. From that, one can obtain
$$\begin{aligned}&\sup \limits _{0\le t\le \tilde{u}}E\left\| x_{\epsilon }(t)-z_{\epsilon }(t)\right\| ^\textsf{p}\\&\quad \le \frac{3^{\textsf{p}-1}}{\big (1-3^{\textsf{p}-1}M_{\mathcal {L}}\big )}\Bigg \{ \Big [\epsilon ^{\textsf{p}}\tilde{u}^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\tilde{M}_T^\textsf{p}\\&\qquad +\,\epsilon ^{\textsf{p}\mathbb {H}}\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}\tilde{u}^{\frac{2\textsf{p}q-\textsf{p}+1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2} \Big )^{\frac{\textsf{p}-2}{2}}\Big )\Big ]\\&\qquad \times \,\alpha \int _0^{\tilde{u}}\big (\vartheta \big [E\left\| x_{\epsilon ,\varsigma } -y_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]\\&\qquad +\,\hat{\vartheta }\big [E\left\| x_{\epsilon ,\varsigma } -y_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]\big )d\varsigma +4^{\textsf{p}-1} \Big [\epsilon ^{\textsf{p}}\tilde{u}^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}\tilde{M}_T^\textsf{p}\\&\qquad +\,\epsilon ^{\textsf{p}\mathbb {H}}\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{M}_T^\textsf{p}\tilde{u}^{\frac{2\textsf{p}q-\textsf{p}+1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big ]\beta + \tilde{M}_T^\textsf{p}\Bigg (\epsilon ^{\textsf{p}}\eta _7+\epsilon ^{\textsf{p}\mathbb {H}}\eta _8\Bigg )\Bigg \}.\\&\quad =\frac{3^{\textsf{p}-1}}{\big (1-3^{\textsf{p}-1}M_{\mathcal {L}}\big )}\Bigg \{\tilde{M}_T^\textsf{p} \Big [\epsilon ^{\textsf{p}}\tilde{u}^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}+\epsilon ^{\textsf{p}\mathbb {H}}\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}}\\&\qquad \times \, \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{u}^{\frac{2\textsf{p}q-\textsf{p}+1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big )\Big ]\alpha \int _0^{\tilde{u}}\big (\vartheta \big [E\left\| x_{\epsilon ,\varsigma }-y_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]\\&\qquad +\hat{\vartheta }\big [E\left\| x_{\epsilon ,\varsigma }-y_{\epsilon ,\varsigma }\right\| ^\textsf{p}\big ]\big )d\varsigma \\&\quad +\,4^{\textsf{p}-1}\tilde{M}_T^\textsf{p}\Big [\Big \{\epsilon ^{\textsf{p}}\tilde{u}^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}+\epsilon ^{\textsf{p}\mathbb {H}}\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}}\\&\quad \times \, \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{u}^{\frac{2\textsf{p}q-\textsf{p}+1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big ) ^{\frac{\textsf{p}-2}{2}} \Big \}\beta + \Bigg (\epsilon ^{\textsf{p}}\eta _7+\epsilon ^{\textsf{p}\mathbb {H}}\eta _8\Bigg )\Big ]\Bigg \}. \end{aligned}$$
Let \(\hat{r} = \epsilon ^{\textsf{p}}\tilde{u}^{\textsf{p}q-1}\Big (\frac{{\textsf{p}-1}}{\textsf{p}q-1}\Big )^{\textsf{p}-1}+\epsilon ^{\textsf{p}\mathbb {H}}\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} \tilde{u}^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}} \tilde{u}^{\frac{2\textsf{p}q-\textsf{p}+1}{2}}\Big (\frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\Big )^{\frac{\textsf{p}-2}{2}}\Big ).\) By Gronwall inequality,
$$\begin{aligned}&\sup \limits _{0\le t \le \tilde{u}}E\left\| x_{\epsilon }(t)-z_{\epsilon }(t)\right\| ^\textsf{p}\\&\quad \le \tilde{M}_T^\textsf{p}\frac{3^{\textsf{p}-1}}{\big (1-3^{\textsf{p}-1}M_{\mathcal {L}}\big )}\Bigg [(\hat{r}\beta +\Bigg (\epsilon ^{\textsf{p}}\eta _7+\epsilon ^{\textsf{p}\mathbb {H}}\eta _8\Bigg )\Bigg ]ex\textsf{p}\left( \frac{3^{\textsf{p}-1}}{\big (1-3^{\textsf{p}-1}M_{\mathcal {L}}\big )}\tilde{M}_T^\textsf{p}\frac{\hat{r}}{t}\alpha \right) . \end{aligned}$$
Set \(0<b <1\) and \(\mathcal {L} >0\) such that for each \(t\in [0,\mathcal {L}\epsilon ^{-b}] \subset \hat{J}\),
$$\begin{aligned} \sup \limits _{0\le t \le \tilde{u}}E\left\| x_{\epsilon }(t)-z_{\epsilon }(t)\right\| ^\textsf{p} \le \eta _9 \mathcal {L}\epsilon ^{1-b}, \end{aligned}$$
$$\begin{aligned} \text {where the constant} \ \ \eta _9\!&=\!\tilde{M}_T^\textsf{p}\frac{3^{\textsf{p}\!-\!1}}{\big (1\!-\!3^{\textsf{p}\!-\!1}M_{\mathcal {L}}\big )}\Bigg [\left( \frac{\hat{r}}{\epsilon ^{1\!-\!b}}\beta \right) \!+\!\Bigg (\epsilon ^{\textsf{p}(1\!-\!b)}\eta _7\!+\!\epsilon ^{\textsf{p}\mathbb {H}(1\!-\!b)}\eta _8\Bigg )\Bigg ]\\&\quad \times \,\exp \left( \frac{3^{\textsf{p}-1}}{\big (1-3^{\textsf{p}-1}M_{\mathcal {L}}\big )}\tilde{M}_T^\textsf{p}\frac{\hat{r}}{t\epsilon ^{b-1}}\alpha \right) . \end{aligned}$$
i.e., \(\tau _1 >0\) and \(\tau _2 >0\) are the given number, put \(\epsilon _1 \in (0,\epsilon _0]\) such that for every \(\epsilon \in (0,\epsilon _1]\) and for each \(t \in [0,\mathcal {L}\epsilon ^{-b}]\),
$$\begin{aligned} \sup \limits _{0 \le t \le \tilde{u}}E\left\| x_{\epsilon }(t) - z_{\epsilon }(t)\right\| ^\textsf{p} \le \tau _1. \end{aligned}$$
(28)
Similarly one can get,
$$\begin{aligned} \sup \limits _{0 \le t \le \tilde{u}}E\left\| y_{\epsilon }(t) - \tilde{z}_{\epsilon }(t)\right\| ^\textsf{p} \le \tau _2. \end{aligned}$$
(29)
Hence the proof.
Proof of Theorem 3.6
For \(0 \le t \le \tilde{T}\), by assumption for x(t) and y(t), one can get
$$\begin{aligned} x(t)&= S_q(t)\big [(\phi _0-\mathcal {N}(0,0,\varphi _0))\big ]+\mathcal {N}^1(t,x_t,y_t)+\int _0^tT_q(t-\varsigma \mathfrak {f}^1(\varsigma ,x_\varsigma ,y_\varsigma )d\varsigma \\ {}&\qquad +\int _0^tT_q(t-\varsigma )\Lambda ^1(\varsigma ,x_\varsigma ,y_\varsigma )d\tilde{Z}_Q^{\mathbb {H}}(\varsigma ).\\ y(t)&= S_q(t)\big [(\hat{\phi }_0-\mathcal {N}(0,0,\hat{\varphi }_0))\big ]+\mathcal {N}^2(t,x_t,y_t)+\int _0^tT_q(t-\varsigma \mathfrak {f}^2(\varsigma ,x_\varsigma ,y_\varsigma )d\varsigma \\&\qquad +\int _0^tT_q(t-\varsigma )\Lambda ^2(\varsigma ,x_\varsigma ,y_\varsigma )d\tilde{Z}_Q^{\mathbb {H}}(\varsigma ). \end{aligned}$$
Through the earlier evaluation,
$$\begin{aligned}&E\left\| x(t)\right\| ^\textsf{p}\\&\quad \le \frac{4^{\textsf{p}-1}}{\left( 1-4^{\textsf{p}-1}M_{\mathcal {L}}^*\right) }\Bigg \{\tilde{M}_RE\left\| \phi _0-\mathcal {N}^1(0,0,\phi _0)\right\| ^\textsf{p}+\tilde{M}_T^\textsf{p} \Bigg [t^{\textsf{p}q-1}\left( \frac{{\textsf{p}-1}}{\textsf{p}q-1}\right) ^{\textsf{p}-1}\\&\qquad +\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} \!-\! \frac{\textsf{p}}{2}} t^{\frac{2\textsf{p}q\!-\!\textsf{p}\!-\!1}{2}}\left( \frac{{\textsf{p}\!-\!2}}{2\textsf{p}q\!-\!\textsf{p}-2}\right) ^{\frac{\textsf{p}\!-\!2}{2}}\Bigg ]\int _0^t \big [\vartheta (E\left\| x_\varsigma \right\| ^\textsf{p})\!+\!\hat{\vartheta }(E\left\| y_\varsigma \right\| ^\textsf{p})\big ]d\varsigma \Bigg \}.\\&E\left\| y(t)\right\| ^\textsf{p}\\&\quad \le \frac{4^{\textsf{p}-1}}{\left( 1-4^{\textsf{p}-1}M_{\mathcal {L}}^*\right) }\Bigg \{\tilde{M}_RE\left\| \hat{\phi }_0-\mathcal {N}^2(0,0,\hat{\phi }i_0)\right\| ^\textsf{p}+\tilde{M}_T^\textsf{p} \Bigg [t^{\textsf{p}q-1}\left( \frac{{\textsf{p}-1}}{\textsf{p}q-1}\right) ^{\textsf{p}-1}\\&\qquad +\tilde{K} C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} \!-\! \frac{\textsf{p}}{2}} t^{\frac{2\textsf{p}q\!-\!\textsf{p}\!-\!1}{2}}\left( \frac{{\textsf{p}\!-\!2}}{2\textsf{p}q\!-\!\textsf{p}\!-\!2}\right) ^{\frac{\textsf{p}\!-\!2}{2}}\Bigg ]\int _0^t \big [\vartheta (E\left\| x_\varsigma \right\| ^\textsf{p})\!+\!\hat{\vartheta }(E\left\| y_\varsigma \right\| ^\textsf{p})\big ]d\varsigma \Bigg \}. \end{aligned}$$
Let
$$\begin{aligned} \vartheta _1(\tilde{u})&=\frac{4^{\textsf{p}-1}}{\left( 1-4^{\textsf{p}-1}\right) }\tilde{M}_T^\textsf{p} \Bigg [t^{\textsf{p}q-1}\left( \frac{{\textsf{p}-1}}{\textsf{p}q-1}\right) ^{\textsf{p}-1}\\&\qquad +\,\tilde{K}C_\mathbb {H}^{\frac{\textsf{p}}{2}} t^{\textsf{p}\mathbb {H} - \frac{\textsf{p}}{2}}t^{\frac{2\textsf{p}q-\textsf{p}-1}{2}}\left( \frac{{\textsf{p}-2}}{2\textsf{p}q-\textsf{p}-2}\right) ^{\frac{\textsf{p}-2}{2}}\Bigg ]\vartheta (\tilde{u}), \end{aligned}$$
here \(\vartheta \) is concave increasing function from \(\mathcal {R}^{+}\) to \(\mathcal {R}^{+}\) satisfying \(\vartheta (0)=0, \vartheta (\tilde{u}) >0\) as \(\tilde{u} >0\) and \(\int _{0^{+}} \frac{d\tilde{u}}{\vartheta (\tilde{u})} = \infty \). Thus, \(\vartheta _1(\tilde{u})\) is clearly, a concave function from \(\mathcal {R}^{+}\) to \(\mathcal {R}^{+}\) for \(\vartheta _1(0)=0, \vartheta _1(\tilde{u}) >\vartheta (\tilde{u})\) as \(0\le \tilde{u} \le 1\) and \(\int _{0^{+}} \frac{d\tilde{u}}{\vartheta _1(\tilde{u})} = \infty \). Now for any \(\epsilon >0\), \(\epsilon _1 = \frac{1}{2} \epsilon \), we have \(\lim \nolimits _{\varsigma \rightarrow 0}\int _{\varsigma }^{\epsilon _1} \dfrac{d\tilde{u}}{\vartheta _1(\tilde{u})} = \infty .\) That is, \(\vartheta _1(\tilde{u})=0\) It is asymptotically if it is stable.
To analyse the stability: Consider there is a positive constant \(\hat{\delta } < \epsilon _1\), so that \(\int _{\varsigma }^{\epsilon _1} \dfrac{d\tilde{u}}{\vartheta _1(\tilde{u})} \ge \tilde{T}\).
$$\begin{aligned} \text { Let} \ \tilde{u}_0&= \frac{4^{\textsf{p}-1}}{\big (1-4^{\textsf{p}-1}\big )}\tilde{M}_RE\left\| \phi _0-\mathcal {N}(0,0,\phi _0)\right\| ^\textsf{p}\\ \tilde{u}(t)&=E\left\| x_t,y_t\right\| ^\textsf{p}, \ v(t) =1, \end{aligned}$$
when \(\tilde{u}_0 \le \hat{\delta } \le \epsilon _1\). (From Corollary 2.3 [47]), one can have
$$\begin{aligned} \int _{\tilde{u}_0}^{\epsilon _1}\frac{d\tilde{u}}{\vartheta _1(\tilde{u})}\ge \int _{\hat{\delta }}^{\epsilon _1}\frac{d\tilde{u}}{\vartheta _1(\tilde{u})}\ge \tilde{T} = \int _0^{\tilde{T}} \varrho (\varsigma )d\varsigma \end{aligned}$$
As, for each \(t \in \hat{J}\), we derived that \(\tilde{u}(t) \le \epsilon _1\) whenever \(\tilde{u}_0 \le \hat{\delta }\). Similarly one can prove the result for y(t) also. Hence, the proof. This result implies that stability of the visual trajectory of the robot driver exists.
Based on the previous analysis, one can drive that the stability of y(t) which represents the visual trajectory fish robot exists if the stability of the visual trajectory of the robot driver exists.
