. Moreover, we have
$$\begin{aligned}& \bigl\Vert \varPhi (t)-\varPhi ^{\epsilon ,\delta }(t) \bigr\Vert _{Y} \\& \quad =\sup_{t \in J}t^{(1- \nu )(1-\mu )} \bigl\Vert \varPhi (t)- \varPhi ^{\epsilon ,\delta }(t) \bigr\Vert \\& \quad \leq \sup_{t \in J}t^{(1-\nu )(1-\mu )} \biggl\{ \biggl\Vert \frac{\mu }{ {\varGamma (\nu (1 - \mu ))}} \int _{0}^{t} \int _{0}^{\delta }{E^{ - 1}} \theta {{(t - s)}^{\nu (1 - \mu ) - 1}} {s^{\mu - 1}} {\varPsi _{\mu }}( \theta )S \bigl({s^{\mu }}\theta \bigr) \\& \qquad {}\times E\bigl[{x_{0}} - q(x)\bigr]\,d \theta \,ds \biggr\Vert \\& \qquad {}+ \mu \biggl\Vert \int _{0}^{t} { \int _{0} ^{\delta }{{E^{ - 1}}\theta {{(t - s)}^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(t - s)}^{\mu }}\theta \bigr)f\bigl(s,x(s)\bigr)\,d\theta \,ds} } \biggr\Vert \\& \qquad {}+ \mu \biggl\Vert \int _{0}^{t} \int _{0} ^{\delta }{{E^{ - 1}}\theta {{(t - s)}^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(t - s)}^{\mu }}\theta \bigr)B{W^{ - 1}}} \\& \qquad {} \times \biggl[ {x_{1}} - \frac{\mu }{{\varGamma (\nu (1 - \mu ))}} \int _{0}^{a} \int _{0}^{\infty }{E^{ - 1}}\theta {{(a - \eta )}^{\nu (1 - \mu ) - 1}} {\eta ^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({\eta ^{\mu }}\theta \bigr) \\& \qquad {}\times E\bigl[ {x_{0}} - q(x) \bigr]\,d\theta \,d\eta \\& \qquad {}- \mu \int _{0}^{a} { \int _{0}^{\infty }{{E^{ - 1}}\theta {{(a - s)} ^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(a - s)}^{\mu }}\theta \bigr)f\bigl(\eta ,x( \eta )\bigr)\,d\theta \,d\eta } } \\& \qquad {}- \mu \int _{0}^{a} { \int _{0}^{\infty }{{E^{ - 1}}\theta {{(a - s)} ^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(a - s)}^{\mu }}\theta \bigr) \biggl\{ { \int _{0}^{\eta }{g\bigl(\eta ,\tau ,x(\tau ),R( \tau )\bigr)\,d\tau } } \biggr\} \,d\theta \,d\eta } } \\& \qquad {} - \mu \int _{0}^{a} { \int _{0}^{\infty }{{E^{ - 1}} \theta {{(a - s)}^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(a - s)}^{ \mu }}\theta \bigr)z(\eta )\,d\theta \,d\eta } } \biggr](s)\,d \theta \,ds \biggr\Vert \\& \qquad {}+ \mu \biggl\Vert \int _{0}^{t} { \int _{0}^{\delta }{{E^{ - 1}}\theta {{(t - s)}^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(t - s)}^{\mu }}\theta \bigr) \int _{0}^{s} {g\bigl(s,\tau ,x(\tau ),R(\tau ) \bigr)\,d\tau } \,d\theta \,ds} } \biggr\Vert \\& \qquad {}+ \mu \biggl\Vert \int _{0}^{t} { \int _{0}^{\delta }{{E^{ - 1}}\theta {{(t - s)}^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(t - s)}^{\mu }}\theta \bigr)z(s)\,d\theta \,ds} } \biggr\Vert \\& \qquad {}+ \mu \biggl\Vert \int _{t - \epsilon }^{t} { \int _{\delta }^{\infty } {{E^{ - 1}}\theta {{(t - s)}^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(t - s)}^{\mu }}\theta \bigr)f\bigl(s,x(s)\bigr)\,d\theta \,ds} } \biggr\Vert \\& \qquad {}+ \mu \biggl\Vert \int _{t - \epsilon }^{t} \int _{\delta }^{\infty } {{E^{ - 1}}\theta {{(t - s)}^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(t - s)}^{\mu }}\theta \bigr)B{W^{ - 1}}} \\& \qquad {} \times \biggl[ {x_{1}} - \frac{\mu }{{\varGamma (\nu (1 - \mu ))}} \int _{0}^{a} \int _{0}^{\infty }{E^{ - 1}}\theta {{(a - \eta )}^{\nu (1 - \mu ) - 1}} {\eta ^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({\eta ^{\mu }}\theta \bigr) \\& \qquad {}\times E\bigl[ {x_{0}} - q(x) \bigr]\,d\theta \,d\eta \\& \qquad {}- \mu \int _{0}^{a} { \int _{0}^{\infty }{{E^{ - 1}}\theta {{(a - s)} ^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(a - s)}^{\mu }}\theta \bigr)f\bigl(\eta ,x( \eta )\bigr)\,d\theta \,d\eta } } \\& \qquad {}- \mu \int _{0}^{a} { \int _{0}^{\infty }{{E^{ - 1}}\theta {{(a - s)} ^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(a - s)}^{\mu }}\theta \bigr) \biggl\{ { \int _{0}^{\eta }{g\bigl(\eta ,\tau ,x(\tau ),R( \tau )\bigr)\,d\tau } } \biggr\} \,d\theta \,d\eta } } \\& \qquad {} - \mu \int _{0}^{a} { \int _{0}^{\infty }{{E^{ - 1}} \theta {{(a - s)}^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(a - s)}^{ \mu }}\theta \bigr)z(\eta )\,d\theta \,d\eta } } \biggr](s)\,d \theta \,ds \biggr\Vert \\& \qquad {}+ \mu \biggl\Vert \int _{t - \epsilon }^{t} { \int _{\delta }^{\infty } {{E^{ - 1}}\theta {{(t - s)}^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(t - s)}^{\mu }}\theta \bigr) \int _{0}^{s} {g\bigl(s,\tau ,x(\tau ),R(\tau ) \bigr)\,d\tau } \,d\theta \,ds} } \biggr\Vert \\& \qquad {} + \mu \biggl\Vert \int _{t - \epsilon }^{t} { \int _{\delta }^{ \infty }{{E^{ - 1}}\theta {{(t - s)}^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl( {{(t - s)}^{\mu }}\theta \bigr)z(s)\,d\theta \,ds} } \biggr\Vert \biggr\} \\& \quad \leq \frac{\mu M \Vert E^{ - 1} \Vert \Vert E \Vert [ \Vert x_{0} \Vert + \Vert q(x) \Vert ] }{ {\varGamma (\nu (1 - \mu ))}} \sup_{t \in J}t^{(1-\nu )(1-\mu )} \int _{0}^{t} {{(t - s)}^{\nu (1 - \mu ) - 1}} {s^{\mu - 1}} \,ds \int _{0} ^{\delta }\theta {\varPsi _{\mu }}( \theta )\,d\theta \\& \qquad {}+ \mu M \bigl\Vert E^{ - 1} \bigr\Vert \sup _{t \in J}t^{(1-\nu )(1-\mu )} \int _{0} ^{t} { {{{(t - s)}^{\mu - 1}} g_{k} (s) \,ds} \int _{0}^{\delta }\theta {\varPsi _{\mu }}( \theta )\,d\theta } \\& \qquad {}+ \mu M \bigl\Vert E^{ - 1} \bigr\Vert \Vert B \Vert \bigl\Vert {W^{ - 1}} \bigr\Vert \sup_{t \in J}t^{(1- \nu )(1-\mu )} \int _{0}^{t} { {{(t - s)}^{\mu - 1}}} \\& \qquad {} \times\biggl[ \Vert x_{1} \Vert + \frac{\mu M \Vert E^{ - 1} \Vert \Vert E \Vert [ \Vert x_{0} \Vert + \Vert q(x) \Vert ] }{\varGamma (\nu (1 - \mu ))} \int _{0}^{a} {{(a - \eta )}^{\nu (1 - \mu ) - 1}} {\eta ^{\mu - 1}} \,d\eta \\& \qquad {}+\mu M \bigl\Vert E^{ - 1} \bigr\Vert \int _{0}^{a} {{{(a - s)}^{\mu - 1}} g_{k} ( \eta ) \,d\eta } +\mu M \bigl\Vert E^{ - 1} \bigr\Vert \int _{0}^{a} {{(a - s)}^{\mu - 1}} h_{k}(\eta )\,d\eta \\& \qquad {} + \mu M \bigl\Vert E^{ - 1} \bigr\Vert \int _{0}^{a} {{(a - s)}^{\mu - 1}}z( \eta )\,d\eta \biggr](s) \,ds \int _{0}^{\delta }\theta {\varPsi _{\mu }}( \theta )\,d\theta \\& \qquad {}+ \mu M \bigl\Vert {E^{ - 1}} \bigr\Vert \sup _{t \in J}t^{(1-\nu )(1-\mu )} \int _{0} ^{t} {(t - s)}^{\mu - 1} h_{k}(s) \,ds \int _{0}^{\delta }\theta {\varPsi _{\mu }}( \theta )\,d\theta \\& \qquad {}+ \mu M \bigl\Vert {E^{ - 1}} \bigr\Vert \sup _{t \in J}t^{(1-\nu )(1-\mu )} \int _{0} ^{t} {(t - s)}^{\mu - 1} z(s) \,ds \int _{0}^{\delta }\theta {\varPsi _{ \mu }}( \theta )\,d\theta \\& \qquad {}+ \mu M \bigl\Vert E^{ - 1} \bigr\Vert \sup _{t \in J}t^{(1-\nu )(1-\mu )} \int _{t-\epsilon } ^{t} { {{{(t - s)}^{\mu - 1}} g_{k} (s) \,ds} \int _{\delta }^{\infty }\theta {\varPsi _{\mu }}( \theta )\,d\theta } \\& \qquad {}+ \mu M \bigl\Vert E^{ - 1} \bigr\Vert \Vert B \Vert \bigl\Vert {W^{ - 1}} \bigr\Vert \sup_{t \in J}t^{(1- \nu )(1-\mu )} \int _{t-\epsilon }^{t} { {{(t - s)}^{\mu - 1}}} \\& \qquad {} \times\biggl[ \Vert x_{1} \Vert + \frac{\mu M \Vert E^{ - 1} \Vert \Vert E \Vert [ \Vert x_{0} \Vert + \Vert q(x) \Vert ] }{\varGamma (\nu (1 - \mu ))} \int _{0}^{a} {{(a - \eta )}^{\nu (1 - \mu ) - 1}} {\eta ^{\mu - 1}} \,d\eta \\& \qquad {}+\mu M \bigl\Vert E^{ - 1} \bigr\Vert \int _{0}^{a} {{{(a - s)}^{\mu - 1}} g_{k} ( \eta ) \,d\eta } +\mu M \bigl\Vert E^{ - 1} \bigr\Vert \int _{0}^{a} {{(a - s)}^{\mu - 1}} h_{k}(\eta )\,d\eta \\& \qquad {} + \mu M \bigl\Vert E^{ - 1} \bigr\Vert \int _{0}^{a} {{(a - s)}^{\mu - 1}}z( \eta )\,d\eta \biggr](s) \,ds \int _{\delta }^{\infty }\theta {\varPsi _{ \mu }}( \theta )\,d\theta \\& \qquad {}+ \mu M \bigl\Vert {E^{ - 1}} \bigr\Vert \sup _{t \in J}t^{(1-\nu )(1-\mu )} \int _{t-\epsilon }^{t} {(t - s)}^{\mu - 1} h_{k}(s) \,ds \int _{\delta } ^{\infty }\theta {\varPsi _{\mu }}( \theta )\,d\theta \\& \qquad {}+ \mu M \bigl\Vert {E^{ - 1}} \bigr\Vert \sup _{t \in J}t^{(1-\nu )(1-\mu )} \int _{t-\epsilon }^{t} {(t - s)}^{\mu - 1} z(s) \,ds \int _{\delta }^{ \infty }\theta {\varPsi _{\mu }}( \theta )\,d\theta . \end{aligned}$$