In this section, we present and prove main results of controllability for a Sobolev-type nonlocal Hilfer fractional differential system with the Clarke subdifferential in Banach spaces in the following form:
$$ \textstyle\begin{cases} D_{0+}^{\nu ,\mu }(Ex(t))+Ax(t) \\ \quad =Bu(t)+f(t,x(t))+\int _{0}^{t}g(t,s,x(s), \int _{0}^{s}{H(s,\tau ,x(\tau ))\,d\tau })\,ds \\ \qquad {}+\partial Z(t,x(t)),\quad t\in J= ( {0,a} ] , \\ I_{0+}^{(1-\nu )(1-\mu )}x(0)+q(x)={x_{0}}, \end{cases} $$
(3.1)
where \(D^{\nu ,\mu }_{0+} \) is the Hilfer fractional derivative, \(0 \leq \nu \leq 1\), \(0 < \mu < 1\), A and E are closed, linear and densely defined operators with domain contained in the Banach space X and ranges contained in the Banach space Y. The state \(x(\cdot )\) takes values in the Banach space X and the control function \(u(\cdot )\) is given in \(L^{2}(J, U)\). The Banach space of admissible control functions with U a Banach space. The symbol B stands for a bounded linear from U into Y. The nonlinear operators \(f : J \times X \rightarrow Y\), \(H : J \times J \times X \rightarrow X\), \(g : J \times J \times X \times X \rightarrow Y\) and \(\partial Z(t, \cdot )\) is the Clarke subdifferential of \(Z(t,\cdot )\).
To establish the result, we need the following additional hypotheses:
-
(H6)
\(f: J \times X \rightarrow Y \) is a continuous function and there exist constants \(N_{1} > 0\) and \(N_{2} > 0\) such that, for all \(t \in J\), \(v_{1}\), \(v_{2}\in X\) we have
$$ \bigl\Vert f(t,v_{1}) - f(t,v_{2}) \bigr\Vert \leq N_{1} \Vert v_{1} - v_{2} \Vert , \qquad N_{2} = \bigl\Vert f(t,0) \bigr\Vert . $$
-
(H7)
\(g: J \times J \times X \times X \rightarrow Y \) is a continuous function and there exist constants \(L_{1} > 0\) and \(L_{2} > 0\) such that, for all \(t,s \in J\), \(v_{1}, v_{2}, w_{1}, w_{2} \in X\) we have
$$\begin{aligned}& \bigl\Vert g(t,s,v_{1},w_{1}) - g(t,s,v_{2},w_{2}) \bigr\Vert \leq L_{1} \bigl[ \Vert v_{1} - v _{2} \Vert + \Vert w_{1} - w_{2} \Vert \bigr] , \\& L_{2} = \bigl\Vert g(t,s,0,0) \bigr\Vert . \end{aligned}$$
-
(H8)
\(H : J \times J \times X \rightarrow X\) is continuous and there exist constants \(L_{3} > 0\), \(L_{4} > 0\), such that for all \(t,s \in J\), \(v_{1}, v_{2} \in X\) we have
$$ \bigl\Vert H(t,s,v_{1}) - H(t,s,v_{2}) \bigr\Vert \leq L_{3} \Vert v_{1} - v_{2} \Vert , \qquad L_{4} = \bigl\Vert H(t,s,0) \bigr\Vert . $$
-
(H9)
The linear operator W from U into E defined by
$$ Wu = \int _{0}^{a} E^{ - 1}P_{\mu }(a-s)Bu(s) \,ds, $$
has an inverse operator \(W^{-1}\) which takes values in \(L^{2}(J, U) \setminus \ker W\), where the kernel space of W is defined by \(\ker W = \{x \in L^{2}(J, U): Wx = 0 \} \) and B is a bounded operator.
Definition 3.1
We say \(x \in C(J,X)\) is a mild solution of the system (3.1) if it satisfies the integral equation
$$\begin{aligned} x(t) =& {E^{ - 1}} {S_{\nu ,\mu }}(t)E \bigl[{x_{0}} - q(x)\bigr] + \int _{0}^{t} {E^{ - 1}} {P_{\mu }}(t - s)f\bigl(s,x(s)\bigr)\,ds + \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} Bu(s)\,ds \\ &{}+ \int _{0}^{t} {E^{ - 1}} {P_{\mu }}(t - s)\biggl\{ \int _{0}^{s} g\bigl(s, \tau ,x(\tau ),R(\tau ) \bigr)\,d\tau \biggr\} \,ds \\ &{}+ \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} z(s)\,ds ,\quad t\in J, \end{aligned}$$
(3.2)
where
$$ R(\tau ) = \int _{0}^{\tau }{H\bigl(\tau ,\eta ,x(\eta )\bigr) \,d\eta }. $$
The proof of mild solution of Eq. (3.1) is similar to the proof of mild solution of Eq. (1.1) in [38].
Definition 3.2
The system (3.1) is said to be controllable on J, if for every \(x_{0},x_{1} \in X\), there exists a control \(u \in L^{2}(J,U)\) such that the mild solution \(x(t)\) of the system (3.1) satisfies \(x(a)= x_{1}\), where \(x_{1}\) and a are the preassigned terminal state and time, respectively.
Theorem 3.1
If the hypotheses (H1)–(H9) are satisfied, then the system (3.1) is controllable on
J
provided that there exists a constant
\(r>0\)
such that
$$\begin{aligned}& M \bigl\Vert {{E^{ - 1}}} \bigr\Vert \biggl( {1 + \frac{{M{a^{\mu }}} \Vert {{E^{ - 1}}} \Vert \Vert B \Vert \Vert {{W^{ - 1}}} \Vert }{{\varGamma ( \mu +1 )}}} \biggr) \biggl[ \frac{ { \Vert {E} \Vert ( { \Vert {x_{0}} \Vert + \Vert q \Vert } )}}{{\varGamma ( {\nu (1 - \mu ) + \mu } )}} \\& \quad {}+ \frac{{M a^{\nu ( \mu - 1 )+1}}}{{\varGamma ( \mu +1 )}} \biggl( {{N_{1}}r + {N_{2}}}+\frac{a}{\mu +1} \biggl( {L_{1}} \biggl( {r + \frac{a}{ \mu +2} ( {{L_{3}}r + {L_{4}}} )} \biggr) + {L_{2}} \biggr)+ \Vert \zeta \Vert + kr \biggr) \biggr] \\& \quad {}+ \frac{{M{a^{\nu (\mu - 1)+1}} }}{{\varGamma ( \mu +1 )}} \bigl\Vert {{E^{ - 1}}} \bigr\Vert \Vert B \Vert \bigl\Vert {{W^{ - 1}}} \bigr\Vert \Vert {x_{1}} \Vert \le r. \end{aligned}$$
Proof
For any \(x \in C(J,X)\subset L^{2}(J,X)\) from Lemma 2.3 we consider the map \(V_{r}: C(J,X)\rightarrow 2^{C(J,X)}\) as follows:
$$\begin{aligned} V_{r}(x) =& \biggl\{ h \in C(J,X) : h(t)={E^{ - 1}} {S_{\nu ,\mu }}(t)E\bigl[ {x_{0}} - q(x)\bigr]+ \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)f\bigl(s,x(s)\bigr)\,ds} \\ &{}+ \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} Bu(s)\,ds + \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s) \int _{0}^{s} {g\bigl(s,\tau ,x(\tau ),R(\tau ) \bigr)\,d\tau \,ds } } \\ &{}+ \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} z(s)\,ds, z\in N(x) \biggr\} , \quad \mbox{for } x \in C(J,X). \end{aligned}$$
We will show \(V_{r}\) has a fixed point using Theorem 2.1. Note \(V_{r}(x)\) is convex from convexity of \(N(x)\). We divide the proof into five steps.
Step 1: \(V_{r}\) maps bounded sets into bounded sets in \(C(J,X)\).
For any \(x \in B_{r}\) and \(\varPhi \in V_{r}(x)\), we choose a \(z \in N(x)\) with
$$\begin{aligned} \varPhi (t) =& {E^{ - 1}} {S_{\nu ,\mu }}(t)E\bigl[{x_{0}} - q(x)\bigr] + \int _{0} ^{t} {{E^{ - 1}} {P_{\mu }}(t - s)f\bigl(s,x(s)\bigr)\,ds + \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} } Bu(s)\,ds \\ &{}+ \int _{0}^{t} {E^{ - 1}} {P_{\mu }}(t - s) \int _{0}^{s} g\bigl(s,\tau ,x( \tau ),R(\tau ) \bigr)\,d\tau \,ds + \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} z(s)\,ds. \end{aligned}$$
Using the assumption (H9) for any arbitrary function \(x(\cdot)\), define the control
$$\begin{aligned} u(t) =& W^{ - 1} \biggl\{ x_{1} - E^{ - 1}S_{\nu ,\mu }(a)E \bigl[x_{0} - q(x)\bigr] - \int _{0}^{a} E^{ - 1}P_{\mu }(a - s)f\bigl(s,x(s)\bigr)\,ds \\ &{}- \int _{0}^{a} E^{ - 1}P_{\mu }(a - s) \int _{0}^{s} g\bigl(s,\tau ,x(\tau ),R( \tau ) \bigr)\,d\tau \,ds - \int _{0}^{a} E^{ - 1}P_{\mu }(a - s) z(s)\,ds \biggr\} (t), \end{aligned}$$
then the operator Φ takes the form
$$\begin{aligned} \varPhi (t) =& {E^{ - 1}} {S_{\nu ,\mu }}(t)E\bigl[{x_{0}} - q(x)\bigr] + \int _{0} ^{t} {{E^{ - 1}} {P_{\mu }}(t - s)f\bigl(s,x(s)\bigr)\,ds} + \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} B{W^{ - 1}} \\ &{}\times \biggl\{ {x_{1}} - {E^{ - 1}} {S_{\nu ,\mu }}(a)E\bigl({x_{0}} - q(x)\bigr) - \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )f\bigl(\eta ,x(\eta )\bigr)\,d\eta } \\ &{} - \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )} \biggl\{ { \int _{0}^{\eta }{g\bigl(\eta ,\tau ,x(\tau ),R( \tau )\bigr)\,d\tau } } \biggr\} \,d\eta \\ &{}- \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )} z(\eta )\,d\eta \biggr\} ( s )\,ds \\ &{}+ \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} \int _{0}^{s} {g\bigl(s,\tau ,x( \tau ),R(\tau )\bigr)\,d\tau \,ds} + \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} z(s)\,ds. \end{aligned}$$
(3.3)
From (H7), (H8) and the Beta function, we have
$$\begin{aligned}& \int _{0}^{t} (t - s)^{\mu -1} \int _{0}^{s} { \biggl\Vert g\biggl(s,\tau ,x(\tau ), \int _{0}^{\tau }{H\bigl(\tau ,\eta ,x(\eta )\bigr) \,d\eta }\biggr)\,d\tau \biggr\Vert } \,ds \\& \quad \le \int _{0}^{t} (t - s)^{\mu -1} \int _{0}^{s} { \biggl( L_{1} \biggl( \Vert x \Vert + \int _{0}^{\tau }{ \bigl\Vert H\bigl(\tau ,\eta ,x(\eta )\bigr) \bigr\Vert \,d\eta }\biggr)+ L_{2}\biggr) \,d\tau } \,ds \\& \quad \le \int _{0}^{t} (t - s)^{\mu -1} \int _{0}^{s} { \biggl( L_{1} \biggl( r + \int _{0}^{\tau }{(L_{3} r + L_{4}) \,d\eta }\biggr)+ L_{2}\biggr) \,d\tau } \,ds \\& \quad \le \int _{0}^{t} (t - s)^{\mu -1} \int _{0}^{s} { \bigl( L_{1} \bigl( r + \tau (L _{3} r + L_{4})\bigr)+ L_{2}\bigr) \,d\tau } \,ds \\& \quad \le \int _{0}^{t} (t - s)^{\mu -1} \biggl[ { \biggl( L_{1} \biggl( s r + \frac{s^{2} }{2}{(L_{3} r + L_{4}) }\biggr)+ s L_{2}\biggr) } \biggr] \,ds \\& \quad \le L_{1} \biggl( r t^{\mu +1} \frac{\varGamma (\mu ) \varGamma (2)}{\varGamma ( \mu +2)}+ \frac{1}{2} t^{\mu +2} \frac{\varGamma (\mu ) \varGamma (3)}{ \varGamma (\mu +3)} {(L_{3} r + L_{4})}\biggr)+ L_{2} t^{\mu +1} \frac{\varGamma (\mu ) \varGamma (2)}{\varGamma (\mu +2)} \\& \quad \le \frac{a^{\mu +1}}{\mu (\mu +1)} \biggl[ L_{1} \biggl( r + \frac{a}{\mu +2}(L _{3} r + L_{4})\biggr)+ L_{2}\biggr]. \end{aligned}$$
From (H5)–(H9), Lemma 2.1 and Hölder’s inequality, we have
$$\begin{aligned} \Vert \varPhi \Vert _{Y} =&\sup_{t \in J}t^{(1-\nu )(1-\mu )} \bigl\Vert \varPhi (t) \bigr\Vert \\ \le& \sup_{t \in J} {t^{( {1 - \nu } )( {1 - \mu } )}} \biggl\{ { \bigl\Vert {{E^{ - 1}}} \bigr\Vert \bigl\Vert {{S_{\nu ,\mu }}(t)} \bigr\Vert \Vert E \Vert \bigl\Vert {{x_{0}} - q(x)} \bigr\Vert } \\ & {}+ \int _{0}^{t} { \bigl\Vert {{E^{ - 1}}} \bigr\Vert \bigl\Vert {{P_{\mu }}(t - s)} \bigr\Vert \bigl\Vert {f\bigl(s,x(s)\bigr)} \bigr\Vert \,ds} + \int _{0}^{t} { \bigl\Vert {{E^{ - 1}}} \bigr\Vert \bigl\Vert {{P_{\mu }}(t - s)} \bigr\Vert } \Vert B \Vert \bigl\Vert {{W^{ - 1}}} \bigr\Vert \\ & {}\times \biggl\Vert {x_{1}} - {E^{ - 1}} {S_{\nu ,\mu }}(a)E\bigl({x_{0}} - q(x)\bigr) - \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )f\bigl(\eta ,x(\eta )\bigr)\,d\eta } \\ & {} - \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )} \biggl\{ { \int _{0}^{\eta }{g\bigl(\eta ,\tau ,x(\tau ),R( \tau )\bigr)\,d\tau } } \biggr\} \,d\eta \\ &{}- \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )} z(\eta )\,d\eta \biggr\Vert ( s )\,ds \\ & {}+ \int _{0}^{t} { \bigl\Vert {{E^{ - 1}}} \bigr\Vert \bigl\Vert {{P_{\mu }}(t - s)} \bigr\Vert } \int _{0}^{s} { \bigl\Vert {g\bigl(s,\tau ,x( \tau ),R(\tau )\bigr)\,d\tau } \bigr\Vert \,ds} \\ &{}+ \int _{0}^{t} { \bigl\Vert {{E^{ - 1}}} \bigr\Vert \bigl\Vert {{P_{\mu }}(t - s)} \bigr\Vert } \bigl\Vert {z(s)} \bigr\Vert \,ds \biggr\} \\ \le& \frac{M}{{\varGamma ( {\nu (1 - \mu ) + \mu } )}} \bigl\Vert {{E^{ - 1}}} \bigr\Vert \Vert E \Vert \bigl( { \Vert {x_{0}} \Vert + \bigl\Vert {q(x)} \bigr\Vert } \bigr) \\ & {}+ \frac{{M{a^{\nu (\mu - 1)+1}}} \Vert {{E^{ - 1}}} \Vert }{{\varGamma ( \mu +1 )}} \\ &{}\times\biggl[ { {{N_{1}}r + {N_{2}}} +\frac{a}{\mu +1} \biggl( {L_{1}} \biggl( {r + \frac{a}{\mu +2} ( {{L_{3}}r + {L_{4}}} )} \biggr) + {L_{2}} \biggr)+ \Vert \zeta \Vert + kr} \biggr] \\ & {}+ \frac{{M{a^{\nu (\mu - 1)+1}} }}{{\varGamma ( \mu +1 )}} \bigl\Vert {{E^{ - 1}}} \bigr\Vert \Vert B \Vert \bigl\Vert {{W^{ - 1}}} \bigr\Vert \Vert {x_{1}} \Vert \\ &{}+ \frac{M^{2}{a^{\mu }}{ \Vert {{E^{ - 1}}} \Vert }^{2} \Vert B \Vert \Vert {{W^{ - 1}}} \Vert \Vert E \Vert }{{\varGamma ( \mu +1 )}{\varGamma ( {\nu (1 - \mu ) + \mu } )}} \bigl( { \Vert {x_{0}} \Vert + \bigl\Vert {q(x)} \bigr\Vert } \bigr) \\ & {}+ \frac{{M^{2}{a^{\nu (\mu - 1)+1}}}{ \Vert {{E^{ - 1}}} \Vert }^{2} \Vert B \Vert \Vert {{W^{ - 1}}} \Vert {a^{\mu }}}{{\varGamma ( \mu +1 )}^{2}} \\ &{}\times\biggl[ { {{N_{1}}r + {N_{2}}} +\frac{a}{\mu +1} \biggl( {L_{1}} \biggl( {r + \frac{a}{ \mu +2} ( {{L_{3}}r + {L_{4}}} )} \biggr) + {L_{2}} \biggr)+ \Vert \zeta \Vert + kr} \biggr] \\ =& \frac{M}{{\varGamma ( {\nu (1 - \mu ) + \mu } )}} \bigl\Vert {{E^{ - 1}}} \bigr\Vert \Vert E \Vert \bigl( { \Vert {x_{0}} \Vert + \bigl\Vert {q(x)} \bigr\Vert } \bigr) \biggl( {1 + \frac{ {M{a^{\mu }}} \Vert {{E^{ - 1}}} \Vert \Vert B \Vert \Vert {{W^{ - 1}}} \Vert }{{\varGamma ( \mu +1 )}}} \biggr) \\ & {}+ \frac{{M{a^{\nu (\mu - 1)+1}}} \Vert {{E^{ - 1}}} \Vert }{{\varGamma ( \mu +1 )}} \\ &{}\times \biggl[ { {{N_{1}}r + {N_{2}}} +\frac{a}{\mu +1} \biggl( {L_{1}} \biggl( {r + \frac{a}{\mu +2} ( {{L_{3}}r + {L_{4}}} )} \biggr) + {L_{2}} \biggr)+ \Vert \zeta \Vert + kr} \biggr] \\ & {}\times \biggl( {1 + \frac{{M{a^{\mu }}} \Vert {{E^{ - 1}}} \Vert \Vert B \Vert \Vert {{W^{ - 1}}} \Vert }{{\varGamma ( \mu +1 )}}} \biggr) + \frac{{M{a^{\nu ( \mu - 1)+1}} }}{{\varGamma ( \mu +1 )}} \bigl\Vert {{E^{ - 1}}} \bigr\Vert \Vert B \Vert \bigl\Vert {{W ^{ - 1}}} \bigr\Vert \Vert {x_{1}} \Vert \\ =& M \bigl\Vert {{E^{ - 1}}} \bigr\Vert \biggl( {1 + \frac{{M{a^{\mu }}} \Vert {{E^{ - 1}}} \Vert \Vert B \Vert \Vert {{W^{ - 1}}} \Vert }{{\varGamma ( \mu +1 )}}} \biggr) \\ &{}\times \biggl[ \frac{{ \Vert {E} \Vert ( { \Vert {x_{0}} \Vert + \Vert q \Vert } )}}{ {\varGamma ( {\nu (1 - \mu ) + \mu } )}} + \frac{{M a^{\nu ( \mu - 1 )+1}}}{ {\varGamma ( \mu +1 )}} \\ &{}\times\biggl( { {{N_{1}}r + {N_{2}}} +\frac{a}{\mu +1} \biggl( {L_{1}} \biggl( {r + \frac{a}{\mu +2} ( {{L_{3}}r + {L _{4}}} )} \biggr) + {L_{2}} \biggr)+ \Vert \zeta \Vert + kr} \biggr) \biggr] \\ & {}+ \frac{{M{a^{\nu (\mu - 1)+1}} }}{{\varGamma ( \mu +1 )}} \bigl\Vert {{E^{ - 1}}} \bigr\Vert \Vert B \Vert \bigl\Vert {{W^{ - 1}}} \bigr\Vert \Vert {x_{1}} \Vert \le r. \end{aligned}$$
Thus \(V_{r}(B_{r}) \) is bounded in \(C(J,X)\).
Step 2: \(\{ V_{r}(x):x \in B_{r} \}\) is equicontinuous (for all \(r>0\)).
For any \(x \in B_{r}\) and \(\varPhi \in V_{r}(x)\) and \(z \in N(x)\) and from Lemma 2.1(ii) and Hölder’s inequality, we have
$$\begin{aligned}& \bigl\Vert \varPhi (t)-\varPhi (0) \bigr\Vert _{Y} \\& \quad =\sup _{t \in J}t^{(1-\nu )(1-\mu )} \bigl\Vert \varPhi (t)-\varPhi (0) \bigr\Vert \\& \quad \leq M \bigl\Vert {{E^{ - 1}}} \bigr\Vert \biggl( {1 + \frac{{M{a^{\mu }}} \Vert {{E^{ - 1}}} \Vert \Vert B \Vert \Vert {{W^{ - 1}}} \Vert }{{\varGamma ( \mu +1 )}}} \biggr) \\& \qquad {}\times \biggl[ \frac{{ \Vert {E} \Vert ( { \Vert {x_{0}} \Vert + \Vert q \Vert } )}}{ {\varGamma ( {\nu (1 - \mu ) + \mu } )}} + \frac{{M a^{\nu ( \mu - 1 )+1}}}{ {\varGamma ( \mu +1 )}} \\& \qquad {}\times\biggl( { {{N_{1}}r + {N_{2}}} +\frac{a}{\mu +1} \biggl( {L_{1}} \biggl( {r + \frac{a}{\mu +2} ( {{L_{3}}r + {L _{4}}} )} \biggr) + {L_{2}} \biggr)+ \Vert \zeta \Vert + kr} \biggr) \biggr] \\& \qquad {}+ \frac{{M{a^{\nu (\mu - 1)+1}} }}{{\varGamma ( \mu +1 )}} \bigl\Vert {{E^{ - 1}}} \bigr\Vert \Vert B \Vert \bigl\Vert {{W^{ - 1}}} \bigr\Vert \Vert {x_{1}} \Vert + \Vert x_{0} \Vert + \Vert q \Vert . \end{aligned}$$
Thus, for all \(\varepsilon >0\) and for sufficiently small \(\delta _{1}>0\), with \(0< t\leq \delta _{1}\), we have \(\| \varPhi (t)-\varPhi (0)\|_{Y}<\frac{\varepsilon }{2} \). Hence, for all \(\varepsilon >0\), \(\forall \tau _{1} , \tau _{2} \in [0,\delta _{1}]\) and \(\forall \varPhi \in V_{r} (B_{r})\), we have \(\| \varPhi (\tau _{2})- \varPhi (\tau _{1})\|_{Y}<\varepsilon \). For any \(x \in B_{r}\), and \(\frac{\delta _{1}}{2}\leq \tau _{1}<\tau _{2}\leq a\), we obtain
$$\begin{aligned}& \bigl\Vert { {\varPhi } (\tau _{2} ) - {\varPhi } (\tau _{1})} \bigr\Vert \\& \quad \le \bigl\Vert {{E^{ - 1}}} \bigr\Vert \biggl\{ { \bigl\Vert { \bigl( {{S_{\nu ,\mu }}(\tau _{2}) - {S_{\nu ,\mu }}(\tau _{1})} \bigr)E \bigl( {{x_{0}} - q(x)} \bigr)} \bigr\Vert } + \biggl\Vert \int _{\tau _{1}}^{\tau _{2}} {{P_{\mu }}(\tau _{2} - s)f\bigl(s,x(s)\bigr)\,ds} \biggr\Vert \\& \qquad {}+ \biggl\Vert { \int _{\tau _{1}}^{\tau _{2}} {P_{\mu }}(\tau _{2} - s)B{W ^{ - 1}}} \biggl\{ {x_{1}} - {E^{ - 1}} {S_{\nu ,\mu }}(a)E\bigl({x_{0}} - q(x) \bigr) \\& \qquad {}- \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )f\bigl(\eta ,x(\eta )\bigr)\,d\eta }- \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )} { \int _{0}^{\eta }{g\bigl(\eta ,\tau ,x(\tau ),R( \tau )\bigr)\,d\tau } } \,d\eta \\& \qquad {}- \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )} z(\eta )\,d\eta \biggr\} ( s )\,ds \biggr\Vert \\& \qquad {}+ \biggl\Vert \int _{\tau _{1}}^{\tau _{2}} {{P_{\mu }}(\tau _{2} - s) \int _{0}^{s} {g\bigl(s,\tau ,x(\tau ),R(\tau ) \bigr)\,d\tau } \,ds} \biggr\Vert + \biggl\Vert \int _{\tau _{1}}^{\tau _{2}} {{P_{\mu }}(\tau _{2} - s)z(s)\,ds} \biggr\Vert \\& \qquad {}+ \biggl\Vert { \int _{0}^{\tau _{1}} \bigl[{P_{\mu }}(\tau _{2} - s)-{P_{\mu }}( \tau _{1} - s)\bigr]f \bigl(s,x(s)\bigr)\,ds} \biggr\Vert \\& \qquad {}+ \biggl\Vert \int _{0}^{\tau _{1}} \bigl[{P_{\mu }}(\tau _{2} - s)-{P_{\mu }}( \tau _{1} - s) \bigr]B{W^{ - 1}} \biggl\{ {x_{1}} - {E^{ - 1}} {S_{\nu ,\mu }}(a)E\bigl( {x_{0}} - q(x)\bigr) \\& \qquad {}- \int _{0}^{a} {E^{ - 1}} {P_{\mu }}(a - \eta )f\bigl( \eta ,x(\eta )\bigr)\,d\eta- \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )} { \int _{0}^{\eta }{g\bigl(\eta ,\tau ,x(\tau ),R( \tau )\bigr)\,d\tau } } \,d\eta \\& \qquad {} - \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )} z(\eta )\,d\eta \biggr\} ( s )\,ds \biggr\Vert \\& \qquad {} + \biggl\Vert { \int _{0}^{\tau _{1}} \bigl[{P_{\mu }}(\tau _{2} - s)- {P_{\mu }}(\tau _{1} - s)\bigr] \int _{0}^{s} {g\bigl(s,\tau ,x(\tau ),R(\tau ) \bigr)\,d\tau } \,ds} \biggr\Vert \\& \qquad {}+ \biggl\Vert { \int _{0}^{\tau _{1}} \bigl[{P_{\mu }}(\tau _{2} - s)-{P_{\mu }}(\tau _{1} - s)\bigr]z(s) \,ds} \biggr\Vert \biggr\} . \end{aligned}$$
(3.4)
From the compactness of \(T(t), t>0\), Lemma 2.1(ii), we see that the right hand side of inequality (3.4) tends to zero as \(\tau _{2} \rightarrow \tau _{1}\). Thus we see that \(\| { ( {\varPhi } )( \tau _{2} ) - ( {\varPhi } )( \tau _{1} )} \| _{Y}\) tends to zero.
For \(\forall \varepsilon >0\), \(\forall \tau _{1}, \tau _{2} \in (0,a]\), \(| \tau _{1}-\tau _{2} |<\delta _{1}\), \(\forall \varPhi \in V_{r}(B_{r}) \) we see that \(\| { ( {\varPhi } )( \tau _{2} ) - ( {\varPhi } )( \tau _{1} )} \|_{Y}<\varepsilon \) independently of \(x \in B_{r}\). Therefore, we deduce that \(\{V_{r}(x): x \in B_{r} \}\) is an equicontinuous family of functions in \(C(J,X)\).
Step 3: \(V_{r}\) is completely continuous.
We prove that, for all \(t \in J\), \(r>0\), the set \(\prod (t)= \{ \varPhi (t): \varPhi \in V_{r}(B_{r}) \}\) is relatively compact in X. Obviously, \(\prod (0)=x_{0} - q(x)\) is compact, so we only need to consider \(t>0\). Let \(0< t< a\) be fixed. For any \(x \in B_{r}\), \(\varPhi \in V_{r}(x)\), we choose \(z \in N(x) \) with
$$\begin{aligned} \varPhi (t) =& {E^{ - 1}} {S_{\nu ,\mu }}(t)E\bigl[{x_{0}} - q(x)\bigr] + \int _{0} ^{t} {{E^{ - 1}} {P_{\mu }}(t - s)f\bigl(s,x(s)\bigr)\,ds} + \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} B{W^{ - 1}} \\ &{}\times \biggl\{ {x_{1}} - {E^{ - 1}} {S_{\nu ,\mu }}(a)E\bigl({x_{0}} - q(x)\bigr) - \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )f\bigl(\eta ,x(\eta )\bigr)\,d\eta } \\ &{}- \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )} \biggl\{ { \int _{0}^{\eta }{g\bigl(\eta ,\tau ,x(\tau ),R( \tau )\bigr)\,d\tau } } \biggr\} \,d\eta \\ &{}- \int _{0}^{a} {{E^{ - 1}} {P_{\mu }}(a - \eta )} z(\eta )\,d\eta \biggr\} ( s )\,ds+ \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} \int _{0}^{s} {g\bigl(s,\tau ,x( \tau ),R(\tau )\bigr)\,d\tau \,ds} \\ &{} + \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} z(s)\,ds,\quad t \in J. \end{aligned}$$
For each \(\epsilon \in (0,t)\), \(t \in (0,a]\), \(x \in B_{r}\), and any \(\delta >0\), we define
$$\begin{aligned} {\varPhi ^{ \epsilon ,\delta }}(t) =& \frac{\mu }{{\varGamma (\nu (1 - \mu ))}} \int _{0}^{t} { \int _{\delta } ^{\infty }{{E^{ - 1}}\theta {{(t - s)}^{\nu (1 - \mu ) - 1}} {s^{ \mu - 1}} {\varPsi _{\mu }}(\theta )S \bigl({s^{\mu }}\theta \bigr)E\bigl[{x_{0}} - q(x)\bigr]\,d \theta \,ds} } \\ &{}+ \mu \int _{0}^{t - \epsilon } { \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} } \\ &{}+ \mu \int _{0}^{t - \epsilon } \int _{\delta }^{\infty }{{E^{ - 1}} \theta {{(t - s)}^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(t - s)}^{ \mu }}\theta \bigr)B{W^{ - 1}}} \\ &{}\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) \\ &{}\times E\bigl[ {x_{0}} - q(x) \bigr]\,d\theta \,d\eta \\ &{}- \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 } } \\ &{}- \mu \int _{0}^{a} \int _{0}^{\infty }{E^{ - 1}}\theta {{(a - s)} ^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(a - s)}^{\mu }}\theta \bigr) \\ &{}\times\biggl\{ { \int _{0}^{\eta }{g\bigl(\eta ,\tau ,x(\tau ),R( \tau )\bigr)\,d\tau } } \biggr\} \,d\theta \,d\eta \\ &{} - \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 \\ &{}+ \mu \int _{0}^{t - \epsilon } { \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} } \\ &{}+ \mu \int _{0}^{t - \epsilon } { \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} } \\ =& \frac{\mu S(\epsilon ^{\mu }\delta ) }{{\varGamma (\nu (1 - \mu ))}} \int _{0}^{t} \int _{\delta }^{\infty }{E^{ - 1}}\theta {{(t - s)}^{ \nu (1 - \mu ) - 1}} {s^{\mu - 1}} {\varPsi _{\mu }}(\theta )S \bigl({s^{\mu }} \theta -\epsilon ^{\mu }\delta \bigr) \\ &{}\times E \bigl[{x_{0}} - q(x)\bigr]\,d\theta \,ds \\ &{}+ \mu S\bigl(\epsilon ^{\mu }\delta \bigr) \int _{0}^{t - \epsilon } { \int _{ \delta }^{\infty }{{E^{ - 1}}\theta {{(t - s)}^{\mu - 1}} {\varPsi _{ \mu }}(\theta )S\bigl({{(t - s)}^{\mu }}\theta - \epsilon ^{\mu }\delta \bigr)f\bigl(s,x(s) \bigr)\,d\theta \,ds} } \\ &{}+ \mu S\bigl(\epsilon ^{\mu }\delta \bigr) \int _{0}^{t - \epsilon } \int _{ \delta }^{\infty }{{E^{ - 1}}\theta {{(t - s)}^{\mu - 1}} {\varPsi _{ \mu }}(\theta )S\bigl({{(t - s)}^{\mu }}\theta -\epsilon ^{\mu }\delta \bigr)B {W^{ - 1}}} \\ &{} \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) \\ &{}\times E\bigl[ {x_{0}} - q(x) \bigr]\,d\theta \,d\eta \\ &{}- \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 } } \\ &{}- \mu \int _{0}^{a} \int _{0}^{\infty }{E^{ - 1}}\theta {{(a - s)} ^{\mu - 1}} {\varPsi _{\mu }}(\theta )S\bigl({{(a - s)}^{\mu }}\theta \bigr) \\ &{}\times\biggl\{ { \int _{0}^{\eta }{g\bigl(\eta ,\tau ,x(\tau ),R( \tau )\bigr)\,d\tau } } \biggr\} \,d\theta \,d\eta \\ &{} - \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 \\ &{}+ \mu S\bigl(\epsilon ^{\mu }\delta \bigr) \int _{0}^{t - \epsilon } \int _{ \delta }^{\infty }{E^{ - 1}}\theta {{(t - s)}^{\mu - 1}} {\varPsi _{ \mu }}(\theta )S\bigl({{(t - s)}^{\mu }}\theta - \epsilon ^{\mu }\delta \bigr) \\ &{}\times\int _{0}^{s} {g\bigl(s,\tau ,x(\tau ),R(\tau ) \bigr)\,d\tau } \,d\theta \,ds \\ &{}+ \mu S\bigl(\epsilon ^{\mu }\delta \bigr) \int _{0}^{t - \epsilon } { \int _{ \delta }^{\infty }{{E^{ - 1}}\theta {{(t - s)}^{\mu - 1}} {\varPsi _{ \mu }}(\theta )S\bigl({{(t - s)}^{\mu }}\theta - \epsilon ^{\mu }\delta \bigr)z(s)\,d\theta \,ds} }. \end{aligned}$$
From the compactness of \(S(\epsilon ^{\mu }\delta )\), \(\epsilon ^{\mu } \delta >0\) and the bounded of \(u(s)\) we see that the set \(\prod_{\epsilon ,\delta }(t)= \{ \varPhi ^{\epsilon ,\delta }(t): \varPhi \in V_{r}(B_{r})\} \) is relatively compact in X for each \(\epsilon \in (0,t)\) and \(\delta >0\). 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}$$
Now we see that \(\| \varPhi (t)-\varPhi ^{\epsilon ,\delta }(t)\|_{Y}\rightarrow 0\) as \(\epsilon \rightarrow 0\), \(\delta \rightarrow 0\). Therefore, the set \(\prod (t)\), \(t>0\) is totally bounded, i.e., relatively compact in X. From the above (and step 2) and the Ascoli–Arzela theorem, we see that \(V_{r}\) is completely continuous.
Step 4: \(V_{r}\) has a closed graph.
Let \(x_{n} \rightarrow x_{*}\) as \(n \rightarrow \infty \) in \(C(J,X)\), \(\varPhi _{n} \in V_{r}(x_{n})\) and \(\varPhi _{n} \rightarrow \varPhi _{*}\) as \(n \rightarrow \infty \) in \(C(J,X)\). We prove that \(\varPhi _{*} \in V_{r}(x_{*})\). Now \(\varPhi _{n} \in V_{r}(x_{n})\), so there exist \(z_{n} \in N(x_{n})\), \(f_{n}=f(t,x_{n}(t))\), \(R_{n}(\tau )= \int _{0}^{\tau }H(\tau ,\eta ,x_{n}(\eta ))\,d\eta \) and \(g_{n}=g(t,s,x_{n}(s),R _{n}(\tau ))\) in \(L^{2}(J,X)\) with
$$\begin{aligned} \varPhi _{n}(t) =& {E^{ - 1}} {S_{\nu ,\mu }}(t)E \bigl[{x_{0}} - q(x)\bigr] + \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)f_{n}\bigl(s,x(s)\bigr)\,ds + \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} } Bu(s)\,ds \\ &{}+ \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s) \int _{0}^{s} {g_{n}\bigl(s, \tau ,x(\tau ),R_{n}(\tau )\bigr)\,d\tau \,ds + } } \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} z_{n}(s)\,ds. \end{aligned}$$
(3.5)
From (H5)–(H8), \(\{z_{n}, f_{n}, g_{n}\}_{n\geq 1}\subseteq L ^{2}(J,X)\) are bounded. Hence we assume that
$$ z_{n} \rightarrow z_{*},\qquad f_{n} \rightarrow f_{*},\qquad g_{n} \rightarrow g _{*},\quad \mbox{weakly in } L^{2}(J,X). $$
(3.6)
From (3.5), (3.6) and compactness of \(P_{\mu }(t)\), we have
$$\begin{aligned} \varPhi _{n}(t) \rightarrow &{E^{ - 1}} {S_{\nu ,\mu }}(t)E\bigl[{x_{0}} - q(x)\bigr] + \int _{0}^{t} {E^{ - 1}} {P_{\mu }}(t - s)f_{*}\bigl(s,x(s)\bigr)\,ds \\ &{}+ \int _{0} ^{t} {E^{ - 1}} {P_{\mu }}(t - s) Bu(s)\,ds+ \int _{0}^{t} {E^{ - 1}} {P_{\mu }}(t - s) \int _{0}^{s} g_{*}\bigl(s, \tau ,x(\tau ),R_{*}(\tau )\bigr)\,d\tau \,ds \\ &{} + \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} z_{*}(s)\,ds. \end{aligned}$$
Note that \(\varPhi _{n} \rightarrow \varPhi _{*}\) in \(C(J,X)\) and \(z_{n} \in N(x_{n}) \). Hence, from Lemma 2.4 we obtain \(z_{*} \in N(x_{*})\) and \(\varPhi _{*} \in V_{r}(x_{*})\), which implies \(V_{r}\) has a closed graph and \(V_{r}\) is u.s.c.
Step 5: A priori estimate.
From steps 1–4, we see that \(V_{r}\) is u.s.c. and is compact convex valued and \(V_{r}(B_{r})\) is a relatively compact set (here \(r>0\)). We now prove that the set \(\varOmega = \{ x \in C(J,X):\lambda x \in V_{r}(x), \lambda >0 \}\) is bounded. For all \(x \in \omega \), there exist \(z \in N(x)\) and f, g in \(L^{2}(J,X)\) with
$$\begin{aligned} x(t) =& \lambda ^{-1}{E^{ - 1}} {S_{\nu ,\mu }}(t)E \bigl[{x_{0}} - q(x)\bigr] + \lambda ^{-1} \int _{0}^{t} {E^{ - 1}} {P_{\mu }}(t - s)f\bigl(s,x(s)\bigr)\,ds \\ &{}+ \lambda ^{-1} \int _{0}^{t} {E^{ - 1}} {P_{\mu }}(t - s) Bu(s)\,ds \\ &{}+ \lambda ^{-1} \int _{0}^{t} {E^{ - 1}} {P_{\mu }}(t - s)\biggl\{ \int _{0} ^{s} g\bigl(s,\tau ,x(\tau ),R(\tau )\bigr)\,d\tau \biggr\} \,ds \\ &{}+ \lambda ^{-1} \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} z(s)\,ds. \end{aligned}$$
(3.7)
Then from assumptions (H5)–(H8), we derive
$$\begin{aligned} \bigl\Vert x(t) \bigr\Vert _{Y} =& \sup_{t\in J} {t^{(1 - \nu )(1 - \mu )}} \bigl\Vert x(t) \bigr\Vert \\ =& \sup {t^{ ( {1 - \nu } ) ( {1 - \mu } )}} \biggl\{ \biggl\Vert \lambda ^{-1}{E^{ - 1}} {S_{\nu ,\mu }}(t)E \bigl[{x_{0}} - q(x)\bigr] + \lambda ^{-1} \int _{0}^{t} {E^{ - 1}} {P_{\mu }}(t - s)f\bigl(s,x(s)\bigr)\,ds \\ &{}+ \lambda ^{-1} \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} Bu(s)\,ds \\ &{}+ \lambda ^{-1} \int _{0}^{t} {E^{ - 1}} {P_{\mu }}(t - s)\biggl\{ \int _{0}^{s} g\bigl(s,\tau ,x(\tau ),R(\tau ) \bigr)\,d\tau \biggr\} \,ds \\ &{}+ \lambda ^{-1} \int _{0}^{t} {{E^{ - 1}} {P_{\mu }}(t - s)} z(s)\,ds \biggr\Vert \biggr\} \\ \leq& \sup {t^{ ( {1 - \nu } ) ( {1 - \mu } )}} \biggl\{ \lambda ^{-1} { \bigl\Vert {{E^{ - 1}}} \bigr\Vert \bigl\Vert {{S _{\nu ,\mu }}(t)} \bigr\Vert \Vert E \Vert \bigl( \Vert {x_{0}} \Vert + \bigl\Vert q(x) \bigr\Vert \bigr) } \\ &{}+ \lambda ^{-1} \int _{0}^{t} { \bigl\Vert {{E^{ - 1}}} \bigr\Vert \bigl\Vert {{P_{\mu }}(t - s)} \bigr\Vert \bigl\Vert {f\bigl(s,x(s)\bigr)} \bigr\Vert \,ds} \\ &{}+ \lambda ^{-1} \int _{0}^{t} { \bigl\Vert {{E^{ - 1}}} \bigr\Vert \bigl\Vert {{P_{\mu }}(t - s)} \bigr\Vert } \Vert B \Vert \bigl\Vert u(s) \bigr\Vert \,ds \\ &{} + \lambda ^{-1} \int _{0}^{t} { \bigl\Vert {{E^{ - 1}}} \bigr\Vert \bigl\Vert {{P_{\mu }}(t - s)} \bigr\Vert } \int _{0}^{s} { \bigl\Vert {g\bigl(s, \tau ,x(\tau ),R(\tau )\bigr)\,d\tau } \bigr\Vert \,ds} \\ &{}+ \lambda ^{-1} \int _{0} ^{t} { \bigl\Vert {{E^{ - 1}}} \bigr\Vert \bigl\Vert {{P_{\mu }}(t - s)} \bigr\Vert } \bigl\Vert {z(s)} \bigr\Vert \,ds \biggr\} \\ \le& \frac{M\lambda ^{-1}}{{\varGamma ( {\nu (1 - \mu ) + \mu } )}} \bigl\Vert {{E^{ - 1}}} \bigr\Vert \Vert E \Vert \bigl( { \Vert {x_{0}} \Vert + \bigl\Vert {q(x)} \bigr\Vert } \bigr) \\ &{}+ \frac{{M{a^{\nu (\mu - 1)+1}}\lambda ^{-1}} \Vert {{E^{ - 1}}} \Vert }{{\varGamma ( \mu +1 )}} \\ &{}\times\biggl[ {{N_{1}}r + {N_{2}}} +\frac{a}{ \mu +1} \biggl( {L_{1}} \biggl( {r + \frac{a}{\mu +2} ( {{L_{3}}r + {L_{4}}} )} \biggr) + {L_{2}} \biggr) \\ &{}+ \Vert \zeta \Vert + kr+ \Vert B \Vert \Vert u \Vert \biggr]. \end{aligned}$$
It follows from (3.7) and \(\lambda ^{-1}<1\) that \(\| x(t) \| _{Y} \leq r\). Hence, \(\| x \|_{C} = \sup_{t\in J} \| x(t) \|_{Y} \leq r \), which implies the set Ω is bounded.
From Theorem 2.1, \(V_{r}\) has a fixed point, i.e., the system (3.1) is controllable and the proof is complete. □