In this section, we state and prove some results on the nonlocal controllability of system (1.1). The discussion is based on the theory of resolvent operators and fixed point theorems. For this purpose, we first make the following assumptions.
- \((H_{AE})\):
-
The pair \((A, E)\) generates an \((\alpha, 1)\)-resolvent family \(\{C_{\alpha, 1}^{E}(t)\}_{t\geq0}\) in X, and
$$M:=\sup_{t\in J} \bigl\Vert C_{\alpha, 1}^{E}(t) \bigr\Vert < +\infty. $$
- \((H_{W})\):
-
The linear operator \(W: L^{2}(J, U)\rightarrow X\) defined by
$$Wu:= \int_{0}^{b}P_{\alpha, 1}^{E}(b-s)Bu(s) \,ds $$
has a linear bounded inverse operator \(W^{-1}\) taking values in \(L^{2}(J, U)\setminus \operatorname{Ker}(W)\), and let \(M_{1}:=\|W^{-1}\|\).
- \((H_{f1})\):
-
\(f: J\times X\rightarrow X\) satisfies the Carathéodory condition, that is, for each \(x\in X\), \(f(\cdot, x): J\rightarrow X\) is strongly measurable; for each \(t\in J\), \(f(t,\cdot ): X\rightarrow X\) is continuous.
- \((H_{f2})\):
-
There is a function \(L_{f}\in L^{1}(J, \mathbb{R}^{+})\) such that
$$\bigl\Vert f(t,x)-f(t,y) \bigr\Vert \leq L_{f}(t) \Vert x-y \Vert , \quad\forall t\in J, x, y\in X. $$
- \((H_{g})\):
-
\(g: C(J, X)\rightarrow X\), and there exists a constant \(L_{g}>0\) such that
$$\bigl\Vert g(x)-g(y) \bigr\Vert \leq L_{g} \Vert x-y \Vert _{C},\quad \forall x,y\in C(J, X). $$
- \((H_{h})\):
-
\(h: C(J, X)\rightarrow X\), and there exists a constant \(L_{h}>0\) such that
$$\bigl\Vert h(x)-h(y) \bigr\Vert \leq L_{h} \Vert x-y \Vert _{C},\quad \forall x,y\in C(J, X). $$
- \((H_{B})\):
-
\(B: U\rightarrow X\) is a linear bounded operator, and let \(M_{B}:=\|B\|\).
By assumption \((H_{W})\), for any \(x_{1}\in X\) and \(x\in C(J, X)\), we define the control \(u_{x}\in L^{2}(J, U)\) as
$$\begin{aligned} u_{x}(t) =&W^{-1} \biggl\{ x_{1}-g(x)-C_{\alpha, 1}^{E}(b) \bigl[x_{0}-g(x) \bigr]-S_{\alpha, 1}^{E}(b) \bigl[y_{0}-h(x) \bigr] \\ &{}- \int_{0}^{b}P_{\alpha, 1}^{E}(b-s)f \bigl(s,x(s)\bigr)\,ds \biggr\} (t),\quad t\in J. \end{aligned}$$
If \(x\in C(J, X)\) is a mild solution of system (1.1) corresponding to the control \(u_{x}\), then by \((H_{W})\) and (2.3) we have
$$\begin{aligned} x(b) =&C_{\alpha, 1}^{E}(b) \bigl[x_{0}-g(x) \bigr]+S_{\alpha, 1}^{E}(b) \bigl[y_{0}-h(x) \bigr] \\ &{}+ \int_{0}^{b}P_{\alpha, 1}^{E}(b-s)f \bigl(s,x(s)\bigr)\,ds+ \int_{0}^{b}P_{\alpha, 1}^{E}(b-s)Bu_{x}(s) \,ds \\ =&x_{1}-g(x), \end{aligned}$$
which implies \(x(b)+g(x)=x_{1}\), and system (1.1) is nonlocally controllable on J. Hence we will now prove that system (1.1) has mild solutions by using resolvent operator theory and fixed point theorems. Define the operator \(Q: C(J, X)\rightarrow C(J, X)\) by
$$\begin{aligned}[b] (Qx) (t)&=C_{\alpha, 1}^{E}(t) \bigl[x_{0}-g(x) \bigr]+S_{\alpha, 1}^{E}(t) \bigl[y_{0}-h(x) \bigr]\\&\quad+ \int_{0}^{t}P_{\alpha, 1}^{E}(t-s) \bigl[f\bigl(s,x(s)\bigr)+Bu_{x}(s) \bigr]\,ds,\quad t\in J.\end{aligned} $$
(3.1)
By Definition 6 the mild solution of system (1.1) is equivalent to the fixed point of Q. We first apply the contraction mapping principle to prove that Q has a fixed point in \(C(J, X)\).
Lemma 7
Assume that conditions\((H_{AE})\), \((H_{W})\), \((H_{f2})\), \((H_{g})\), and\((H_{h})\)are satisfied. Then for all\(x, y\in C(J, X)\)and\(t\in J\), we have
$$\bigl\Vert u_{x}(t)-u_{y}(t) \bigr\Vert \leq M_{1} \biggl[(1+M)L_{g}+MbL_{h}+ \frac{Mb^{\alpha -1}}{\varGamma(\alpha)} \Vert L_{f} \Vert _{L^{1}} \biggr] \Vert x-y \Vert _{C}. $$
Proof
For any \(x, y\in C(J, X)\) and \(t\in J\), by the definition of \(u_{x}\) and \(u_{y}\) we have
$$\begin{aligned} & \bigl\Vert u_{x}(t)-u_{y}(t) \bigr\Vert \\ &\quad\leq M_{1} \biggl[ (1+M) \bigl\Vert g(x)-g(y) \bigr\Vert +Mb \bigl\Vert h(x)-h(y) \bigr\Vert \\ &\qquad{}+\frac{Mb^{\alpha -1}}{\varGamma(\alpha)} \int_{0}^{b} \bigl\Vert f\bigl(s,x(s)\bigr)-f \bigl(s,y(s)\bigr) \bigr\Vert \,ds \biggr] \\ &\quad\leq M_{1} \biggl[ (1+M)L_{g} \Vert x-y \Vert _{C}+MbL_{h} \Vert x-y \Vert _{C}\\ &\qquad{}+\frac{Mb^{\alpha -1}}{\varGamma(\alpha)} \int_{0}^{b}L_{f}(s) \bigl\Vert x(s)-y(s) \bigr\Vert \,ds \biggr] \\ &\quad\leq M_{1} \biggl[(1+M)L_{g}+MbL_{h}+ \frac{Mb^{\alpha-1}}{\varGamma(\alpha )} \Vert L_{f} \Vert _{L^{1}} \biggr] \Vert x-y \Vert _{C}. \end{aligned}$$
This completes the proof. □
Theorem 1
Let assumptions\((H_{AE})\), \((H_{W})\), \((H_{f1})\), \((H_{f2})\), \((H_{g})\), \((H_{h})\), and\((H_{B})\)hold. Then system (1.1) is nonlocally controllable onJ, provided that
$$ M^{*}:= \biggl\{ \frac{MM_{1}M_{B}b^{\alpha}}{\varGamma(\alpha)}L_{g}+M \biggl( 1+ \frac{MM_{1}M_{B}b^{\alpha}}{\varGamma(\alpha)} \biggr) \biggl[ L_{g}+bL_{h}+ \frac{b^{\alpha-1}}{\varGamma(\alpha)} \Vert L_{f} \Vert _{L^{1}} \biggr] \biggr\} < 1. $$
(3.2)
Proof
For any \(x, y\in C(J, X)\) and \(t\in J\), by (3.1) and Lemma 7 we have
$$\begin{aligned} & \bigl\Vert (Qx) (t)-(Qy) (t) \bigr\Vert \\ &\quad\leq M \bigl\Vert g(x)-g(y) \bigr\Vert +Mb \bigl\Vert h(x)-h(y) \bigr\Vert \\ &\qquad{}+\frac{Mb^{\alpha-1}}{\varGamma(\alpha)} \biggl[ \int_{0}^{t} \bigl\Vert f\bigl(s,x(s)\bigr)-f \bigl(s,y(s)\bigr) \bigr\Vert \,ds+ M_{B} \int_{0}^{t} \bigl\Vert u_{x}(s)-u_{y}(s) \bigr\Vert \,ds \biggr] \\ &\quad\leq \biggl( ML_{g}+MbL_{h}+ \frac{Mb^{\alpha-1}}{\varGamma(\alpha)} \Vert L_{f} \Vert _{L^{1}} \biggr) \Vert x-y \Vert _{C} \\ &\qquad{}+\frac{MM_{1}M_{B}b^{\alpha}}{\varGamma(\alpha)} \biggl[ (1+M)L_{g}+MbL_{h}+ \frac{Mb^{\alpha-1}}{\varGamma(\alpha)} \Vert L_{f} \Vert _{L^{1}} \biggr] \Vert x-y \Vert _{C} \\ &\quad=M^{*} \Vert x-y \Vert _{C}. \end{aligned}$$
From (3.2) it follows that \(M^{*}<1\). Hence by the contraction mapping principle, Q has a unique fixed point x in \(C(J, X)\) satisfying \(x(b)+g(x)=x_{1}\). In other words, system (1.1) is nonlocally controllable on J. □
Remark 1
If \(E=I\), where I denotes the identity operator in X, and \(h(x)\equiv0\) for all \(x\in C(J, X)\), then Theorem 1 is a natural extension of Theorem 3.1 in [10], because we delete the compactness condition \((H_{4})\) in [10].
The Lipschitz condition \((H_{f2})\) of the nonlinear term f is difficult to verify in applications. If we apply more weaker conditions on f, we can also prove the controllability results for system (1.1). For \(r>0\), set \(\varOmega_{r}:=\{x\in C(J, X): \|x\|_{C}\leq r\}\).
Lemma 8
Assume that conditions\((H_{AE})\), \((H_{W})\), \((H_{g})\), \((H_{h})\), and\((H_{f2})'\)are satisfied, where
- \((H_{f2})'\):
-
For each\(r>0\), there is a function\(\varphi_{r}\in L^{1}(J, \mathbb{R}^{+})\)satisfying\(\lim_{r\rightarrow\infty}\frac{\| \varphi_{r}\|_{L^{1}}}{r}=\sigma<\infty\)such that
$$\sup_{ \Vert x \Vert \leq r} \bigl\Vert f(t,x) \bigr\Vert \leq \varphi_{r}(t),\quad \forall t\in J. $$
Then for any\(x\in\varOmega_{r}\)and\(t\in J\), we have
$$ \bigl\Vert u_{x}(t) \bigr\Vert \leq\mathfrak{C}+M_{1} \biggl[ (1+M)L_{g}r+MbL_{h}r+\frac {Mb^{\alpha-1}}{\varGamma(\alpha)} \Vert \varphi_{r} \Vert _{L^{1}} \biggr], $$
(3.3)
where\(\mathfrak{C}:=M_{1} [ \|x_{1}\|+M\|x_{0}\|+Mb\|y_{0}\|+(1+M)\| g(0)\|+Mb\|h(0)\| ]\).
Proof
Applying assumptions \((H_{AE})\), \((H_{W})\), \((H_{g})\), \((H_{h})\), and \((H_{f2})'\), by direct calculation we can easily prove that \(u_{x}\) satisfies inequality (3.3). So we omit the details. □
Remark 2
If f satisfies the linear growth conditions, for example, \(f(t,x)\leq a_{1}(t)x+a_{2}(t)\), \(t\in J\), \(x\in X\), where \(a_{1}, a_{2}\in L^{1}(J, \mathbb {R})\), then condition \((H_{f2})'\) holds when we choose \(\varphi_{r}(t)=\| a_{1}(t)\|r+\|a_{2}(t)\|\).
Lemma 9
Assume that the conditions\((H_{AE})\), \((H_{W})\), \((H_{f1})\), \((H_{f2})'\), \((H_{g})\), \((H_{h})\)and\((H_{B})\)are satisfied. Then the operatorQ, defined as in (3.1), maps\(\varOmega_{r}\)into itself for some\(r>0\)provided that
$$ \frac{MM_{1}M_{B}b^{\alpha}}{\varGamma(\alpha)}L_{g}+M \biggl( 1+\frac {MM_{1}M_{B}b^{\alpha}}{\varGamma(\alpha)} \biggr) \biggl( L_{g}+bL_{h}+\frac {b^{\alpha-1}}{\varGamma(\alpha)}\sigma \biggr)< 1. $$
(3.4)
Moreover, \(Q: \varOmega_{r}\rightarrow\varOmega_{r}\)is continuous.
Proof
It is obvious that the operator \(Q: C(J, X)\rightarrow C(J, X)\) is continuous under these assumptions. Hence we just prove \(Q(\varOmega_{r})\subset\varOmega_{r}\) for some \(r>0\). If this were not true, then for any \(r>0\), there would be \(x\in\varOmega_{r}\) such that \(r<\|(Qx)(t)\|\) for all \(t\in J\). By Lemma 7 and (3.1) we have
$$\begin{aligned} r < & \bigl\Vert (Qx) (t) \bigr\Vert \\ \leq& M\bigl( \Vert x_{0} \Vert + \bigl\Vert g(x) \bigr\Vert \bigr)+Mb\bigl( \Vert y_{0} \Vert + \bigl\Vert h(x) \bigr\Vert \bigr) \\ &{}+\frac{Mb^{\alpha-1}}{\varGamma(\alpha)} \int_{0}^{t} \bigl\Vert f\bigl(s,x(s)\bigr) \bigr\Vert \, ds+M_{B}\frac{Mb^{\alpha-1}}{\varGamma(\alpha)} \int_{0}^{t} \bigl\Vert u_{x}(s) \bigr\Vert \,ds \\ \leq&M\bigl( \Vert x_{0} \Vert +L_{g}r+ \bigl\Vert g(0) \bigr\Vert \bigr)+Mb\bigl( \Vert y_{0} \Vert +L_{h}r+ \bigl\Vert h(0) \bigr\Vert \bigr) \\ &{}+\frac{Mb^{\alpha-1}}{\varGamma(\alpha)} \Vert \varphi_{r} \Vert _{L^{1}}+M_{B}\frac{Mb^{\alpha-1}}{\varGamma(\alpha)} \int_{0}^{t} \bigl\Vert u_{x}(s) \bigr\Vert \, ds \\ \leq&M\bigl( \Vert x_{0} \Vert + \bigl\Vert g(0) \bigr\Vert \bigr)+Mb\bigl( \Vert y_{0} \Vert + \bigl\Vert h(0) \bigr\Vert \bigr)+\frac{MM_{B}b^{\alpha }}{\varGamma(\alpha)}\mathfrak{C} \\ &{}+ML_{g}r+MbL_{h}r \\ &{}+\frac{Mb^{\alpha-1}}{\varGamma(\alpha)} \Vert \varphi_{r} \Vert _{L^{1}}\frac{MM_{1}M_{B}b^{\alpha}}{\varGamma(\alpha)}\biggl[(1+M)L_{g}r+MbL_{h}r+ \frac {Mb^{\alpha-1}}{\varGamma(\alpha)} \Vert \varphi_{r} \Vert _{L^{1}} \biggr]. \end{aligned}$$
Dividing both sides by r and taking the lower limit as \(r\rightarrow \infty\), we obtain
$$1\leq\frac{MM_{1}M_{B}b^{\alpha}}{\varGamma(\alpha)}L_{g}+ \biggl( 1+\frac {MM_{1}M_{B}b^{\alpha}}{\varGamma(\alpha)} \biggr) \biggl( ML_{g}+MbL_{h}+\frac {Mb^{\alpha-1}}{\varGamma(\alpha)} \sigma \biggr), $$
which is a contradiction to (3.4). Thus there is \(r>0\) such that \(Q(\varOmega_{r})\subset\varOmega_{r}\). □
Theorem 2
Let assumptions\((H_{AE})\), \((H_{W})\), \((H_{f1})\), \((H_{f2})'\), \((H_{g})\), \((H_{h})\), \((H_{B})\), and\((H_{f3})\)hold, where
- \((H_{f3})\):
-
For\(t\in[0,b]\), the set\(V_{\varepsilon}:= \{ P_{\alpha, 1}^{E}(t-s)[f(s, x(s))+Bu(s)]: x\in\varOmega_{r}, s\in[0, t-\varepsilon], \varepsilon\in(0,t) \}\)is compact.
Then system (1.1) is nonlocally controllable onJwhen (3.4) is satisfied.
Proof
We define two operators \(Q_{1}, Q_{2}: C(J, X)\rightarrow C(J, X)\) by
$$ (Q_{1}x) (t)=C_{\alpha, 1}^{E}(t) \bigl[x_{0}-g(x)\bigr]+S_{\alpha, 1}^{E}(t) \bigl[y_{0}-h(x)\bigr],\quad t\in J, $$
(3.5)
and
$$ (Q_{2}x) (t)= \int_{0}^{t}P_{\alpha, 1}^{E}(t-s) \bigl[f\bigl(s, x(s)\bigr)+Bu(s)\bigr]\,ds,\quad t\in J. $$
(3.6)
Then by (3.1) we know that \(Q=Q_{1}+Q_{2}\). By \((H_{AE})\), \((H_{g})\), and \((H_{h})\) it is clear that
$$ \bigl\Vert (Q_{1}x) (t)-(Q_{1}y) (t) \bigr\Vert \leq M(L_{g}+bL_{h}) \Vert x-y \Vert _{C},\quad \forall x, y\in C(J, X). $$
(3.7)
Next, we prove that the set \(V:=\{Q_{2}x: x\in\varOmega_{r}\}\) is relatively compact in \(C(J, X)\). To apply the Ascoli–Arzelà theorem, we prove that \(V:=\{Q_{2}x: x\in\varOmega_{r}\}\) is equicontinuous in \(C(J, X)\) and \(V(t):=\{(Q_{2}x)(t): x\in\varOmega_{r}\}\) is relatively compact in X. For any \(0\leq t_{1}< t_{2}\leq b\) and \(x\in\varOmega_{r}\), by Lemmas 2 and 8 we have
$$\begin{aligned} \bigl\Vert (Q_{2}x) (t_{2})-(Q_{2}x) (t_{1}) \bigr\Vert \leq& \biggl\Vert \int_{0}^{t_{1}}\bigl[P_{\alpha, 1}^{E}(t_{2}-s)-P_{\alpha, 1}^{E}(t_{1}-s) \bigr] \bigl[f\bigl(s, x(s)\bigr)+Bu(s)\bigr]\,ds \biggr\Vert \\ &{}+ \biggl\Vert \int_{t_{1}}^{t_{2}}P_{\alpha, 1}^{E}(t_{2}-s) \bigl[f\bigl(s, x(s)\bigr)+Bu(s)\bigr]\,ds \biggr\Vert \\ \leq& \int_{0}^{t_{1}} \bigl\Vert f\bigl(s, x(s) \bigr)+Bu(s) \bigr\Vert \,ds\sup_{s\in[0,t_{1}]} \bigl\Vert P_{\alpha, 1}^{E}(t_{2}-s)-P_{\alpha, 1}^{E}(t_{1}-s) \bigr\Vert \\ &{}+\frac{Mb^{\alpha-1}}{\varGamma(\alpha)} \int_{t_{1}}^{t_{2}} \bigl\Vert f\bigl(s, x(s) \bigr)+Bu(s) \bigr\Vert \,ds \\ \rightarrow&0 \end{aligned}$$
as \(t_{2}-t_{1}\rightarrow0\), which implies that the set V is equicontinuous in \(C(J, X)\).
Let
$$\bigl(Q_{2}^{\varepsilon}x\bigr) (t)= \int_{0}^{t-\varepsilon}P_{\alpha, 1}^{E}(t-s) \bigl[f\bigl(s, x(s)\bigr)+Bu(s)\bigr]\,ds,\quad t\in J. $$
By the assumption \((H_{f3})\), \(\overline{\operatorname{conv}(V_{\varepsilon})}\) is also a compact set, where \(\overline{\operatorname{conv}(V_{\varepsilon})}\) means the convex closure of \(V_{\varepsilon}\). By the mean value theorem for Bochner integrals, we deduce that \((Q_{2}^{\varepsilon}x)(t)\in (t-\varepsilon)\overline{\operatorname{conv}(V_{\varepsilon})}\) for \(t\in J\). So the set \(V_{2}^{\varepsilon}(t):=\{(Q_{2}^{\varepsilon}x)(t): x\in\varOmega _{r}\}\) is relatively compact in X. Moreover, for any \(x\in\varOmega_{r}\), we have
$$\begin{aligned} \bigl\Vert (Q_{2}x) (t)-\bigl(Q_{2}^{\varepsilon}x \bigr) (t) \bigr\Vert \leq&\frac{Mb^{\alpha-1}}{\varGamma(\alpha)} \int_{t-\varepsilon }^{t} \bigl\Vert f\bigl(s, x(s) \bigr)+Bu(s) \bigr\Vert \,ds \\ \leq&\frac{Mb^{\alpha-1}}{\varGamma(\alpha)} \int_{t-\varepsilon }^{t}\varphi_{r}(s)\,ds+ \frac{MM_{B}b^{\alpha-1}M^{**}}{\varGamma(\alpha )}\varepsilon \\ \rightarrow&0 \end{aligned}$$
as \(\varepsilon\rightarrow0^{+}\), where \(M^{**}:=\mathfrak{C}+M_{1} [ (1+M)L_{g}r+MbL_{h}r+\frac{Mb^{\alpha-1}}{\varGamma(\alpha)}\|\varphi _{r}\|_{L^{1}} ]\). Thus the set \(V(t):=\{(Q_{2}x)(t): x\in\varOmega_{r}\}\) is relatively compact in X. By the Ascoli–Arzelà theorem the set V is relatively compact. Hence \(\gamma_{C}(V)=\gamma_{C}(Q_{2}(\varOmega_{r}))=0\).
At last, by the properties of H-MNC and because of \(M(L_{g}+bL_{h})<1\), we obtain that
$$\begin{aligned} \gamma_{C}\bigl(Q(\varOmega_{r})\bigr) \leq& \gamma_{C}\bigl(Q_{1}(\varOmega_{r})\bigr)+ \gamma _{C}\bigl(Q_{2}(\varOmega_{r})\bigr) \\ \leq& M(L_{g}+bL_{h})\gamma_{C}( \varOmega_{r}) \\ < &\gamma_{C}(\varOmega_{r}), \end{aligned}$$
which implies that \(Q: \varOmega_{r}\rightarrow\varOmega\) is a condensing mapping. By Sadovskii’s fixed point theorem (see Lemma 6) Q has at least one fixed point x in \(\varOmega_{r}\), which is the mild solution of system (1.1) satisfying \(x(b)+g(x)=x_{1}\). Therefore system (1.1) is nonlocally controllable. □
H-MNC condition is another important tool guaranteeing the compactness of the solution operator. In what follows, we assume that f satisfies the following H-MNC condition:
- \((H_{f4})\):
-
There exists a constant \(L_{1}>0\) such that
$$\gamma\bigl(f(t, D_{0})\bigr)\leq L_{1} \gamma(D_{0}),\quad t\in J, $$
for every countable subset \(D_{0}\subset X\).
Lemma 10
LetXbe a separable Hilbert space. Assume that conditions\((H_{AE})\), \((H_{W})\), \((H_{f2})'\), \((H_{f4})\), \((H_{g})\), and\((H_{h})\)hold. Then
$$\gamma \bigl( \bigl\{ u_{x}(s): x\in D_{0}\bigr\} \bigr)\leq M_{1} \biggl[ (1+M)L_{g}+MbL_{h}+ \frac{2Mb^{\alpha}L_{1}}{\varGamma(\alpha)} \biggr]\gamma _{C}(D_{0}),\quad s\in J, $$
where\(D_{0}\subset\varOmega_{r}\)is a countable subset of\(\varOmega_{r}\).
Proof
By Lemma 5 we obtain that
$$\begin{aligned} \gamma \bigl( \bigl\{ u_{x}(s): x\in D_{0}\bigr\} \bigr) \leq& M_{1} \bigl( (1+M)L_{g}\gamma(D_{0})+MbL_{h} \gamma(D_{0}) \bigr) \\ &{}+\frac{MM_{1}b^{\alpha-1}}{\varGamma(\alpha)} \int_{0}^{b}L_{1}\gamma \bigl(D_{0}(s)\bigr)\,ds \\ \leq&M_{1} \biggl[ (1+M)L_{g}+MbL_{h}+ \frac{Mb^{\alpha}L_{1}}{\varGamma(\alpha )} \biggr]\gamma_{C}(D_{0}). \end{aligned}$$
The proof is completed. □
Theorem 3
LetXbe a separable Hilbert space. Assume that assumptions\((H_{AE})\), \((H_{W})\), \((H_{f1})\), \((H_{f2})'\), \((H_{f4})\), \((H_{g})\), \((H_{h})\), and\((H_{B})\)are satisfied. If the inequality conditions (3.4) and
$$\frac{2MM_{B}b^{\alpha}M_{1}}{\varGamma(\alpha)}L_{g}+\biggl(1+\frac {2MM_{B}b^{\alpha}M_{1}}{\varGamma(\alpha)}\biggr) (ML_{g}+MbL_{h})+\frac{2Mb^{\alpha }L_{1}}{\varGamma(\alpha)}\biggl(1+ \frac{MM_{B}b^{\alpha}M_{1}}{\varGamma(\alpha)}\biggr)< 1 $$
hold, then system (1.1) is nonlocally controllable onJ.
Proof
Define two operators \(Q_{1}\) and \(Q_{2}\) as in (3.5) and (3.6), respectively. By the properties of H-MNC and (3.7) we easily obtain that
$$ \gamma_{C}\bigl(Q_{1}(\varOmega_{r})\bigr)\leq M(L_{g}+bL_{h})\gamma_{C}( \varOmega_{r}). $$
(3.8)
On the other hand, since \(Q_{2}(\varOmega_{r})\subset\varOmega_{r}\) and the set \(Q_{2}(\varOmega_{r})\) is equicontinuous in \(C(J, X)\), by Lemmas 3 and 4 there is a countable set \(D_{0}\subset\varOmega_{r}\) such that
$$ \gamma_{C} \bigl( Q_{2}(\varOmega_{r}) \bigr) \leq2 \gamma_{C} \bigl( Q_{2}(D_{0}) \bigr)=2 \max_{t\in J} \gamma \bigl( Q_{2}(D_{0}) (t) \bigr). $$
(3.9)
Applying assumption \((H_{f4})\) and Lemma 10, we have
$$\begin{aligned} \gamma \bigl( Q_{2}(D_{0}) (t) \bigr) =&\gamma \biggl( \biggl\{ \int _{0}^{t}P_{\alpha, 1}^{E}(t-s) \bigl[f\bigl(s, x(s)\bigr)+Bu_{x}(s)\bigr]\,ds: x\in D_{0}\biggr\} \biggr) \\ \leq&\frac{Mb^{\alpha-1}L_{1}}{\varGamma(\alpha)} \int_{0}^{t}\gamma \bigl(D_{0}(s) \bigr)\,ds+\frac{MM_{B}b^{\alpha-1}}{\varGamma(\alpha)} \int_{0}^{t}\gamma \bigl(\bigl\{ u_{x}(s): x\in D_{0}\bigr\} \bigr)\,ds \\ \leq&\frac{Mb^{\alpha}L_{1}}{\varGamma(\alpha)}\gamma_{C}(D_{0})+ \frac {MM_{B}b^{\alpha}M_{1}}{\varGamma(\alpha)}\biggl[(1+M)L_{g}+MbL_{h}+ \frac{Mb^{\alpha }L_{1}}{\varGamma(\alpha)}\biggr]\gamma_{C}(D_{0}). \end{aligned}$$
This, together with (3.9), gives
$$\begin{aligned}[b] \gamma_{C} \bigl( Q_{2}(\varOmega_{r}) \bigr) &\leq\frac{2Mb^{\alpha }L_{1}}{\varGamma(\alpha)}\gamma_{C}(D_{0})\\&\quad+ \frac{2MM_{B}b^{\alpha }M_{1}}{\varGamma(\alpha)}\biggl[(1+M)L_{g}+MbL_{h}+ \frac{Mb^{\alpha}L_{1}}{\varGamma (\alpha)}\biggr]\gamma_{C}(D_{0}).\end{aligned} $$
(3.10)
Combining (3.8) and (3.10), because of \(\gamma _{C}(D_{0})\leq\gamma_{C}(\varOmega_{r})\), we obtain that
$$\begin{aligned} \gamma_{C} \bigl( Q(\varOmega_{r}) \bigr) =& \gamma_{C} \bigl( Q_{1}(\varOmega_{r}) \bigr)+ \gamma_{C} \bigl( Q_{2}(\varOmega_{r}) \bigr) \\ \leq& \biggl[ \frac{2MM_{B}b^{\alpha}M_{1}}{\varGamma(\alpha )}L_{g}+\biggl(1+ \frac{2MM_{B}b^{\alpha}M_{1}}{\varGamma(\alpha)}\biggr) (ML_{g}+MbL_{h}) \\ &{}+\frac{2Mb^{\alpha}L_{1}}{\varGamma(\alpha)}\biggl(1+\frac{MM_{B}b^{\alpha }M_{1}}{\varGamma(\alpha)}\biggr) \biggr]\gamma_{C}(\varOmega_{r}) \\ < &\gamma_{C}(\varOmega_{r}). \end{aligned}$$
Thus we conclude that \(Q: \varOmega_{r}\rightarrow\varOmega_{r}\) is a condensing mapping. By Sadovskii’s fixed point theorem, Q has at least one fixed point x in \(\varOmega_{r}\), which is the mild solution of system (1.1) satisfying \(x(b)+g(x)=x_{1}\). Therefore system (1.1) is nonlocally controllable. □