We present our main results of the paper in this section. We need the definition of complete controllability of the system (1).

### Definition 3.1

The system (1) is said to be completely controllable on the interval *I* if \(\mathcal{R}(T,\phi ) =X\), where \(\mathcal{R}(T,\phi )=\{x_{T}(\phi ,u)(0):u(\cdot )\in L^{2}(I,U)\}\).

To prove the main results, we impose the following hypotheses:

\((P_{1})\) The function \(h:[0,T]\times C \rightarrow X\) is continuous and there exists a constant \(\beta \in (0,1)\) and *L*, \(L_{1}\), for any \(x,y\in C\), \(A^{\beta }h(\cdot , x)\) is strongly measurable and \(A^{\beta }h(t,\cdot )\) satisfies the Lipchitz condition \(\|A^{\beta }h(t, x)-A^{\beta }h(t, y)\|\leq L\|x-y\|\) and the inequality \(\|A^{\beta }h(t, x)\|\leq L_{1}(\|x\|+1)\).

\((P_{2})\) The nonlinear function \(f:I\times C\times U \rightarrow X\) is continuous and there exists a constant \(L_{2}>0\) such that

$$\bigl\Vert f(t,\varphi ,u) \bigr\Vert \leq L_{2}\bigl(1+ \Vert \varphi \Vert _{C}+ \Vert u \Vert \bigr),\quad (t,\varphi ,u) \in I\times C\times U. $$

\((P_{3})\) The linear fractional control system (11) is completely controllable.

\((P_{4})\) The nonlinear function \(f(t,x_{t}, u(t))\) satisfies the Lipschitz condition, that is, there exists a constant \(L_{3}\) such that

$$\bigl\Vert f(t,\varphi _{1},u_{1})-f(t,\varphi _{2},u_{2}) \bigr\Vert \leq L_{3} \bigl( \Vert \varphi _{1}-\varphi _{2} \Vert _{C}+ \Vert u_{1}-u_{2} \Vert \bigr),\quad (\varphi _{1},u_{1}), (\varphi _{2},u_{2}) \in C \times U. $$

Define an operator Φ on \(C(I,C)\times C(I,U)\) as

$$ \Phi (x,u)=(z,v) $$

(13)

with the norm \(\|(x,u)\|=\|x_{t} \|_{C}+\|u \|\), \((x,u)\in C(I,C)\times C(I,U)\), \(t\in I\), where

$$\begin{aligned}& v(t)=B^{*}T_{q}^{*}(T-t) \bigl( \Gamma _{0}^{T} \bigr)^{-1}p(x,u), \end{aligned}$$

(14)

$$\begin{aligned}& \begin{aligned} &z(t)= S_{q}(t) \bigl[\phi (0)-h(0,x_{0}) \bigr] + h(t,x_{t})+ \int _{0}^{t} (t-s)^{q-1} AT_{q}(t-s)h(s,x_{s})\,ds \\ &\hphantom{z(t)=}{}+ \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) \bigl(Bv(s)+f \bigl(s,x_{s},u(s) \bigr) \bigr)\,ds, \quad t\in [0,T], \\ & z_{0}(\theta )= \phi (\theta ), \quad -r\leq \theta \leq 0, \end{aligned} \end{aligned}$$

(15)

\(p(x,u)= x_{T}-S_{q}(T)[ \phi (0) -h(0,x_{0})]-h(T,x_{T})-\int _{0}^{T} (T-s)^{q-1} AT_{q}(T-s)h(s,x_{s})\,ds - \int _{0}^{T} (T-s)^{q-1} T_{q}(T-s) f(s,x_{s},u(s))\,ds\).

It will be shown that the system (1) is completely controllable on *I* if the operator Φ has a fixed point in \(C(I,C)\times C(I,U)\).

### Theorem 3.1

*Assume that the hypotheses* \((P_{1})\)*–*\((P_{4})\) *are satisfied*. *Then*, *the problem* (1) *has a unique mild solution in* \(C([-r,T];X)\) *provided that*

$$\begin{aligned} \bigl\vert A^{-\beta } \bigr\vert L+ \biggl(1+ \frac{M M_{B} T^{q}}{\Gamma (1+q)} \biggr) \biggl( \frac{ d\Gamma (1+\beta )c_{1-\beta }L T^{q\beta }}{\beta \Gamma (1+q\beta )} + \frac{dM T^{q} L_{3} }{\Gamma (1+q)} \biggr) + \frac{M T^{q} L_{3} }{\Gamma (1+q)} < 1, \end{aligned}$$

(16)

*where* \(\beta \in (0,1)\), \(M_{B}=|B|\), \(d=\frac{M_{B} Mq}{\gamma \Gamma (1+q)}\).

### Proof

Obviously, \(x \in C([-r,T];X)\) is a mild solution of the system (1) if and only if the operator Φ has a fixed point in \(C(I,C)\times C(I,U)\). Therefore, it is sufficient to prove that Φ has a fixed point in \(C(I,C)\times C(I,U)\). We first show that Φ maps \(C(I,C)\times C(I,U)\) into itself. Based on Lemma 2.3 and the condition \((P_{1})\), we have

$$\begin{aligned} & \biggl\Vert \int _{0}^{t} (t-s)^{q-1} AT_{q}(t-s)h(s,x_{s})\,ds \biggr\Vert \\ &\quad \leq \int _{0}^{t} (t-s)^{q-1} A^{1-\beta }T_{q}(t-s) A^{\beta }h(s,x_{s})\,ds \\ &\quad \leq \int _{0}^{t} (t-s)^{q-1} \frac{q\Gamma (1+\beta )c_{1-\beta }}{(t-s)^{q(1-\beta )} \Gamma (1+q\beta )} L_{1} \bigl( \Vert x_{s} \Vert _{C} +1 \bigr)\,ds \\ &\quad \leq \frac{\Gamma (1+\beta )c_{1-\beta }}{\beta \Gamma (1+q\beta )} L_{1} \bigl( \Vert x_{t} \Vert _{C} +1 \bigr) T^{q\beta },\quad \beta \in (0,1). \end{aligned}$$

(17)

According to Lemma 2.2(i) and the hypothesis \((P_{2})\), we have

$$ \biggl\Vert \int _{0}^{T} (T-s)^{q-1} T_{q}(T-s) f \bigl(s,x_{s},u(s) \bigr)\,ds \biggr\Vert \leq \frac{MT^{q}}{\Gamma (1+q)}L_{2} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr). $$

(18)

By using (17) and (18), Lemmas 2.1 and 2.4, and hypothesis \((P_{3})\), it can be shown that there exist two constants \(C_{1},C_{2}>0\) such that

$$\begin{aligned} \bigl\Vert v(t) \bigr\Vert ={}& \biggl\Vert B^{*}T_{q}^{*}(T-t) \bigl(\Gamma _{0}^{T} \bigr)^{-1} \biggl( x_{T}-S_{q}(T) \bigl[ \phi (0) -h(0,x_{0}) \bigr]-h(T,x_{T}) \\ & {}- \int _{0}^{T} (T-s)^{q-1} AT_{q}(T-s)h(s,x_{s})\,ds \\ &{} - \int _{0}^{T} (T-s)^{q-1} T_{q}(T-s) f \bigl(s,x_{s},u(s) \bigr)\,ds \biggr) \biggr\Vert \\ \leq{} & \frac{M_{B} Mq}{\gamma \Gamma (1+q)} \biggl[ \vert x_{T} \vert +M \Vert \phi \Vert +M \bigl\vert A^{- \beta } \bigr\vert L_{1} \bigl( \Vert \phi \Vert +1 \bigr) + \bigl\vert A^{-\beta } \bigr\vert L_{1} \bigl( \Vert x_{T} \Vert +1 \bigr) \\ &{}+ \frac{\Gamma (1+\beta )c_{1-\beta }}{\beta \Gamma (1+q\beta )} L_{1} \bigl( \Vert x_{t} \Vert _{C} +1 \bigr) T^{q\beta }+\frac{MT^{q}}{\Gamma (1+q)}L_{2} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr) \biggr] \\ \leq {}& C_{1} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr) \end{aligned}$$

(19)

and

$$\begin{aligned} \bigl\Vert z(t) \bigr\Vert ={}& \biggl\Vert S_{q}(t) \bigl[\phi (0)-h(0,x_{0}) \bigr] + h(t,x_{t})+ \int _{0}^{t} (t-s)^{q-1} AT_{q}(t-s)h(s,x_{s})\,ds \\ &{}+ \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) \bigl(Bv(s)+f \bigl(s,x_{s},u(s) \bigr) \bigr)\,ds \biggr\Vert \\ \leq{} & M \Vert \phi \Vert +M \bigl\vert A^{-\beta } \bigr\vert L_{1} \bigl( \Vert \phi \Vert +1 \bigr) + \bigl\vert A^{-\beta } \bigr\vert L_{1} \bigl( \Vert x_{t} \Vert _{C}+1 \bigr) \\ &{}+ \frac{\Gamma (1+\beta )c_{1-\beta }}{\beta \Gamma (1+q\beta )} L_{1} \bigl( \Vert x_{t} \Vert _{C} +1 \bigr) T^{q\beta } \\ &{}+ \frac{M M_{B} T^{q}}{\Gamma (1+q)} C_{1} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr)+ \frac{MT^{q}}{\Gamma (1+q)}L_{2} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr) \\ \leq {}& C_{2} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr), \end{aligned}$$

(20)

where \(\beta \in (0,1)\). It follows from (15), (19) and (20) that there exists a constant \(C_{3}\) such that

$$ \bigl\Vert \Phi (x,u) \bigr\Vert = \Vert z \Vert _{C([-r,T];X)} + \Vert v \Vert \leq C_{3} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr), $$

(21)

which means that Φ maps \(C(I,C)\times C(I,U)\) into itself.

We next prove that the operator Φ is a contraction mapping on \(C(I,C)\times C(I,U)\). For any \((x,u),(y,w)\in C(I,C)\times C(I,U)\), it holds that

$$\begin{aligned} & \bigl\Vert \Phi (x,u)-\Phi (y,w) \bigr\Vert \\ &\quad = \Vert v_{1}-v_{2} \Vert + \Vert z_{1}-z_{2} \Vert _{C([-r,T];X)} \\ &\quad \leq \Vert v_{1}-v_{2} \Vert + \bigl\Vert h(t,x_{t})-h(t,y_{t}) \bigr\Vert + \biggl\Vert \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) B \bigl(v_{1}(s)-v_{2}(s) \bigr)\,ds \biggr\Vert \\ &\qquad {}+ \bigg\| \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) ( f \bigl(s,x_{s},u(s)- f \bigl(s,y_{s},w(s) \bigr) \bigr) \,ds\bigg\| \\ &\quad = I_{1}+I_{2}+I_{3}+I_{4}. \end{aligned}$$

(22)

By hypotheses \((P_{1})\)–\((P_{4})\), Lemma 2.2(i), and (17) and (18), we have

$$\begin{aligned} I_{1}=& \Vert v_{1}-v_{2} \Vert \\ =& \biggl\Vert B^{*}T_{q}^{*}(T-t) \bigl( \Gamma _{0}^{T} \bigr)^{-1} \bigg( \int _{0}^{T} (T-s)^{q-1} AT_{q}(T-s) \bigl(h(s,x_{s})-h(s,y_{s}) \bigr) \,ds \\ &+ \int _{0}^{T} (T-s)^{q-1}T_{q}(T-s) \bigl( f \bigl(s,x_{s},u(s)- f \bigl(s,y_{s},w(s) \bigr) \bigr)\,ds \bigr) \biggr\Vert \\ \leq & \frac{M_{B} Mq}{\gamma \Gamma (1+q)} \biggl( \frac{\Gamma (1+\beta )c_{1-\beta }L T^{q\beta }}{\beta \Gamma (1+q\beta )} \Vert x_{t}-y_{t} \Vert _{C} + \frac{M T^{q} L_{3} }{\Gamma (1+q)} \bigl( \Vert x_{t}-y_{t} \Vert _{C}+ \Vert u-w \Vert \bigr) \biggr) \\ \leq & \biggl( \frac{ d\Gamma (1+\beta )c_{1-\beta }L T^{q\beta }}{\beta \Gamma (1+q\beta )} + \frac{dM T^{q} L_{3} }{\Gamma (1+q)} \biggr) \bigl( \Vert x-y \Vert _{C([-r,T];X)}+ \Vert u-w \Vert \bigr), \end{aligned}$$

(23)

where \(d=\frac{M_{B} Mq}{\gamma \Gamma (1+q)}\). The condition \((P_{1})\) implies

$$\begin{aligned} I_{2}=& \bigl\Vert h(t,x_{t})-h(t,y_{t}) \bigr\Vert \leq \bigl\vert A^{-\beta } \bigr\vert L \Vert x-y \Vert _{C([-r,T];X)}. \end{aligned}$$

(24)

Based on Lemma 2.2(i) and (23), one can obtain

$$\begin{aligned} I_{3}={}& \biggl\Vert \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) B \bigl(v_{1}(s)-v_{2}(s) \bigr)\,ds \biggr\Vert \\ \leq{} & \frac{M M_{B} T^{q}}{\Gamma (1+q)} \Vert v_{1}-v_{2} \Vert \\ \leq {}& \frac{M M_{B} T^{q}}{\Gamma (1+q)} \biggl( \frac{ d\Gamma (1+\beta )c_{1-\beta }L T^{q\beta }}{\beta \Gamma (1+q\beta )} + \frac{dM T^{q} L_{3} }{\Gamma (1+q)} \biggr) \bigl( \Vert x-y \Vert _{C([-r,T];X)}+ \Vert u-w \Vert \bigr). \end{aligned}$$

(25)

Similar to the discussion of \(I_{1}\), we obtain

$$\begin{aligned} I_{4}={}& \biggl\Vert \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) \bigl( f \bigl(s,x_{s},u(s)- f \bigl(s,y_{s},w(s) \bigr) \bigr)\bigr)\,ds \biggr\Vert \\ \leq{} & \frac{M T^{q} L_{3} }{\Gamma (1+q)} \bigl( \Vert x-y \Vert _{C([-r,T];X)}+ \Vert u-w \Vert \bigr). \end{aligned}$$

(26)

Then, (22)–(26) imply

$$\begin{aligned} & \bigl\Vert \Phi (x,u)-\Phi (y,w) \bigr\Vert _{C([-r,T];X)} \\ &\quad \leq \biggl[ \bigl\vert A^{-\beta } \bigr\vert L+ \biggl(1+ \frac{M M_{B} T^{q}}{\Gamma (1+q)} \biggr) \biggl( \frac{ d\Gamma (1+\beta )c_{1-\beta }L T^{q\beta }}{\beta \Gamma (1+q\beta )} \biggr) \\ &\qquad {} +\frac{M T^{q} L_{3} }{\Gamma (1+q)} \biggr] \bigl( \Vert x-y \Vert _{C([-r,T];X)}+ \Vert u-w \Vert \bigr). \end{aligned}$$

(27)

In view of (16), we obtain that Φ is a contraction. Consequently, Φ has a fixed point in \(C(I,C)\times C(I,U)\) by the Banach fixed-point theorem, which is a mild solution of the system (1). This completes the proof. □

### Theorem 3.2

*If all the assumptions of Theorem *3.1*hold*, *then the system* (1) *is completely controllable on* *I*.

### Proof

Let \((\bar{x}(\cdot ),\bar{u})\) be a fixed point of the operator Φ in (13), that is

$$ \Phi \bigl(\bar{x} (\cdot ),\bar{u} \bigr)= \bigl(\bar{x} (\cdot ),\bar{u} \bigr), $$

(28)

where

$$ \begin{aligned} &\bar{x}(t)= S_{q}(t) \bigl[ \phi (0)-h(0,x_{0}) \bigr] + h(t,\bar{x}_{t})+ \int _{0}^{t} (t-s)^{q-1} AT_{q}(t-s)h(s,\bar{x}_{s})\,ds \\ &\hphantom{\bar{x}(t)=}{}+ \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) \bigl(B\bar{u}(s)+f \bigl(s, \bar{x}_{s},\bar{u}(s) \bigr) \bigr)\,ds, \quad t \in [0,T], \\ & \bar{x}_{0}(\theta )= \phi (\theta ), \quad -r\leq \theta \leq 0 \end{aligned} $$

(29)

and the control function

$$ \bar{u}(t)=B^{*}T_{q}^{*}(T-t) \bigl(\Gamma _{0}^{T} \bigr)^{-1} p(\bar{x}, \bar{u} ), $$

(30)

here \(p(\bar{x},\bar{u})=x_{T}-S_{q}(T)[\phi (0)-h(0,x_{0})] + h(T, x_{T})+ \int _{0}^{T} (T-s)^{q-1} AT_{q}(T-s)h(s,\bar{x}_{s})\,ds - \int _{0}^{T} (T-s)^{q-1}T_{q}(T-s) f(s,\bar{x}_{s},\bar{u}(s))\,ds\).

According to Theorem 3.1, any fixed point of Φ is a mild solution of the system (1). Then, by (12), (29) and (30), we have

$$\begin{aligned} \bar{x}(T) ={}& x_{T}- p(\bar{x},\bar{u})+ \int _{0}^{T} (T-s)^{q-1} T_{q}(T-s) B\bar{u}(s)\,ds \\ ={}& x_{T}- p(\bar{x},\bar{u})+ \int _{0}^{T} (T-s)^{q-1} T_{q}(T-s)B B^{*}T_{q}^{*}(T-s) \bigl( \Gamma _{0}^{T} \bigr)^{-1} p(\bar{x},\bar{u})\,ds \\ ={}& x_{T}- p(\bar{x},\bar{u} ) + p(\bar{x},\bar{u}) \\ ={}& x_{T}. \end{aligned}$$

(31)

Thus, the system (1) is approximately controllable on *I* by Definition 3.1. The proof is completed. □