In order to demonstrate the main results, we outline the following assumptions:
- (\(\mathrm{A}_{1}\)):
-
The operator \(\mathbb{T}(t)\) is a compact for all \(t>0\),
- (\(\mathrm{A}_{2}\)):
-
The function \(f: [0, T] \times E \times \mathcal{V}\rightarrow E\) is a Carathéodory function, that is,
- (\(\mathrm{F}_{1}\)):
-
For each \(t \in [0,T]\), the function \(f(t, \cdot , \cdot ):E \rightarrow E\) is continuous,
- (\(\mathrm{F}_{2}\)):
-
For each \(u\in E\), the function \(f(\cdot , u, v):[0,T] \rightarrow E\) is measurable.
- (\(\mathrm{A}_{3}\)):
-
For any \(k>0\), there exist \(\alpha p> 1\) and function \(m_{k} \in L^{p}([0,T],E)\) such that for any \(u\in E\) and \(v \in \mathcal{V}\) satisfying \(\|u\| \leq k\),
$$ \bigl\Vert f(t,u, v) \bigr\Vert \leq m_{k}(t), \quad \text{ a.e. } t\in [0,T], $$
and there exists \(L>0\) such that
$$ \liminf_{k \rightarrow \infty } \frac{ \Vert m_{k} \Vert _{L^{p}}}{k}=L. $$
- (\(\mathrm{A}_{4}\)):
-
The function \(g: [0,T] \times E \rightarrow E\) is continuous and there exist a positive constant \(\beta \in (0,1)\) and \(M_{1}, M_{2} >0\) such that
$$ \bigl\Vert \mathcalligra{A}^{\beta } g(t,u) - \mathcalligra{A}^{\beta } g(t,w) \bigr\Vert \leq M_{1} \Vert u-w \Vert $$
and
$$ \bigl\Vert \mathcalligra{A}^{\beta } g(t,u) \bigr\Vert \leq M_{2} \bigl( \Vert u \Vert +1 \bigr). $$
- (\(\mathrm{A}_{5}\)):
-
The function f is a locally Lipschitz continuous with respect to \(\mathcal{V}\), i.e., for all \(t \in [0, T]\) and \(u_{1}, u_{2} \in E\), there exists constant \(L_{f}>0\) such that
$$ \bigl\Vert f\bigl(t,u_{1}(t), v\bigr)-f\bigl(t,u_{2}(t),v \bigr) \bigr\Vert \leq L_{f} \Vert u_{1}-u_{2} \Vert . $$
The following existence of mild solutions for the problem (1) will be proved by using Krasnoselskii’s fixed point theorem.
Theorem 4.1
Assume (\(\mathrm{A}_{1}\))–(\(\mathrm{A}_{4}\)) are true. Then, the problem (1) has at least one mild solution provided that
$$ \begin{aligned}[b] &(M+1)M_{2} \bigl\Vert \mathcalligra{A}^{-\beta } \bigr\Vert + \frac{C_{1-\beta } \Gamma (1+\beta )}{\beta \Gamma (1+ \alpha \beta )}M_{2} \bigl( \omega (T)- \omega (0) \bigr) ^{\alpha \beta } \\ &\quad {} + \frac{M L}{\Gamma (\alpha )} \sup_{0 \leq t \leq T} \bigl\lvert \omega ^{\prime }(t)\bigr\rvert ^{\frac{1}{p}} \biggl\lbrace \frac{ ( \omega (T)- \omega (0) ) ^{\frac{\alpha p-1}{p-1}}}{\frac{\alpha p-1}{p-1}} \biggr\rbrace ^{\frac{p-1}{p}} < 1 \end{aligned} $$
(10)
and
$$ \biggl( ( M +1 ) \bigl\Vert \mathcalligra{A}^{-\beta } \bigr\Vert + \frac{C_{1-\beta } \Gamma (1+\beta )}{ \beta \Gamma (1+ \alpha \beta )} \bigl( \omega (T)- \omega (0) \bigr) ^{\alpha \beta } \biggr) M_{1} < 1. $$
(11)
Proof
For any \(k>0\), we let \(B_{k}= \lbrace {u \in C([0, T], E): \|u (t)\| \leq k \text{ for all } t \in [0,T]} \rbrace \). Then, \(B_{k}\) is a bounded closed convex subset of \(C([0, T], E)\). For each positive k, we define two operators \(\mathcal{F}_{1}\) and \(\mathcal{F}_{2}\) on \(B_{k}\) as follows:
$$\begin{aligned} ( \mathcal{F}_{1}u ) (t) &= {\mathcal{Q}^{\alpha ;\omega }(t,0)} \bigl( u_{0} -g(0,u_{0}) \bigr) +g\bigl(t,u(t)\bigr) \\ &\quad {} + \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}}\omega ^{\prime }(\tau ) {\mathcalligra{A} \mathcal{R}^{\alpha ; \omega }(t,\tau )}g\bigl(\tau ,u(\tau )\bigr) \,\mathrm{d} { \tau } \end{aligned}$$
and
$$ ( \mathcal{F}_{2}u ) (t) = \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) { \mathcal{R}^{\alpha ;\omega }(t,\tau )}f\bigl(\tau ,u(\tau ),v(\tau )\bigr) \, \mathrm{d} {\tau }, $$
for \(t \in [0, T]\).
From Lemma 3.3(v), for \(t \in [0, T]\), we obtain
$$\begin{aligned}& \biggl\Vert \int _{0}^{t} { \bigl( \omega (t)- \omega ( \tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) { \mathcalligra{A} \mathcal{R}^{\alpha ;\omega }(t,\tau )}g\bigl(\tau ,u(\tau )\bigr) \,\mathrm{d} { \tau } \biggr\Vert \\& \quad \leq \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \omega ^{\prime }(\tau ) \bigl\Vert { \mathcalligra{A}\mathcal{R}^{ \alpha ;\omega }(t,\tau )}g\bigl(\tau ,u(\tau )\bigr) \bigr\Vert \,\mathrm{d} {\tau } \\& \quad = \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \omega ^{\prime }(\tau ) \bigl\Vert { \mathcalligra{A}^{1- \beta } \mathcal{R}^{\alpha ;\omega }(t,\tau )} \mathcalligra{A}^{\beta }g\bigl(\tau ,u( \tau )\bigr) \bigr\Vert \,\mathrm{d} {\tau } \\& \quad \leq \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \omega ^{\prime }(\tau ) \frac{\alpha C_{1-\beta } \Gamma (1+\beta )}{\Gamma (1+ \alpha \beta ) ( \omega (t) - \omega (\tau ) )^{\alpha (1-\beta )} } \bigl\Vert \mathcalligra{A}^{\beta }g\bigl(\tau ,u(\tau )\bigr) \bigr\Vert \,\mathrm{d} {\tau } \\& \quad \leq \frac{\alpha C_{1-\beta } \Gamma (1+\beta )}{\Gamma (1+ \alpha \beta )}M_{2} \bigl( 1+ \Vert u \Vert \bigr) \int _{0}^{t}{ \bigl( \omega (t)- \omega ( \tau ) \bigr) ^{\alpha \beta -1}} \omega ^{\prime }(\tau ) \,\mathrm{d} { \tau } \\& \quad = \frac{\alpha C_{1-\beta } \Gamma (1+\beta )}{\Gamma (1+ \alpha \beta )}M_{2} \bigl( 1+ \Vert u \Vert \bigr) \frac{ ( \omega (t)- \omega (0) ) ^{\alpha \beta }}{\alpha \beta } \\& \quad \leq \frac{\alpha C_{1-\beta } \Gamma (1+\beta )}{\Gamma (1+ \alpha \beta )}M_{2} ( 1+k ) \frac{ ( \omega (T)- \omega (0) ) ^{\alpha \beta }}{\alpha \beta }. \end{aligned}$$
Thus, \(\|{ ( \omega (t)- \omega (\tau ) ) ^{\alpha -1}} \omega ^{ \prime }(\tau ) {\mathcalligra{A}\mathcal{R}^{\alpha ;\omega }(t,\tau )}g( \tau ,u(\tau )) \|\) is Lebesgue integrable with respect to \(\tau \in [0, t]\) for all \(t\in [0,T]\). From Theorem 2.16 (Bochner’s theorem), we obtain that \(\|{ ( \omega (t)- \omega (\tau ) ) ^{\alpha -1}} \omega ^{ \prime }(\tau ) {\mathcalligra{A}\mathcal{R}^{\alpha ;\omega }(t,\tau )}g( \tau ,u(\tau )) \,\mathrm{d}{\tau }\|\) is Bochner integrable with respect to \(\tau \in [0, t]\) for all \(t\in [0,T]\).
Similarly, by Lemma 3.3(i), we also obtain
$$\begin{aligned}& \biggl\Vert \int _{0}^{t} { \bigl( \omega (t)- \omega ( \tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) { \mathcal{R}^{\alpha ; \omega }(t,\tau )}f\bigl(\tau ,u(\tau ),v(\tau )\bigr) \, \mathrm{d} {\tau } \biggr\Vert \\& \quad \leq \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \omega ^{\prime }(\tau ) \bigl\Vert { \mathcal{R}^{\alpha ;\omega }(t, \tau )}f\bigl(\tau ,u(\tau ),v(\tau )\bigr) \bigr\Vert \,\mathrm{d} {\tau } \\& \quad \leq \frac{M}{\Gamma (\alpha )} \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) \bigl\Vert f\bigl( \tau ,u(\tau ),v(\tau )\bigr) \bigr\Vert \,\mathrm{d} {\tau } \\& \quad \leq \frac{M}{\Gamma (\alpha )} \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) \bigl\Vert f\bigl( \tau ,u(\tau ),v(\tau )\bigr) \bigr\Vert \,\mathrm{d} {\tau } \\& \quad \leq \frac{M}{\Gamma (\alpha )} \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) m_{k}( \tau ) \,\mathrm{d} {\tau } \\& \quad \leq \frac{M \Vert m_{k} \Vert _{L^{p}}}{\Gamma (\alpha )} \biggl\lbrace \int _{0}^{t} \bigl[ { \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \omega ^{\prime }(\tau ) \bigr]^{\frac{p}{p-1}} \,\mathrm{d} {\tau } \biggr\rbrace ^{\frac{p-1}{p}} \\& \quad \leq \frac{M \Vert m_{k} \Vert _{L^{p}}}{\Gamma (\alpha )} \biggl\lbrace \sup_{0 \leq t \leq T} \bigl\lvert \omega ^{\prime }(t)\bigr\rvert ^{ \frac{1}{p-1}} \int _{0}^{t} { \bigl( \omega (t)- \omega ( \tau ) \bigr) ^{\frac{ ( \alpha -1 ) p}{p-1}}} \omega ^{\prime }( \tau ) \,\mathrm{d} { \tau } \biggr\rbrace ^{\frac{p-1}{p}} \\& \quad = \frac{M \Vert m_{k} \Vert _{L^{p}}}{\Gamma (\alpha )} \sup_{0 \leq t \leq T} \bigl\lvert \omega ^{\prime }(t)\bigr\rvert ^{\frac{1}{p}} \biggl\lbrace \frac{ ( \omega (t)- \omega (0) ) {\frac{\alpha p-1}{p-1}}}{1+\frac{ ( \alpha -1 ) p}{p-1}} \biggr\rbrace ^{\frac{p-1}{p}} \\& \quad \leq \frac{M \Vert m_{k} \Vert _{L^{p}}}{\Gamma (\alpha )} \sup_{0 \leq t \leq T} \bigl\lvert \omega ^{\prime }(t)\bigr\rvert ^{\frac{1}{p}} \biggl\lbrace \frac{ ( \omega (T)- \omega (0) ) ^{\frac{\alpha p-1}{p-1}} }{1+\frac{ ( \alpha -1 ) p}{p-1}} \biggr\rbrace ^{ \frac{p-1}{p}} . \end{aligned}$$
Thus, \(\| { ( \omega (t)- \omega (\tau ) ) ^{\alpha -1}} \omega ^{ \prime }(\tau ) {\mathcal{R}^{\alpha ;\omega }(t,\tau )}f(\tau ,u( \tau ),v(\tau )) \|\) is Lebesgue integrable with respect to \(\tau \in [0, t]\) for all \(t\in [0,T]\). From Theorem 2.16 (Bochner’s theorem), it follows that \(\| { ( \omega (t)- \omega (\tau ) ) ^{\alpha -1}} \omega ^{ \prime }(\tau ) {\mathcal{R}^{\alpha ;\omega }(t,\tau )}f(\tau ,u( \tau ),v(\tau )) \|\) is Bochner integrable with respect to \(\tau \in [0, t]\) for all \(t\in [0,T]\).
The proof will be separated into three parts.
Step 1: \(\mathcal{F}_{1}u+\mathcal{F}_{2}w \in B_{k}\) whenever \(u, w \in B_{k}\).
We assume that for each \(k>0\), there exist \(u_{k}, w_{k} \in B_{k} \) such that
$$ \bigl\Vert ( \mathcal{F}_{1}u_{k} ) (t) + ( \mathcal{F}_{2}w_{k} ) (t) \bigr\Vert >k,\quad \text{for $t \in [0,T]$}.$$
According to (\(\mathrm{A}_{3}\)) and Lemma 3.3(i), it follows that
$$\begin{aligned} k &< \bigl\Vert ( \mathcal{F}_{1}u_{k} ) (t) + ( \mathcal{F}_{2}w_{k} ) (t) \bigr\Vert \\ &\leq \bigl\Vert {\mathcal{Q}^{\alpha ;\omega }(t,0)} \bigl( u_{0} -g(0,u_{0}) \bigr) \bigr\Vert + \bigl\Vert g \bigl(t,u_{k}(t)\bigr) \bigr\Vert \\ &\quad {} + \biggl\Vert { \int _{0}^{t}{ \bigl( \omega (t)- \omega ( \tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) {\mathcalligra{A} \mathcal{R}^{\alpha ;\omega }(t,\tau )}g\bigl(\tau ,u_{k}(\tau )\bigr) \,\mathrm{d} {\tau }} \biggr\Vert \\ &\quad {} + \biggl\Vert \int _{0}^{t}{ \bigl( \omega (t)- \omega ( \tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) { \mathcal{R}^{ \alpha ;\omega }(t,\tau )}f\bigl(\tau ,w_{k}(\tau ),v( \tau )\bigr) \,\mathrm{d} { \tau } \biggr\Vert \\ &\leq M \bigl( \Vert u_{0} \Vert + \bigl\Vert \mathcalligra{A}^{-\beta } \mathcalligra{A}^{ \beta } g(0,u_{0}) \bigr\Vert \bigr) + \bigl\Vert \mathcalligra{A}^{-\beta } \mathcalligra{A}^{ \beta } g(t,u_{k}) \bigr\Vert \\ &\quad {} + \biggl\Vert { \int _{0}^{t}{ \bigl( \omega (t)- \omega ( \tau ) \bigr) ^{\alpha -1}}\omega ^{\prime }(\tau ) {\mathcalligra{A} \mathcal{R}^{\alpha ;\omega }(t,\tau )}g\bigl(\tau ,u_{k}(\tau )\bigr) \,\mathrm{d} {\tau }} \biggr\Vert \\ &\quad {} + \biggl\Vert \int _{0}^{t}{ \bigl( \omega (t)- \omega ( \tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) { \mathcal{R}^{ \alpha ;\omega }(t,\tau )}f\bigl(\tau ,w_{k}(\tau ),v( \tau )\bigr) \,\mathrm{d} { \tau } \biggr\Vert \\ &\leq M \bigl( \Vert u_{0} \Vert + \bigl\Vert \mathcalligra{A}^{-\beta } \bigr\Vert M_{2}(k+1) \bigr) + \bigl\Vert \mathcalligra{A}^{-\beta } \bigr\Vert M_{2}(k+1) \\ &\quad {} + \frac{\alpha C_{1-\beta } \Gamma (1+\beta )}{\Gamma (1+ \alpha \beta )}M_{2} ( 1+k ) \frac{ ( \omega (T)- \omega (0) ) ^{\alpha \beta }}{\alpha \beta } \\ &\quad {} + \frac{M \Vert m_{k} \Vert _{L^{p}}}{\Gamma (\alpha )} \sup_{0 \leq t \leq T} \bigl\lvert \omega ^{\prime }(t)\bigr\rvert ^{\frac{1}{p}} \biggl\lbrace \frac{ ( \omega (T)- \omega (0) ) ^{\frac{\alpha p-1}{p-1}}}{\frac{\alpha p-1}{p-1}} \biggr\rbrace ^{\frac{p-1}{p}} . \end{aligned}$$
Multiplying to both sides by \(\frac{1}{k}\) and taking the limit inferior as \(k \rightarrow \infty \), we get
$$\begin{aligned} 1 & \leq M \biggl( \liminf_{k \rightarrow \infty } \frac{ \Vert u_{0} \Vert }{k} + \bigl\Vert \mathcalligra{A}^{-\beta } \bigr\Vert M_{2} \biggr) + \bigl\Vert \mathcalligra{A}^{-\beta } \bigr\Vert M_{2} \\ &\quad {} + \frac{\alpha C_{1-\beta } \Gamma (1+\beta )}{\Gamma (1+ \alpha \beta )}M_{2} \frac{ ( \omega (T)- \omega (0) ) ^{\alpha \beta }}{\alpha \beta } \\ &\quad {} + \liminf_{k \rightarrow \infty } \frac{M \Vert m_{k} \Vert _{L^{p}}}{k\Gamma (\alpha )} \sup _{0 \leq t \leq T} \bigl\lvert \omega ^{\prime }(t)\bigr\rvert ^{\frac{1}{p}} \biggl\lbrace \frac{ ( \omega (T)- \omega (0) ) ^{\frac{\alpha p-1}{p-1}}}{\frac{\alpha p-1}{p-1}} \biggr\rbrace ^{\frac{p-1}{p}} \\ & = (M+1)M_{2} \bigl\Vert \mathcalligra{A}^{-\beta } \bigr\Vert \\ &\quad {} + \frac{\alpha C_{1-\beta } \Gamma (1+\beta )}{\Gamma (1+ \alpha \beta )}M_{2} \frac{ ( \omega (T)- \omega (0) ) ^{\alpha \beta }}{\alpha \beta } \\ &\quad {} + \frac{M L}{\Gamma (\alpha )} \sup_{0 \leq t \leq T} \bigl\lvert \omega ^{\prime }(t)\bigr\rvert ^{\frac{1}{p}} \biggl\lbrace \frac{ ( \omega (T)- \omega (0) ) ^{\frac{\alpha p-1}{p-1}}}{1+\frac{ ( \alpha -1 ) p}{p-1}} \biggr\rbrace ^{\frac{p-1}{p}} < 1, \end{aligned}$$
which is contradiction.
Step 2: \(\mathcal{F}_{1}\) is a contraction on \(B_{k}\).
For arbitrary \(u, w \in B_{k}\), we have
$$\begin{aligned}& \bigl\Vert (\mathcal{F}_{1} u ) (t)- ( \mathcal{F}_{1} w ) (t) \bigr\Vert \\& \quad \leq \bigl\Vert {\mathcal{Q}^{\alpha ;\omega }(t,0)} \bigl( g\bigl(0,u(0) \bigr) - g\bigl(0,w(0)\bigr) \bigr) \bigr\Vert + \bigl\Vert g\bigl(t,u(t) \bigr)-g\bigl(t,w(t)\bigr) \bigr\Vert \\& \qquad {} + \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \omega ^{\prime }(\tau ) \bigl\Vert {\mathcalligra{A}\mathcal{R}^{\alpha ;\omega }(t, \tau )}g\bigl( \tau ,u(\tau )\bigr) - {\mathcalligra{A}\mathcal{R}^{\alpha ;\omega }(t, \tau )}g\bigl( \tau ,w(\tau )\bigr) \bigr\Vert \,\mathrm{d} {\tau } \\& \quad \leq M \bigl\Vert \mathcalligra{A}^{-\beta } \bigl( \mathcalligra{A}^{\beta } g\bigl(0,u(0)\bigr) - \mathcalligra{A}^{\beta } g\bigl(0,w(0)\bigr) \bigr) \bigr\Vert \\& \qquad {} + \bigl\Vert \mathcalligra{A}^{-\beta } \bigl( \mathcalligra{A}^{\beta } g\bigl(t,u(t)\bigr) - \mathcalligra{A}^{\beta } g\bigl(t,w(t)\bigr) \bigr) \bigr\Vert \\& \qquad {} + \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \omega ^{\prime }(\tau ) \bigl\Vert {\mathcalligra{A}^{1-\beta }\mathcal{R}^{\alpha ; \omega }(t, \tau )\mathcalligra{A}^{\beta }} \bigl( g\bigl(\tau ,u(\tau )\bigr) - g\bigl( \tau ,w(\tau )\bigr) \bigr) \bigr\Vert \,\mathrm{d} {\tau } \\& \quad \leq M \bigl\Vert \mathcalligra{A}^{-\beta } \bigr\Vert M_{1} \Vert u-w \Vert + \bigl\Vert \mathcalligra{A}^{- \beta } \bigr\Vert M_{1} \Vert u-w \Vert \\& \qquad {} + \frac{\alpha C_{1-\beta } \Gamma (1+\beta )}{\Gamma (1+ \alpha \beta )}M_{1} \frac{ ( \omega (T)- \omega (0) ) ^{\alpha \beta }}{\alpha \beta } \Vert u-w \Vert \\& \quad = \biggl( ( M +1 ) \bigl\Vert \mathcalligra{A}^{-\beta } \bigr\Vert + \frac{C_{1-\beta } \Gamma (1+\beta )}{ \beta \Gamma (1+ \alpha \beta )} \bigl( \omega (T)- \omega (0) \bigr) ^{\alpha \beta } \biggr) M_{1} \Vert u-w \Vert \end{aligned}$$
for \(t \in [0, T]\). According to (11) of Theorem 4.1, we obtain that \(\mathcal{F}_{1}\) is a contraction.
Step 3: \(\mathcal{F}_{2}\) is a completely continuous operator.
Firstly, we claim that \(\mathcal{F}_{2}\) is continuous on \(B_{k}\). Let \(\{u_{n}\} \subset B_{k}\) be such that \(u_{n} \rightarrow u\in B_{k} \) as \(n \rightarrow \infty \). For \(t \in [0,T]\), by Assumptions (\(\mathrm{A}_{2}\)) and (\(\mathrm{A}_{3}\)), we have
$$ f\bigl(t,u_{n}(t),v(t)\bigr) \rightarrow f\bigl(t,u(t),v(t)\bigr)\quad \text{as } n \rightarrow \infty $$
and
$$ \bigl\Vert f\bigl(t,u_{n}(t),v(t)\bigr) - f\bigl(t,u(t),v(t)\bigr) \bigr\Vert \leq 2m_{k}(t)\quad \text{for all } n \in \mathbb{N}. $$
Using the Lebesgue dominated convergence theorem, for any \(t \in [0,T]\), we obtain
$$ \begin{aligned} &\bigl\Vert ( \mathcal{F}_{2}u_{n} ) (t) - ( \mathcal{F}_{2}u ) (t) \bigr\Vert \\ &\quad \leq \int _{0}^{t} \bigl(\omega (t)-\omega (\tau ) \bigr)^{\alpha -1} \omega ^{\prime }(\tau ) \bigl\Vert \mathcal{R}^{\alpha ;\omega }(t,\tau ) \bigl[f\bigl(t,u_{n}(\tau ),v( \tau )\bigr) -f\bigl(t,u(\tau ),v(\tau )\bigr) \bigr] \bigr\Vert \,\mathrm{d} { \tau } \\ &\quad \leq \frac{M}{\Gamma (\alpha )} \int _{0}^{t} \bigl(\omega (t)- \omega (\tau ) \bigr)^{\alpha -1} \omega ^{\prime }(\tau ) \bigl\Vert f \bigl(t,u_{n}( \tau ),v(\tau )\bigr) -f\bigl(t,u(\tau ),v(\tau )\bigr) \bigr\Vert \,\mathrm{d} {\tau } \rightarrow 0 \end{aligned} $$
as \(n \rightarrow \infty \). This implies that \(\| ( \mathcal{F}_{2}u_{n} )(t) - ( \mathcal{F}_{2}u )(t)\|_{C} \rightarrow 0 \) as \(n \rightarrow \infty \). Hence \(\mathcal{F}_{2}\) is continuous.
Next, we prove the equicontinuity of \(\mathcal{F}_{2}(B_{k})\). For any \(u \in B_{k}\), we have for \(0 \leq t_{1} < t_{2} \leq T\),
$$\begin{aligned}& \bigl\Vert ( \mathcal{F}_{2}u ) (t_{2}) - ( \mathcal{F}_{2}u ) (t_{1}) \bigr\Vert \\ & \quad \leq \biggl\Vert \int _{0}^{t_{2}}{ \bigl( \omega (t_{2})- \omega ( \tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) {\mathcal{R}^{ \alpha ;\omega }(t_{2},{\tau })}f\bigl(t,u(\tau ),v(\tau )\bigr) \,\mathrm{d} {\tau } \\ & \qquad {} - \int _{0}^{t_{1}}{ \bigl( \omega (t_{1})- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) {\mathcal{R}^{\alpha ; \omega }(t_{1},{\tau })}f\bigl(t,u(\tau ),v(\tau )\bigr) \,\mathrm{d} {\tau } \biggr\Vert \\ & \quad = \biggl\Vert \int _{0}^{t_{1}}{ \bigl( \omega (t_{2})- \omega ( \tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) {\mathcal{R}^{ \alpha ;\omega }(t_{2},{\tau })}f\bigl(t,u(\tau ),v(\tau )\bigr) \,\mathrm{d} {\tau } \\ & \qquad {} + \int _{t_{1}}^{t_{2}}{ \bigl( \omega (t_{2})- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) {\mathcal{R}^{\alpha ; \omega }(t_{2},{\tau })}f\bigl(t,u(\tau ),v(\tau )\bigr) \,\mathrm{d} {\tau } \\ & \qquad {} + \int _{0}^{t_{1}}{ \bigl( \omega (t_{1})- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) {\mathcal{R}^{\alpha ; \omega }(t_{2},{\tau })}f\bigl(t,u(\tau ),v(\tau )\bigr) \,\mathrm{d} {\tau } \\ & \qquad {} - \int _{0}^{t_{1}}{ \bigl( \omega (t_{1})- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) {\mathcal{R}^{\alpha ; \omega }(t_{2},{\tau })}f\bigl(t,u(\tau ),v(\tau )\bigr) \,\mathrm{d} {\tau } \\ & \qquad {} - \int _{0}^{t_{1}}{ \bigl( \omega (t_{1})- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) {\mathcal{R}^{\alpha ; \omega }(t_{1},{\tau })}f\bigl(t,u(\tau ),v(\tau )\bigr) \,\mathrm{d} {\tau } \biggr\Vert \\ & \quad \leq \biggl\Vert \int _{t_{1}}^{t_{2}}{ \bigl( \omega (t_{2})- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) { \mathcal{R}^{\alpha ;\omega }(t_{2},{\tau })}f\bigl(t,u(\tau ),v(\tau )\bigr) \,\mathrm{d} {\tau } \biggr\Vert \\ & \qquad {} + \biggl\Vert \int _{0}^{t_{1}}{ \bigl[ \bigl( \omega (t_{2})- \omega (\tau ) \bigr) ^{\alpha -1} - \bigl( \omega (t_{1})- \omega ( \tau ) \bigr) ^{\alpha -1} \bigr] } \omega ^{\prime }(\tau ) { \mathcal{R}^{\alpha ;\omega }(t_{2},{\tau })}f\bigl(t,u(\tau ),v(\tau )\bigr) \,\mathrm{d} {\tau } \biggr\Vert \\ & \qquad {} +\,\, \biggr\Vert \int _{0}^{t_{1}}{ \bigl( \omega (t_{1})- \omega ( \tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) \bigl[ { \mathcal{R}^{\alpha ;\omega }(t_{2},{ \tau })} - {\mathcal{R}^{\alpha ; \omega }(t_{1},{\tau })} \bigr] f \bigl(t,u(\tau ),v(\tau )\bigr) \,\mathrm{d} {\tau } \biggr\Vert \\ & \quad =: I_{1}+I_{2}+I_{3}. \end{aligned}$$
By Lemma 3.3, we obtain that
$$ I_{1} \leq \frac{M \Vert m_{k} \Vert _{L^{p}}}{\Gamma (\alpha )} \sup_{0 \leq t \leq T} \bigl\lvert \omega ^{\prime }(t)\bigr\rvert ^{\frac{1}{p}} \biggl\lbrace \frac{ ( \omega (t_{2})- \omega (t_{1}) ) ^{1+\frac{ ( \alpha -1 ) p}{p-1}}}{1+\frac{ ( \alpha -1 ) p}{p-1}} \biggr\rbrace ^{\frac{p-1}{p}} $$
and
$$\begin{aligned} I_{2} & \leq \frac{M \Vert m_{k} \Vert _{L^{p}}}{\Gamma (\alpha )} \\ &\quad {} \times \sup_{0 \leq t \leq T} \bigl\lvert \omega ^{\prime }(t) \bigr\rvert ^{\frac{1}{p}} \biggl\lbrace \frac{ ( \omega (t_{1}) ) ^{1+\frac{ ( \alpha -1 ) p}{p-1}} - ( \omega (t_{2}) ) ^{1+\frac{ ( \alpha -1 ) p}{p-1}} - ( \omega (t_{2})- \omega (t_{1}) ) ^{1+\frac{ ( \alpha -1 ) p}{p-1}} }{1+\frac{ ( \alpha -1 ) p}{p-1}} \biggr\rbrace ^{ \frac{p-1}{p}} \end{aligned}$$
and hence \(I_{1} \rightarrow 0\) and \(I_{2} \rightarrow 0\) as \(t_{2} \rightarrow t_{1}\). For \(t_{1}=0\) and \(0< t_{2}\leq T\), it easy to see that \(I_{4}=0\). Thus, for any \(\varepsilon \in (0,t_{1})\), we have
$$\begin{aligned} I_{3} &\leq \biggl\Vert \int _{0}^{t_{1}-\varepsilon }{ \bigl( \omega (t_{1})- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{ \prime }(\tau ) \bigl[ {\mathcal{R}^{\alpha ;\omega }(t_{2},{ \tau })} - { \mathcal{R}^{\alpha ;\omega }(t_{1},{\tau })} \bigr] f \bigl(t,u(\tau ),v( \tau )\bigr) \,\mathrm{d} {\tau } \biggr\Vert \\ &\quad {} + \biggl\Vert \int _{t_{1}-\varepsilon }^{t_{1}}{ \bigl( \omega (t_{1})- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }( \tau ) \bigl[ {\mathcal{R}^{\alpha ;\omega }(t_{2},{ \tau })} - { \mathcal{R}^{\alpha ;\omega }(t_{1},{\tau })} \bigr] f \bigl(t,u(\tau ),v( \tau )\bigr) \,\mathrm{d} {\tau } \biggr\Vert \\ & \leq \frac{M \Vert m_{k} \Vert _{L^{p}}}{\Gamma (\alpha )} \\ &\quad {} \times \sup_{0 \leq t \leq T} \bigl\lvert \omega ^{\prime }(t) \bigr\rvert ^{\frac{1}{p}} \biggl\lbrace \frac{ ( \omega (t_{1})- \omega (0) ) ^{1+\frac{ ( \alpha -1 ) p}{p-1}} - ( \omega (t_{1})- \omega (t_{1}-\varepsilon ) ) ^{1+\frac{ ( \alpha -1 ) p}{p-1}}}{1+\frac{ ( \alpha -1 ) p}{p-1}} \biggr\rbrace ^{\frac{p-1}{p}} \\ &\quad {} \times \sup_{0\leq s< t _{1}-\varepsilon } \bigl\Vert {\mathcal{R}^{ \alpha ;\omega }(t_{2},{ \tau })} - {\mathcal{R}^{\alpha ;\omega }(t_{1},{ \tau })} \bigr\Vert \\ &\quad {} + \frac{2M \Vert m_{k} \Vert _{L^{p}}}{\Gamma (\alpha )} \biggl\lbrace \frac{ ( \omega (t_{1})- \omega (t_{1}-\varepsilon ) ) ^{1+\frac{ ( \alpha -1 ) p}{p-1}}}{1+\frac{ ( \alpha -1 ) p}{p-1}} \biggr\rbrace ^{\frac{p-1}{p}} . \end{aligned}$$
Therefore \(I_{3} \rightarrow 0\) as \(t_{2} \rightarrow t_{1}\) and \(\varepsilon \rightarrow 0\) by Lemma 3.3, (iii) and (iv). It follows that
$$ \bigl\| ( \mathcal{F}_{2}u )(t_{2}) - ( \mathcal{F}_{2}u )(t_{1})\bigr\| \rightarrow 0\quad \text{independently of } u \in B_{k}\text{ as }t_{2} \rightarrow t_{1}, $$
which means that \(\mathcal{F}_{2}(B_{k})\) is equicontinuous.
Now, we will prove that \(N(t)= \lbrace { ( \mathcal{F}_{2} u ) (t) : u \in B_{k}} \rbrace \) is relatively compact in E for all \(t \in [0,T]\). Notice that \(N(0)\) is relatively compact in E. Fix \(t \in (0, T]\), then, for every \(\varepsilon >0\) and \(\delta >0\), we define an operator \(\mathcal{F}_{2}^{\varepsilon , \delta }\) on \(B_{k}\) as
$$\begin{aligned} & \bigl( \mathcal{F}_{2}^{\varepsilon , \delta }u \bigr) (t) \\ &\quad = \alpha \int _{0}^{t-\varepsilon } \int _{\delta }^{\infty } \theta \phi _{\alpha }( \theta ) { \bigl( \omega (t)- \omega (\tau ) \bigr) ^{\alpha -1}} \\ &\qquad {} \times \mathbb{T} \bigl( { \bigl( \omega (t)- \omega (0) \bigr)^{ \alpha } \theta } \bigr) f\bigl(\tau , u(\tau ), v(\tau )\bigr) \omega ^{\prime }( \tau )\,\mathrm{d} \theta \,\mathrm{d} {\tau } \\ &\quad = \alpha \int _{0}^{t-\varepsilon } \int _{\delta }^{\infty } \theta \phi _{\alpha }( \theta ) { \bigl( \omega (t)- \omega (\tau ) \bigr) ^{\alpha -1}} \\ &\qquad {} \times \mathbb{T} \bigl( { \bigl( \omega (t)- \omega (0) \bigr)^{ \alpha } \theta }+ \varepsilon ^{\alpha }\delta - \varepsilon ^{\alpha } \delta \bigr) f\bigl(\tau , u(\tau ), v(\tau )\bigr) \omega ^{\prime }(\tau ) \,\mathrm{d} \theta \,\mathrm{d} {\tau } \\ & \quad = \alpha \int _{0}^{t-\varepsilon } \int _{\delta }^{\infty } \theta \phi _{\alpha }( \theta ) { \bigl( \omega (t)- \omega (\tau ) \bigr) ^{\alpha -1}} \\ & \qquad {}\times \bigl[ \mathbb{T} \bigl( \varepsilon ^{\alpha }\delta \bigr) \mathbb{T} \bigl( { \bigl( \omega (t)- \omega (0) \bigr)^{\alpha } \theta }- \varepsilon ^{\alpha }\delta \bigr) \bigr] f\bigl(\tau , u(\tau ), v(\tau ) \bigr) \omega ^{\prime }(\tau )\,\mathrm{d} \theta \,\mathrm{d} {\tau } \\ &\quad = \alpha \mathbb{T} \bigl( \varepsilon ^{\alpha }\delta \bigr) \int _{0}^{t-\varepsilon } \int _{\delta }^{\infty } \theta \phi _{ \alpha }( \theta ) { \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \\ &\qquad {} \times \mathbb{T} \bigl( { \bigl( \omega (t)- \omega (0) \bigr)^{\alpha } \theta }- \varepsilon ^{\alpha }\delta \bigr) f\bigl(\tau , u(\tau ), v( \tau )\bigr) \omega ^{\prime }(\tau ) \,\mathrm{d} \theta \,\mathrm{d} { \tau }, \end{aligned}$$
where \(u \in B_{k}\).
By the compactness of \(\mathbb{T}(\varepsilon ^{\alpha } \delta )\) for \(\varepsilon ^{\alpha } \delta >0\), it follows that the set \(N_{\varepsilon , \delta }(t)= \lbrace ( \mathcal{F}_{2}^{ \varepsilon , \delta } u ) (t) : u \in B_{k} \rbrace \) is relatively compact in E for all \(\varepsilon >0\) and \(\delta >0\). Furthermore, for any \(u \in B_{k}\), we have
$$\begin{aligned}& \bigl\Vert ( \mathcal{F}_{2}u ) (t)- \bigl( \mathcal{F}_{2}^{\varepsilon , \delta }u \bigr) (t) \bigr\Vert \\& \quad = \alpha \biggl\Vert \int _{0}^{t} \int _{0}^{\delta } \theta \phi _{ \alpha }( \theta ) { \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \omega ^{\prime }(\tau ) \mathbb{T} \bigl( { \bigl( \omega (t)- \omega (0) \bigr)^{\alpha } \theta } \bigr) f\bigl(\tau , u( \tau ), v(\tau )\bigr) \, \mathrm{d} \theta \,\mathrm{d} {\tau } \\& \qquad {} + \int _{0}^{t} \int _{\delta }^{\infty } \theta \phi _{\alpha }( \theta ) { \bigl( \omega (t)- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) \mathbb{T} \bigl( { \bigl( \omega (t)- \omega (0) \bigr)^{\alpha } \theta } \bigr) f\bigl(\tau , u(\tau ), v( \tau )\bigr)\, \mathrm{d} \theta \,\mathrm{d} {\tau } \\& \qquad {} + \int _{0}^{t-\varepsilon } \int _{\delta }^{\infty } \theta \phi _{\alpha }( \theta ) { \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \\& \qquad {}\times\mathbb{T} \bigl( { \bigl( \omega (t)- \omega (0) \bigr)^{ \alpha } \theta } \bigr) f \bigl(\tau , u(\tau ), v(\tau )\bigr) \omega ^{\prime }( \tau )\,\mathrm{d} \theta \,\mathrm{d} {\tau } \biggr\Vert \\& \quad \leq \alpha \biggl\Vert \int _{0}^{t} \int _{0}^{\delta } \theta \phi _{\alpha }( \theta ) { \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \omega ^{\prime }(\tau ) \mathbb{T} \bigl( { \bigl( \omega (t)- \omega (0) \bigr)^{\alpha } \theta } \bigr) f\bigl(\tau , u( \tau ), v(\tau )\bigr) \, \mathrm{d} \theta \,\mathrm{d} {\tau } \biggr\Vert \\& \qquad {} + \alpha \biggl\Vert \int _{t - \varepsilon }^{t} \int _{ \delta }^{\infty } \theta \phi _{\alpha }( \theta ) { \bigl( \omega (t)- \omega (\tau ) \bigr) ^{\alpha -1}} \omega ^{\prime }(\tau ) \\& \qquad {}\times \mathbb{T} \bigl( { \bigl( \omega (t)- \omega (0) \bigr)^{\alpha } \theta } \bigr) f\bigl(\tau , u(\tau ), v(\tau )\bigr) \, \mathrm{d} \theta \,\mathrm{d} {\tau } \biggr\Vert \\& \quad \leq \alpha M \Vert m_{k} \Vert _{L^{p}} \biggl\lbrace \sup_{0 \leq s\leq t} \bigl\lvert \omega ^{\prime }(\tau )\bigr\rvert ^{\frac{1}{p-1}} \int _{0}^{t} { \bigl( \omega (t)- \omega ( \tau ) \bigr) ^{\frac{(\alpha -1)p}{p-1}}} \omega ^{\prime }(\tau ) \,\mathrm{d} { \tau } \biggr\rbrace ^{ \frac{p-1}{p}} \biggl( \int _{0}^{\delta } \theta \phi _{\alpha }( \theta )\,\mathrm{d} \theta \biggr) \\& \qquad {} + \alpha M \Vert m_{k} \Vert _{L^{p}} \biggl\lbrace \sup_{t- \varepsilon \leq s\leq t} \bigl\lvert \omega ^{\prime }(t)\bigr\rvert ^{ \frac{1}{p-1}} \int _{t- \varepsilon }^{t} { \bigl( \omega (t)- \omega ( \tau ) \bigr) ^{\frac{(\alpha -1)p}{p-1}}} \omega ^{\prime }( \tau ) \,\mathrm{d} { \tau } \biggr\rbrace ^{\frac{p-1}{p}} \\& \qquad {} \times \biggl( \int _{0}^{\delta } \theta \phi _{\alpha }( \theta )\,\mathrm{d} \theta \biggr) \\& \quad = \alpha M \Vert m_{k} \Vert _{L^{p}} \sup _{0 \leq s\leq t} \bigl\lvert \omega ^{ \prime }(\tau )\bigr\rvert ^{\frac{1}{p}} \biggl\lbrace \frac{{ ( \omega (t)- \omega (0) ) ^{1+\frac{ ( \alpha -1 ) p}{p-1}}}}{1+\frac{ ( \alpha -1 ) p}{p-1}} \biggr\rbrace ^{\frac{p-1}{p}} \biggl( \int _{0}^{\delta } \theta \phi _{ \alpha }( \theta )\,\mathrm{d} \theta \biggr) \\& \qquad {} + \alpha M \Vert m_{k} \Vert _{L^{p}} \sup _{t-\varepsilon \leq s\leq t} \bigl\lvert \omega ^{\prime }(t)\bigr\rvert ^{\frac{1}{p}} \biggl\lbrace \frac{{ ( \omega (t)- \omega (t- \varepsilon ) ) ^{1+\frac{ ( \alpha -1 ) p}{p-1}}}}{1+\frac{ ( \alpha -1 ) p}{p-1}} \biggr\rbrace ^{\frac{p-1}{p}} \\& \qquad {} \times \biggl( \int _{0}^{\delta } \theta \phi _{\alpha }( \theta )\,\mathrm{d} \theta \biggr) \\& \quad \rightarrow 0\quad \text{as } \varepsilon , \delta \rightarrow 0^{+} . \end{aligned}$$
Hence, there are relatively compact sets arbitrary close to the set \(N(t)\) for \(t>0\). Therefore, we conclude that \(N(t)\) is relatively compact in E. It follows that the set \(\mathcal{F}_{2}(B_{k})\) is relatively compact in \(C([0,T],E)\) by Arzelá–Ascoli theorem. This implies that \(\mathcal{F}_{2}\) is a completely continuous by the continuity of \(\mathcal{F}_{2}\) and relatively compactness of \(\mathcal{F}_{2}(B_{k})\). Hence, Krasnoselskii’s fixed point theorem implies that \(\mathcal{F}_{1} + \mathcal{F}_{2}\) has a fixed point \(u^{*}\) in \(B_{k}\), which is a mild solution of (1). □
Next, we prove a uniqueness result by means of the Banach contraction theorem.
Theorem 4.2
Assume that (\(\mathrm{A}_{3}\))–(\(\mathrm{A}_{5}\)) are satisfied and condition (10) of Theorem 4.1holds. Then, the problem (1) has a unique mild solution if
$$ \begin{aligned}[b] &( M+1 ) M_{1} \bigl\Vert \mathcalligra{A}^{-\beta } \bigr\Vert + \frac{ C_{1-\beta } \Gamma (1+\beta )}{ \beta \Gamma (1+ \alpha \beta )}M_{1} \bigl( \omega (T)- \omega (0) \bigr) ^{\alpha \beta } \\ &\quad {}+ \frac{M}{\Gamma (\alpha +1)} L_{f} \bigl( \omega (T) - \omega (0) \bigr)^{\alpha } < 1. \end{aligned} $$
(12)
Proof
For \(u \in B_{k}\), we define the operator \(\mathcal{G}\) on \(B_{k}\) by
$$ \begin{aligned} ( \mathcal{G}u ) (t)&= {\mathcal{Q}^{\alpha ; \omega }(t,0)} \bigl( u_{0} -g(0,u_{0}) \bigr) +g\bigl(t,u(t)\bigr) \\ &\quad {} + \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \omega ^{\prime }(\tau ) {\mathcalligra{A} \mathcal{R}^{\alpha ; \omega }(t,\tau )}g\bigl(s,u(\tau )\bigr) \,\mathrm{d} {\tau } \\ &\quad {} + \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}}\omega ^{\prime }(\tau ) {\mathcal{R}^{\alpha ;\omega }(t, \tau )}f\bigl(s,u(\tau ),v(\tau )\bigr) \,\mathrm{d} {\tau }. \end{aligned} $$
Notice that it is enough to show the uniqueness of a fixed point of \(\mathcal{G}\) on \(B_{k}\). According to (10), we know that \(\mathcal{G}\) is an operator from \(B_{k}\) into itself.
For any \(u, u^{*} \in B_{k}\) and \(t \in [0,T]\), according to (\(\mathrm{A}_{3}\))–(\(\mathrm{A}_{5}\)), we have
$$\begin{aligned}& \bigl\Vert ( \mathcal{G}u ) (t) - \bigl( \mathcal{G}u^{*} \bigr) (t) \bigr\Vert \\& \quad \leq \bigl\Vert {\mathcal{Q}^{\alpha ;\omega }(t,0)} \bigl( g\bigl(0,u(0) \bigr) - g\bigl(0,u^{*}(0)\bigr) \bigr) \bigr\Vert + \bigl\Vert g \bigl(t,u(t)\bigr)-g\bigl(t,u^{*}(t)\bigr) \bigr\Vert \\& \qquad {} + \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \omega ^{\prime }(\tau ) \\& \qquad {} \times \bigl\Vert \mathcalligra{A}^{1-\beta } \mathcal{R}^{\alpha ; \omega }(t, \tau ) \bigl[ \mathcalligra{A}^{\beta }g\bigl(s,u(\tau )\bigr) - \mathcalligra{A}^{\beta }g\bigl(s,u^{*}(\tau )\bigr) \bigr] \bigr\Vert \,\mathrm{d} {\tau } \\& \qquad {} + \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}}\omega ^{\prime }(\tau ) \\& \qquad {} \times \bigl\Vert {\mathcal{R}^{\alpha ;\omega }(t,\tau )} \bigl[ f\bigl(s,u( \tau ),v(\tau )\bigr) - f\bigl(s,u^{*}(\tau ),v(\tau )\bigr) \bigr] \bigr\Vert \,\mathrm{d} {\tau } \\& \quad \leq M \bigl\Vert \mathcalligra{A}^{-\beta } \bigl( \mathcalligra{A}^{\beta } g\bigl(0,u(0)\bigr) - \mathcalligra{A}^{\beta } g\bigl(0,u^{*}(0) \bigr) \bigr) \bigr\Vert \\& \qquad {} + \bigl\Vert \mathcalligra{A}^{-\beta } \bigl( \mathcalligra{A}^{\beta } g\bigl(t,u(t)\bigr) - \mathcalligra{A}^{\beta } g\bigl(t,u^{*}(t) \bigr) \bigr) \bigr\Vert \\& \qquad {} + \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}} \omega ^{\prime }(\tau ) \frac{\alpha C_{1-\beta }}{ ( \omega (t)- \omega (\tau ) ) ^{\alpha ( 1-\beta ) }} \frac{\Gamma (1+\beta )}{\Gamma (1+\alpha \beta )} \\& \qquad {} \times \bigl\Vert \mathcalligra{A}^{\beta }g\bigl(s,u(\tau )\bigr) - \mathcalligra{A}^{\beta }g\bigl(s,u^{*}(\tau )\bigr) \bigr\Vert \,\mathrm{d} {\tau } \\& \qquad {} + \int _{0}^{t}{ \bigl( \omega (t)- \omega (\tau ) \bigr) ^{ \alpha -1}}\omega ^{\prime }(\tau ) \\& \qquad {} \times \frac{M}{\Gamma (\alpha )} \bigl\Vert f\bigl(s,u(\tau ),v( \tau )\bigr) - f\bigl(s,u^{*}(\tau ),v(\tau )\bigr) \bigr\Vert \,\mathrm{d} {\tau } \\& \quad \leq M \bigl\Vert \mathcalligra{A}^{-\beta } \bigr\Vert M_{1} \bigl\Vert u-u^{*} \bigr\Vert + \bigl\Vert \mathcalligra{A}^{-\beta } \bigr\Vert M_{1} \bigl\Vert u-u^{*} \bigr\Vert \\& \qquad {} + \frac{\alpha C_{1-\beta } \Gamma (1+\beta )}{\Gamma (1+ \alpha \beta )}M_{1} \frac{ ( \omega (T)- \omega (0) ) ^{\alpha \beta }}{\alpha \beta } \bigl\Vert u-u^{*} \bigr\Vert \\& \qquad {} + \frac{M}{\Gamma (\alpha +1)} L_{f} \bigl( \omega (T) - \omega (0) \bigr)^{\alpha } \bigl\Vert u-u^{*} \bigr\Vert \\& \quad = \biggl( ( M+1 ) M_{1} \bigl\Vert \mathcalligra{A}^{-\beta } \bigr\Vert + \frac{ C_{1-\beta } \Gamma (1+\beta )}{ \beta \Gamma (1+ \alpha \beta )}M_{1} \bigl( \omega (T)- \omega (0) \bigr) ^{\alpha \beta } \\& \qquad {} + \frac{M}{\Gamma (\alpha +1)} L_{f} \bigl( \omega (T) - \omega (0) \bigr)^{\alpha } \biggr) \bigl\Vert u-u^{*} \bigr\Vert . \end{aligned}$$
This implies that \(\mathcal{G}\) is a contraction map satisfying (12). Hence the uniqueness of a fixed point of the map \(\mathcal{G}\) on \(B_{k}\) follows from the Banach contraction principle. □