Let \(\mathcal{C}_{\mathcal{A}} = C([0,T],\mathbb{R})\) be a Banach space of all continuous functions from \([0,T]\) to \(\mathbb{R}\), endowed with the norm defined by
$$\|x\|_{\mathcal{C_{A}}}=\max\bigl\{ \|x\|,\bigl\| D_{\omega}^{\nu}x\bigr\| \bigr\} , $$
where \(\|x\|= \sup_{t\in[0,T]} |x(t)|\) and \(\|D_{\omega}^{\nu}x\|= \sup_{t\in[0,T]} |D_{\omega}^{\nu}x(t)|\). Define the operator \(\mathcal{A}:\mathcal{C}_{\mathcal{A}}\rightarrow\mathcal {C}_{\mathcal{A}}\) by
$$\begin{aligned} (\mathcal{A}x) (t) :=&\frac{1}{\Gamma_{q}(\alpha)} \int_{0}^{t}(t-qs)^{(\alpha -1)}f \bigl(s,x(s),D_{w}^{\nu}x(s)\bigr)\,d_{q}s \\ &{}-\frac{t^{\alpha-2}(t-\eta)}{\Gamma_{q}(\alpha)\int _{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s } \\ &{}\times \int_{0}^{T} \int_{0}^{s}g(r) (T-ps)^{(\beta-1)}(s-qr)^{(\alpha -1)}f \bigl(r,x(r),D_{w}^{\nu}x(r)\bigr)\,d_{q}r\,d_{p}s \\ &{}+ \frac{ t^{\alpha-2} \int_{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha -2}(t-s) \,d_{p}s }{\eta^{\alpha-2}\Gamma_{q}(\alpha) \int _{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s } \\ &{}\times \biggl[ \int_{0}^{\eta}(\eta-qs)^{(\alpha-1)}f \bigl(s,x(s),D_{w}^{\nu}x(s)\bigr)\,d_{q}s- \Gamma_{q}(\alpha) \rho(x) \biggr]. \end{aligned}$$
(3.1)
Observe that the problem (1.3) has solutions if and only if the operator \(\mathcal{A}\) has fixed points.
Now, we are in the position to establish the main results. Our first result is based on Banach’s fixed point theorem.
Theorem 3.1
Assume that a functional
\(\rho\in C([0,T],\mathbb{R})\rightarrow \mathbb{R}\), \(f:[0,T]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb {R}\)
and
\(g:[0,T]\rightarrow\mathbb{R}\)
are continuous functions satisfying the following conditions:
- (H1):
-
There exist positive numbers
\(L_{1}\), \(L_{2}\)
such that, for each
\(t\in[0,T]\)
and
\(x,y\in{\mathcal{C}}_{\mathcal{A}}\),
$$\bigl|f\bigl(t,x,D_{w}^{\nu}x\bigr)-f\bigl(t,y,D_{w}^{\nu}y\bigr)\bigr|\leq L_{1}|x-y|+L_{2}\bigl|D_{w}^{\nu}x-D_{w}^{\nu}y\bigr|. $$
- (H2):
-
There exists a positive number
τ
such that, for each
\(x,y\in{\mathcal{C}}_{\mathcal{A}}\),
$$\bigl|\rho(x)-\rho(y)\bigr|\leq\tau\|x-y\|_{\mathcal{C_{A}}}. $$
- (H3):
-
For each
\(t\in[0,T]\), \(0< n< g(t)< N\).
- (H4):
-
\(\Theta:=\lambda (\Omega+\Lambda )+\frac{\tau\Gamma _{q}(\alpha+1)}{\eta^{2}}\Lambda<1\),
where
$$\begin{aligned} &\lambda=\max \{L_{1}+L_{2}\}, \\ &\Omega=\frac{T^{\alpha}}{\Gamma_{q}(\alpha+1)}+\frac{N |T-\eta| T^{\alpha}\Gamma_{p}(\alpha+1)}{ n\vert T[\alpha-1]_{p}-\eta[\alpha+\beta-1]_{p}\vert [\alpha+\beta ]_{p}\Gamma_{q}(\alpha+1) \Gamma_{p}(\alpha-1)}, \\ &\Lambda=\frac{N T^{\alpha-1} \eta^{2} \vert [\alpha-1]_{p}-[\alpha +\beta-1]_{p} \vert }{n \vert T [\alpha-1]_{p}-\eta[\alpha+\beta -1]_{p}\vert \Gamma_{q}(\alpha+1)}. \end{aligned}$$
(3.2)
Then the boundary value problem (1.3) has a unique solution.
Proof
We transform the boundary value problem (1.3) into a fixed point problem \(x=\mathcal{A}x\), where \(\mathcal{A}:\mathcal{C}_{\mathcal{A}}\rightarrow\mathcal{C}_{\mathcal {A}}\) is defined by (3.1). Assuming that \(\sup_{t\in[0,T]}{|f(t,0,0)|} = M\) and \(\sup_{x\in{\mathcal{C}}_{\mathcal{A}}}{|\rho(x)|} = K\), we choose a constant R satisfied with
$$ R\geq\frac{M (\Omega+\Lambda )+\frac{\Gamma_{q}(\alpha+1)}{\eta ^{2}}K \Lambda}{1-\Theta}. $$
(3.3)
Now, we will show that \(\mathcal{A}B_{R}\subset B_{R}\), where \(B_{R} = \{x \in \mathcal{C}_{\mathcal{A}}: \|x\|_{\mathcal{C_{A}}} \leq R\}\). For all \(x, y\in\mathcal{C}_{\mathcal{A}}\) and for each \(t\in[0,T]\), we have
$$\begin{aligned} &|\mathcal{A}x| \\ &\quad\leq \frac{1}{\Gamma_{q}(\alpha)} \int_{0}^{t}(t-qs)^{(\alpha-1)}\bigl(\bigl|f \bigl(s,x(s),D_{w}^{\nu}x(s)\bigr)-f(s,0,0)\bigr|+\bigl|f(s,0,0)\bigr| \bigr)\,d_{q}s \\ &\qquad{}+ \frac{t^{\alpha-2}(t-\eta)}{\Gamma_{q}(\alpha) \int _{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s } \int _{0}^{T} \int_{0}^{s}g(r) (T-ps)^{(\beta-1)} \\ &\qquad{}\times(s-qr)^{(\alpha-1)}\bigl(\bigl|f\bigl(r,x(r),D_{w}^{\nu}x(r)\bigr)-f(r,0,0)\bigr|+\bigl|f(r,0,0)\bigr|\bigr)\,d_{q}r\,d_{p}s \\ &\qquad{}+ \frac{ t^{\alpha-2} \int_{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha -2}(t-s) \,d_{p}s }{\eta^{\alpha-2}\Gamma_{q}(\alpha) \int _{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s } \biggl[\Gamma _{q}(\alpha) \bigl(\bigl| \rho(x)-\rho(0)\bigr|+\bigl|\rho(0)\bigr| \bigr) \\ &\qquad{}+ \int_{0}^{\eta}(\eta-qs)^{(\alpha-1)}\bigl(\bigl|f \bigl(s,x(s),D_{w}^{\nu}x(s)\bigr)-f(s,0,0)\bigr|+\bigl|f(s,0,0)\bigr| \bigr)\,d_{q}s \biggr] \\ &\quad\leq \bigl(L_{1}\|x\|+L_{2}\bigl\| D_{\omega}^{\nu}x\bigr\| +M\bigr) \biggl[\frac{t^{\alpha }}{\Gamma_{q}(\alpha+1)} \biggr] +\bigl(L_{1}\|x \|+L_{2}\bigl\| D_{\omega}^{\nu}x\bigr\| +M\bigr) \\ &\qquad{}\times \biggl[\frac{N t^{\alpha-2} T^{2} |t-\eta|\Gamma_{p}(\alpha+1)}{ n \Gamma_{p}(\alpha-1)\Gamma_{q}(\alpha+1)[\alpha+\beta]_{p} |T [\alpha -1]_{p}-\eta[\alpha+\beta-1]_{p}|} \biggr] \\ &\qquad{}+\bigl(L_{1}\|x\|+L_{2}\bigl\| D_{\omega}^{\nu}x\bigr\| +M\bigr) \biggl[\frac{ N \eta^{2} t^{\alpha -2} | t [\alpha+\beta-1]_{p}- T [\alpha-1]_{p}| }{n \Gamma_{q}(\alpha +1) |T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}|} \biggr] \\ &\qquad{}+\bigl(\tau\|x\|_{\mathcal{C_{A}}}+K\bigr) \biggl[\frac{ N t^{\alpha-2} | t [\alpha+\beta-1]_{p}- T [\alpha-1]_{p}| }{n |T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}|} \biggr] \\ &\quad\leq \bigl(\lambda \|x\|_{\mathcal{C_{A}}}+M\bigr) \biggl\{ \frac{N T^{\alpha} |T-\eta|\Gamma_{p}(\alpha+1)}{ n \Gamma_{p}(\alpha-1)\Gamma_{q}(\alpha+1)[\alpha+\beta]_{p} \vert T [\alpha-1]_{p}-\eta[\alpha+\beta-1]_{p}\vert } \\ &\qquad{} + \frac{T^{\alpha}}{\Gamma_{q}(\alpha+1)}+\frac{ N \eta^{2} T^{\alpha -1} |[\alpha+\beta-1]_{p}- [\alpha-1]_{p}| }{n \Gamma_{q}(\alpha+1) \vert T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}\vert } \biggr\} \\ &\qquad{}+\bigl(\tau\|x\|_{\mathcal{C_{A}}}+K\bigr) \biggl\{ \frac{ N T^{\alpha-1} |[\alpha+\beta-1]_{p}- [\alpha-1]_{p}| }{n \vert T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}\vert } \biggr\} \\ &\quad= R\Theta+M (\Omega+\Lambda ) +\frac{\Gamma_{q}(\alpha+1)}{\eta ^{2}} K\Lambda \end{aligned}$$
and
$$\begin{aligned} &\bigl|D_{w}^{\nu}\mathcal{A}x\bigr|\\ &\quad= \bigl|D_{w} I_{w}^{1-\nu}\mathcal{A}x\bigr| \\ &\quad= \biggl| D_{w} \biggl\{ -\frac{\int_{0}^{T}\int_{0}^{s}g(r)(T-ps)^{(\beta-1)} (s-qr)^{\alpha-1} f(r,x(r),D_{w}^{\nu} x(r)) \,d_{q}r\,d_{p}s}{\Gamma_{w}(1-\nu )\Gamma_{q}(\alpha) \int_{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s} \\ &\qquad{}\times \int_{0}^{t}(t-ws)^{(-\nu)} s^{\alpha-2}(s-\eta) \,d_{w}s \\ &\qquad{}\times \frac{\int_{0}^{\eta}(\eta-qs)^{\alpha-1} f(s,x(s),D_{w}^{\nu} x(s)) \,d_{w}s-\Gamma_{q}(\alpha) \rho(x)}{ \eta^{\alpha-2} \Gamma_{w}(1-\nu) \Gamma _{q}(\alpha) \int_{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s } \\ &\qquad{}\times \biggl[ \int_{0}^{t}(t-ws)^{(-\nu)}s^{\alpha-1} \,d_{w} \int _{0}^{T}g(s) (T-ps)^{(\beta-1)}s^{\alpha-2} \,d_{p}s \\ &\qquad{}- \int_{0}^{t}(t-ws)^{(-\nu)}s^{\alpha-2} \,d_{w}s \int _{0}^{T}g(s) (T-ps)^{(\beta-1)}s^{\alpha-1} \,d_{p}s \biggr] \\ &\qquad{}+\frac{1}{\Gamma_{q}(\alpha)\Gamma_{w}(1-\nu)} \int_{0}^{t} \int _{0}^{s}(t-ws)^{(-\nu)} (s-qr)^{\alpha-1}f\bigl(r,x(r),D_{w}^{\nu} x(r)\bigr) \,d_{q}r\,d_{w}s \biggr\} \biggr| \\ &\quad=\frac{ [\int_{0}^{wt}(wt-ws)^{(-\nu)} s^{\alpha-2}(s-\eta) \,d_{w}s - \int_{0}^{t}(t-ws)^{(-\nu)} s^{\alpha-2}(s-\eta) \,d_{w}s ]}{ (1-w) t \Gamma_{w}(1-\nu)\Gamma_{q}(\alpha) \int_{0}^{T}g(s)(T-ps)^{(\beta -1)} s^{\alpha-2}(s-\eta) \,d_{p}s} \\ &\qquad{}\times \biggl[ \int_{0}^{T} \int_{0}^{s}g(r) (T-ps)^{(\beta-1)} (s-qr)^{\alpha-1} \bigl(|f\bigl(r,x,D_{w}^{\nu}x \bigr)-f(r,0,0)|+|f(r,0,0)|\bigr) \,d_{q}r\,d_{p}s \biggr] \\ &\qquad{}+ \frac{\int_{0}^{\eta}(\eta-qs)^{\alpha-1} (|f(s,x,D_{w}^{\nu}x)-f(s,0,0)|+|f(s,0,0)|) \,d_{w}s+\Gamma_{q}(\alpha) (|\rho(x)-\rho (0)|+|\rho(0)| )}{ (1-w) t \eta^{\alpha-2} \Gamma_{w}(1-\nu) \Gamma_{q}(\alpha) \int_{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s } \\ &\qquad{}\times \biggl[ \biggl| \int_{0}^{wt}(wt-ws)^{(-\nu)}s^{\alpha-1} \,d_{w} - \int _{0}^{t}(t-ws)^{(-\nu)}s^{\alpha-1} \,d_{w} \biggr| \\ &\qquad{}\times \int_{0}^{T}g(s) (T-ps)^{\beta-1}s^{\alpha-2} \,d_{p}s + \biggl| \int _{0}^{wt}(wt-ws)^{(-\nu)}s^{\alpha-2} \,d_{w} \\ &\qquad{}- \int_{0}^{t}(t-ws)^{(-\nu)}s^{\alpha-2} \,d_{w} \biggr| \int _{0}^{T}g(s) (T-ps)^{\beta-1}s^{\alpha-1} \,d_{p}s \biggr] \\ &\qquad{}+\frac{1}{(1-w) t \Gamma_{q}(\alpha)\Gamma_{w}(1-\nu)} \biggl| \int _{0}^{wt} \int_{0}^{s}(wt-ws)^{(-\nu)} (s-qr)^{\alpha-1} \\ &\qquad{}\times \bigl(\bigl|f\bigl(r,x,D_{w}^{\nu}x \bigr)-f(r,0,0)\bigr|+\bigl|f(r,0,0)\bigr|\bigr) \,d_{q}r\,d_{w}s \\ &\qquad{}- \int_{0}^{t} \int_{0}^{s}(t-ws)^{(-\nu)} (s-qr)^{\alpha-1}\bigl(\bigl|f\bigl(r,x,D_{w}^{\nu}x \bigr)-f(r,0,0)\bigr|+\bigl|f(r,0,0)\bigr|\bigr) \,d_{q}r\,d_{w}s \biggr| \\ &\quad\leq \bigl(L_{1}\|x\|+L_{2} \|D_{\omega}^{\nu}x\|+M\bigr) \\ &\qquad{}\times \biggl[ \frac{N T t^{\alpha-\nu-2}\Gamma_{p}(\alpha+1)}{ n \Gamma_{q}(\alpha+1)[\alpha+\beta]_{p} \Gamma_{p}(\alpha-1) |[\alpha -1]_{p}-T [\alpha+\beta-1]_{p}|} \biggr] \\ &\qquad{}\times \biggl| \frac{t \Gamma_{w}(\alpha) (1-w^{\alpha-\nu})}{(1-w)\Gamma _{w}(\alpha-\nu+1)} -\frac{\Gamma_{w}(\alpha-1) (1-w^{\alpha-\nu-1})}{(1-w)\Gamma_{w}(\alpha -\nu)} \biggr| \\ &\qquad{}+ \biggl[\bigl(L_{1}\|x\|+L_{2}\|D_{\omega}^{\nu}x\|+M\bigr) \\ &\qquad{}\times \biggl(\frac{N \eta^{2} t^{\alpha-\nu-2}}{ nT \Gamma_{q}(\alpha+1) \Gamma_{p}(\alpha-1) \bigl|[\alpha-1]_{p}-T [\alpha +\beta-1]_{p}\bigr|} \biggr) \\ &\qquad{}+\bigl(\tau\|x\|_{\mathcal{C_{A}}}+K\bigr) \biggl(\frac{N t^{\alpha-\nu-2}}{ nT \Gamma_{p}(\alpha-1) |[\alpha-1]_{p}-T [\alpha+\beta-1]_{p}|} \biggr) \biggr] \\ &\qquad{}\times \biggl| \frac{t \Gamma_{w}(\alpha) (1-w^{\alpha-\nu}) [\alpha+\beta -1]_{p}}{(1-w)\Gamma_{w}(\alpha-\nu+1)} -\frac{\Gamma_{w}(\alpha-1) (1-w^{\alpha-\nu-1}) T [\alpha -1]_{p}}{(1-w)\Gamma_{w}(\alpha-\nu)} \biggr| \\ &\qquad{}+\bigl(L_{1}\|x\|+L_{2}\|D_{\omega}^{\nu}x\|+M\bigr) \biggl[\frac{t^{\alpha-\nu} \Gamma_{w}(\alpha+1)}{\Gamma_{q}(\alpha+1)\Gamma_{w}(\alpha-\nu+2)} \biggr] \biggl\vert \frac{1-w^{\alpha-\nu+1}}{1-w} \biggr\vert \\ &\quad\leq \bigl(\lambda \|x\|_{\mathcal{C_{A}}}+M\bigr) \biggl\{ \biggl[ \frac{N T^{\alpha-\nu-1}\Gamma_{p}(\alpha+1)}{ n \Gamma_{q}(\alpha+1)[\alpha+\beta]_{p} \Gamma_{p}(\alpha-1) |[\alpha -1]_{p}-T [\alpha+\beta-1]_{p}|} \biggr] \\ &\qquad{}\times \biggl| \frac{T \Gamma_{w}(\alpha) (1-w^{\alpha-\nu})}{(1-w)\Gamma _{w}(\alpha-\nu+1)} -\frac{\Gamma_{w}(\alpha-1) (1-w^{\alpha-\nu-1})}{(1-w)\Gamma_{w}(\alpha -\nu)} \biggr| \\ &\qquad{}+ \biggl[\frac{N \eta^{2} T^{\alpha-\nu-1}}{ n \Gamma_{q}(\alpha+1) \Gamma_{p}(\alpha-1) |[\alpha-1]_{p}-T [\alpha +\beta-1]_{p}|} \biggr] \\ &\qquad{}\times \biggl| \frac{\Gamma_{w}(\alpha) (1-w^{\alpha-\nu}) [\alpha+\beta -1]_{p}}{(1-w)\Gamma_{w}(\alpha-\nu+1)} -\frac{\Gamma_{w}(\alpha-1) (1-w^{\alpha-\nu-1}) [\alpha -1]_{p}}{(1-w)\Gamma_{w}(\alpha-\nu)} \biggr| \\ &\qquad{}+ \biggl[\frac{T^{\alpha-\nu} \Gamma_{w}(\alpha+1)}{\Gamma_{q}(\alpha +1)\Gamma_{w}(\alpha-\nu+2)} \biggr]\biggl\vert \frac{1-w^{\alpha-\nu +1}}{1-w}\biggr\vert \biggr\} \\ &\qquad{}+\bigl(\tau\|x\|_{\mathcal{C_{A}}}+K\bigr) \biggl[\frac{N T^{\alpha-\nu-1}}{ n \Gamma_{p}(\alpha-1) |[\alpha-1]_{p}-T [\alpha+\beta-1]_{p}|} \biggr] \\ &\qquad{}\times \biggl| \frac{\Gamma_{w}(\alpha) (1-w^{\alpha-\nu}) [\alpha+\beta -1]_{p}}{(1-w)\Gamma_{w}(\alpha-\nu+1)} -\frac{\Gamma_{w}(\alpha-1) (1-w^{\alpha-\nu-1}) [\alpha -1]_{p}}{(1-w)\Gamma_{w}(\alpha-\nu)} \biggr| \\ &\quad= (\lambda R+M) \biggl\{ \biggl(\frac{T^{\alpha}}{\Gamma_{q}(\alpha +1)} \biggr) \biggl[ \frac{1}{T^{\nu}}\cdot\frac{\Gamma_{w}(\alpha+1)}{\Gamma_{w}(\alpha-\nu +2)}\cdot\biggl\vert \frac{1-w^{\alpha-\nu+1}}{1-w} \biggr\vert \biggr] \\ &\qquad{}+ \biggl(\frac{N T^{\alpha} |T-\eta|\Gamma_{p}(\alpha+1)}{ n \Gamma_{p}(\alpha-1)\Gamma_{q}(\alpha+1)[\alpha+\beta]_{p} \vert T [\alpha-1]_{p}-\eta[\alpha+\beta-1]_{p}\vert } \biggr) \\ &\qquad{}\times \biggl[ \frac{1}{T^{\nu+1}} \cdot\frac{\vert T [\alpha-1]_{p}-\eta [\alpha+\beta-1]_{p}\vert }{|T [\alpha+\beta-1]_{p}-[\alpha-1]_{p}|} \\ &\qquad{}\times\frac{1}{|T-\eta|} \biggl| T \frac{\Gamma_{w}(\alpha)}{\Gamma_{w}(\alpha -\nu+1)}\cdot\frac{(1-w^{\alpha-\nu})}{(1-w)} - \frac{\Gamma_{w}(\alpha-1)}{\Gamma_{w}(\alpha-\nu)}\cdot\frac{(1-w^{\alpha -\nu-1})}{(1-w)} \biggr| \biggr] \\ &\qquad{} + \biggl(\frac{ N \eta^{2} T^{\alpha-1} |[\alpha-1]_{p}-[\alpha+\beta -1]_{p}| }{n \Gamma_{q}(\alpha+1) \vert T [\alpha-1]_{p}- \eta [\alpha +\beta-1]_{p}\vert } \biggr) \\ &\qquad{}\times \biggl[\frac{1}{T^{\nu}} \cdot\frac{\vert T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}\vert }{|T [\alpha+\beta-1]_{p}-[\alpha-1]_{p}|}\cdot \frac{1}{|[\alpha+\beta-1]_{p}- [\alpha-1]_{p}|} \\ &\qquad{}\times \biggl| \frac{\Gamma_{w}(\alpha)}{\Gamma_{w}(\alpha-\nu+1)}\cdot\frac {(1-w^{\alpha-\nu})}{(1-w)} [\alpha+ \beta-1]_{p}-\frac{\Gamma_{w}(\alpha -1)}{\Gamma_{w}(\alpha-\nu)}\cdot\frac{(1-w^{\alpha-\nu-1})}{(1-w)} [ \alpha-1]_{p} \biggr| \biggr] \biggr\} \\ &\qquad{} +(\tau R+K) \biggl\{ \biggl(\frac{ N T^{\alpha-1} |[\alpha-1]_{p}-[\alpha+\beta-1]_{p}| }{n \vert T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}\vert } \biggr) \\ &\qquad{}\times \biggl[\frac{1}{T^{\nu}} \cdot\frac{\vert T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}\vert }{|T [\alpha+\beta-1]_{p}-[\alpha-1]_{p}|}\cdot \frac{1}{|[\alpha+\beta-1]_{p}- [\alpha-1]_{p}|} \\ &\qquad{}\times \biggl| \frac{\Gamma_{w}(\alpha)}{\Gamma_{w}(\alpha-\nu+1)}\cdot\frac {(1-w^{\alpha-\nu})}{(1-w)} [\alpha+ \beta-1]_{p}-\frac{\Gamma_{w}(\alpha -1)}{\Gamma_{w}(\alpha-\nu)}\cdot\frac{(1-w^{\alpha-\nu-1})}{(1-w)} [ \alpha-1]_{p} \biggr| \biggr] \biggr\} \\ &\quad< R\Theta+M\Omega+ \biggl(M+\frac{\Gamma_{q}(\alpha+1)}{\eta^{2}} \biggr)\Lambda. \end{aligned}$$
Therefore, we obtain \(\|{\mathcal{A}}x\|_{\mathcal{C_{A}}}\leq R\) and hence \(\mathcal{A}B_{R}\subset B_{R}\).
Next, we will show that \(\mathcal{A}\) is a contraction. Denote
$${\mathcal{S}}\bigl[t,x,y,D_{w}^{\nu}x,D_{w}^{\nu}y\bigr]= \bigl|f\bigl(t,x(t),D_{w}^{\nu}x(t)\bigr)-f \bigl(t,y(t),D_{w}^{\nu}y(t)\bigr) \bigr|. $$
For all \(x, y\in\mathcal{C}_{\mathcal{A}}\) and for each \(t\in[0,T]\), we have
$$\begin{aligned} &|\mathcal{A}x-\mathcal{A}y| \\ &\quad\leq \biggl|\frac{1}{\Gamma_{q}(\alpha)} \int_{0}^{t}(t-qs)^{(\alpha-1)}{\mathcal{S}} \bigl[s,x,y,D_{w}^{\nu}x,D_{w}^{\nu}y \bigr] \,d_{q}s \\ &\qquad{}-\frac{t^{\alpha-2}(t-\eta)}{\Gamma_{q}(\alpha) \int _{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s } \int _{0}^{T} \int_{0}^{s} g(r) (T-ps)^{(\beta-1)} \\ &\qquad{}\times(s-qr)^{(\alpha-1)} {\mathcal{S}}\bigl[r,x,y,D_{w}^{\nu}x,D_{w}^{\nu}y\bigr] \,d_{q}r\,d_{p}s \\ &\qquad{}+ \frac{ t^{\alpha-2} \int_{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha -2}(t-s) \,d_{p}s }{\eta^{\alpha-2}\Gamma_{q}(\alpha) \int _{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s } \\ &\qquad{}\times \biggl( \int_{0}^{\eta}(\eta-qs)^{(\alpha-1)} {\mathcal {S}} \bigl[s,x,y,D_{w}^{\nu}x,D_{w}^{\nu}y \bigr] \,d_{q}s-\Gamma_{q}(\alpha) \bigl| \rho(x)-\rho (y)\bigr| \biggr) \biggr| \\ &\quad\leq \lambda\|x-y\|_{\mathcal{C_{A}}} \biggl\{ \frac{T^{\alpha}}{\Gamma _{q}(\alpha+1)}+ \frac{N T^{\alpha} |T-\eta|\Gamma_{p}(\alpha+1)}{ n \Gamma_{p}(\alpha-1)\Gamma_{q}(\alpha+1)[\alpha+\beta]_{p} \vert T [\alpha-1]_{p}-\eta[\alpha+\beta-1]_{p}\vert } \\ &\qquad{} + \frac{ N \eta^{2} T^{\alpha-1} |[\alpha+\beta-1]_{p}- [\alpha-1]_{p}| }{n \Gamma_{q}(\alpha+1) \vert T [\alpha-1]_{p}- \eta [\alpha+\beta -1]_{p}\vert } \biggr\} \\ &\qquad{} + \tau\|x-y\|_{\mathcal{C_{A}}} \biggl\{ \frac{ N T^{\alpha-1} |[\alpha+\beta-1]_{p}- [\alpha-1]_{p}| }{n \vert T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}\vert } \biggr\} \\ &\quad=\|x-y\|_{\mathcal{C_{A}}}\Theta \end{aligned}$$
and
$$\begin{aligned} &\bigl|D_{w}^{\nu}\mathcal{A}x-D_{w}^{\nu} \mathcal{A}y\bigr| \\ &\quad= \bigl|D_{w} \bigl(I_{w}^{1-\nu } \mathcal{A}x-I_{w}^{1-\nu}\mathcal{A}y\bigr)\bigr| \\ &\quad= \biggl| D_{w} \biggl\{ -\frac{\int_{0}^{T}\int_{0}^{s}g(r)(T-ps)^{(\beta-1)} (s-qr)^{\alpha-1} {\mathcal{S}}[r,x,y,D_{w}^{\nu}x,D_{w}^{\nu}y] \,d_{q}r\,d_{p}s}{ \Gamma_{w}(1-\nu)\Gamma_{q}(\alpha) \int_{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s} \\ &\qquad{}\times \int_{0}^{t}(t-ws)^{(-\nu)} s^{\alpha-2}(s-\eta) \,d_{w}s \\ &\qquad+ \frac{\int_{0}^{\eta}(\eta-qs)^{\alpha-1} {\mathcal {S}}[s,x,y,D_{w}^{\nu}x,D_{w}^{\nu}y] \,d_{w}s-\Gamma_{q}(\alpha) | \rho(x)-\rho (y)|}{ \eta^{\alpha-2} \Gamma_{w}(1-\nu) \Gamma_{q}(\alpha) \int _{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s } \\ &\qquad{}\times \biggl[ \int_{0}^{t}(t-ws)^{(-\nu)}s^{\alpha-1} \,d_{w} \int _{0}^{T}g(s) (T-ps)^{(\beta-1)}s^{\alpha-2} \,d_{p}s \\ &\qquad{}- \int_{0}^{t}(t-ws)^{(-\nu)}s^{\alpha-2} \,d_{w}s \int _{0}^{T}g(s) (T-ps)^{(\beta-1)}s^{\alpha-1} \,d_{p}s \biggr] \\ &\qquad{}+\frac{1}{\Gamma_{q}(\alpha)\Gamma_{w}(1-\nu)} \int_{0}^{t} \int _{0}^{s}(t-ws)^{(-\nu)} (s-qr)^{\alpha-1} {\mathcal{S}}\bigl[r,x,y,D_{w}^{\nu}x,D_{w}^{\nu}y\bigr] \,d_{q}r\,d_{w}s \biggr\} \biggr| \\ & \quad\leq \lambda\|x-y\|_{\mathcal{C_{A}}} \biggl\{ \frac{T^{\alpha}}{\Gamma _{q}(\alpha+1)}+ \frac{N T^{\alpha} |T-\eta|\Gamma_{p}(\alpha+1)}{ n \Gamma_{p}(\alpha-1)\Gamma_{q}(\alpha+1)[\alpha+\beta]_{p} \vert T [\alpha-1]_{p}-\eta[\alpha+\beta-1]_{p}\vert } \\ &\qquad{} + \frac{ N \eta^{2} T^{\alpha-1} |[\alpha+\beta-1]_{p}- [\alpha-1]_{p}| }{n \Gamma_{q}(\alpha+1) \vert T [\alpha-1]_{p}- \eta [\alpha+\beta -1]_{p}\vert } \biggr\} \\ &\qquad{} + \tau\|x-y\|_{\mathcal{C_{A}}} \biggl\{ \frac{ N T^{\alpha-1} |[\alpha+\beta-1]_{p}- [\alpha-1]_{p}| }{n \vert T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}\vert } \biggr\} \\ &\quad=\|x-y\|_{\mathcal{C_{A}}}\Theta. \end{aligned}$$
Thus, \(\|{\mathcal{A}}x-{\mathcal{A}}y\|_{\mathcal{C_{A}}}\leq \Theta\|x-y\|_{\mathcal{C_{A}}}\). From (H4), we see that \(\mathcal{A}\) is a contraction.
Hence, the conclusion of the theorem follows by Banach’s contraction mapping principle. This completes the proof. □
Our second result is based on the following Krasnoselskii fixed point theorem.
Theorem 3.2
(Krasnoselskii fixed point theorem) [30]
Let
K
be a bounded closed convex and nonempty subset of a Banach space
X. Let
A, B
be operators such that
-
(i)
\(Ax+By \in K\)
whenever
\(x,y \in K\),
-
(ii)
A
is compact and continuous,
-
(iii)
B
is a contraction mapping.
Then there exists
\(z \in K\)
such that
\(z = Az+Bz\).
Theorem 3.3
(Arzela-Ascoli theorem) [30]
Let
\(D\subseteq\mathbb{R}^{n}\)
be a bounded domain, \(K\subseteq C(\overline{D},\mathbb{R})\)
be bounded and the following property of equicontinuity holds. For every
\(\epsilon>0\), there exists
\(\delta>0\), so that
$$\|x-y\|< \delta\quad\Rightarrow\quad\bigl|u(x)-u(y)\bigr|< \epsilon,\quad \forall x,y \in \overline{D}, \forall u\in K. $$
Then
K̅
is compact.
Theorem 3.4
Assume that (H2)-(H3) hold. In addition, \(f:[0,T]\times\mathbb {R}\times\mathbb{R}\rightarrow\mathbb{R}\)
is a continuous function satisfying the following condition:
- (H5):
-
For all
\((t,x,D_{w}^{\nu}x)\in[0,T]\times \mathbb{R}\times \mathbb{R}\), with
\(\mu\in C([0,T], \mathbb{R}^{+})\),
$$\bigl|f\bigl(t,x,D_{w}^{\nu}x\bigr)\bigr|\leq\mu(t). $$
If
$$\begin{aligned} \Phi:=\|\mu\|(\Omega+\Lambda)+\frac{\alpha\tau}{\eta^{2}}\Lambda < 1, \end{aligned}$$
(3.4)
then the boundary value problem (1.3) has at least one solution on
\([0,T]\).
Proof
Set \(\sup_{t\in[0,T]}|\mu(t)|=\| \mu\|\), and choose a constant
In view of Lemma 2.4, we define the operators \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\) on the ball \(B_{R}=\{x\in\mathcal {C}_{\mathcal{A}}:\|x\|_{\mathcal{C_{A}}}\leq R\}\) by
$$\begin{aligned}& \begin{aligned}[b] (\mathcal{A}_{1}x) (t) ={}&{-}\frac{t^{\alpha-2}(t-\eta)}{\Gamma_{q}(\alpha)\int _{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s } \\ &{}\times \int_{0}^{T} \int_{0}^{s}g(r) (T-ps)^{(\beta-1)}(s-qr)^{(\alpha -1)}f \bigl(r,x(r),D_{w}^{\nu}x(r)\bigr)\,d_{q}r\,d_{p}s \\ &{}+ \frac{ t^{\alpha-2} \int_{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha -2}(t-s) \,d_{p}s }{\eta^{\alpha-2}\Gamma_{q}(\alpha) \int _{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s } \\ &{}\times \biggl( \int_{0}^{\eta}(\eta-qs)^{(\alpha-1)}f \bigl(s,x(s),D_{w}^{\nu}x(s)\bigr)\,d_{q}s- \Gamma_{q}(\alpha) \rho(x) \biggr), \end{aligned} \end{aligned}$$
(3.6)
$$\begin{aligned}& (\mathcal{A}_{2}x) (t) =\frac{1}{\Gamma_{q}(\alpha)} \int_{0}^{t}(t-qs)^{(\alpha -1)}f \bigl(s,x(s),D_{w}^{\nu}x(s)\bigr)\,d_{q}s. \end{aligned}$$
(3.7)
For all \(x, y \in B_{R}\), by computing directly, we have
$$\begin{aligned} &|\mathcal{A}_{1}x+\mathcal{A}_{2}y| \\ &\quad\leq\|\mu\| \biggl\{ \frac{T^{\alpha}}{\Gamma_{q}(\alpha+1)}+\frac{N T^{\alpha} |T-\eta|\Gamma_{p}(\alpha+1)}{ n \Gamma_{p}(\alpha-1)\Gamma_{q}(\alpha+1)[\alpha+\beta]_{p} \vert T [\alpha-1]_{p}-\eta[\alpha+\beta-1]_{p}\vert } \\ &\qquad{} + \frac{ N \eta^{2} T^{\alpha-1} |[\alpha+\beta-1]_{p}- [\alpha-1]_{p}| }{n \Gamma_{q}(\alpha+1) \vert T [\alpha-1]_{p}- \eta [\alpha+\beta -1]_{p}\vert } \biggr\} + \tau \biggl\{ \frac{ N T^{\alpha-1} |[\alpha+\beta-1]_{p}- [\alpha -1]_{p}| }{n \vert T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}\vert } \biggr\} \\ &\quad= \|\mu\|(\Omega+\Lambda)+\frac{\alpha\tau}{\eta^{2}}\Lambda \\ &\quad=\Phi \leqslant R. \end{aligned}$$
Similarly to the proof above and Theorem 3.1, we obtain \(\|D_{w}^{\nu}\mathcal{A}_{1}x+D_{w}^{\nu}\mathcal{A}_{2}y \|< R\), and hence \(\|\mathcal{A}_{1}x+\mathcal{A}_{2}y\|_{\mathcal{C_{A}}}< R\). Therefore,
$$\mathcal{A}_{1}x+\mathcal{A}_{2}y\in B_{R}. $$
The condition (3.3) implies that \(\mathcal{A}_{2}\) is a contraction mapping.
Next, we will show that \(\mathcal{A}_{1}\) is compact and continuous. Continuity of f coupled with the assumption (H4) implies that the operator \(\mathcal{A}_{1}\) is continuous and uniformly bounded on \(B_{R}\). For \(t_{1}, t_{2} \in[0,T]\) with \(t_{1}< t_{2}\), we have
$$\begin{aligned} &\bigl|\mathcal{A}_{1}x(t_{2})-\mathcal{A}_{1}x(t_{1})\bigr| \\ &\quad\leq \bigl( \bigl|t_{2}^{\alpha-2}-t_{1}^{\alpha-2}\bigr|+ \eta \bigl|t_{2}^{\alpha -1}-t_{1}^{\alpha-1}\bigr| \bigr) \\ &\qquad{}\times \frac{\int_{0}^{T}\int_{0}^{s}g(r)(T-ps)^{(\beta-1)}(s-qr)^{(\alpha -1)}f(r,x(r),D_{w}^{\nu}x(r))\,d_{q}r\,d_{p}s}{\Gamma_{q}(\alpha)\int _{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s } \\ &\qquad{}+ \bigl|t_{2}^{\alpha-2}-t_{1}^{\alpha-2} \bigr| \biggl( \int_{0}^{\eta}(\eta -qs)^{(\alpha-1)}f \bigl(s,x(s),D_{w}^{\nu}x(s)\bigr)\,d_{q}s+ \Gamma_{q}(\alpha) \rho (x) \biggr) \\ &\qquad{}\times \frac{ \int_{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(t-s) \,d_{p}s }{\eta^{\alpha-2}\Gamma_{q}(\alpha) \int_{0}^{T}g(s)(T-ps)^{(\beta-1)} s^{\alpha-2}(s-\eta) \,d_{p}s } \\ &\quad\leq\|\mu\| \biggl\{ \frac{ ( |t_{2}^{\alpha-2}-t_{1}^{\alpha-2}|+\eta |t_{2}^{\alpha-1}-t_{1}^{\alpha-1}| ) N T^{2} \Gamma_{p}(\alpha+1)}{ n \Gamma_{p}(\alpha-1)\Gamma_{q}(\alpha+1)[\alpha+\beta]_{p} \vert T [\alpha-1]_{p}-\eta[\alpha+\beta-1]_{p}\vert } \\ &\qquad{} + \frac{ N \eta^{2} T \vert t_{2}^{\alpha-2}-t_{1}^{\alpha-2} \vert \vert [\alpha+\beta-1]_{p}- [\alpha-1]_{p}\vert }{n \Gamma_{q}(\alpha+1) \vert T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}\vert } \biggr\} \\ &\qquad{} + \tau \biggl\{ \frac{ N T \vert t_{2}^{\alpha-2}-t_{1}^{\alpha-2} \vert \vert [\alpha+\beta-1]_{p}- [\alpha-1]_{p}\vert }{n \vert T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}\vert } \biggr\} . \end{aligned}$$
Similarly to the proof above and Theorem 3.1, we obtain
$$\begin{aligned} & \bigl|D_{w}^{\mu}\mathcal{A}_{1}x(t_{2})-D_{w}^{\mu}\mathcal{A}_{1}x(t_{1}) \bigr| \\ &\quad< \bigl|\mathcal{A}_{1}x(t_{2})-\mathcal{A}_{1}x(t_{1})\bigr| \\ &\quad\leq\|\mu\| \biggl\{ \frac{ ( |t_{2}^{\alpha-2}-t_{1}^{\alpha-2}|+\eta |t_{2}^{\alpha-1}-t_{1}^{\alpha-1}| ) N T^{2} \Gamma_{p}(\alpha+1)}{ n \Gamma_{p}(\alpha-1)\Gamma_{q}(\alpha+1)[\alpha+\beta]_{p} \vert T [\alpha-1]_{p}-\eta[\alpha+\beta-1]_{p}\vert } \\ &\qquad{} + \frac{ N \eta^{2} T \vert t_{2}^{\alpha-2}-t_{1}^{\alpha-2} \vert \vert [\alpha+\beta-1]_{p}- [\alpha-1]_{p}\vert }{n \Gamma_{q}(\alpha+1) \vert T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}\vert } \biggr\} \\ &\qquad{}+ \tau \biggl\{ \frac{ N T \vert t_{2}^{\alpha-2}-t_{1}^{\alpha-2} \vert \vert [\alpha+\beta-1]_{p}- [\alpha-1]_{p}\vert }{n \vert T [\alpha-1]_{p}- \eta [\alpha+\beta-1]_{p}\vert } \biggr\} . \end{aligned}$$
Actually, as \(|t_{2}-t_{1}|\rightarrow0\), the right-hand side of the above inequality tends to be zero. So \(\mathcal{A}_{1}\) is relatively compact on \(B_{R}\). Hence, by the Arzela-Ascoli theorem, \(\mathcal{A}_{1}\) is compact on \(B_{R}\).
Therefore, all the assumptions of Theorem (3.2) are satisfied and the conclusion of Theorem 3.2 implies that the boundary value problem (1.3) has at least one solution on \([0,T]\). This completes the proof. □