In order to apply a fixed point theorem of the alternative for contractions on a generalized complete metric space to achieve our main result, we want to collect the following facts.
Definition 4.1
For a nonempty set V, a function \(d:V\times V\rightarrow [0,\infty ]\) is called a generalized metric on V if and only if d satisfies
- ⋄:
-
\(d(v_{1},v_{2})=0\) if and only if \(v_{1}=v_{2}\);
- ⋄:
-
\(d(v_{1},v_{2})=d(v_{2},v_{1})\) for all \(v_{1},v _{2}\in V\);
- ⋄:
-
\(d(v_{1},v_{3})\leq d(v_{1},v_{2})+d(v_{2},v_{3})\) for all \(v_{1}, v_{2}, v_{3}\in V\).
Lemma 4.1
([9])
Let
\((V,d)\)
be a generalized complete metric space. Assume that
\(T:V\rightarrow V\)
is a strictly contractive operator with the Lipschitz constant
\(L<1\). If there exists a
\(k\geq 0\)
such that
\(d(T^{k+1} v,T ^{k} v)<\infty \)
for some
v
in
V, then the followings statements are true:
- (\(B_{1}\)):
-
The sequence
\(\{T^{n} v\}\)
converges to a fixed point
\(v^{*}\)
of
T;
- (\(B_{2}\)):
-
The unique fixed point of
T
is
\(v^{*}\in V^{*}= \{u\in V \textrm{ such that } d(T^{k} v,u)<\infty \}\);
- (\(B_{3}\)):
-
If
\(u\in V^{*}\), then
\(d(u,v^{*})\leq \frac{1}{1-L}d(Tu,u)\).
We can introduce some assumptions as follows:
-
\((H_{1})\)
:
-
\(f\in C(J\times \mathbb{R}\times \mathbb{R},\mathbb{R})\).
-
\((H_{2})\)
:
-
There exists a positive constant
\(L_{f}\)
such that
$$\bigl\vert f(t,u_{1},\bar{u_{1}})-f(t,u_{2}, \bar{u_{2}}) \bigr\vert \leq L_{f_{1}} \vert u_{1}-u_{2} \vert +\bar{L}_{f_{2}} \vert \bar{u_{1}}-\bar{u_{2}} \vert , \quad \textit{for each }t\in J\textit{ and all }u_{1}, u_{2}\in \mathbb{R}. $$
-
\((H_{3})\)
:
-
\(g_{k}\in C((s_{k-1},t_{k}]\times \mathbb{R}, \mathbb{R})\)
and there are positive constant
\(L_{gk}\), \(k=1,2,\dots ,m\)
such that
$$\bigl\vert g_{k} (t,u_{1})-g_{k} (t,u_{2}) \bigr\vert \leq L_{gk} \vert u_{1}-u_{2} \vert , \quad \textit{for each }t\in (s_{k-1},t_{k}],\textit{ and all }u_{1},u_{2} \in \mathbb{R}. $$
-
\((H_{4})\)
:
-
Let
\(\varphi \in C(J,\mathbb{R}_{+})\)
be a nondecreasing function. There exists
\(c_{\varphi }> 0\)
such that
$$ \biggl( \int _{0}^{t}\bigl(\varphi (s)\bigr)^{\frac{1}{p}} \,ds \biggr)^{p}\leq C_{\varphi }\varphi (t)\quad \textit{for each } t\in J. $$
(4.1)
Theorem 4.2
Suppose that
\((H_{1})\)
and
\((H_{2})\)
are satisfied and also a function
\(y\in PC(J,\mathbb{R})\)
satisfies (3.1). Then there exists a unique solution
x
of Eq. (1.1) such that
$$ x(t)= \textstyle\begin{cases} \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {}-\frac{\lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds +\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ \quad {} -\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s,x(s),{^{c}}D_{0,t}^{\alpha }( {^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} +\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} -\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds - (\frac{ \Delta (\theta \eta ^{p}-\varGamma (p+1))t^{\beta }}{\varGamma (p+1)\varGamma ( \beta +1)}-1 )x, \\ \quad t\in (0,s_{0}], \\ \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds\\ \quad {} -\frac{ \lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds \\ \quad {}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )}\int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s,x(s),{^{c}}D _{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {}-\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )}\int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ \quad {}-\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds + (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-\lambda ) \\ \quad {}\times \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1}f(s,x(s),{^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{ \beta }+\lambda ))x(s)\,ds \\ \quad {}- (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )\frac{ \lambda }{\varGamma {\beta }} \int _{o}^{t_{k}}(t_{k}-s)^{\beta -1}x(s)\,ds \\ \quad {}- (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )g _{k} (t_{k},x(t_{k}) ), \quad t\in (t_{k},s_{k}], k=1,2, \dots ,m, \\ g_{k}(t_{k},x(t_{k})), \quad t\in (s_{k-1},t_{k}], k=1,2,\dots ,m \end{cases} $$
(4.2)
and
$$\begin{aligned} \bigl\vert y(t)-x(t) \bigr\vert \leq & \biggl\{ \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{\alpha + \beta -r} +\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t^{\beta -r} \\ &{} +\frac{\Delta C_{\varphi }(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T ^{\alpha +\beta -r}\\ &{} +\frac{\lambda \Delta C_{\varphi }(t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \biggl( \frac{1-r}{\beta -r} \biggr)^{1-r}T^{\beta -r} \\ &{} +\frac{\theta \Delta C_{\varphi }(t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl( \frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &{} +\frac{C_{\varphi }\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r} \eta ^{\beta +p-r} \\ &{} + \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &{}\times\frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r}t_{k}^{\alpha +\beta -r} \\ &{} + \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\\ &{}\times\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+1 \biggr\} \\ &{}\times \biggl(\frac{\varphi (t)+\psi }{1-M} \biggr) \end{aligned}$$
(4.3)
for all
\(t\in J\)
if
\(0<\alpha <\beta <1\), with
$$ M=\max \{M_{1},M_{2}\}< 1, $$
(4.4)
where
$$\begin{aligned} M_{1} =&\max \biggl\{ \frac{1}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r} (L_{f_{1}}C_{\varphi }+\bar{L}_{f _{2}} C_{\varphi } )s_{k}^{\alpha +\beta -r}\\ &{} +\frac{\lambda C_{ \varphi }\varphi (t)}{\varGamma {\beta }} \biggl( \frac{1-r}{\beta -r} \biggr)^{1-r}s_{k}^{\beta -r} \\ &{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r} (L _{f_{1}}C_{\varphi }+\bar{L}_{f_{2}} C_{\varphi } )T^{\alpha + \beta -r} \\ &{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r} (L_{f_{1}}C_{\varphi }+\bar{L}_{f_{2}} C_{\varphi } ) \eta ^{\alpha +\beta +p-r} \\ &{}+\frac{C_{\varphi }\varphi (t)\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{ \beta +p-r} \biggr)^{1-r}\eta ^{\beta +p-r}\\ &{} + \biggl(\Delta \frac{(t_{k} ^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}- \varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &{} \times \frac{1}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha + \beta -r} \biggr)^{1-r} (L_{f_{1}}C_{\varphi }+\bar{L}_{f_{2}} C_{ \varphi } )t_{k}^{\alpha +\beta -r} \\ &{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })C_{\varphi } \varphi (t)}{\varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T^{\beta -r}\\ &{} + \biggl(\Delta \frac{(t_{k}^{\beta }-t^{ \beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{ \varGamma (p+1)} \biggr)-1 \biggr) \\ &{} \times \biggl( \frac{\lambda C_{\varphi }\varphi (t)}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+L_{gk} \biggr) \textit{ such that } k=1,2,\dots ,m \biggr\} , \end{aligned}$$
$$\begin{aligned} M_{2} =&\max \biggl\{ \frac{L_{f_{1}}}{\varGamma (\alpha +\beta +1)}s_{k} ^{\alpha +\beta } +\frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta +1)}s _{k}^{\alpha +\beta } + \frac{\lambda }{\varGamma ({\beta }+1)}s_{k}^{ \beta }\\ &{} +\frac{\Delta (t_{k}^{\beta }-t^{\beta })L_{f_{1}}}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +1)}T^{\alpha +\beta } \\ &{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })\bar{L}_{f_{2}}}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +1)}T^{\alpha +\beta } +\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +1)}T ^{\beta }\\ &{} + \frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })L_{f_{1}}}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p+1)}\eta ^{\alpha +\beta +p} \\ &{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })\bar{L}_{f_{2}}}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p+1)}\eta ^{\alpha +\beta +p} +\frac{ \theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta +p+1)}\eta ^{\beta +p} \\ &{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &{}\times\biggl( \frac{L_{f_{1}}}{\varGamma (\alpha +\beta +1)}t_{k}^{\alpha +\beta }+ \frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta +1)}t_{k}^{ \alpha +\beta } \biggr) \\ &{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \\ &{}\times\biggl(\frac{\lambda }{\beta \varGamma {\beta }} t_{k}^{\beta }+L_{gk} \biggr)\textit{ such that }k=0,1,\dots ,m \biggr\} . \end{aligned}$$
Proof
Consider the space of piecewise continuous functions
$$ V= \bigl\{ g:J\rightarrow \mathbb{R} \textrm{ such that } g\in PC(J, \mathbb{R}) \bigr\} , $$
endowed with the generalized metric on V, defined by
$$\begin{aligned} d(g,h) =&\inf \bigl\{ C_{1}+C_{2}\in [0,+\infty ] \\ &{} \text{such that } \bigl\vert g(t)-h(t) \bigr\vert \leq (C_{1}+ C_{2}) \bigl(\varphi (t)+ \psi \bigr) \textrm{ for all } t\in J \bigr\} , \end{aligned}$$
(4.5)
where
$$\begin{aligned} C_{1}\in \bigl\{ C\in [0,\infty ] \text{ such that } \bigl\vert g(t)-h(t) \bigr\vert \leq C \varphi (t)\text{ for all } t\in (t_{k},s_{k}], k=0,1, \dots ,m \bigr\} \end{aligned}$$
and
$$\begin{aligned} C_{2}\in \bigl\{ C\in [0,\infty ] \text{ such that } \bigl\vert g(t)-h(t) \bigr\vert \leq C\psi \text{ for all } t\in (s_{k-1},t_{k}], k=1,2, \dots ,m \bigr\} . \end{aligned}$$
It is easy to verify that \((V,d)\) is a complete generalized metric space [19].
Define an operator \(\varLambda :V\rightarrow V\) by
$$ (\varLambda x) (t)= \textstyle\begin{cases} \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {}-\frac{\lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds +\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ \quad {} -\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s,x(s),{^{c}}D_{0,t}^{\alpha }( {^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} +\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} -\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds - (\frac{ \Delta (\theta \eta ^{p}-\varGamma (p+1))t^{\beta }}{\varGamma (p+1)\varGamma ( \beta +1)}-1 )x_{0}, \\ \quad t\in (0,s_{0}], \\ \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} -\frac{\lambda }{\varGamma {\beta }}\int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds -\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )}\int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds \\ \quad {} +\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )}\int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s,x(s),{^{c}}D _{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} -\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)}\int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,x(s), {^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{\beta }+\lambda ))x(s)\,ds \\ \quad {} +\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)}\int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds + (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-\lambda ) \\ \quad {} \times \frac{1}{\varGamma (\alpha +\beta )}\int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1}f(s,x(s),{^{c}}D_{0,t}^{\alpha }({^{c}}D_{0,t}^{ \beta }+\lambda ))x(s)\,ds \\ \quad {} - (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1}x(s)\,ds \\ \quad {} - (\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} (\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} )-1 )g _{k}(t_{k},x(t_{k})),\\ \quad t\in (t_{k},s_{k}], k=1,2,\dots ,m, \\ g_{k}(t_{k},x(t_{k})), \quad t\in (s_{k-1},t_{k}], k=1,2,\dots ,m \end{cases} $$
(4.6)
for all x belongs to V and \(t\in J\). Obviously, according to \((H_{1})\), Λ is a well-defined operator.
Next we shall verify that Λ is strictly contractive on V. Note that according to definition of \((V,d)\), for any \(g,h\in V\), it is possible to find \(C_{1},C_{2},C_{3},C_{4}\in [0,\infty ]\) such that
$$ \bigl\vert g(t)-h(t) \bigr\vert \leq \textstyle\begin{cases} C_{1}\varphi (t),\quad t\in (t_{k},s_{k}], k=0,\dots ,m, \\ C_{2}\psi , \quad t\in (s_{k-1},t_{k}],k=1,\dots ,m, \end{cases} $$
(4.7)
and
$$\begin{aligned} &\bigl\vert {^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)- {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s) \bigr\vert \\ &\quad \leq \textstyle\begin{cases} C_{3}\zeta (t)\leq C_{1}\varphi (t),\quad t\in (t_{k},s_{k}], k=0, \dots ,m, \\ C_{4}\zeta (t)\leq C_{2}\psi , \quad t\in (s_{k-1},t_{k}], k=1, \dots ,m. \end{cases}\displaystyle \end{aligned}$$
From the definition of Λ in Eq. (4.6), \((H_{2})\), \((H_{3})\) and (4.7), we obtain that
Case 1. For \(t\in [0,s_{0}]\),
$$\begin{aligned} &\bigl\vert (\varLambda g) (t)-(\varLambda h) (t) \bigr\vert \\ &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{ \beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{ \beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{ \beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha + \beta -1} \\ &\qquad{} \times \bigl[L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \bigr]\,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl[L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \bigr]\,ds \\ &\qquad{}+\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \\ &\qquad{} \times \bigl[L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \bigr]\,ds \\ &\qquad{}+\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\quad =\frac{L_{f_{1}}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{L_{f_{1}}\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{ \bar{L}_{f_{2}}\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta )} \\ &\qquad{}\times \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{ \alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha }\bigl( {^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{L_{f_{1}}\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad {}+\frac{\bar{L}_{f_{2}}\theta \Delta t^{\beta }}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \\ &\qquad{}\times \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \bigl\vert {^{c}}D _{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\quad \leq \frac{L_{f_{1}}C_{1}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds +\frac{\lambda C_{1}}{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}C_{1}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad {}+\frac{L_{f_{1}}C_{1}\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{ \alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda C_{1}\Delta t^{\beta }}{\varGamma (\beta )\varGamma ( \beta +1)} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad {}+\frac{ \bar{L}_{f_{2}} C_{1}\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{L_{f_{1}} C_{1}\theta \Delta t^{\beta }}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}} C_{1}\theta \Delta t^{\beta }}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{C_{1}\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1) \varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert \varphi (s) \bigr\vert \,ds \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{L_{f_{1}} C_{1}}{\varGamma (\alpha +\beta )} \biggl( \int _{0} ^{t}(t-s)^{\frac{\alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0} ^{t} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\bar{L}_{f_{2}}C_{1}}{\varGamma (\alpha +\beta )} \biggl( \int _{0} ^{t}(t-s)^{\frac{\alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0} ^{t} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\lambda C_{1}}{\varGamma {\beta }} \biggl( \int _{o}^{t}(t-s)^{\frac{ \beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{t} \bigl(\varphi (s) \bigr)^{ \frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{L_{f_{1}} C_{1}\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \biggl( \int _{0}^{T}(T-s)^{\frac{\alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{T} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\bar{L}_{f_{2}} C_{1}\Delta t^{\beta }}{\varGamma (\beta +1) \varGamma (\alpha +\beta )} \biggl( \int _{0}^{T}(T-s)^{\frac{\alpha + \beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{T} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{C_{1}\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma ( \beta +1)} \biggl( \int _{0}^{T}(T-s)^{\frac{\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{T} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{L_{f_{1}} C_{1}\theta \Delta t^{\beta }}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \biggl( \int _{0}^{\eta }(\eta -s)^{\frac{ \alpha +\beta +p-1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{\eta }\bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\bar{L}_{f_{2}} C_{1}\theta \Delta t^{\beta }}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +p)} \biggl( \int _{0}^{\eta }(\eta -s)^{\frac{ \alpha +\beta +p-1}{1-r}}\,ds \biggr)^{1-r} \bigl(\varphi (s) \bigr)^{ \frac{1}{r}}\,ds )^{r} \\ &\qquad{}+\frac{\theta \Delta \lambda t^{\beta }C_{1}}{\varGamma (\beta +1) \varGamma (\beta +p)} \biggl( \int _{0}^{\eta }(\eta -s)^{ \frac{\beta +p-1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{\eta } \bigl( \varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\quad \leq \frac{L_{f_{1}} C_{1}C_{\varphi }\varphi (t)}{\varGamma (\alpha + \beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{\alpha + \beta -r} +\frac{\bar{L}_{f_{2}} C_{1}C_{\varphi }\varphi (t)}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{ \alpha +\beta -r} \\ &\qquad{}+\frac{\lambda C_{1}C_{\varphi }\varphi (t)}{\varGamma {\beta }} \biggl(\frac{r-1}{ \beta -r} \biggr)^{1-r}t^{\beta -r} +\frac{L_{f_{1}} C_{1}C_{\varphi } \varphi (t)\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T^{\alpha +\beta -r} \\ &\qquad{}+\frac{\bar{L}_{f_{2}} C_{1}C_{\varphi }\varphi (t)\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha + \beta -r} \biggr)^{1-r}T^{\alpha +\beta -r} +\frac{C_{1}C_{\varphi } \varphi (t)\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T^{\beta -r} \\ &\qquad{}+\frac{L_{f_{1}} C_{1}C_{\varphi }\varphi (t)\theta \Delta t^{ \beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{ \alpha +\beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &\qquad{}+\frac{\bar{L}_{f_{2}} C_{1}C_{\varphi }\varphi (t)\theta \Delta t ^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{ \alpha +\beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &\qquad{}+\frac{\theta \Delta \lambda t^{\beta }C_{1}C_{\varphi }\varphi (t)}{ \varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r} \eta ^{\beta +p-r} \\ &\quad \leq \biggl\{ \frac{1}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r} (L_{f_{1}} +\bar{L}_{f_{2}} )s _{0}^{\alpha +\beta -r}+\frac{\lambda \Delta t^{\beta }}{\varGamma ( \beta )\varGamma (\beta +1)} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T^{ \beta -r} \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \biggl(\frac{r-1}{\beta -r} \biggr)^{1-r}s _{0}^{\beta -r} +\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r} (L _{f_{1}}+\bar{L}_{f_{2}} )T^{\alpha +\beta -r} \\ &\qquad{}+\frac{\theta \Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha + \beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r} (L_{f _{1}} +\bar{L}_{f_{2}} )\eta ^{\alpha +\beta +p-r} \\ &\qquad{}+\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r}\eta ^{\beta +p-r} \biggr\} C_{1} C_{\varphi }\varphi (t). \end{aligned}$$
Case 2. For \(t\in (s_{k-1},t_{k}]\), we have
$$ \bigl\vert (\varLambda g)t-(\varLambda h)t \bigr\vert = \bigl\vert g_{k}\bigl(t,g(t)\bigr)-g_{k}\bigl(t,h(t)\bigr) \bigr\vert \leq L_{gk} \bigl\vert g(t)-h(t) \bigr\vert \leq L_{gk}C_{2}\psi . $$
Case 3. For \(t\in (t_{k},s_{k}]\) and \(s\in (t_{k},s_{k}]\),
$$\begin{aligned} &\bigl\vert (\varLambda g) (t)-(\varLambda h) (t) \bigr\vert \\ &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds+\frac{ \Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha + \beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)\bigr))-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{ \beta }+ \lambda \bigr)h(s)\bigr)) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \bigl\vert g\bigl(t _{k},g(t_{k})\bigr)-g \bigl(t_{k},h(t_{k})\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha + \beta -1} \\ &\qquad{} \times \bigl(L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D _{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s) \bigr\vert \bigr)\,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl(L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D _{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s) \bigr\vert \bigr)\,ds \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \\ &\qquad{} \times \bigl(L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D _{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s) \bigr\vert \bigr)\,ds \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1} \\ &\qquad{} \times [L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \bigl\vert g\bigl(t _{k},g(t_{k})\bigr)-g \bigl(t_{k},h(t_{k})\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad =\frac{L_{f_{1}}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds+\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\Delta L_{f_{1}} (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{\Delta \bar{L}_{f_{2}}(t_{k}^{\beta }-t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta )} \\ &\qquad{}\times \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{ \alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha }\bigl( {^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{L_{f_{1}}\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}\theta \Delta (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \\ &\qquad{}\times \int _{0}^{\eta }(\eta -s)^{ \alpha +\beta +p-1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{ \beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl( \frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{}\times \frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta )} \int _{0}^{t _{k}}(t_{k}-s)^{\alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D _{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t} ^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{L_{f_{1}}}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t _{k}-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \bigl\vert g\bigl(t _{k},g(t_{k})\bigr)-g \bigl(t_{k},h(t_{k})\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{L_{f_{1}}C_{1}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds +\frac{\bar{L}_{f_{2}}C_{1}}{ \varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda C_{1}}{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds +\frac{\Delta L_{f_{1}}C_{1} (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{ \alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\Delta \bar{L}_{f_{2}}C_{1}(t_{k}^{\beta }-t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha + \beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda C_{1}\Delta (t_{k} ^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \int _{0}^{T}(T-s)^{ \beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{L_{f_{1}}C_{1}\theta \Delta (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{ \alpha +\beta +p-1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}C_{1}\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }( \eta -s)^{\alpha +\beta +p-1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta C_{1}\Delta \lambda (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{L_{f_{1}}C_{1}}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t _{k}-s)^{\alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{\bar{L}_{f_{2}}C_{1}}{\varGamma (\alpha +\beta )} \int _{0} ^{t_{k}}(t_{k}-s)^{\alpha +\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda C_{1}}{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert \varphi (s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)L _{gk} \bigl\vert g(t_{k})-h(t_{k})) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{L_{f_{1}} C_{1}}{\varGamma (\alpha +\beta )} \biggl( \int _{0} ^{t}(t-s)^{\frac{\alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0} ^{t}\bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\bar{L}_{f_{2}} C_{1}}{\varGamma (\alpha +\beta )} \biggl( \int _{0}^{t}(t-s)^{\frac{\alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{t} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\lambda C_{1}}{\varGamma {\beta }} \biggl( \int _{o}^{t}(t-s)^{\frac{ \beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{t}\bigl(\varphi (s)\bigr)^{ \frac{1}{r}} \,ds \biggr)^{r} \\ &\qquad{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })L_{f_{1}} C_{1}}{\varGamma ( \beta +1)\varGamma (\alpha +\beta )} \biggl( \int _{0}^{T}(T-s)^{\frac{ \alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{T}\bigl(\varphi (s)\bigr)^{ \frac{1}{r}} \,ds \biggr)^{r} \\ &\qquad{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })\bar{L}_{f_{2}} C_{1}}{ \varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl( \int _{0}^{T}(T-s)^{\frac{ \alpha +\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{T} \bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })C_{1}}{\varGamma ( \beta +1)\varGamma (\beta )} \biggl( \int _{0}^{T}(T-s)^{ \frac{\beta -1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{T}\bigl(\varphi (s)\bigr)^{ \frac{1}{r}} \,ds \biggr)^{r} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })L_{f_{1}} C_{1}}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl( \int _{0}^{\eta }( \eta -s)^{\frac{\alpha +\beta +p-1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0} ^{\eta }\bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })\bar{L}_{f_{2}} C _{1}}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl( \int _{0}^{ \eta }(\eta -s)^{\frac{\alpha +\beta +p-1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{\eta }\bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+\frac{C_{1}\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\beta +p)} \biggl( \int _{0}^{\eta }(\eta -s)^{\frac{ \beta +p-1}{1-r}}\,ds \biggr)^{1-r} \biggl( \int _{0}^{\eta }\bigl(\varphi (s) \bigr)^{ \frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{}\times\frac{L_{f_{1}} C_{1}}{\varGamma (\alpha +\beta )} \biggl( \int _{0} ^{t_{k}}(t_{k}-s)^{\frac{\alpha +\beta -1}{1-r}} \,ds \biggr)^{1-r} \biggl( \int _{0}^{t_{k}}\bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{}\times\frac{\bar{L}_{f_{2}} C_{1}}{\varGamma (\alpha +\beta )} \biggl( \int _{0}^{t_{k}}(t_{k}-s)^{\frac{\alpha +\beta -1}{1-r}} \,ds \biggr)^{1-r} \biggl( \int _{0}^{t_{k}}\bigl(\varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \\ &\qquad{}\times \frac{\lambda C_{1}}{\varGamma {\beta }} \biggl( \int _{o}^{t_{k}}(t _{k}-s)^{\frac{\beta -1}{1-r}} \,ds \biggr)^{1-r} \biggl( \int _{0}^{t_{k}}\bigl( \varphi (s) \bigr)^{\frac{1}{r}}\,ds \biggr)^{r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)L _{gk}C_{2}\psi \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{L_{f_{1}} C_{1} C_{\varphi }\varphi (t)}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{\alpha + \beta -r} +\frac{\bar{L}_{f_{2}} C_{1} C_{\varphi }\varphi (t)}{ \varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t ^{\alpha +\beta -r} \\ &\qquad{}+\frac{\lambda C_{1}C_{\varphi }\varphi (t)}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t^{\beta -r}+ \frac{\Delta (t_{k}^{\beta }-t^{ \beta })L_{f_{1}} C_{1}C_{\varphi }\varphi (t)}{\varGamma (\beta +1) \varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T ^{\alpha +\beta -r} \\ &\qquad{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })\bar{L}_{f_{2}} C_{1}C_{ \varphi }\varphi (t)}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r}T^{\alpha +\beta -r} \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })C_{1}C_{\varphi } \varphi (t)}{\varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T^{\beta -r} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })L_{f_{1}} C_{1}C _{\varphi }\varphi (t)}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })\bar{L}_{f_{2}} C _{1}C_{\varphi }\varphi (t)}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &\qquad{}+\frac{C_{1}C_{\varphi }\varphi (t)\theta \Delta \lambda (t_{k}^{ \beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{ \beta +p-r} \biggr)^{1-r}\eta ^{\beta +p-r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{L_{f_{1}} C_{1}C_{\varphi }\varphi (t)}{\varGamma (\alpha + \beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t_{k}^{\alpha + \beta -r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{\bar{L}_{f_{2}} C_{1}C_{\varphi }\varphi (t)}{\varGamma ( \alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t_{k} ^{\alpha +\beta -r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{\lambda C_{1}C_{\varphi }\varphi (t)}{\varGamma {\beta }} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}t_{k}^{\beta -r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)L _{gk}C_{2}\psi \\ &\quad \leq \biggl\{ \frac{1}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r} (L_{f_{1}}+\bar{L}_{f_{2}} )s_{0} ^{\alpha +\beta -r} +\frac{\lambda }{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}s_{0}^{\beta -r} \\ &\qquad{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r} (L _{f_{1}}+\bar{L}_{f_{2}} )T^{\alpha +\beta -r} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r} (L_{f_{1}}+\bar{L}_{f_{2}} )\eta ^{\alpha +\beta +p-r} \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r} \eta ^{\beta +p-r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{ \varGamma (\beta +1)} \biggl( \frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{} \times \frac{1}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha + \beta -r} \biggr)^{1-r} (L_{f_{1}}+\bar{L}_{f_{2}} )t_{k}^{\alpha +\beta -r} + \frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T^{ \beta -r} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \biggl(\frac{\lambda }{\varGamma {\beta }} \biggl( \frac{1-r}{\beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+L_{gk} \biggr) \biggr\} C_{\varphi } \\ &\qquad{}\times (C_{1}+C_{2} ) \bigl( \varphi (t)+\psi \bigr). \end{aligned}$$
Also, for \(t\in (t_{k},s_{k}]\) and \(s\in (s_{k-1},t_{k}]\), we have
$$\begin{aligned} &\bigl\vert (\varLambda g) (t)-(\varLambda h) (t) \bigr\vert \\ &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1} \\ &\qquad{} \times \bigl\vert f\bigl(s,g(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr)-f \bigl(s,h(s),{^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \bigl\vert g\bigl(t _{k},g(t_{k})\bigr)-g \bigl(t_{k},h(t_{k})\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha + \beta -1} \\ &\qquad{} \times \bigl[L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D _{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)h(s) \bigr\vert \bigr]\,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds +\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \\ &\qquad{} \times [L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \\ &\qquad{} \times [L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1} \\ &\qquad{} \times [L_{f_{1}} \bigl\vert g(s)-h(s) \bigr\vert + \bar{L}_{f_{2}} \bigl\vert {^{c}}D _{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \bigl\vert g\bigl(t _{k},g(t_{k})\bigr)-g \bigl(t_{k},h(t_{k})\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad =\frac{L_{f_{1}}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\Delta L_{f_{1}} (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1) \varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\Delta \bar{L}_{f_{2}}(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)\varGamma (\alpha +\beta )} \\ &\qquad{}\times \int _{0}^{T}(T-s)^{\alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)- {^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{L_{f_{1}}\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}\theta \Delta (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \\ &\qquad{}\times \int _{0}^{\eta }(\eta -s)^{ \alpha +\beta +p-1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{} + \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl( \frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{}\times \frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta )} \int _{0}^{t _{k}}(t_{k}-s)^{\alpha +\beta -1} \bigl\vert {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D _{0,t}^{\beta }+\lambda \bigr)g(s)-{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t} ^{\beta }+\lambda \bigr)h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{}\times \frac{L_{f_{1}}}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t _{k}-s)^{\alpha +\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{\beta -1} \bigl\vert g(s)-h(s) \bigr\vert \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \bigl\vert g\bigl(t _{k},g(t_{k})\bigr)-g \bigl(t_{k},h(t_{k})\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{L_{f_{1}}C_{2}\psi }{\varGamma (\alpha +\beta )} \int _{0} ^{t}(t-s)^{\alpha +\beta -1}\,ds + \frac{\bar{L}_{f_{2}}C_{2}\psi }{ \varGamma (\alpha +\beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1}\,ds \\ &\qquad{}+ \frac{ \lambda C_{2}\psi }{\varGamma {\beta }} \int _{0}^{t}(t-s)^{\beta -1}\,ds +\frac{\Delta L_{f_{1}}C_{2}\psi (t_{k}^{\beta }-t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha + \beta -1}\,ds \\ &\qquad{}+ \frac{\Delta \bar{L}_{f_{2}}C_{2}\psi (t_{k}^{\beta }-t ^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{ \alpha +\beta -1}\,ds \\ &\qquad{}+\frac{\lambda C_{2}\psi \Delta (t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1}\,ds \\ &\qquad{}+ \frac{L _{f_{1}}C_{2}\psi \theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}\,ds \\ &\qquad{}+\frac{\bar{L}_{f_{2}}C_{2}\psi \theta \Delta (t_{k}^{\beta }- t ^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \int _{0}^{ \eta }(\eta -s)^{\alpha +\beta +p-1}\,ds \\ &\qquad{}+ \frac{\theta C_{2}\psi \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}\,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{L_{f_{1}}C_{2}\psi }{\varGamma (\alpha +\beta )} \int _{0} ^{t_{k}}(t_{k}-s)^{\alpha +\beta -1} \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{\bar{L}_{f_{2}}C_{2}\psi }{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t_{k}-s)^{\alpha +\beta -1} \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda C_{2}\psi }{\varGamma {\beta }} \int _{0}^{t_{k}}(t_{k}-s)^{ \beta -1} \,ds \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)L _{gk} \bigl\vert g(t_{k})-h(t_{k})) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\quad \leq \frac{L_{f_{1}} C_{2}\psi }{\varGamma (\alpha +\beta )(\alpha + \beta )}t^{\alpha +\beta } +\frac{\bar{L}_{f_{2}} C_{2}\psi }{\varGamma (\alpha +\beta )(\alpha +\beta )}t^{\alpha +\beta } +\frac{\lambda C _{2}\psi }{\beta \varGamma {\beta }}t^{\beta } \\ &\qquad{}+\frac{\Delta (t_{k}^{ \beta }-t^{\beta })L_{f_{1}} C_{2}\psi }{\varGamma (\beta +1)(\alpha + \beta )\varGamma (\alpha +\beta )}T^{\alpha +\beta } +\frac{\Delta (t_{k}^{\beta }-t^{\beta })\bar{L}_{f_{2}} C_{2} \psi }{\varGamma (\beta +1)(\alpha +\beta )\varGamma (\alpha +\beta )}T^{ \alpha +\beta } \\ &\qquad{} +\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta }) C _{2}\psi }{\varGamma (\beta +1)\beta \varGamma (\beta )}T^{\beta } +\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })L_{f_{1}} C_{2} \psi }{\varGamma (\beta +1)(\alpha +\beta +p)\varGamma (\alpha +\beta +p)} \eta ^{\alpha +\beta +p} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{ \beta })\bar{L}_{f_{2}} C_{2}\psi }{\varGamma (\beta +1)(\alpha +\beta +p) \varGamma (\alpha +\beta +p)}\eta ^{\alpha +\beta +p} \\ &\qquad{}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta }) C_{2} \psi }{\varGamma (\beta +1)(\beta +p)\varGamma (\beta +p)}\eta ^{\beta +p} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda C_{2}\psi }{\beta \varGamma {\beta }} t_{k}^{\beta } \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{L_{f_{1}} C_{2}\psi }{(\alpha +\beta )\varGamma (\alpha + \beta )}t_{k}^{\alpha +\beta } \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{\bar{L}_{f_{2}} C_{2}\psi }{(\alpha +\beta )\varGamma ( \alpha +\beta )}t_{k}^{\alpha +\beta } \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)L _{gk} C_{2}\psi \\ &\quad \leq \biggl\{ \frac{L_{f_{1}}}{\varGamma (\alpha +\beta +1)}s_{k}^{ \alpha +\beta }+ \frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta +1)}s _{k}^{\alpha +\beta } +\frac{\lambda }{\varGamma ({\beta }+1)}s_{k}^{ \beta } \\ &\qquad{}+ \frac{\Delta (t_{k}^{\beta }-t^{\beta })L_{f_{1}}}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +1)}T^{\alpha +\beta } \\ &\qquad{}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })\bar{L}_{f_{2}}}{\varGamma ( \beta +1)\varGamma (\alpha +\beta +1)}T^{\alpha +\beta }+\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +1)}T ^{\beta } \\ &\qquad{}+ \frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })L_{f_{1}}}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p+1)}\eta ^{\alpha +\beta +p} \\ &\qquad{}+\frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })\bar{L}_{f_{2}}}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p+1)}\eta ^{\alpha +\beta +p} +\frac{ \theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta +p+1)}\eta ^{\beta +p} \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad{}\times \biggl( \frac{L_{f_{1}}}{\varGamma (\alpha +\beta +1)}t_{k}^{\alpha +\beta }+ \frac{\bar{L}_{f_{2}}}{\varGamma (\alpha +\beta +1)}t_{k}^{ \alpha +\beta } \biggr) \\ &\qquad{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \biggl(\frac{\lambda }{\beta \varGamma {\beta }} t_{k}^{\beta }+L_{gk} \biggr) \biggr\} \\ &\qquad{}\times(C_{1}+C_{2}) \bigl(\varphi (t)+\psi \bigr). \end{aligned}$$
From above, we have
$$ \bigl\vert (\varLambda g) (t)-(\varLambda h) (t) \bigr\vert \leq M(C_{1}+C_{2}) \bigl(\varphi (t)+\psi \bigr), \quad t\in [0, \tau ], $$
that is,
$$ d(\varLambda g,\varLambda h)\leq M(C_{1}+C_{2}) \bigl(\varphi (t)+\psi \bigr). $$
Hence, we conclude that
$$ d(\varLambda g,\varLambda h)\leq Md(g,h), \quad \text{for any } g,h\in V. $$
Since condition (4.4) is strictly contractive, continuity property is thus shown. Now we take \(g_{0}\in V\). From the piecewise continuity property of \(g_{0}\) and \(\varLambda g_{0}\), it follows that there exists a constant \(0< G_{1}<\infty \) such that
$$\begin{aligned} &\bigl\vert (\varLambda g_{0}) (t)-g_{0}(t) \bigr\vert \\ &\quad \leq \biggl\vert \frac{1}{\varGamma (\alpha + \beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s,g_{0}(s),{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g_{0}(s)\,ds \\ &\qquad {}-\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds \\ &\qquad {}-\frac{\Delta t^{\beta }}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s,g_{0}(s),{^{c}}D_{0,t}^{ \alpha } \bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g_{0}(s) \,ds \\ &\qquad {}+\frac{\lambda \Delta t^{\beta }}{\varGamma (\beta )\varGamma (\beta +1)} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds+ \frac{\theta \Delta t^{\beta }}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \\ &\qquad {} \times \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,g_{0}(s), {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g_{0}(s)\,ds \\ &\qquad {}-\frac{\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma ( \beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds \\ &\qquad {}+ \biggl(\frac{\Delta (\theta \eta ^{p}-\varGamma (p+1))t^{\beta }}{ \varGamma (p+1)\varGamma (\beta +1)}+1 \biggr)x_{0}-g_{0}(t) \biggr\vert \\ &\quad \leq G_{1} \varphi (t)\leq G_{1}\bigl(\varphi (t)+\psi \bigr), \quad t\in (0,s_{0}]. \end{aligned}$$
There exists a constant \(0< G_{2}<\infty \) such that
$$\begin{aligned} &\bigl\vert (\varLambda g_{0}) (t)-g_{0}(t) \bigr\vert = \bigl\vert g_{k}\bigl(t,g_{0}(t)\bigr)-g_{0}(t) \bigr\vert \leq G_{2} \psi \leq G_{2}\bigl(\varphi (t)+\psi \bigr), \\ &\quad t\in (s_{k-1},t_{k}], k=1,2,\dots ,m. \end{aligned}$$
Also we can find a constant \(0< G_{3}<\infty \) such that
$$\begin{aligned} &\bigl\vert (\varLambda g_{0}) (t)-g_{0}(t) \bigr\vert \\ &\quad \leq \biggl\vert \frac{1}{\varGamma (\alpha + \beta )} \int _{0}^{t}(t-s)^{\alpha +\beta -1}f(s,g_{0}(s),{^{c}}D_{0,t} ^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+\lambda \bigr)g_{0}(s)\,ds \\ &\qquad {}-\frac{\lambda }{\varGamma {\beta }} \int _{o}^{t}(t-s)^{\beta -1}x(s)\,ds \\ &\qquad {}+\frac{\Delta (t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta )} \int _{0}^{T}(T-s)^{\alpha +\beta -1}f(s,g_{0}(s), {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g_{0}(s)\,ds \\ &\qquad {}-\frac{\lambda \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1) \varGamma (\beta )} \int _{0}^{T}(T-s)^{\beta -1}x(s)\,ds- \frac{\theta \Delta (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \\ &\qquad {} \times \int _{0}^{\eta }(\eta -s)^{\alpha +\beta +p-1}f(s,g_{0}(s), {^{c}}D_{0,t}^{\alpha }\bigl({^{c}}D_{0,t}^{\beta }+ \lambda \bigr)g_{0}(s)\,ds \\ &\qquad {}+\frac{\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma ( \beta +1)\varGamma (\beta +p)} \int _{0}^{\eta }(\eta -s)^{\beta +p-1}x(s)\,ds \\ &\qquad {}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \\ &\qquad {}\times \frac{1}{\varGamma (\alpha +\beta )} \int _{0}^{t_{k}}(t_{k}-s)^{ \alpha +\beta -1}f(s,g_{0}(s),{^{c}}D_{0,t}^{\alpha } \bigl({^{c}}D_{0,t} ^{\beta }+\lambda \bigr)g_{0}(s)\,ds \\ &\qquad {}- \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{ \lambda }{\varGamma {\beta }} \int _{o}^{t_{k}}(t_{k}-s)^{\beta -1}x(s) \,ds \\ &\qquad {}- \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)g _{k}(t_{k})-g_{0}(t) \biggr\vert , \quad t\in (t_{k},s_{k}], \\ &\bigl\vert (\varLambda g_{0}) (t)-g_{0}(t) \bigr\vert \leq G_{3}\varphi (t)\leq G_{3}\bigl(\varphi (t)+\psi \bigr), \quad t\in (t_{k},s_{k}], k=1,2,\dots ,m. \end{aligned}$$
Since f, \(g_{k}\) and \(g_{0}\) are bounded on J and \(\varphi (\cdot )>0\), Eq. (4.5) implies that \(d(\varLambda g_{0},g_{0})<\infty \).
By using Banach fixed point theorem, there exists a continuous function \(x:J\rightarrow \mathbb{R}\) such that \(\varLambda ^{n}g_{0}\rightarrow x\) in \((V,d)\) as \(n\rightarrow \infty \) and \(\varLambda x=y_{0}\), that is, x satisfies Eq. (4.2) for every \(t\in J\).
Now we show that \(\{g\in V \textrm{ such that } d(g_{0},g)<\infty \}=V\). For any \(g\in V\), since g and \(g_{0}\) are bounded on J and \(\min_{t\in J}\varphi (t)>0\), there exists a constant \(0< C_{g}<\infty \) such that \(|g_{0}(t)-g(t)|\leq C_{g}(\varphi (t)+\psi)\), for any \(t\in J\). Hence, we have \(d(g_{0},g)<\infty \) for all \(g\in V\), that is, \(\{g\in V \textrm{ such that } d(g_{0},g)<\infty \}=V\). Thus, we determine that x is the unique continuous function satisfying Eq. (4.2). Using (3.2) and \((H_{4})\), we can write
$$\begin{aligned} d(y,\varLambda y) \leq & \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{\alpha +\beta -r} +\frac{ \lambda C_{\varphi }}{\varGamma {\beta }} \biggl( \frac{1-r}{\beta -r} \biggr)^{1-r}t^{\beta -r} \\ &{}+\frac{\Delta C_{\varphi }(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T^{\alpha +\beta -r}\\ &{}+\frac{\lambda \Delta C_{\varphi }(t _{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}T^{\beta -r} \\ &{}+\frac{\theta \Delta C_{\varphi }(t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha + \beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &{}+\frac{C_{\varphi }\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r}\eta ^{\beta +p-r} \\ &{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r}t_{k}^{\alpha +\beta -r} \\ &{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+1. \end{aligned}$$
Summarizing, we have
$$\begin{aligned} d(y,x) \leq & \frac{d(\varLambda y,y)}{1-M} \\ \leq & \biggl(\frac{1}{1-M} \biggr) \biggl\{ \frac{C_{\varphi }}{\varGamma ( \alpha +\beta )} \biggl( \frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{ \alpha +\beta -r} +\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t^{\beta -r} \\ &{}+\frac{\Delta C_{\varphi }(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T^{\alpha +\beta -r}\\ &{}+\frac{\lambda \Delta C_{\varphi }(t _{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}T^{\beta -r} \\ &{}+\frac{\theta \Delta C_{\varphi }(t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha + \beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &{}+\frac{C_{\varphi }\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r}\eta ^{\beta +p-r} \\ &{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r}t_{k}^{\alpha +\beta -r} \\ &{}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} +\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+1 \biggr\} . \end{aligned}$$
This shows that (4.3) is true for \(t\in J\). □
Here, we give an example to illustrate our main result.
Example 4.3
$$ \textstyle\begin{cases} {^{c}}D_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )x(t)\\ \quad = \frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t ^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )x(t)} \\ \qquad {} +\int _{0}^{t}\frac{(t-s)^{\frac{3}{2}}}{\varGamma {\frac{5}{2}}} (\frac{ \vert x(s) \vert }{8+e^{s}+s^{2}} )\,ds, \quad t\in (0,1]\cup (2,3], \\ x(t)=\frac{x(t)}{(3+t^{2})(1+ \vert x(t) \vert )}, \quad t\in (1,2], \\ x(0)=\frac{\sqrt{2}}{3}, \qquad x(1)=\frac{5}{6}\int _{0}^{\frac{1}{4}}\frac{(\frac{1}{4}-s)}{\varGamma \frac{4}{3}}\,ds \quad 0< \eta < 1 \end{cases} $$
(4.8)
and
$$ \textstyle\begin{cases} \vert {^{c}}D_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )y(t)-\frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )y(t)}{8+e^{t}+t ^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )y(t)} \\ \quad {} -\int _{0}^{t}\frac{(t-s)^{\frac{3}{2}}}{\varGamma {\frac{5}{2}}} (\frac{ \vert y(s) \vert }{8+e^{s}+s^{2}} )\,ds \vert \leq e^{t}, \quad t\in (0,1]\cup (2,3], \\ \vert y(t)-\frac{y(t)}{(3+t^{2})(1+ \vert x(t) \vert )} \vert \leq 1, \quad t\in (1,2]. \end{cases} $$
Let \(J=[0,3]\), \(\alpha =\beta =\frac{1}{2}\), \(r=\frac{1}{3}\), \(\Delta =-2.70\), \(\theta =\frac{5}{6}\), \(p=\frac{4}{3}\), \(\eta = \frac{1}{4}\) and \(0=t_{0}< s_{0}=1< t_{1}=2< s_{1}=\tau =3\). Denote \(f(t,x(t))=\frac{|x(t)|}{8+e^{t}+t^{2}}\) with \(L_{f}=\frac{1}{9}\) for \(t\in (0,1]\cup (2,3]\) and \(g_{1}(t,x(t))= \frac{x(t)}{(3+t^{2})(1+|x(t)|)}\) with \(L_{g_{k}}=\frac{1}{4}\) for \(t\in (1,2]\). Putting \(\psi =1\), \(L_{f_{1}}=\bar{L}_{f_{2}}= \frac{1}{4}\)
\(\varphi (t)=e^{t}\) and \(c_{\varphi }=1\), we have \((\int _{0}^{t}(e^{s})^{3}\,ds )^{\frac{1}{3}}\leq e^{t}\) and let \(M_{1}\approx -0.5900\), \(M_{2}\approx 0.9713\), so \(M=0.9713 < 1\).
By Theorem 4.2, there exists a unique solution \(x:[0,3]\rightarrow \mathbb{R}\) such that
$$ x(t)= \textstyle\begin{cases} \int _{0}^{t} (\frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ( {^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )x(t)})\,ds -0.0846\int _{o}^{t}(t-s)^{\frac{-1}{2}}x(s)\,ds \\ \quad {} +0.0650 t^{\frac{1}{2}}\int _{0}^{1} (\frac{ \vert x(t) \vert + {{^{c}}D} _{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D _{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)} )\,ds \\ \quad {} -0.0901 t^{\frac{1}{2}}\int _{0}^{1}(1-s)^{\frac{-1}{2}}x(s)\,ds \\ \quad {} -0.7454 \sqrt{t}\int _{0}^{\frac{1}{4}}(\frac{1}{4}-s)^{ \frac{4}{3}} (\frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ( {^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )x(t)} )\,ds \\ \quad {} +0.1415\sqrt{t}\int _{0}^{\frac{1}{4}}(\frac{1}{4}-s)^{\frac{5}{6}}x(s)\,ds + (0.9476\sqrt{t}+1 )x_{0}, \quad t\in [0,1], \\ \int _{0}^{t} (\frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ( {^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )x(t)} )\,ds \\ \quad {} -0.0846\int _{o}^{t}(t-s)^{\frac{-1}{2}}x(s)\,ds \\ \quad {} -1.0650(\sqrt{2}-\sqrt{t})\int _{0}^{1} (\frac{ \vert x(t) \vert + {{^{c}}D} _{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D _{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)} )\,ds \\ \quad {} +0.0901(\sqrt{2}-\sqrt{t})\int _{0}^{1}(1-s)^{\frac{-1}{2}}x(s)\,ds \\ \quad {} +0.7454(\sqrt{2}-\sqrt{t})\int _{0}^{\frac{1}{4}}(\frac{1}{4}-s)^{ \frac{4}{3}} (\frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ( {^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+ \frac{3}{20} )x(t)} )\,ds \\ \quad {} -0.1415(\sqrt{2}-\sqrt{t})\int _{0}^{\frac{1}{4}}(\frac{1}{4}-s)^{ \frac{5}{6}}x(s))\,ds \\ \quad {} + (0.9476(\sqrt{2}-\sqrt{t})-\frac{3}{20} )\int _{0}^{2} (\frac{ \vert x(t) \vert + {{^{c}}D}_{0,t}^{\frac{1}{2}} ({^{c}}D_{0,t} ^{\frac{1}{2}}+\frac{3}{20} )x(t)}{8+e^{t}+t^{2}+ {{^{c}}D}_{0,t} ^{\frac{1}{2}} ({^{c}}D_{0,t}^{\frac{1}{2}}+\frac{3}{20} )x(t)} )\,ds \\ \quad {} - (0.9476(\sqrt{2}-\sqrt{t})-1 )0.846 \int _{o}^{2}(2-s)^{ \frac{-1}{2}}x(s)\,ds \\ \quad {} - (0.9476(\sqrt{2}-\sqrt{t})-1 ) \frac{x(t)}{(3+t^{2})(1+ \vert x(t) \vert )}, \quad t\in (2,3] \\ \frac{x(t)}{(3+t^{2})(1+ \vert x(t) \vert )}, \quad t\in (1,2]. \end{cases} $$
Then
$$\begin{aligned} &\bigl\vert y(t)-x(t) \bigr\vert \\ &\quad \leq \biggl\{ \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl( \frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{\alpha + \beta -r} +\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t^{\beta -r} \\ &\qquad {}+\frac{\Delta C_{\varphi }(t_{k}^{\beta }-t^{\beta })}{\varGamma ( \beta +1)\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T^{\alpha +\beta -r} +\frac{\lambda \Delta C_{\varphi }(t _{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}T^{\beta -r} \\ &\qquad {}+\frac{\theta \Delta C_{\varphi }(t_{k}^{\beta }- t^{\beta })}{ \varGamma (\beta +1)\varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha + \beta +p-r} \biggr)^{1-r}\eta ^{\alpha +\beta +p-r} \\ &\qquad {}+\frac{C_{\varphi }\theta \Delta \lambda (t_{k}^{\beta }- t^{\beta })}{\varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r}\eta ^{\beta +p-r} \\ &\qquad {}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl(\frac{1-r}{ \alpha +\beta -r} \biggr)^{1-r}t_{k}^{\alpha +\beta -r} \\ &\qquad {}+ \biggl(\Delta \frac{(t_{k}^{\beta }-t^{\beta })}{\varGamma (\beta +1)} \biggl(\frac{\theta \eta ^{p} -\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr)\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+1 \biggr\} \biggl(\frac{\varphi (t)+ \psi }{1-M} \biggr), \end{aligned}$$
which can further be reduced to
$$\begin{aligned} &\bigl\vert y(t)-x(t) \bigr\vert \\ &\quad \leq \biggl\{ \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl( \frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t^{\alpha + \beta -r} +\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{ \beta -r} \biggr)^{1-r}t^{\beta -r} \\ &\qquad {}-\frac{\Delta C_{\varphi }t^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T^{ \alpha +\beta -r} -\frac{\lambda \Delta C_{\varphi }t^{\beta }}{ \varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T ^{\beta -r} \\ &\qquad {}-\frac{\theta \Delta C_{\varphi }t^{\beta }}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r} \eta ^{\alpha +\beta +p-r} \\ &\qquad {}-\frac{C_{\varphi }\theta \Delta \lambda t^{\beta }}{\varGamma (\beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r} \eta ^{\beta +p-r} \\ &\qquad {}- \biggl(\frac{\Delta t^{\beta }}{\varGamma (\beta +1)} \biggl(\frac{ \theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{C _{\varphi }}{\varGamma (\alpha +\beta )} \biggl( \frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}t_{k}^{\alpha +\beta -r} \\ &\qquad {}- \biggl(\frac{\Delta t^{\beta }}{\varGamma (\beta +1)} \biggl(\frac{ \theta \eta ^{p} -\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \frac{ \lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}t_{k}^{\beta -r}+1 \biggr\} \biggl( \frac{\varphi (t)+\psi }{1-M} \biggr). \end{aligned}$$
This implies
$$\begin{aligned} &\bigl\vert y(t)-x(t) \bigr\vert \\ &\quad \leq \biggl\{ \frac{C_{\varphi }}{\varGamma (\alpha +\beta )} \biggl( \frac{1-r}{\alpha +\beta -r} \biggr)^{1-r} \tau ^{\alpha +\beta -r} +\frac{\lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}\tau ^{\beta -r} \\ &\qquad {}-\frac{\Delta C_{\varphi }\tau ^{\beta }}{\varGamma (\beta +1)\varGamma ( \alpha +\beta )} \biggl(\frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}T^{ \alpha +\beta -r} -\frac{\lambda \Delta C_{\varphi }\tau ^{\beta }}{ \varGamma (\beta +1)\varGamma (\beta )} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}T ^{\beta -r} \\ &\qquad {}-\frac{\theta \Delta C_{\varphi }\tau ^{\beta }}{\varGamma (\beta +1) \varGamma (\alpha +\beta +p)} \biggl(\frac{1-r}{\alpha +\beta +p-r} \biggr)^{1-r} \eta ^{\alpha +\beta +p-r} \\ &\qquad {}-\frac{C_{\varphi }\theta \Delta \lambda \tau ^{\beta }}{\varGamma ( \beta +1)\varGamma (\beta +p)} \biggl(\frac{1-r}{\beta +p-r} \biggr)^{1-r} \eta ^{\beta +p-r} \\ &\qquad {}- \biggl(\frac{\Delta \tau ^{\beta }}{\varGamma (\beta +1)} \biggl(\frac{ \theta \eta ^{p}-\varGamma (p+1)}{\varGamma (p+1)} \biggr)-\lambda \biggr) \frac{C _{\varphi }}{\varGamma (\alpha +\beta )} \biggl( \frac{1-r}{\alpha +\beta -r} \biggr)^{1-r}\tau ^{\alpha +\beta -r} \\ &\qquad {}- \biggl(\frac{\Delta \tau ^{\beta }}{\varGamma (\beta +1)} \biggl(\frac{ \theta \eta ^{p} -\varGamma (p+1)}{\varGamma (p+1)} \biggr)-1 \biggr) \frac{ \lambda C_{\varphi }}{\varGamma {\beta }} \biggl(\frac{1-r}{\beta -r} \biggr)^{1-r}\tau ^{\beta -r}+1 \biggr\} \biggl( \frac{\varphi (t)+\psi }{1-M} \biggr). \end{aligned}$$
Plugging-in the values, we have
$$\begin{aligned} &\bigl\vert y(t)-x(t) \bigr\vert \\ &\quad \leq \biggl\{ \frac{1}{\varGamma (\frac{1}{2}+ \frac{1}{2})} \biggl( \frac{1-\frac{1}{3}}{\frac{1}{2}+\frac{1}{2}- \frac{1}{3}} \biggr)^{(1-\frac{1}{3})}3^{(\frac{1}{2}+\frac{1}{2}- \frac{1}{3})} +\frac{(0.15)}{\varGamma {\frac{1}{2}}} \biggl(\frac{1- \frac{1}{3}}{\frac{1}{2}-\frac{1}{3}} \biggr)^{(1-\frac{1}{3})}3^{( \frac{1}{2}-\frac{1}{3})} \\ &\qquad {}-\frac{(-2.7) 3^{\frac{1}{2}}}{\varGamma (\frac{1}{2}+1)\varGamma ( \frac{1}{2}+\frac{1}{2})} \biggl(\frac{1-\frac{1}{3}}{\frac{1}{2}+ \frac{1}{2}-\frac{1}{3}} \biggr)^{(1-\frac{1}{3})}3^{(\frac{1}{2}+ \frac{1}{2}-\frac{1}{3})}\\ &\qquad {}-\frac{(0.15)(-2.7) 3^{\frac{1}{2}}}{\varGamma (\frac{1}{2}+1)\varGamma (\frac{1}{2})} \biggl(\frac{1-\frac{1}{3}}{ \frac{1}{2}-\frac{1}{3}} \biggr)^{(1-\frac{1}{3})}3^{(\frac{1}{2}- \frac{1}{3})} \\ &\qquad {}-\frac{(0.833)(-2.7) 3^{\frac{1}{2}}}{\varGamma (\frac{1}{2}+1)\varGamma (\frac{1}{2}+\frac{1}{2}+\frac{4}{3})} \biggl(\frac{1-\frac{1}{3}}{ \frac{1}{2}+\frac{1}{2}+\frac{4}{3}-\frac{1}{3}} \biggr)^{1-\frac{1}{3}}(0.25)^{ \frac{1}{2}+\frac{1}{2}+\frac{4}{3}-\frac{1}{3}} \\ &\qquad {}-\frac{(0.833)(-2.7)(0.15) 3^{\frac{1}{2}}}{\varGamma (\frac{1}{2}+1) \varGamma (\frac{1}{2}+\frac{4}{3})} \biggl(\frac{1-\frac{1}{3}}{ \frac{1}{2}+\frac{4}{3}-\frac{1}{3}} \biggr)^{1-\frac{1}{3}}(0.25)^{ \frac{1}{2}+\frac{4}{3}-\frac{1}{3}} \\ &\qquad {}- \biggl(\frac{(-2.7) 3^{\frac{1}{2}}}{\varGamma (\frac{1}{2}+1)} \biggl(\frac{(0.833)(0.25)^{ \frac{4}{3}}-\varGamma (\frac{4}{3}+1)}{\varGamma (\frac{4}{3}+1)} \biggr)-(0.15) \biggr)\\ &\qquad {}\times \frac{1}{\varGamma (\frac{1}{2}+\frac{1}{2})} \biggl(\frac{1- \frac{1}{3}}{\frac{1}{2}+\frac{1}{2}-\frac{1}{3}} \biggr)^{(1- \frac{1}{3})}3^{(\frac{1}{2}+\frac{1}{2}-\frac{1}{3})} \\ &\qquad {}- \biggl(\frac{(-2.7) 3^{\frac{1}{2}}}{\varGamma (\frac{1}{2}+1)} \biggl(\frac{(0.833)(0.25)^{ \frac{4}{3}} -\varGamma (\frac{4}{3}+1)}{\varGamma (\frac{4}{3}+1)} \biggr)-1 \biggr) \frac{(0.15) }{\varGamma {\frac{1}{2}}} \biggl(\frac{1-\frac{1}{3}}{ \frac{1}{2}-\frac{1}{3}} \biggr)^{1-\frac{1}{3}}3^{(\frac{1}{2}- \frac{1}{3})}+1 \biggr\} \\ &\qquad {}\times \biggl(\frac{e^{t}+1}{1-0.9714} \biggr) \\ &\quad \leq 5.4846 \biggl(\frac{e^{t}+1}{0.0286} \biggr) \\ &\quad \leq 191.769 \bigl(e^{t}+1 \bigr), \end{aligned}$$
thus problem (4.8) is Ulam–Hyers–Rassias stable.