In this part, we give important definitions and auxiliary lemmas pertinent to our main results.
Let \(J:= [ 0,T ]\) and \(J^{\prime }:= ( 0,T ] \). Let \(\mathcal{\mathbb{R} }=\mathcal{C} ( J ) \) be the Banach space of continuous functions \(\mathfrak{u}:J^{\prime }\rightarrow \mathbb{R} \) with the norm \(\Vert \mathfrak{u} \Vert =\max \{ \vert \mathfrak{u}(\sigma ) \vert :\sigma \in J\}\). Clearly, \(\mathcal{\mathbb{R} }\) is a Banach space with this norm, and hence the product space \(\mathcal{\mathbb{R} }\times \mathcal{\mathbb{R} }\) is also a Banach space with the norm
$$ \bigl\Vert ( \mathfrak{u},\vartheta ) \bigr\Vert = \Vert \mathfrak{u} \Vert + \Vert \vartheta \Vert . $$
We define the space \(\mathcal{PC} ( J ) \) of piecewise continuous functions \(\mathfrak{u}:J^{\prime }\rightarrow \mathbb{R} \) by
$$ \mathcal{PC} ( J ) =\left \{ \textstyle\begin{array}{c} \mathfrak{u}:J^{\prime }\rightarrow \mathbb{R} ;\mathfrak{u}(\sigma )\in \mathcal{C} ( ( \sigma _{k},\sigma _{k+1} ] ,\mathbb{R} ) ;k=0,1,\ldots,m, \\ \mathfrak{u}(\sigma _{k}^{+})\text{ and }\mathfrak{u}(\sigma _{k}^{-})\text{ exist}\text{ }\text{ with }\mathfrak{u}(\sigma _{k}^{+})= \mathfrak{u}(\sigma _{k}^{-})\text{ for }k=0,1,\ldots,m\end{array}\displaystyle \right \} . $$
Obviously, \(\mathcal{PC} ( J ) \) is a Banach space endowed with the norm
$$ \Vert \mathfrak{u} \Vert _{\mathcal{PC} ( J ) }= \underset{\sigma \in J}{\max } \bigl\vert \mathfrak{u}(\sigma ) \bigr\vert . $$
Define the product space \(\mathcal{B}=\mathcal{PC} ( J ) \times \mathcal{PC} ( J ) \) with the norm
$$ \bigl\Vert ( \mathfrak{u},\vartheta ) \bigr\Vert _{ \mathcal{B}}= \Vert \mathfrak{u} \Vert _{\mathcal{PC} ( J ) }+ \Vert \vartheta \Vert _{\mathcal{PC} ( J ) } $$
for \(( \mathfrak{u},\vartheta ) \in \mathcal{B}\).
Definition 2.1
([36])
Let \(\mathfrak{y}>0\) and \(f\in L_{1} ( J ) \). Then the generalized RL fractional integral of a function f of order \(\mathfrak{y}\) with respect to φ is defined as
$$ \mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi }f(\sigma )= \frac{1}{\Gamma (\mathfrak{y})} \int _{0}^{\sigma }\varphi ^{\prime }(s) \bigl( \varphi (\sigma )-\varphi (s) \bigr) ^{\mathfrak{y}-1}f(s)\,ds. $$
Definition 2.2
([41])
Let \(n-1<\mathfrak{y}<n\in \mathbb{N} \), and let \(f,\varphi \in \mathcal{PC}^{n} ( J ) \). Then the generalized Hilfer fractional derivative of a function f of order \(\mathfrak{y}\) and type \(0\leq \mathfrak{p}\leq 1\) with respect to φ is defined as
$$\begin{aligned} ^{H}\mathcal{D}_{0^{+}}^{\mathfrak{y},\mathfrak{p},\varphi }f(\sigma ) =&\mathcal{I}_{0^{+}}^{\mathfrak{p}(n-\mathfrak{y});\varphi }f_{ \varphi }^{[n]}\mathcal{I}_{0^{+}}^{ ( 1-\mathfrak{p} ) ( n- \mathfrak{y} ) ,\varphi }f(\sigma ) \\ =&\mathcal{I}_{0^{+}}^{\mathfrak{p}(n-\mathfrak{y});\varphi }f_{ \varphi }^{[n]} \mathcal{I}_{0^{+}}^{n-\mathfrak{\gamma },\varphi }f( \sigma ) \\ =&\mathcal{I}_{0^{+}}^{\mathfrak{p}(n-\mathfrak{y});\varphi } \mathcal{D}_{a^{+}}^{\mathfrak{\gamma };\varphi }f( \sigma ),\quad \mathfrak{\gamma }=\mathfrak{y}+n \mathfrak{p}-\mathfrak{yp}, \end{aligned}$$
where
$$ \mathcal{D}_{0^{+}}^{\mathfrak{\gamma };\varphi }f(\sigma )=f_{ \varphi }^{[n]} \mathcal{I}_{0^{+}}^{(1-\mathfrak{p})(n-\mathfrak{y}); \varphi }f(\sigma ),\quad \text{and} \quad f_{\varphi }^{[n]}= \biggl( \frac{1}{\varphi ^{\prime }(\sigma )} \frac{d}{d\sigma } \biggr) ^{n}. $$
Lemma 2.3
([41] )
Let \(\mathfrak{\gamma }=\mathfrak{y}+\mathfrak{p}-\mathfrak{yp}\), \(\mathfrak{y}>0\), \(\mathfrak{p}>0\), and \(u\in \mathcal{PC}_{1-\mathfrak{\gamma };\varphi }^{\mathfrak{\gamma }} ( J ) \). Then
$$ \mathcal{I}_{0^{+}}^{\mathfrak{\gamma };\varphi }\mathcal{D}_{0^{+}}^{\mathfrak{\gamma };\varphi }u= \mathcal{I}_{0^{+}}^{\mathfrak{y}; \varphi }\textit{ }^{H} \mathcal{D}_{0^{+}}^{\mathfrak{y},\mathfrak{p};\varphi }u\quad \textit{and} \quad \mathcal{D}_{0^{+}}^{\mathfrak{\gamma };\varphi } \mathcal{I}_{0^{+}}^{\mathfrak{y};\varphi }u= \mathcal{D}_{0^{+}}^{\mathfrak{p}(1- \mathfrak{y});\varphi }u. $$
Theorem 2.4
([41] )
Let \(0\leq \mathfrak{\gamma }<\mathfrak{y}\) and \(u\in \mathcal{PC} ( J ) \). Then \(\mathcal{I}_{0^{+}}^{\mathfrak{y};\varphi }\ u(0)=\underset{\sigma \rightarrow 0^{+}}{\lim }I_{0^{+}}^{\mathfrak{y};\varphi }u(\sigma )=0\).
Lemma 2.5
([36, 41])
Let \(\mathfrak{y},\mathfrak{p}>0\) and \(\delta >0\). Then
$$\begin{aligned}& \mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi }\mathcal{I}_{0^{+}}^{ \mathfrak{p},\varphi }f( \sigma )=\mathcal{I}_{0^{+}}^{\mathfrak{y}+\mathfrak{p}, \varphi }f(\sigma ), \\& \mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi } \bigl( \varphi (\sigma )- \varphi (0) \bigr) ^{\delta -1}= \frac{\Gamma (\mathfrak{\gamma })}{\Gamma (\mathfrak{y}+\mathfrak{\gamma })} \bigl( \varphi (\sigma )- \varphi (0) \bigr) ^{\mathfrak{y}+\delta -1}, \end{aligned}$$
and
$$ ^{H}\mathcal{D}_{0^{+}}^{\mathfrak{y},\mathfrak{p},\varphi } \bigl( \varphi ( \sigma )-\varphi (0) \bigr) ^{\mathfrak{\gamma }-1}=0, \quad \mathfrak{\gamma }=\mathfrak{y}+n\mathfrak{p}-\mathfrak{yp}. $$
Lemma 2.6
([41])
If \(f\in \mathcal{PC}^{n} ( J ) \), \(n-1<\mathfrak{y}<n\), and \(0\leq \mathfrak{p}\leq 1\), then
$$ \mathcal{I}_{0^{+}}^{\mathfrak{y};\varphi }\textit{ }^{H} \mathcal{D}_{0^{+}}^{\mathfrak{y},\mathfrak{p},\varphi }f(\sigma )=f(\sigma )-\sum _{k=1}^{n} \frac{ ( \varphi (\sigma )-\varphi (0) ) ^{\mathfrak{\gamma }-k}}{\Gamma (\mathfrak{\gamma }-k+1)}f_{\varphi }^{ [ n-k ] } \mathcal{I}_{a^{+}}^{(1-\mathfrak{p})(n-\mathfrak{y});\varphi }f(0), $$
and
$$ ^{H}\mathcal{D}_{0^{+}}^{\mathfrak{y},\mathfrak{p},\varphi } \mathcal{I}_{0^{+}}^{\mathfrak{y};\varphi }f(\sigma )=f(\sigma ). $$
Lemma 2.7
([31] (Leray–Schauder alternative))
Let \(\Xi :\mathcal{X}\rightarrow \mathcal{X}\) be a completely continuous operator, and let \(\digamma (\Xi )= \{ y\in \mathcal{X}:y=\xi \Xi (y),\xi \in [ 0,1 ] \} \). Then either the set \(\digamma (\Xi )\) is unbounded, or Ξ has at least one fixed point.
Theorem 2.8
([29] (Banach fixed point theorem))
Let \(\mathcal{X}\) be a Banach space, let \(K\subset \mathcal{X}\) be closed, and let \(\Xi :K\rightarrow K\) be a strict contraction, that is, \(\Vert \Xi (x)-\Xi (y) \Vert \leq L \Vert x-y \Vert \) for some \(0< L<1\) and all \(x,y\in K\). Then Ξ has a fixed point in K.
Lemma 2.9
Let \(\mathfrak{\gamma }=\mathfrak{y}+\mathfrak{p}-\mathfrak{yp}\), \(\mathfrak{y}\in ( 0,1 ) \), \(\mathfrak{p}\in [ 0,1 ] \), and let \(\varpi :J^{\prime }\rightarrow \mathbb{R} \) be a continuous function. Then \(\mathfrak{u}\in \mathcal{PC}^{\mathfrak{\gamma }} ( J ) \) satisfies
$$ \textstyle\begin{cases} \mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }} \mathfrak{u}(\sigma )=\varpi (\sigma ),&\sigma \in J:= [ 0,T ] ,\sigma \neq \sigma _{k},k=1,\ldots,m, \\ \Delta \mathfrak{u} \vert _{\sigma =\sigma _{k}}=Z_{k} \mathfrak{u}(\sigma _{k}^{-}),&k=1,\ldots,m,\\ \mathfrak{u}(T)=w \end{cases} $$
(2.1)
if and only if \(\mathfrak{u}\) satisfies the following integral equations:
$$ \mathfrak{u}(\sigma )= \textstyle\begin{cases} \frac{ ( \varphi (\sigma )-\varphi (0) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (0) ) ^{\mathfrak{\gamma }-1}} [ w-\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( T ) ] +\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( \sigma ) ,\quad \sigma \in [ 0,\sigma _{1} ] , \\ \sum_{i=1}^{k+1} \frac{ ( \varphi (\sigma _{i})-\varphi (\sigma _{i-1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{i-1}) ) ^{\mathfrak{\gamma }-1}} [ w-\mathcal{I}_{i-1^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( T ) ] +\sum_{i=1}^{k}\mathcal{I}_{\sigma _{i-1}^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( \sigma _{i} ) \\ \quad {}+\mathcal{I}_{\sigma _{k}^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( \sigma ) +\sum_{i=1}^{k}Z_{i}\mathfrak{u}(\sigma _{i}^{-}), \quad \sigma \in ( \sigma _{k},\sigma _{k+1} ] , k=1,\ldots,m.\end{cases} $$
(2.2)
Proof
First, let \(\mathfrak{u}\in \mathcal{PC}^{\mathfrak{\gamma }} ( J ) \) be a solution of problem (2.1). We prove that \(\mathfrak{u}\) is a solution of (2.2).
If \(\sigma \in [ 0,\sigma _{1} ] \), then \(\mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }} \mathfrak{u}(\sigma )=\varpi (\sigma )\), \([ \sigma ] =0\). Taking the operator \(\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi }\) on both sides of the first equation in (2.1) and using Lemma 2.6, we have
$$ \mathfrak{u}(\sigma )= \frac{ ( \varphi (\sigma )-\varphi (0) ) ^{\mathfrak{\gamma }-1}}{\Gamma (\mathfrak{\gamma })}\mathcal{I}_{0^{+}}^{1-\mathfrak{\gamma },\varphi } \mathfrak{u}(0)+\mathcal{I}_{0^{+}}^{ \mathfrak{y},\varphi }\varpi (s) ( \sigma ) . $$
(2.3)
By the terminal condition we have
$$ \mathcal{I}_{0^{+}}^{1-\mathfrak{\gamma },\varphi }\mathfrak{u}(0)= \frac{\Gamma (\mathfrak{\gamma })}{ ( \varphi (T)-\varphi (0) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{0^{+}}^{\mathfrak{y}, \varphi } \varpi (s) ( T ) \bigr] . $$
(2.4)
Putting (2.4) into (2.3), we get
$$ \mathfrak{u}(\sigma )= \frac{ ( \varphi (\sigma )-\varphi (0) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (0) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] +\mathcal{I}_{0^{+}}^{ \mathfrak{y},\varphi }\varpi (s) ( \sigma ) . $$
This means
$$ \mathfrak{u}\bigl(\sigma _{1}^{-}\bigr)= \frac{ ( \varphi (\sigma _{1})-\varphi (0) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (0) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{0^{+}}^{\mathfrak{y}, \varphi } \varpi (s) ( T ) \bigr] +\mathcal{I}_{0^{+}}^{ \mathfrak{y},\varphi }\varpi (s) ( \sigma _{1} ) . $$
Since \(\mathfrak{u}(\sigma _{1}^{-})=\mathfrak{u}(\sigma _{1}^{+})-Z_{1}\mathfrak{u}(\sigma _{1}^{-})\), we get
$$ \mathfrak{u}\bigl(\sigma _{1}^{+}\bigr)= \frac{ ( \varphi (\sigma _{1})-\varphi (0) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (0) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{0^{+}}^{\mathfrak{y}, \varphi } \varpi (s) ( T ) \bigr] +\mathcal{I}_{0^{+}}^{ \mathfrak{y},\varphi }\varpi (s) ( \sigma _{1} ) +Z_{1}\mathfrak{u}\bigl( \sigma _{1}^{-}\bigr). $$
If \(\sigma \in ( \sigma _{1},\sigma _{2} ] \), then \(\mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }}\mathfrak{u}( \sigma )=\varpi (\sigma )\), \([ \sigma ] =\sigma _{1}\), and \(\mathfrak{u}(\sigma )\) is given by
$$\begin{aligned} \mathfrak{u}(\sigma ) =&\mathfrak{u}\bigl(\sigma _{1}^{+} \bigr)+ \frac{ ( \varphi (\sigma )-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{1}^{+}}^{\mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] +\mathcal{I}_{\sigma _{1}^{+}}^{\mathfrak{y}, \varphi }\varpi (s) ( \sigma ) \\ =&\frac{ ( \varphi (\sigma _{1})-\varphi (0) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (0) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] + \frac{ ( \varphi (\sigma )-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{1}^{+}}^{\mathfrak{y}, \varphi } \varpi (s) ( T ) \bigr] \\ &{}+\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( \sigma _{1} ) +\mathcal{I}_{\sigma _{1}^{+}}^{\mathfrak{y}, \varphi }\varpi (s) ( \sigma ) +Z_{1}\mathfrak{u}\bigl(\sigma _{1}^{-} \bigr). \end{aligned}$$
This means that
$$\begin{aligned} \mathfrak{u}\bigl(\sigma _{2}^{-}\bigr) =& \frac{ ( \varphi (\sigma _{1})-\varphi (0) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (0) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{0^{+}}^{\mathfrak{y}, \varphi } \varpi (s) ( T ) \bigr] \\ &{}+ \frac{ ( \varphi (\sigma _{2})-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{1}^{+}}^{ \mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( \sigma _{1} ) +\mathcal{I}_{\sigma _{1}^{+}}^{\mathfrak{y}, \varphi }\varpi (s) ( \sigma _{2} ) +Z_{1}\mathfrak{u}\bigl( \sigma _{1}^{-}\bigr). \end{aligned}$$
Since \(\mathfrak{u}(\sigma _{2}^{-})=\mathfrak{u}(\sigma _{2}^{+})-Z_{2}\mathfrak{u}(\sigma _{2}^{-})\), we get
$$\begin{aligned} \mathfrak{u}\bigl(\sigma _{2}^{+}\bigr) =& \frac{ ( \varphi (\sigma _{1})-\varphi (0) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (0) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{0^{+}}^{\mathfrak{y}, \varphi } \varpi (s) ( T ) \bigr] \\ &{}+ \frac{ ( \varphi (\sigma _{2})-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{1}^{+}}^{ \mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( \sigma _{1} ) +\mathcal{I}_{\sigma _{1}^{+}}^{\mathfrak{y}, \varphi }\varpi (s) ( \sigma _{2} ) +Z_{1}\mathfrak{u}\bigl( \sigma _{1}^{-}\bigr)+Z_{2}\mathfrak{u}\bigl(\sigma _{2}^{-}\bigr). \end{aligned}$$
If \(\sigma \in ( \sigma _{2},\sigma _{3} ] \), then \(\mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }}\mathfrak{u}( \sigma )=\varpi (\sigma )\), \([ \sigma ] =\sigma _{2}\), and \(\mathfrak{u}(\sigma )\) is given by
$$\begin{aligned} \mathfrak{u}(\sigma ) =&\mathfrak{u}\bigl(\sigma _{2}^{+} \bigr)+ \frac{ ( \varphi (\sigma )-\varphi (\sigma _{2}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{2}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{2}^{+}}^{\mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] +\mathcal{I}_{\sigma _{2}^{+}}^{\mathfrak{y}, \varphi }\varpi (s) ( \sigma ) \\ =&\frac{ ( \varphi (\sigma _{1})-\varphi (0) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (0) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+ \frac{ ( \varphi (\sigma _{2})-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{1}^{+}}^{ \mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+ \frac{ ( \varphi (\sigma )-\varphi (\sigma _{2}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{2}) ) ^{1\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{2}^{+}}^{\mathfrak{y}, \varphi } \varpi (s) ( T ) \bigr] \\ &{}+\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( \sigma _{1} ) +\mathcal{I}_{\sigma _{1}^{+}}^{\mathfrak{y}, \varphi }\varpi (s) ( \sigma _{2} ) +\mathcal{I}_{\sigma _{2}^{+}}^{ \mathfrak{y},\varphi }\varpi (s) ( \sigma ) +Z_{1}\mathfrak{u}\bigl( \sigma _{1}^{-} \bigr)+Z_{2}\mathfrak{u}\bigl(\sigma _{2}^{-} \bigr). \end{aligned}$$
This means that
$$\begin{aligned} \mathfrak{u}\bigl(\sigma _{3}^{-}\bigr) =& \frac{ ( \varphi (\sigma _{1})-\varphi (0) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (0) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{0^{+}}^{\mathfrak{y}, \varphi } \varpi (s) ( T ) \bigr] \\ &{}+ \frac{ ( \varphi (\sigma _{2})-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{1}^{+}}^{ \mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+ \frac{ ( \varphi (\sigma _{3})-\varphi (\sigma _{2}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{2}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{2}^{+}}^{ \mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( \sigma _{1} ) +\mathcal{I}_{\sigma _{1}^{+}}^{\mathfrak{y}, \varphi }\varpi (s) ( \sigma _{2} ) +\mathcal{I}_{\sigma _{2}^{+}}^{ \mathfrak{y},\varphi }\varpi (s) ( \sigma _{3} ) \\ &{}+Z_{1}\mathfrak{u}\bigl(\sigma _{1}^{-} \bigr)+Z_{2}\mathfrak{u}\bigl(\sigma _{2}^{-} \bigr). \end{aligned}$$
After impulse \(( \mathfrak{u}(\sigma _{3}^{-})=\mathfrak{u}(\sigma _{3}^{+})-Z_{3} \mathfrak{u}(\sigma _{3}^{-}) ) \), we get
$$\begin{aligned} \mathfrak{u}\bigl(\sigma _{3}^{-}\bigr) =& \frac{ ( \varphi (\sigma _{1})-\varphi (0) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (0) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{0^{+}}^{\mathfrak{y}, \varphi } \varpi (s) ( T ) \bigr] \\ &{}+ \frac{ ( \varphi (\sigma _{2})-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{1}^{+}}^{ \mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+ \frac{ ( \varphi (\sigma _{3})-\varphi (\sigma _{2}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{2}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{2}^{+}}^{ \mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( \sigma _{1} ) +\mathcal{I}_{\sigma _{1}^{+}}^{\mathfrak{y}, \varphi }\varpi (s) ( \sigma _{2} ) +\mathcal{I}_{\sigma _{2}^{+}}^{ \mathfrak{y},\varphi }\varpi (s) ( \sigma _{3} ) \\ &{}+Z_{1}\mathfrak{u}\bigl(\sigma _{1}^{-} \bigr)+Z_{2}\mathfrak{u}\bigl(\sigma _{2}^{-} \bigr)+Z_{3} \mathfrak{u}\bigl(\sigma _{3}^{-} \bigr). \end{aligned}$$
If \(\sigma \in ( \sigma _{3},\sigma _{4} ] \), then \(\mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }}\mathfrak{u}( \sigma )=\varpi (\sigma )\), \([ \sigma ] =\sigma _{3}\), and \(\mathfrak{u}(\sigma )\) is given by
$$\begin{aligned} \mathfrak{u}(\sigma ) =&\mathfrak{u}\bigl(\sigma _{3}^{+} \bigr)+ \frac{ ( \varphi (\sigma )-\varphi (\sigma _{3}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{3}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{3}^{+}}^{\mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] +\mathcal{I}_{\sigma _{3}^{+}}^{\mathfrak{y}, \varphi }\varpi (s) ( \sigma ) \\ =&\frac{ ( \varphi (\sigma _{1})-\varphi (0) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (0) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+ \frac{ ( \varphi (\sigma _{2})-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{1}^{+}}^{ \mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+ \frac{ ( \varphi (\sigma _{3})-\varphi (\sigma _{2}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{2}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{2}^{+}}^{ \mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+ \frac{ ( \varphi (\sigma )-\varphi (\sigma _{3}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{3}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{3}^{+}}^{\mathfrak{y}, \varphi } \varpi (s) ( T ) \bigr] \\ &{}+\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( \sigma _{1} ) +\mathcal{I}_{\sigma _{1}^{+}}^{\mathfrak{y}, \varphi }\varpi (s) ( \sigma _{2} ) +\mathcal{I}_{\sigma _{2}^{+}}^{ \mathfrak{y},\varphi }\varpi (s) ( \sigma _{3} ) +\mathcal{I}_{\sigma _{3}^{+}}^{ \mathfrak{y},\varphi } \varpi (s) ( \sigma ) \\ &{}+Z_{1}\mathfrak{u}\bigl(\sigma _{1}^{-} \bigr)+Z_{2}\mathfrak{u}\bigl(\sigma _{2}^{-} \bigr)+Z_{3} \mathfrak{u}\bigl(\sigma _{3}^{-} \bigr). \end{aligned}$$
Assume that
$$\begin{aligned} \mathfrak{u}\bigl(\sigma _{k}^{+}\bigr) =& \frac{ ( \varphi (\sigma _{1})-\varphi (0) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (0) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{0^{+}}^{\mathfrak{y}, \varphi } \varpi (s) ( T ) \bigr] \\ &{}+ \frac{ ( \varphi (\sigma _{2})-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{1}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{1}^{+}}^{ \mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+\cdots + \frac{ ( \varphi (\sigma _{k})-\varphi (\sigma _{k-1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{k-1}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{k-1}^{+}}^{ \mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+\mathcal{I}_{0^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( \sigma _{1} ) +\mathcal{I}_{\sigma _{1}^{+}}^{\mathfrak{y}, \varphi }\varpi (s) ( \sigma _{2} ) +\cdots +\mathcal{I}_{ \sigma _{k-1}^{+}}^{\mathfrak{y},\varphi } \varpi (s) ( \sigma _{k} ) \\ &{}+Z_{1}\mathfrak{u}\bigl(\sigma _{1}^{-} \bigr)+Z_{2}\mathfrak{u}\bigl(\sigma _{2}^{-} \bigr)+ \cdots +Z_{k}\mathfrak{u}\bigl(\sigma _{k}^{-} \bigr). \end{aligned}$$
Then, inductively, for \(\sigma \in ( \sigma _{k},\sigma _{k+1} ] \), we have \(\mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }}\mathfrak{u}(\sigma )=\varpi (\sigma )\), \([ \sigma ] =\sigma _{k} \), and \(\mathfrak{u}(\sigma )\) is given by
$$\begin{aligned} \mathfrak{u}(\sigma ) =&\mathfrak{u}\bigl(\sigma _{k}^{+} \bigr)+ \frac{ ( \varphi (\sigma )-\varphi (\sigma _{k}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{k}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{\sigma _{k}^{+}}^{\mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] +\mathcal{I}_{\sigma _{k}^{+}}^{\mathfrak{y}, \varphi }\varpi (s) ( \sigma ) \\ =&\sum_{i=1}^{k+1} \frac{ ( \varphi (\sigma _{i})-\varphi (\sigma _{i-1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{i-1}) ) ^{\mathfrak{\gamma }-1}} \bigl[ w-\mathcal{I}_{i-1^{+}}^{\mathfrak{y},\varphi } \varpi (s) ( T ) \bigr] \\ &{}+\sum_{i=1}^{k} \mathcal{I}_{\sigma _{i-1}^{+}}^{\mathfrak{y}, \varphi }\varpi (s) ( \sigma _{i} ) +\mathcal{I}_{\sigma _{k}^{+}}^{\mathfrak{y},\varphi }\varpi (s) ( \sigma ) +\sum_{i=1}^{k}Z_{i}\mathfrak{u}\bigl(\sigma _{i}^{-}\bigr). \end{aligned}$$
Thus (2.2) is satisfied.
Conversely, assume that \(\mathfrak{u}\) satisfies equation (2.2).
Case 1: \(\sigma \in [ 0,\sigma _{1} ] \).
Replacing σ by T in (2.2), we get \(\mathfrak{u}(T)=w\). On the other hand, applying \(\mathcal{D}_{0^{+}}^{\mathfrak{\gamma };\varphi }\) to both sides of (2.2) and using Lemma 2.3, we get
$$ \mathcal{D}_{0^{+}}^{\mathfrak{\gamma };\varphi }\mathfrak{u}(\sigma )=\mathcal{D}_{0^{+}}^{\mathfrak{p}(1-\mathfrak{y});\varphi }\varpi ( \sigma ). $$
(2.5)
Since \(\mathfrak{u}\in \mathcal{PC}^{\mathfrak{\gamma }} ( J ) \), by definition of \(\mathcal{PC}^{\mathfrak{\gamma }} ( J ) \) we have \(\mathcal{D}_{0^{+}}^{\mathfrak{\gamma };\varphi }\mathfrak{u}\in \mathcal{PC} ( J ) \). So, (2.5) implies
$$ \mathcal{D}_{0^{+}}^{\mathfrak{\gamma };\varphi }\mathfrak{u}(\sigma )=\mathcal{D}_{0^{+}}^{\mathfrak{p}(1-\mathfrak{y});\varphi }\varpi ( \sigma )\in \mathcal{PC} ( J ) . $$
For \(\varpi \in \mathcal{PC} ( J ) \), it is obvious that \(\mathcal{I}_{0^{+}}^{1-\mathfrak{p}(1-\mathfrak{y});\varphi }\varpi \in \mathcal{PC}^{1} ( J ) \). Hence ϖ and \(\mathcal{I}_{0^{+}}^{1-\mathfrak{p}(1-\mathfrak{y});\varphi }\varpi \) satisfy the conditions of Theorem 2.6. Now, applying \(\mathcal{I}_{0^{+}}^{\mathfrak{p}(1-\mathfrak{y});\varphi }\) to both sides of (2.5) and using Theorem 2.6, we get
$$ ^{H}\mathcal{D}_{0^{+}}^{\mathfrak{y},\mathfrak{p};\varphi } \mathfrak{u}(\sigma )=\varpi (\sigma )- \frac{\mathcal{I}_{0^{+}}^{1-\mathfrak{p}(1-\mathfrak{y});\varphi }\varpi (0)}{\Gamma (\mathfrak{p}(1-\mathfrak{y}))}\bigl(\varphi (\sigma )-\varphi (0)\bigr)^{\mathfrak{p}(1-\mathfrak{y})-1}. $$
(2.6)
By Theorem 2.4 we have \(\mathcal{I}_{0^{+}}^{1-\mathfrak{p}(1-\mathfrak{y});\varphi }\varpi (0)=0\). Hence (2.6) becomes
$$ ^{H}\mathcal{D}_{0^{+}}^{\mathfrak{y},\mathfrak{p};\varphi } \mathfrak{u}(\sigma )=\varpi (\sigma ),\quad \sigma \in J. $$
Case 2: \(\sigma \in ( \sigma _{k},\sigma _{k+1} ] \).
By the same technique as in case 1 we can easily prove case 2. □
Lemma 2.10
Let \(\mathfrak{\gamma }=\mathfrak{y}+\mathfrak{p}-\mathfrak{yp}\) be such that \(\mathfrak{y}\in ( 0,1 ) \), \(\mathfrak{p}\in [ 0,1 ] \), and let \(f,g:J^{\prime }\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) be continuous functions. If \(( \mathfrak{u},\vartheta ) \in \mathcal{B}\) satisfies problem (1.1), then by Lemma 2.9, \(( \mathfrak{u},\vartheta ) \) satisfies the following integral equations:
$$ \textstyle\begin{cases} \mathfrak{u}(\sigma )= \textstyle\begin{cases} \sum_{0< \sigma _{k}< \sigma } \frac{ ( \varphi (\sigma _{k})-\varphi (\sigma _{k-1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{k-1}) ) ^{\mathfrak{\gamma }-1}} [ w_{1}-\mathcal{I}_{\sigma _{k-1}^{+}}^{\mathfrak{y},\varphi }f(s,\mathfrak{u}(s), \mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }}\vartheta (s)) ( T ) ] \\ \quad {}+\sum_{0< \sigma _{k}< \sigma }\mathcal{I}_{\sigma _{k-1}^{+}}^{ \mathfrak{y},\varphi }f(s,\mathfrak{u}(s),\mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }}\vartheta (s)) ( \sigma _{k} ) + \mathcal{I}_{\sigma _{k}^{+}}^{\mathfrak{y},\varphi }f(s,\mathfrak{u}(s), \mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }}\vartheta (s)) ( \sigma ) \\ \quad {}+\sum_{0< \sigma _{k}< \sigma }Z_{k}\mathfrak{u}(\sigma _{k}^{-}),\quad \sigma \in ( \sigma _{k},\sigma _{k+1} ] ,k=1,\ldots,m,\end{cases}\displaystyle \\ \vartheta (\sigma )= \textstyle\begin{cases} \sum_{0< \sigma _{k}< \sigma } \frac{ ( \varphi (\sigma _{k})-\varphi (\sigma _{k-1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{k-1}) ) ^{\mathfrak{\gamma }-1}} [ w_{2}-\mathcal{I}_{\sigma _{k-1}^{+}}^{\mathfrak{y},\varphi }g(s,\mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }}\mathfrak{u}(s),\vartheta (s)) ( T ) ] \\ \quad {}+\sum_{0< \sigma _{k}< \sigma }\mathcal{I}_{\sigma _{k-1}^{+}}^{ \mathfrak{y},\varphi }g(s,\mathcal{D}_{ [ \sigma ] }^{ \mathfrak{y,p,\varphi }}\mathfrak{u}(s),\vartheta (s)) ( \sigma _{k} ) + \mathcal{I}_{\sigma _{k}^{+}}^{\mathfrak{y},\varphi }g(s,\mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }}\mathfrak{u}(s),\vartheta (s)) ( \sigma ) \\ \quad {}+\sum_{0< \sigma _{k}< \sigma }Z_{k}\mathfrak{u}(\sigma _{k}^{-}), \quad \sigma \in ( \sigma _{k},\sigma _{k+1} ] ,k=1,\ldots,m.\end{cases}\displaystyle \end{cases} $$
Consider the continuous operator \(\Xi :\mathcal{B}\rightarrow \mathcal{B}\) defined by
$$ \Xi ( \mathfrak{u},\vartheta ) (\sigma )= \bigl( \Xi _{1} ( \mathfrak{u},\vartheta ) (\sigma ),\Xi _{2} ( \vartheta , \mathfrak{u} ) (\sigma ) \bigr) , $$
(2.7)
where
$$ \Xi _{1} ( \mathfrak{u},\vartheta ) (\sigma )= \textstyle\begin{cases} \sum_{0< \sigma _{k}< \sigma } \frac{ ( \varphi (\sigma _{k})-\varphi (\sigma _{k-1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{k-1}) ) ^{\mathfrak{\gamma }-1}} [ w_{1}-\mathcal{I}_{\sigma _{k-1}^{+}}^{\mathfrak{y},\varphi }f(s,\mathfrak{u}(s), \mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }}\vartheta (s)) ( T ) ] \\ \quad {}+\sum_{0< \sigma _{k}< \sigma }\mathcal{I}_{\sigma _{k-1}^{+}}^{ \mathfrak{y},\varphi }f(s,\mathfrak{u}(s),\mathcal{D}_{0}^{ \mathfrak{y,p,\varphi }}\vartheta (s)) ( \sigma _{k} )\\ \quad {} +\mathcal{I}_{\sigma _{k}^{+}}^{\mathfrak{y},\varphi }f(s,\mathfrak{u}(s),\mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }}\vartheta (s)) ( \sigma ) \\ \quad {}+\sum_{0< \sigma _{k}< \sigma }Z_{k}\mathfrak{u}(\sigma _{k}^{-}), \quad \sigma \in ( \sigma _{k},\sigma _{k+1} ] , k=1,\ldots,m,\end{cases} $$
(2.8)
and
$$ \Xi _{2} ( \vartheta ,\mathfrak{u} ) (\sigma )= \textstyle\begin{cases} \sum_{0< \sigma _{k}< \sigma } \frac{ ( \varphi (\sigma _{k})-\varphi (\sigma _{k-1}) ) ^{\mathfrak{\gamma }-1}}{ ( \varphi (T)-\varphi (\sigma _{k-1}) ) ^{\mathfrak{\gamma }-1}} [ w_{2}-\mathcal{I}_{\sigma _{k-1}^{+}}^{\mathfrak{y},\varphi }g(s,\mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }}\mathfrak{u}(s),\vartheta (s)) ( T ) ] \\ \quad {}+\sum_{0< \sigma _{k}< \sigma }\mathcal{I}_{\sigma _{k-1}^{+}}^{ \mathfrak{y},\varphi }g(s,\mathcal{D}_{ [ \sigma ] }^{ \mathfrak{y,p,\varphi }}\mathfrak{u}(s),\vartheta (s)) ( \sigma _{k} ) \\ \quad {}+ \mathcal{I}_{\sigma _{k}^{+}}^{\mathfrak{y},\varphi }g(s,\mathcal{D}_{ [ \sigma ] }^{\mathfrak{y,p,\varphi }}\mathfrak{u}(s),\vartheta (s)) ( \sigma ) \\ \quad +\sum_{0< \sigma _{k}< \sigma }Z_{k}\mathfrak{u}(\sigma _{k}^{-}), \quad \sigma \in ( \sigma _{k},\sigma _{k+1} ] , k=1,\ldots,m.\end{cases} $$
(2.9)
Note that the fixed points of the operator Ξ are solutions of problem (1.1).