In this part, we investigate conditions for the existence of at least one solution to HFDEs (1).
Lemma 2
If \(\kappa \in L(\mathscr{J})\), then the solution \(\phi \in AC(\mathscr{J})\) of linear BVPs with nonlinear integral BCs
$$ \textstyle\begin{cases} {}_{0}^{\mathit{ABC}} \mathrm{D}^{\omega }_{t}\phi (t)= \kappa (t), \quad 0< \omega \leq {1}, t\in \mathscr{J}, \\ \phi (0)= \int _{0}^{\xi } \frac{(\xi \eta )^{\omega 1}}{\Gamma (\omega )}\mathscr{H} (\eta , \phi (\eta ) )\,d\eta , \end{cases} $$
is given by
$$ \begin{aligned} \phi (t) ={}& \int _{0}^{\xi } \frac{(\xi \eta )^{\omega 1}}{\Gamma (\omega )}\mathscr{H} \bigl(\eta , \phi (\eta ) \bigr)\,d\eta +\frac{(1\omega )}{\mathcal{N}(\omega )}\kappa (t) \\ &{}+ \frac{\omega }{\mathcal{N}(\omega )\Gamma (\omega )} \int _{0}^{t}(t \eta )^{\omega 1}\kappa ( \eta )\,d\eta . \end{aligned} $$
(2)
Proof 1
Thanks to Lemma 1, we can easily obtain result (2).
Corollary 1
In view of Lemma 2, the solution of nonlinear integral BVPs (1) is given by
$$ \begin{aligned} \phi (t)={}&\mathscr{F} \bigl(t,\phi (t) \bigr)+ \frac{(1\omega )}{\mathcal{N}(\omega )}\mathscr{G} \bigl(t,\phi (t) \bigr) + \int _{0}^{\xi }\frac{(\xi \eta )^{\omega 1}}{\Gamma (\omega )} \mathscr{H} \bigl(\eta ,\phi (\eta ) \bigr)\,d\eta \\ &{}+\frac{\omega }{\mathcal{N}(\omega )\Gamma (\omega )} \int _{0}^{t}(t \eta )^{\omega 1} \mathscr{G} \bigl(\eta ,\phi (\eta ) \bigr)\,d\eta . \end{aligned} $$
(3)
To derive the interrelated results, we need the following hypothesis to hold:
 \((A_{1})\):

For constants \(L_{\mathscr{F}}, L_{\mathscr{G}}, L_{\mathscr{H}}>0\), we have for any \(\phi , \bar{\phi }\in \mathscr{Z}\)
$$\begin{aligned}& \bigl\vert \mathscr{F} \bigl(t,\phi (t) \bigr)\mathscr{F} \bigl(t,\bar{\phi }(t) \bigr) \bigr\vert \leq L_{ \mathscr{F}} \bigl\vert \phi (t)\bar{\phi }(t) \bigr\vert , \\& \bigl\vert \mathscr{G} \bigl(t,\phi (t) \bigr)\mathscr{G} \bigl(t,\bar{\phi }(t) \bigr) \bigr\vert \leq L_{ \mathscr{G}} \bigl\vert \phi (t)\bar{\phi }(t) \bigr\vert , \end{aligned}$$
and
$$ \bigl\vert \mathscr{H} \bigl(t,\phi (t) \bigr)\mathscr{H} \bigl(t,\bar{\phi }(t) \bigr) \bigr\vert \leq L_{ \mathscr{H}} \bigl\vert \phi (t)\bar{\phi }(t) \bigr\vert . $$
 \((A_{2})\):

For any real constants \(C_{\mathscr{G}}, M_{\mathscr{G}}>0\), we have
$$ \bigl\vert \mathscr{G} \bigl(t,\phi (t) \bigr) \bigr\vert \leq C_{\mathscr{G}} \bigl\vert \phi (t) \bigr\vert +M_{ \mathscr{G}}. $$
 \((A_{3})\):

Also, for real constants \(C_{\mathscr{H}}>0\), \(M_{\mathscr{H}}>0\), we have
$$ \bigl\vert \mathscr{H} \bigl(t,\phi (t) \bigr) \bigr\vert \leq C_{\mathscr{H}} \bigl\vert \phi (t) \bigr\vert +M_{ \mathscr{H}}. $$
On using (3), we define the operator \(\mathscr{A}: \mathscr{Z}\rightarrow \mathscr{Z}\) by
$$ \begin{aligned} \mathscr{A}\phi (t)={}& \mathscr{F} \bigl(t,\phi (t) \bigr)+ \frac{(1\omega )}{\mathcal{N}(\omega )}\mathscr{G} \bigl(t,\phi (t) \bigr) + \int _{0}^{\xi }\frac{(\xi \eta )^{\omega 1}}{\Gamma (\omega )} \mathscr{H} \bigl(\eta ,\phi (\eta ) \bigr)\,d\eta \\ &{}+\frac{\omega }{\mathcal{N}(\omega )\Gamma (\omega )} \int _{0}^{t}(t \eta )^{\omega 1} \mathscr{G} \bigl(\eta ,\phi (\eta ) \bigr)\,d\eta , \quad t\in \mathscr{J}. \end{aligned} $$
(4)
Theorem 2
Under hypothesis \((A_{1})\), the considered problem (1) has a unique solution if
$$ \Upsilon = \biggl(L_{\mathscr{F}}+ \frac{(1\omega )}{\mathcal{N}(\omega )}L_{\mathscr{G}} + \frac{L_{\mathscr{H}}\xi ^{\omega }}{\Gamma (\omega +1)} \biggl(1+ \frac{\omega }{\mathcal{N}(\omega )} \biggr) \biggr)< 1. $$
Proof 2
If \(\phi , \bar{\phi }\in \mathscr{Z}\), from (4) we have
$$\begin{aligned} \bigl\Vert \mathscr{A}(\phi )\mathscr{A}(\bar{\phi }) \bigr\Vert _{\mathscr{Z}} =&\max_{t \in \mathscr{J}} \bigl\vert \mathscr{A}\phi (t)\mathscr{A}\bar{\phi }(t) \bigr\vert \\ =&\max_{t\in \mathscr{J}} \biggl\vert \mathscr{F} \bigl(t,\phi (t) \bigr)+ \frac{(1\omega )}{\mathcal{N}(\omega )}\mathscr{G} \bigl(t,\phi (t) \bigr) \\ &{}+ \int _{0}^{\xi }\frac{(\xi \eta )^{\omega 1}}{\Gamma (\omega )} \mathscr{H} \bigl(\eta ,\phi (\eta ) \bigr)\,d\eta \\ &{}+\frac{\omega }{\mathcal{N}(\omega )\Gamma (\omega )} \int _{0}^{t}(t \eta )^{\omega 1} \mathscr{G} \bigl(\eta ,\phi (\eta ) \bigr)\,d\eta \\ &{}\mathscr{F} \bigl(t,\bar{\phi }(t) \bigr) \frac{(1\omega )}{\mathcal{N}(\omega )}\mathscr{G} \bigl(t,\bar{\phi }(t) \bigr) \\ &{} \int _{0}^{\xi }\frac{(\xi \eta )^{\omega 1}}{\Gamma (\omega )} \mathscr{H} \bigl(\eta ,\bar{\phi }(\eta ) \bigr)\,d\eta \\ &{}\frac{\omega }{\mathcal{N}(\omega )\Gamma (\omega )} \int _{0}^{t}(t \eta )^{\omega 1} \mathscr{G} \bigl(\eta ,\bar{\phi }(\eta ) \bigr)\,d\eta \biggr\vert \\ \leq & L_{\mathscr{F}} \Vert \phi \bar{\phi } \Vert _{\mathscr{Z}}+ \frac{(1\omega )}{\mathcal{N}(\omega )}L_{\mathscr{G}} \Vert \phi  \bar{\phi } \Vert _{\mathscr{Z}} \\ &{}+\frac{L_{\mathscr{H}}\xi ^{\omega }}{\Gamma (\omega +1)} \Vert \phi  \bar{\phi } \Vert _{\mathscr{Z}} + \frac{\omega L_{\mathscr{H}}\xi ^{\omega }}{\mathcal{N}(\omega )\Gamma (\omega +1)} \Vert \phi \bar{\phi } \Vert _{\mathscr{Z}}. \end{aligned}$$
Hence, we obtain
$$ \bigl\Vert \mathscr{A}(\phi )\mathscr{A}(\bar{\phi }) \bigr\Vert _{ \mathscr{Z}} \leq \biggl(L_{\mathscr{F}}+ \frac{(1\omega )}{\mathcal{N}(\omega )}L_{\mathscr{G}} + \frac{L_{\mathscr{H}}\xi ^{\omega }}{\Gamma (\omega +1)} \biggl(1+ \frac{\omega }{\mathcal{N}(\omega )} \biggr) \biggr) \Vert \phi \bar{\phi } \Vert _{ \mathscr{Z}}. $$
Thus we get
$$ \bigl\Vert \mathscr{A}(\phi )\mathscr{A}(\bar{\phi }) \bigr\Vert _{\mathscr{Z}}\leq \Upsilon \Vert \phi \bar{\phi } \Vert _{\mathscr{Z}}.$$
(5)
Therefore, \(\mathscr{A}\) is a contraction and the said problem (1) has unique solutions.
Thanks to Theorem 1, we derive the next result.
Theorem 3
Thanks to hypotheses \((A_{1})\) and \((A_{2})\), the HFDE (1) has at least one solution if
$$ L_{\mathscr{F}}+\frac{(1\omega )}{\mathcal{N}(\omega )}L_{ \mathscr{G}}< 1. $$
Proof 3
Here we define two operators F and G from (4) as
$$\begin{aligned}& F\phi (t)= \mathscr{F} \bigl(t,\phi (t) \bigr)+ \frac{(1\omega )}{\mathcal{N}(\omega )}\mathscr{G} \bigl(t,\phi (t) \bigr), \\& \begin{aligned}[t] G\phi (t)={}& \int _{0}^{\xi } \frac{(\xi \eta )^{\omega 1}}{\Gamma (\omega )}\mathscr{H} \bigl(\eta , \phi (\eta ) \bigr)\,d\eta \\ &{}+\frac{\omega }{\mathcal{N}(\omega )\Gamma (\omega )} \int _{0}^{t}(t \eta )^{\omega 1} \mathscr{G} \bigl(\eta ,\phi (\eta ) \bigr)\,d\eta . \end{aligned} \end{aligned}$$
(6)
Step 1. Let \(\phi ,\bar{\phi }\in \mathscr{Z}\), we have from (6)
$$ \bigl\vert F\phi (t)F\bar{\phi }(t) \bigr\vert \leq \bigl\vert \mathscr{F} \bigl(t,\phi (t) \bigr) \mathscr{F} \bigl(t,\bar{\phi }(t) \bigr) \bigr\vert + \biggl\vert \frac{(1\omega )}{\mathcal{N}(\omega )} \bigl(\mathscr{G} \bigl(t,\phi (t) \bigr) \mathscr{G} \bigl(t,\bar{\phi }(t) \bigr) \bigr) \biggr\vert . $$
On simplification, we obtain
$$ \Vert F\phi F\bar{\phi } \Vert _{\mathscr{Z}} \leq L_{\mathscr{F}} \Vert \phi  \bar{\phi } \Vert _{\mathscr{Z}} +\frac{(1\omega )}{\mathcal{N}(\omega )}L_{ \mathscr{G}} \Vert \phi \bar{\phi } \Vert _{\mathscr{Z}}. $$
We have
$$ \Vert F\phi F\bar{\phi } \Vert _{\mathscr{Z}} \leq \biggl(L_{\mathscr{F}}+ \frac{(1\omega )}{\mathcal{N}(\omega )}L_{\mathscr{G}} \biggr) \Vert \phi  \bar{\phi } \Vert _{\mathscr{Z}}. $$
Thus F is contraction.
Step 2. To derive the required condition in respect of G, let \(\mathscr{B}=\{\phi \in \mathscr{Z}; \\phi \_{\mathscr{Z}}\leq \rho \}\), then from (6) we can do
$$\begin{aligned}& \begin{aligned} \bigl\vert G\phi (t) \bigr\vert &\leq \biggl\vert \int _{0}^{\xi } \frac{(\xi \eta )^{\omega 1}}{\Gamma (\omega )}\mathscr{H} \bigl(\eta , \phi (\eta ) \bigr)\,d\eta \biggr\vert + \biggl\vert \frac{\omega }{\mathcal{N}(\omega )\Gamma (\omega )} \int _{0}^{t}(t \eta )^{\omega 1} \mathscr{G} \bigl(\eta ,\phi (\eta ) \bigr)\,d\eta \biggr\vert \\ &\leq \int _{0}^{\xi } \frac{(\xi \eta )^{\omega 1}}{\Gamma (\omega )} \bigl\vert \mathscr{H} \bigl(\eta , \phi (\eta ) \bigr) \bigr\vert \,d\eta + \frac{\omega }{\mathcal{N}(\omega )\Gamma (\omega )} \int _{0}^{t}(t \eta )^{\omega 1} \bigl\vert \mathscr{G} \bigl(\eta ,\phi (\eta ) \bigr) \bigr\vert \,d\eta , \end{aligned} \\& \Vert G\phi \Vert _{\mathscr{Z}} \leq \frac{\xi ^{\omega }}{\Gamma (\omega +1)} \bigl(C_{\mathscr{H}} \Vert \phi \Vert _{\mathscr{Z}}+M_{\mathscr{H}} \bigr) + \frac{\omega \xi ^{\omega }}{\mathcal{N}(\omega )\Gamma (\omega +1)} \bigl(C_{\mathscr{G}} \Vert \phi \Vert _{\mathscr{Z}}+M_{\mathscr{G}} \bigr), \\& \Vert G\phi \Vert _{\mathscr{Z}} \leq \frac{\xi ^{\omega }}{\Gamma (\omega +1)} \biggl(C_{\mathscr{H}}\rho +M_{ \mathscr{H}}+\frac{\omega }{\mathcal{N}(\omega )} (C_{\mathscr{G}} \rho +M_{\mathscr{G}} ) \biggr)=\Lambda , \\& \Vert G\phi \Vert _{\mathscr{Z}} \leq \Lambda . \end{aligned}$$
Thus G is bounded. Further, as \(\mathscr{H}\), \(\mathscr{G}\) are continuous, therefore G is also.
Step 3. For equicontinuity, let \({t}_{1}<{t}_{2}\in \mathscr{J}\), we have
$$\begin{aligned} \bigl\vert G\phi (t_{2})G\phi (t_{1}) \bigr\vert \leq & \biggl\vert \frac{\omega }{\mathcal{N}(\omega )\Gamma (\omega )} \int _{0}^{t_{2}}(t_{2} \eta )^{\omega 1}\mathscr{G} \bigl(\eta ,\phi (\eta ) \bigr)\,d\eta \\ &{}\frac{\omega }{\mathcal{N}(\omega )\Gamma (\omega )} \int _{0}^{t_{1}}(t_{1} \eta )^{\omega 1}\mathscr{G} \bigl(\eta ,\phi (\eta ) \bigr)\,d\eta \biggr\vert \\ \leq & \biggl\vert \frac{\omega }{\mathcal{N}(\omega )\Gamma (\omega )} \int _{0}^{t_{1}} \bigl[(t_{2}\eta )^{\omega 1}(t_{2}\eta )^{\omega 1} \bigr] \mathscr{G} \bigl(\eta ,\phi (\eta ) \bigr)\,d\eta \biggr\vert \\ &{}+ \biggl\vert \frac{\omega }{\mathcal{N}(\omega )\Gamma (\omega )} \int _{t_{1}}^{t_{2}}(t_{2} \eta )^{\omega 1}\mathscr{G} \bigl(\eta ,\phi (\eta ) \bigr)\,d\eta \biggr\vert \\ \leq &\frac{\omega }{\mathcal{N}(\omega )\Gamma (\omega +1)} (C_{ \mathscr{G}}\rho +M_{\mathscr{G}} ) \bigl(t_{1}^{\omega }t_{2}^{ \omega } \bigr). \end{aligned}$$
(7)
Obviously, from (7), we see that \(t_{1}\rightarrow t_{2}\), the righthand side of the above inequality goes to zero, therefore \(\G\phi (t_{2})G\phi (t_{1})\_{\mathscr{Z}}\rightarrow 0\) as \(t_{1}\rightarrow t_{2}\). As the operator Gϕ is continuous and bounded, so it is uniformly continuous. Also \(G(\mathscr{B})\subset \mathscr{B}\) is compact. Thanks to Arzelá–Ascoli theorem, the operator G fulfills all the conditions of complete continuity. Hence HFDE (1) has at least one solution.