For the convenience of the reader, we present here some basic definitions and lemmas, which are used throughout this paper.
Definition 1
([19])
Let \(\overline{J}=[a, b]\) be a finite interval on the half-axis \(\mathbb{R}^{+}\), \(C (\overline{J})\) be the Banach space of all continuous functions from J̅ into \(\mathbb{R}^{+}\) with the norm \(\|y\|_{C} = \max_{t \in \overline{J}} | y(t)|\) and the parameters \(\rho >0\), \(0\leq \gamma < 1\).
-
(1)
The weighted space \(C_{\gamma , \rho} (\overline{J})\) of continuous functions y on \((a, b]\) is defined by
$$ C_{ \gamma ,\rho} (\overline{J}) = \biggl\{ y : (a, b] \to \mathbb{R} : \biggl( \frac{ t^{\rho}- a^{\rho}}{ \rho} \biggr)^{\gamma} y(t) \in C(\overline{J}) \biggr\} ,$$
with the norm
$$ \Vert y \Vert _{ C_{\gamma , \rho}}= \biggl\Vert \biggl( \frac{t^{\rho}- a^{\rho}}{ \rho} \biggr)^{\gamma} y(t) \biggr\Vert _{C} = \max _{t\in \overline{J}} \biggl\vert \biggl( \frac{ t^{\rho}- a^{\rho}}{\rho} \biggr)^{ \gamma} y(t) \biggr\vert ,$$
where \(C_{ 0,\rho} (\overline{J})= C (\overline{J})\).
-
(2)
Let \(\delta _{\rho}= ( t^{\rho}\frac{\mathrm{d}}{\mathrm{d}t} )\). For \(n \in \mathbb{N}\), we denote by \(C^{n}_{ \delta _{\rho}, \gamma} (\overline{J})\) the Banach space of functions y that are continuously differentiable on J̅, with operator \(\delta _{\rho}\), up to order \((n - 1)\) and that have the derivative \(\delta ^{n}_{\rho} y\) of order n on \((a, b]\) such that \(\delta ^{n}_{ \rho} y\in C_{\gamma , \rho}(\overline{J})\), that is,
$$ C^{n}_{ \delta _{\rho},\gamma} (\overline{J}) = \bigl\{ y : (a, b] \to \mathbb{R} : \delta ^{k}_{\rho} \in C(\overline{J}), k=0, 1 \dots , n-1, \delta ^{n}_{\rho} y\in C_{ \gamma ,\rho} ( \overline{J}) \bigr\} ,$$
where \(n \in \mathbb{N}\), with the norms
$$ \begin{aligned} & \Vert y \Vert _{C^{n}_{\delta _{\rho}} } = \sum _{k=0}^{n} \max_{t\in \overline{J} } \bigl\vert \delta ^{k}_{ \rho} g(x) \bigr\vert , \\ & \Vert y \Vert _{C^{n}_{\delta _{\rho},\gamma} } = \sum_{k=0}^{n-1} \bigl\Vert \delta ^{k}_{ \rho} g \bigr\Vert _{C} + \bigl\Vert \delta ^{n}_{\rho} y \bigr\Vert _{C_{ \gamma ,\rho}}. \end{aligned} $$
For \(n=0\), we have \(C^{0}_{ \delta _{\rho},\gamma} (\overline{J})= C_{\gamma , \rho} ( \overline{J})\).
Definition 2
([17, 18])
The generalized left-sided fractional integral \({}^{\rho}\mathcal{I}^{\alpha}_{a^{+}} [y](\cdot )\) of order \(\alpha \in \mathbbm{ }\mathbb{C}\), (\(Re(\alpha ) >0\)) is defined for \(y \in C^{1}_{\gamma}(\overline{J})\) by
$$ {}^{\rho} \mathcal{I}^{\alpha}_{a^{+}} [y] (t) = \frac { \rho ^{1-\alpha} }{ \Gamma (\alpha ) } \int _{a}^{t} \bigl( t^{\rho} - \xi ^{\rho} \bigr)^{ \alpha -1 } \xi ^{\rho -1} y(\xi ) \,\mathrm{d} \xi , $$
(5)
for \(t > a\) and \(\rho > 0\), provided the integral exists. Similarly, the right-sided fractional integral \({}^{\rho} \mathcal{I}^{\alpha}_{b^{-}} [y] (\cdot )\) is defined by
$$ {}^{\rho} \mathcal{I}^{\alpha}_{b^{-}} [y] (t) = \frac { \rho ^{1-\alpha} }{ \Gamma (\alpha )} \int _{t}^{b} \bigl( t^{\rho} - \xi ^{\rho} \bigr)^{\alpha -1} \xi ^{\rho -1} y(\xi ) \,\mathrm{d} \xi , \quad \forall t< b. $$
(6)
Definition 3
([17, 18])
Let \(\alpha \in \mathbbm{ }\mathbb{C}\), with \(Re(\alpha ) \geq 0\), \(n = [Re(\alpha )] + 1\) and \(\rho >0\). The generalized fractional derivatives, corresponding to the generalized fractional integrals (5) and (6), are defined for \(0 \leq a < t < b \leq \infty \) and \(y \in C^{1}_{\gamma}(\overline{J})\) by
$$ {}^{\rho} \mathcal{D}_{a^{+}}^{\alpha } [y] (t) = \frac{ \rho ^{ \alpha -n-1} }{\Gamma (n-\alpha )} \biggl( t^{1-\rho} \frac{\mathrm{d} }{{\mathrm {d}} t} \biggr)^{n} \int _{a}^{t} \bigl( t^{\rho} - \xi ^{\rho} \bigr)^{ n - \alpha +1} \xi ^{ \rho -1} y(\xi ) \,\mathrm{d}\xi , $$
(7)
and
$$ {}^{\rho} \mathcal{D}_{b^{-}}^{\alpha } [y] (t)= \frac{\rho ^{\alpha -n-1} }{ \Gamma (n-\alpha )} \biggl( - t^{ 1 - \rho} \frac{\mathrm{d}}{\mathrm{d} t} \biggr)^{n} \int _{t}^{b} \bigl( t^{\rho} - \xi ^{\rho} \bigr)^{n-\alpha +1} \xi ^{\rho -1} y( \xi ) \,\mathrm{d}\xi , $$
(8)
if the integrals exist.
Definition 4
([36])
Let order α and type β satisfy \(0<\alpha \leq 1\) and \(0\leq \beta \leq 1\). The H-\(\mathrm{K}\mathbb{FD}\) (left sided / right sided), with respect to t, with \(\rho >0\) of a function \(y \in C_{1-\gamma , \rho} (\overline{J})\) is defined by
$$\begin{aligned} {}^{\rho} \mathcal{D}_{a^{\pm}}^{ \alpha , \beta } [y] (t)&= \biggl( \pm ^{ \rho} \mathcal{I}_{ a^{\pm}}^{ \beta ( 1- \alpha )} \biggl(t^{\rho -1} \frac{\mathrm{d}}{\mathrm{d} t} \biggr) {}^{\rho} \mathcal{I}^{ ( 1 - \beta ) ( 1 - \alpha )}_{ a^{\pm}} [y] \biggr) (t) \\ &= \bigl( \pm {}^{\rho} \mathcal{I}_{a^{\pm}}^{ \beta ( 1 - \alpha )} \delta _{\rho} {}^{\rho} \mathcal{I}^{(1-\beta ) ( 1 - \alpha )}_{a^{\pm}} [y] \bigr) (t), \end{aligned}$$
(9)
where \({}^{\rho} \mathcal{I}_{a^{\pm}}^{\eta}\) is the generalized fractional integral given in Definition 2.
Properties 1
([17])
We recall some properties of \({}^{\rho}\mathcal{D}_{a^{+}}^{\alpha , \beta}\) as follows:
-
P1)
The operator \({}^{\rho}\mathcal{D}_{a^{+}}^{\alpha , \beta}\) can be written as
$$ {}^{\rho}\mathcal{D}_{a^{+}}^{\alpha ,\beta} = {}^{\rho} \mathcal{I}_{a^{+}}^{\beta ( 1 - \alpha )} \delta _{\rho} {}^{\rho} \mathcal{I}^{1-\gamma}_{a^{+}} = {}^{\rho} \mathcal{I}_{a^{+}}^{\beta (1-\alpha )} {}^{\rho} \mathcal{D}^{\gamma}_{a^{+}},$$
where \(\gamma = \alpha + \beta (1-\alpha )\).
-
P2)
The fractional derivative \({}^{\rho}\mathcal{D}_{a^{+}}^{\alpha , \beta}\) is an interpolator of the following fractional derivatives:
-
Hilfer (\(\rho \to 1\)),
-
Hilfer–Hadamard (\(\rho \to 0\)),
-
generalized (\(\beta = 0\)),
-
generalized Caputo-type (\(\beta = 1\)),
-
Riemann–Liouville (\(\beta = 0\), \(\rho \to 1\)),
-
Hadamard (\(\beta = 0\), \(\rho \to 0\)),
-
Caputo (\(\beta = 1\), \(\rho \to 1\)),
-
Caputo–Hadamard (\(\beta = 1\), \(\rho \to 0\)),
-
Liouville (\(\beta = 0\), \(\rho \to 1\), \(a = 0\)),
-
Weyl (\(\beta = 0\), \(\rho \to 1\), \(a =-\infty \)).
First, we state the following key lemma.
Lemma 5
([37])
Let \({}^{\rho} \mathcal{I}_{a^{+}}^{\alpha}\) and \({}^{\rho}\mathcal{D}_{a^{+}}^{\alpha} \), as defined in Eqs. (5) and (7), respectively, for \(t> a\). Then, for \(\alpha \geq 0\) and \(\zeta >0\), we have
$$ {}^{\rho} \mathcal{I}_{a^{+}}^{\alpha} \biggl[ \biggl( \frac{t^{\rho} - a^{\rho}}{\rho} \biggr)^{\zeta -1} \biggr](t) = \frac{\Gamma (\zeta )}{\Gamma (\alpha +\zeta )} \biggl( \frac{t^{\rho}-a^{\rho}}{ \rho} \biggr)^{\alpha +\zeta -1}, $$
and
$$ {}^{\rho}\mathcal{D}_{a^{+}}^{ \alpha} \biggl[ \biggl( \frac{t^{\rho} - a^{\rho}}{ \rho} \biggr)^{\zeta -1} \biggr](t) = 0, $$
for almost all \(\alpha \in (0,1)\).
Theorem 6
([6, 17])
Let \(\alpha >0\), \(\beta >0\), \(1\leq p\leq \infty \), \(0< a < b <\infty \), and \(\rho , c \in \mathbb{R}\), \(\rho \geq c\), Then, for \(y \in C^{p}_{c}(J)\),
$$ {}^{\rho}\mathcal{I}_{a^{+}}^{\alpha}\, {}^{\rho} \mathcal{I}_{a^{+}}^{\beta}[y] = {}^{\rho} \mathcal{I}_{a^{+}}^{\alpha +\beta} [y], \qquad {}^{\rho} \mathcal{D}_{a^{+}}^{\alpha} {}^{\rho}\mathcal{D}_{a^{+}}^{\beta}[y] = {}^{\rho} \mathcal{D}_{a^{+}}^{\alpha +\beta}[y].$$
Lemma 7
([36])
Let \(0<\alpha <1\), \(0\leq \gamma <1\). If \(y\in C_{\gamma}(\overline{J})\) and \(y\in C^{1}_{\gamma}(\overline{J})\), then
$$ {}^{\rho} \mathcal{I}_{a^{+}}^{\alpha} {}^{\rho} \mathcal{D}_{a^{+}}^{\alpha} [y] (t) = y(t) - \frac{ {}^{\rho}\mathcal{I}_{a^{+}}^{1-\alpha}[y](a) }{ \Gamma (\alpha )} \biggl( \frac{ t^{\rho}-a^{\rho}}{\rho} \biggr)^{\alpha -1},\quad \forall x \in J= (a, b). $$
Lemma 8
([17])
Let \(\alpha >0\), \(0\leq \gamma <1\), and \(y\in C_{\gamma}( \overline{J})\). Then,
$$ {}^{\rho} \mathcal{D}_{a^{+}}^{ \alpha} {}^{\rho} \mathcal{I}_{a^{+}}^{\alpha} [y](t) = y(t), $$
for each \(t\in J\).
Lemma 9
([36])
Let \(0< a< b<\infty \), \(\alpha >0\), \(0\leq \gamma <1\), and \(y \in C_{\gamma ,\rho}(\overline{J})\). If \(\alpha >\gamma \), then \({}^{\rho}\mathcal{I}_{a^{+}}^{\alpha} [y]\) is continuous on J̅ and
$$ {}^{\rho}\mathcal{I}_{a^{+}}^{\alpha} [y] (a) = \lim _{t\to a^{+}} {}^{\rho} \mathcal{I}_{a^{+}}^{ \alpha} [y](t)=0.$$
Throughout the remainder of this paper, we consider the following function spaces defined in [36]. We consider the parameters α, β, γ, and μ satisfying \(\gamma = \alpha +\beta -\alpha \beta \), for \(0 \leq \mu <1\),
$$\begin{aligned}& C_{1-\gamma ,\rho}^{\alpha ,\beta} (\overline{J})= \bigl\{ y\in C_{1- \gamma ,\rho} (\overline{J}), {}^{\rho}\mathcal{D}_{a^{+}}^{\alpha , \beta} [y] \in C_{\mu ,\rho} (\overline{J}) \bigr\} , \end{aligned}$$
(10)
$$\begin{aligned}& C_{1-\gamma ,\rho}^{\gamma} (\overline{J}) = \bigl\{ y\in C_{1-\gamma , \rho} (\overline{J}), {}^{\rho}\mathcal{D}_{a^{+}}^{\gamma} [y] \in C_{1 - \gamma ,\rho} (\overline{J}) \bigr\} , \end{aligned}$$
(11)
and \(C_{1-\gamma ,\rho}^{\gamma} (\overline{J}) \subset C_{1-\gamma ,\rho}^{\alpha ,\beta} (\overline{J})\).
Lemma 10
([36])
Let \(0<\alpha <1\), \(0\leq \beta \leq 1\), and \(\gamma = \alpha +\beta (1-\alpha )\). If \(y\in C_{1-\gamma}^{\gamma} (\overline{J})\), then
$$ {}^{\rho}\mathcal{I}_{a^{+}}^{\gamma} {}^{\rho} \mathcal{D}_{a^{+}}^{\gamma} [y] = {}^{\rho} \mathcal{I}_{a^{+}}^{\alpha} {}^{\rho} \mathcal{D}_{a^{+}}^{\alpha ,\beta} [y], $$
(12)
and
$$ {}^{\rho}\mathcal{D}_{a^{+}}^{\gamma} {}^{\rho} \mathcal{I}_{a^{+}}^{\alpha}[y] = {}^{\rho} \mathcal{D}_{a^{+}}^{ \beta ( 1 - \alpha )} [y]. $$
(13)
Lemma 11
([36])
Let \(g \in L^{1}(J)\). If \({}^{\rho}\mathcal{D}_{a^{+}}^{\beta (1-\alpha )} [y]\) exists on \(L^{1}(J)\), then
$$ {}^{\rho}\mathcal{D}_{a^{+}}^{\alpha ,\beta} {}^{\rho} \mathcal{I}_{a^{+}}^{\alpha} [y] = {}^{\rho}\mathcal{I}_{a^{+}}^{ \beta (1-\alpha )} {}^{\rho} \mathcal{D}_{a^{+}}^{ \beta (1-\alpha )} [y].$$
Lemma 12
([36])
Let \(0<\alpha <1\), \(0\leq \beta \leq 1\), and \(\gamma = \alpha +\beta (1-\alpha )\). If \(y\in C_{1-\gamma} (\overline{J})\) and \({}^{\rho}\mathcal{I}_{a^{+}}^{1-\beta (1-\alpha )}\in C^{1}_{1- \gamma} (\overline{J})\), then \({}^{\rho}\mathcal{D}_{a^{+}}^{\alpha ,\beta} {}^{\rho} \mathcal{I}_{a^{+}}^{\alpha}\) exists on J̅ and
$$ {}^{\rho}\mathcal{D}_{a^{+}}^{\alpha ,\beta} {}^{\rho} \mathcal{I}_{a^{+}}^{\alpha} [y](t)=y(t),\quad \forall t\in \overline{J}. $$
(14)
The following key theorems are used in the remainder of the paper.
Theorem 13
([36])
Let \(\gamma = \alpha + \beta ( 1 - \alpha )\), where \(0<\alpha <1\) and \(0\leq \beta \leq 1\). If \(g: (a,b] \times \mathbb{R} \to \mathbbm{ }\mathbb{R}\) is a function such that \(g(\cdot , y(\cdot ) ) \in C_{1-\gamma ,\rho} (\overline{J})\) for any \(y \in C_{1-\gamma ,\rho}\). A function \(y \in C^{\gamma}_{ 1-\gamma ,\rho} (\overline{J})\) is the solution of the fractional initial-value problem
$$ \textstyle\begin{cases} {}^{\rho} \mathcal{D}_{a^{+}}^{ \alpha , \beta} [y](t)= g(t, y(t)), \\ {}^{\rho}\mathcal{I}_{a^{+}}^{ 1-\gamma} [y](a)=c, \end{cases} $$
if and only if y satisfies the following equation
$$ y(t) = \frac{c}{ \Gamma (\gamma )} \biggl( \frac{t^{\rho} - a^{\rho}}{\rho} \biggr)^{\gamma -1} + \frac{1}{ \Gamma (\alpha )} \int _{a}^{t} \biggl( \frac{t^{\rho} - \xi ^{\rho}}{ \rho} \biggr)^{ \alpha -1} \xi ^{\rho -1} g\bigl(\xi , y(\xi )\bigr) \,\mathrm{d} \xi . $$
Theorem 14
(Banach’s fixed-point theorem [38])
Let \(\mathcal{Y}\) be a nonempty closed subset of a Banach space \(\mathfrak{X}\) and \(\mathcal{F} : \mathcal{Y} \to \mathcal{Y}\) be a contraction operator. Then, there is a unique \(y\in \mathcal{Y}\) with \(\mathcal{F}(y) = y\).
Theorem 15
(Schauder’s fixed-point theorem [38])
Let \(\mathcal{Y}\) be a nonempty closed subset of a Banach space \(\mathfrak{X}\) and \(\mathcal{F}: \mathcal{Y} \to \mathcal{Y}\) be a continuous mapping such that \(\mathcal{F}(\mathcal{Y}) \subset \mathfrak{X}\) is relatively compact. Then, \(\mathcal{F}\) has at least one fixed point in \(\mathcal{Y}\).
Theorem 16
(Arzelà–Ascoli theorem [38])
A subset \(\mathcal{Y}\) of \(C(\mathfrak{X})\) is relatively compact iff it is closed, bounded and equicontinuous.