**1. Introduction**

The fractional-order differential equation is the oldest theory in the field of science and engineering. This theory has been used over the years, as the outcomes were found to be important in the field of economics, control theory and material sciences see [1–4]. Because of the nonlocal property of fractional-order differential equation, researchers are allowed to select the most appropriate

operator and use it in order to get a better description of the complex phenomena in the real world. The generalization of classical calculus are the fractional calculus. Nevertheless, there are various definitions of fractional integrals and derivatives of arbitrary order with different types of operator. Recently, Furati et al. [5] proposed a Hilfer fractional derivatives which interpolates with Riemann-Liouville and Caputo fractional derivatives. These fractional operator provide an extra degree of freedom when choosing the initial condition. Furthermore, models based on this operator provide an excellent results compared with the integer-order derivatives, for example, we refer the interesting reader to see [6–18].

Qualitative analysis of fractional differential equations plays a vital role in the field of fractional differential equations. However, many researchers studied the existence and uniqueness of solution of differential equation with different types of fractional integral and derivatives. More recently, motivated by classical Riemann-Liouville, Caputo fractional derivative, Hilfer-fractional derivative, *ψ*-Riemann-Liouville integral and *ψ*-Caputo fractional derivatives, Sousa and Oliveira [19] initiated an interesting fractional differential operator called *ψ*-Hilfer fractional derivatives, that is a fractional derivative of a function with respect to another function *ψ*. These fractional derivatives generalized the aforementioned fractional derivatives and integrals. The main advantages of these operator is the freedom of choice of the function *ψ* and its merge and acquire the properties of the aforementioned fractional operators. Results based on these setting can be found in [18–34]. The Ulam-Hyers stability point of view, is the vital and special type of stability that attracts many researchers in the field of mathematical analysis. Moreover, the Ulam-Hyers and Ulam-Hyers-Rassias stability of linear, implicit and nonlinear fractional differential equations were examined in [17,35–49].

Pantograph differential equations are a special class of delay differential equation arising in deterministic situations and are of the form:

$$\begin{cases} \mathbf{g}'(\mathbf{s}) = \mathbf{k}\mathbf{g}(\mathbf{s}) + \log(\lambda \mathbf{s}), \qquad \mathbf{s} \in [0, b], \; b > 0, \; 0 < \lambda < 1, \\\mathbf{g}(\mathbf{0}) = \mathbf{g}\_0. \end{cases} \tag{1}$$

The pantograph is a device used in electric trains to collects electric current from the overload lines. This equation was modeled by Ockendon and Tayler [50]. Pantograph equation play a vital role in physics, pure and applied mathematics, such as control systems, electrodynamics, probability, number theory, and quantum mechanics. Motivated by their importance, a lot of researchers generalized these equation in to various forms and introduced the solvability aspect of such problems both theoretically and numerically, (for more details see [16,51–57] and references therein). However, very few works have been proposed with respect to pantograph fractional differential equations.

In [48], the authors considered an implicit fractional differential equations with nonlocal condition described by:

$$\begin{cases} D\_{0^{+}}^{a,\emptyset}w(\tau) = f(\tau, w(\tau), D\_{0^{+}}^{a,\emptyset}w(\tau)), & \tau \in I = [0, T],\\ I\_{0^{+}}^{1-\gamma}w(0) = \sum\_{i=1}^{m} c\_{i}w(\eta\_{i}), & a \le \gamma = a + \beta - a\beta, \eta\_{i} \in [0, T], \end{cases} \tag{2}$$

where *Dα*,*<sup>β</sup>* <sup>0</sup><sup>+</sup> (·) is the Hilfer fractional derivative of order (0 < *α* < 1) and type 0 ≤ *β* ≤ 1. The existence and uniqueness results were obtained by applying Schaefer's fixed point theorem and Banach's contraction principle. Moreover, the authors discussed the stability analysis via Gronwall's lemma. Sousa and Oliveira [47] discussed the existence, uniqueness and Ulam-Hyers-Rassias stability for a class of *ϕ*-Hilfer fractional differential equations described by:

$$\begin{cases} \, \, ^H \mathfrak{D}\_{a^+}^{a,\beta;\mathfrak{p}} \mathfrak{g}(t) = f(t, \mathfrak{g}(t) \, ^H \mathfrak{D}\_{a^+}^{a,\beta;\mathfrak{p}} \mathfrak{g}(t)), & t \in \mathcal{J} = [a, T]\_\prime\\ \mathfrak{D}\_{a^+}^{1-\gamma;\mathfrak{p}} \mathfrak{g}(a) = \mathfrak{g}\_{a\prime} & a \le \gamma = a + \beta - a\beta, \quad T > a, \end{cases} \tag{3}$$

where *<sup>H</sup>*D*α*,*β*;*<sup>ϕ</sup>* <sup>0</sup><sup>+</sup> (·) is the *ϕ*-Hilfer fractional derivative of order (0 < *α* ≤ 1) and operator (0 ≤ *β* ≤ 1), I1−*γ*;*<sup>ϕ</sup>* <sup>0</sup><sup>+</sup> (·), is the Riemann-Liouville fractional integral of order 1 − *γ*, with respect to the function *ϕ*, *<sup>f</sup>* : [*a*, *<sup>T</sup>*] <sup>×</sup> <sup>R</sup><sup>2</sup> <sup>→</sup> <sup>R</sup> is a continuous function. Recently Harikrishman et. al [58] established existence and uniqueness of nonlocal initial value problem for fractional pantograph differential equation involving *ψ*-Hilfer fractional derivative of the form:

$$\begin{cases} \, \mathbf{^{H}l^{a,\emptyset;\mathfrak{P}}} \boldsymbol{v}(s) = f(s, \boldsymbol{v}(s), \boldsymbol{v}(\lambda s)), & s \in (a, b], \quad s > a, \quad 0 < \lambda < 1, \\\, \mathbf{1}\_{a^{+}}^{1-\gamma;\mathfrak{P}} \boldsymbol{v}(a) = \sum\_{j=1}^{k} c\_{i} \boldsymbol{v}(\tau\_{j}), & \tau\_{j} \in (a, b], \quad \gamma = a + \beta - a\beta, \end{cases} \tag{4}$$

where *<sup>H</sup>*D *α*,*β*;*φ <sup>a</sup>*<sup>+</sup> (·) is the *ψ*-Hilfer fractional derivative of order 0 < *α* < 1 and type 0 ≤ *β* ≤ 1, I 1−*γ*;*ψ <sup>a</sup>*<sup>+</sup> (·), is the Riemann-Liouville fractional integral of order 1 − *γ*, with respect to the continuous function *ψ* such that *ψ* (·) <sup>&</sup>gt; 0, *<sup>f</sup>* <sup>∈</sup> *<sup>C</sup>*(*<sup>t</sup>* <sup>∈</sup> (*a*, *<sup>b</sup>*], <sup>R</sup>2, <sup>R</sup>).

Motivated by the papers [21,47,48] and some familiar results on fractional pantograph differential equations [16,52,55,58]. We discuss the existence and uniqueness of the solution of the implicit pantograph fractional differential equations involving *φ*-Hilfer fractional derivatives. Furthermore, the Ulam-Hyers and generalized Ulam-Hyers stability are also discussed. The implicit pantograph fractional differential equations involving *φ*-Hilfer fractional derivatives is of the form

$$\begin{cases} ^H\mathcal{D}\_{0^+}^{r,p;\Phi}z(t) = f(t, z(t), z(\gamma t)\_\* {}^H\mathcal{D}\_{0^+}^{r,p;\Phi}z(\gamma t)), & t \in \mathcal{J} = (0, T], \; 0 < \gamma < 1, \\ \mathcal{Z}\_{0^+}^{1-q;\Phi}z(0^+) = \sum\_{i=1}^m b\_i \mathcal{Z}\_{0^+}^{q;\Phi}z(\zeta\_i), & r \le q = r + p - rp, \end{cases} \tag{5}$$

where *<sup>H</sup>*D*r*,*p*;*<sup>φ</sup>* <sup>0</sup><sup>+</sup> (·) is the generalized *φ*-Hilfer fractional derivatives of order (0 < *r* < 1) and type (<sup>0</sup> <sup>≤</sup> *<sup>p</sup>* <sup>≤</sup> <sup>1</sup>), <sup>I</sup>1−*q*;*<sup>φ</sup>* <sup>0</sup><sup>+</sup> (·) and <sup>I</sup>*ρ*;*<sup>φ</sup>* <sup>0</sup><sup>+</sup> (·) are *φ*-Riemann-Lioville fractional integral of order 1 − *q* and *ρ* > 0 respectively with respect to the continuous function *φ* such that *φ* (·) <sup>=</sup> 0, *<sup>f</sup>* : (0, *<sup>T</sup>*] <sup>×</sup> <sup>R</sup><sup>3</sup> <sup>→</sup> <sup>R</sup> is a given continuous function, *<sup>T</sup>* <sup>&</sup>gt; 0, *bi* <sup>∈</sup> <sup>R</sup> and *<sup>ξ</sup><sup>i</sup>* ∈ J satisfying 0 <sup>&</sup>lt; *<sup>ξ</sup>*<sup>1</sup> <sup>≤</sup> *<sup>ξ</sup>*<sup>2</sup> ≤ ··· ≤ *<sup>ξ</sup><sup>m</sup>* <sup>&</sup>lt; *<sup>T</sup>* for *i* = 1, 2, ··· , *m*.

As far as we know, to the best of our understanding, results of Ulam-Hyers and generalized Ulam-Hyers stability with respect to the pantograph differential equation are very few and in fact most authors discuss existence and uniqueness, while we study existence, uniqueness and stability analysis for a class of implicit pantograph fractional differential equations with *φ*-Hilfer derivatives and nonlocal Riemann-Liouville fractional integral condition.

This paper contributes to the growth of qualitative analysis of fractional differential equation in particular pantograph fractional differential equation when *φ*-Hilfer fractional derivatives involved and the nonlocal initial condition proposed in this paper generalized the following initial conditions:


In addition, we notice that the function *f*(*s*, *v*(*s*), *v*(*λs*)), *s* ∈ (*a*, *b*], 0 < *λ* < 1, defined in Equation (4) is not well-define for some choices of *λ*.

Therefore, the paper is organized as follows: In Section 2, it recalls some basic and fundamental definitions and lemmas. In Section 3, we prove existence and uniqueness of the proposed problem (5). Ulam-Hyers and generalized Ulam-Hyers stability for the proposed problem were discussed in Section 4. While in Section 5, two examples were given to illustrate the applicability of our results. Lastly, the conclusion part of the paper is given in Section 6.

## **2. Preliminaries**

This section will recall some useful prerequisites facts, definitions and some fundamental lemmas with respect to fractional differential equations.

Throughout the paper, we denote <sup>C</sup>[<sup>J</sup> , <sup>R</sup>] the Banach space of all continuous functions from <sup>J</sup> into R with the norm defined by [1]

$$\|f\| = \sup\_{t \in \mathcal{J}} \{|f(t)|\}.$$

The weighted space <sup>C</sup>*q*,*φ*[<sup>J</sup> , <sup>R</sup>] of continuous function *<sup>f</sup>* on the interval [*a*, *<sup>T</sup>*] is defined by

$$\mathcal{L}\_{q,\phi}[\mathcal{J}, \mathbb{R}] = \{ f(t) : (a, T] : (\phi(t) - \phi(0))^q f(t) \in \mathcal{C}[\mathcal{J}, \mathbb{R}] \}\_{\prime \prime}$$

with the norm

$$\|f\|\_{\mathcal{E}\_{\theta,\theta}[\mathcal{T},\mathbb{R}]} = \| (\phi(t) - \phi(0))^q f(t) \| = \max | (\phi(t) - \phi(0))^q f(t) : t \in \mathcal{T} \vert.$$

Moreover, for each *<sup>n</sup>* <sup>∈</sup> <sup>N</sup> and 0 <sup>≤</sup> *<sup>q</sup>* <sup>&</sup>lt; 1 with *<sup>q</sup>* <sup>=</sup> *<sup>r</sup>* <sup>+</sup> *<sup>p</sup>* <sup>−</sup> *rp*

$$\mathcal{C}^{\mathfrak{n}}\_{q\mathcal{A}}[\mathcal{J}, \mathbb{R}] = \{ f^n \in \mathcal{C}\_{q\mathcal{A}}[\mathcal{J}, \mathbb{R}] \},$$

$$\mathcal{C}^{r,p}\_{q\gg}[\mathcal{I},\mathbb{R}] = \{ f \in \mathcal{C}\_{q\gg}[\mathcal{I},\mathbb{R}] \, : \, \mathcal{D}^{r,p\diamond \Phi}\_{0^{+}} \in \mathcal{C}\_{q\gg}[\mathcal{I},\mathbb{R}] \}.$$

Indeed, for *n* = 0, we have

$$\mathcal{C}^{0}\_{\mathfrak{q};\Phi}[\mathcal{J},\mathbb{R}] = \mathcal{C}\_{\mathfrak{q};\Phi}[\mathcal{J},\mathbb{R}],$$

with the norm

$$||f||\_{\mathcal{C}^{n}\_{q;\phi}[\mathcal{I},\mathbb{R}]} = \sum\_{k=0}^{n-1} ||f^{k}||\_{\mathcal{C}[\mathcal{I},\mathbb{R}]} + ||f^{n}||\_{\mathcal{C}^{n}\_{q;\phi}[\mathcal{I},\mathbb{R}]} \cdot \mathbb{I}$$

Furthermore, we present the following space <sup>C</sup>*r*,*<sup>p</sup>* <sup>1</sup>−*q*;*φ*[<sup>J</sup> , <sup>R</sup>] and <sup>C</sup>*<sup>q</sup>* <sup>1</sup>−*q*;*φ*[<sup>J</sup> , <sup>R</sup>] defined as:

$$\mathcal{L}\_{1-q;\phi}^{r,p}[\mathcal{J},\mathbb{R}] = \{ f \in \mathcal{C}\_{1-q;\phi}[\mathcal{J},\mathbb{R}], \mathcal{D}\_{0^+}^{r,p;\phi} \in \mathcal{C}\_{1-q;\phi}[\mathcal{J},\mathbb{R}] \}$$

and

$$\mathcal{C}\_{1-q;\mathfrak{\phi}}^q[\mathcal{I},\mathbb{R}] = \{ f \in \mathcal{C}\_{1-q;\mathfrak{\phi}}[\mathcal{J},\mathbb{R}], \mathcal{D}\_{0^+}^{q;\mathfrak{\phi}} \in \mathcal{C}\_{1-q;\mathfrak{\phi}}[\mathcal{J},\mathbb{R}] \}.$$

$$\text{Clearly, } \mathcal{C}^{q}\_{1-q;\phi}[\mathcal{I}, \mathbb{R}] \subset \mathcal{C}^{r,p}\_{1-q;\phi}[\mathcal{I}, \mathbb{R}].$$

**Definition 1** ([1])**.** *Let* (0, *<sup>b</sup>*] *be a finite or infinite interval on the half-axis* <sup>R</sup>+*, and <sup>φ</sup>*(*ξ*) <sup>≥</sup> <sup>0</sup> *be monotone function on* (*a*, *b*] *whose φ* (*ξ*) *is continuous on* (0, *b*)*. The φ-Hilfer Riemann-Liouville fractional integral of order r* <sup>∈</sup> <sup>R</sup><sup>+</sup> *of function w is defined by*

$$(\mathcal{Z}\_{0^{+}}^{r\beta}w)(\xi) = \frac{1}{\Gamma(r)} \int\_{0^{+}}^{\overline{\xi}} \phi'(s)(\phi(\xi) - \phi(s))w(s)ds, \quad \overline{\xi} > 0,\tag{6}$$

*where* Γ(·) *represent the Gamma function.*

**Definition 2** ([5])**.** *Let n* − 1 < *r* < *n,* 0 ≤ *p* ≤ 1*. The left-sided Hilfer fractional derivative of order r and parameter p of function w is defined by*

$$\mathcal{D}\_{0^{+}}^{r,p}w(\xi) = \left(\mathcal{Z}\_{0^{+}}^{p(n-r)}\mathcal{D}^{n}\mathcal{Z}\_{0^{+}}^{(1-p)(n-r)}w\right)(\xi),\tag{7}$$

*where* <sup>D</sup>*<sup>n</sup>* <sup>=</sup> *<sup>d</sup> dξ n .* The following Definition generalized Euqation (7).

**Definition 3** ([19])**.** *Let <sup>f</sup>* , *<sup>φ</sup>* ∈ C*n*[<sup>J</sup> , <sup>R</sup>] *be two functions such that <sup>φ</sup>*(*ξ*) <sup>≥</sup> <sup>0</sup> *and <sup>φ</sup>* (*t*) = 0 *for all <sup>ξ</sup>* <sup>∈</sup> [<sup>J</sup> , <sup>R</sup>] *and <sup>n</sup>* <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>r</sup>* <sup>&</sup>lt; *<sup>n</sup> with <sup>n</sup>* <sup>∈</sup> <sup>N</sup>*. The left-side <sup>φ</sup>-Hilfer fractional derivative of a function <sup>w</sup> of order r and type* (0 ≤ *p* ≤ 1) *is defined by*

$$\mathcal{T}\_{0^{+}}^{H} \mathcal{D}\_{0^{+}}^{r, p; \mathfrak{\phi}} w(\mathfrak{f}) = \mathcal{T}\_{0^{+}}^{p(n-r); \mathfrak{\phi}} \left( \frac{1}{\mathfrak{\phi}'(\mathfrak{f})} \frac{d}{d\mathfrak{f}} \right)^{n} \mathcal{T}\_{0^{+}}^{(1-p)(n-r); \mathfrak{\phi}} w(\mathfrak{f}). \tag{8}$$

The following lemma shows the semigroup properties of *φ*-Hilfer fractional integral and derivative.

**Lemma 1** ([5])**.** *Let r* <sup>≥</sup> <sup>0</sup>*,* <sup>0</sup> <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>1</sup> *and w* <sup>∈</sup> *<sup>L</sup>*1[<sup>J</sup> , <sup>R</sup>]*. Then*

$$T\_{0^{+}}^{r\circ\Phi}T\_{0^{+}}^{r\circ\Phi}w(\xi) = T\_{0^{+}}^{r+p\circ\Phi}w(\xi)\_{r}$$

*a.e ξ* ∈ J *. In particular, if w* ∈ C*q*,*φ*[<sup>J</sup> , <sup>R</sup>] *and w* ∈ C[<sup>J</sup> , <sup>R</sup>]*, then*

$$T\_{0^{+}}^{r\circ\Phi}T\_{0^{+}}^{p\circ\Phi}w(\xi) = T\_{0^{+}}^{r+p\circ\Phi}w(\xi),$$

*for all <sup>ξ</sup>* <sup>∈</sup> (0, *<sup>T</sup>*] *and <sup>H</sup>*D*r*;*<sup>φ</sup>*

$${}^{H}\mathcal{D}\_{0^{+}}^{\tau\Phi}\mathcal{T}\_{0^{+}}^{\tau\Phi}w(\xi) = w(\xi)\_{\prime}$$

*for all ξ* ∈ J *.*

The composition of the *φ*-Hilfer fractional integral and derivative operator is given by the following lemmas.

**Lemma 2** ([21])**.** *Let r* <sup>≥</sup> <sup>0</sup>*,* <sup>0</sup> <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>1</sup> *and q* <sup>=</sup> *<sup>r</sup>* <sup>+</sup> *<sup>p</sup>* <sup>−</sup> *rp. If w*(*ξ*) ∈ C*<sup>q</sup>* <sup>1</sup>−*q*[<sup>J</sup> , <sup>R</sup>]*, then*

$$\mathcal{I}\_{0^+}^{q\phi\_H} \mathcal{D}\_{0^+}^{q;\phi} w(\xi) = \mathcal{I}\_{0^+}^{r\phi\_H} \mathcal{D}\_{0^+}^{r,p\phi} w(\xi)$$

*and*

$${}^{H}\mathcal{D}\_{0^{+}}^{q;\phi}\mathcal{T}\_{0^{+}}^{r\phi}w(\mathfrak{f}) = {}^{H}\mathcal{D}\_{0^{+}}^{p(1-r);\phi}w(\mathfrak{f})\,.$$

**Lemma 3** ([6,19])**.** *If w* ∈ C*n*[<sup>J</sup> , <sup>R</sup>] *and let n* <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>r</sup>* <sup>&</sup>lt; *n,* <sup>0</sup> <sup>≤</sup> *<sup>p</sup>* <sup>≤</sup> <sup>1</sup> *and q* <sup>=</sup> *<sup>r</sup>* <sup>+</sup> *<sup>p</sup>* <sup>−</sup> *rp. Then*

$$\mathcal{Z}\_{0^{+}}^{r;\phi\_{H}}\mathcal{D}\_{0^{+}}^{r,p;\phi}w(\xi) = w(\xi) - \sum\_{k=1}^{n} \frac{(\phi(\xi) - \phi(0))^{q-k}}{\Gamma(q-k+1)} w\_{\phi}^{[n-k]} \mathcal{Z}\_{0^{+}}^{(1-p)(n-r);\phi} w(0),$$

*for all ξ* ∈ J . *Moreover, if* 0 < *r* < 1, *we have*

$$\mathcal{T}\_{0^{+}}^{r; \Phi H} \mathcal{D}\_{0^{+}}^{r, p; \Phi} w(\xi) = w(\xi) - \frac{(\phi(\xi) - \phi(0))^{q-1}}{\Gamma(q)} \mathcal{T}\_{0^{+}}^{(1-p)(1-r); \Phi} w(0) \dots$$

*In addition, if w* ∈ C1−*q*;*φ*[<sup>J</sup> , <sup>R</sup>] *and* <sup>I</sup>1−*q*;*<sup>φ</sup>* <sup>0</sup><sup>+</sup> *<sup>w</sup>* ∈ C<sup>1</sup> <sup>1</sup>−*q*;*φ*[<sup>J</sup> , <sup>R</sup>], *then*

$$\mathcal{Z}\_{0^{+}}^{q\phi\_{H}}\mathcal{D}\_{0^{+}}^{q;\phi}w(\xi) = w(\xi) - \frac{(\phi(\xi) - \phi(0))^{q-1}}{\Gamma(q)}\mathcal{Z}\_{0^{+}}^{(1-q);\phi}w(0),$$

*for all* 0 < *q* < 1 *and t* ∈ J .

**Lemma 4** ([6])**.** *Let r* <sup>&</sup>gt; <sup>0</sup>*,* <sup>0</sup> <sup>≤</sup> *<sup>q</sup>* <sup>&</sup>lt; <sup>1</sup> *and w* ∈ C*q*;*φ*[<sup>J</sup> , <sup>R</sup>]*. If r* <sup>&</sup>gt; *q, then* <sup>I</sup>*r*;*<sup>φ</sup>* <sup>0</sup><sup>+</sup> *<sup>w</sup>* ∈ C[<sup>J</sup> , <sup>R</sup>] *and*

$$\mathcal{T}\_{0^{+}}^{r\phi}w(0) = \lim\_{\vec{\xi} \to 0} \mathcal{T}\_{0^{+}}^{r\phi}w(\vec{\xi}) = 0.$$

**Lemma 5** ([21])**.** *Let r* <sup>&</sup>gt; <sup>0</sup>*,* <sup>0</sup> <sup>≤</sup> *<sup>p</sup>* <sup>≤</sup> <sup>1</sup> *and q* <sup>=</sup> *<sup>r</sup>* <sup>+</sup> *<sup>p</sup>* <sup>−</sup> *rp. If w* ∈ C*<sup>q</sup>* <sup>1</sup>−*q*;*φ*[<sup>J</sup> , <sup>R</sup>]*, then*

$$\mathcal{T}^{q;\phi}\_{0^{+}}\mathcal{D}^{q;\phi}\_{0^{+}}w(\emptyset) = \mathcal{T}^{r\phi}\_{0^{+}}\mathcal{D}^{r,p;\phi}\_{0^{+}}w(\emptyset).$$

*and*

$$\mathcal{D}\_{0^+}^{H} \mathcal{D}\_{0^+}^{q \phi} \mathcal{Z}\_{0^+}^{r \phi} w(\mathfrak{f}) = \mathcal{D}\_{0^+}^{q(1-r)\phi} w(\mathfrak{f}) .$$

**Lemma 6.** *Let f* <sup>∈</sup> *<sup>L</sup>*1(<sup>J</sup> ) *such that* <sup>D</sup>*p*(1−*r*);*<sup>φ</sup>* <sup>0</sup><sup>+</sup> *<sup>w</sup>* <sup>∈</sup> *<sup>L</sup>*1(<sup>J</sup> ) *exists, then*

$$
\mathcal{D}\_{0^+}^{r,p;\phi} \mathcal{T}\_{0^+}^{r;\phi} w(\xi) = \mathcal{T}\_{0^+}^{p(1-r);\phi} \mathcal{D}\_{0^+}^{p(1-r);\phi} w(\xi).
$$

Next, we take into account some important properties of *φ*-fractional derivative and integral operator as follows:

**Proposition 1** ([1])**.** *Let ξ* > 0*, r* ≥ 0 *and s* > 0*. Then, φ-fractional integral and derivative of a power function are given by*

$${}^{H}\mathcal{D}\_{0^{+}}^{r\phi}(\phi(\xi)-\phi(0))^{s-1} = \frac{\Gamma(s)}{\Gamma(s-r)}(\phi(\xi)-\phi(0))^{r+s-1}$$

*and*

$$\mathcal{Z}\_{0^{+}}^{r,\phi}(\phi(\xi^{\mathfrak{z}}) - \phi(0))^{s-1} = \frac{\Gamma(s)}{\Gamma(s+r)}(\phi(\xi^{\mathfrak{z}}) - \phi(0))^{r+s-1}.$$

*Furthermore, if* 0 < *r* < 1*, then*

$$\prescript{H}{}{\mathcal{D}}\_{0^{+}}^{r\phi} (\phi(\xi) - \phi(0))^{r-1} = 0.$$

**Theorem 1** ([19])**.** *If w* ∈ C1[<sup>J</sup> , <sup>R</sup>]*,* <sup>0</sup> <sup>&</sup>lt; *<sup>r</sup>* <sup>&</sup>lt; <sup>1</sup> *and* <sup>0</sup> <sup>≤</sup> *<sup>p</sup>* <sup>≤</sup> <sup>1</sup>*. Then we have the followings:* (*i*) *<sup>H</sup>*D*r*,*p*;*<sup>φ</sup>* <sup>0</sup><sup>+</sup> <sup>I</sup>*r*;*<sup>φ</sup>* <sup>0</sup><sup>+</sup> *w*(*ξ*) = *w*(*ξ*)*.* (*ii*) <sup>I</sup>*r*;*<sup>φ</sup>* <sup>0</sup><sup>+</sup> *<sup>H</sup>*D*r*,*p*;*<sup>φ</sup>* <sup>0</sup><sup>+</sup> *<sup>w</sup>*(*ξ*) = *<sup>w</sup>*(*ξ*) <sup>−</sup> (*φ*(*ξ*)−*φ*(0))*q*−<sup>1</sup> <sup>Γ</sup>(*q*) <sup>I</sup>(1−*p*)(1−*r*);*<sup>φ</sup>* <sup>0</sup><sup>+</sup> *w*(*ξ*)*.*

**Lemma 7** ([6])**.** *Let <sup>h</sup>* : J × <sup>R</sup> <sup>→</sup> <sup>R</sup> *such that for any <sup>z</sup>* ∈ C1−*q*;*φ*[<sup>J</sup> , <sup>R</sup>], *<sup>h</sup>* ∈ C1−*q*;*φ*[<sup>J</sup> , <sup>R</sup>]*. A function <sup>z</sup>* ∈ C*<sup>q</sup>* <sup>1</sup>−*q*;*φ*[<sup>J</sup> , <sup>R</sup>] *is a solution of the fractional initial value problem:*

$$\begin{cases} ^H D\_{0^+}^{r,p;\Phi} z(t) = h(t), \quad 0 < r \le 1, \quad 0 \le p \le 1, \\\ I\_{0^+}^{1-q,\Phi} z(0^+) = z\_0 \in \mathbb{R}, \quad q = r + p - rp, \end{cases}$$

*if and only if z satisfies the following integral equation,*

$$z(t) = \frac{z\_0}{\Gamma(q)} (\phi(t) - \phi(0))^{q-1} + \frac{1}{\Gamma(r)} \int\_{0^+}^t \phi'(s) (\phi(t) - \phi(s))^{r-1} h(s) ds.$$
