Generalized Substantial Fractional Operators and Well-Posedness of Cauchy Problem

Abstract

In this work, we focus on substantial fractional integral and differential operators which play an important role in modeling anomalous diffusion. We introduce a new generalized substantial fractional integral. Generalizations of fractional substantial derivatives are also introduced in both Riemann–Liouville and Caputo sense. Furthermore, we analyze fundamental properties of these operators. Finally, we consider a class of generalized substantial fractional differential equations and discuss the existence, uniqueness and continuous dependence of solutions on initial data.

1 Introduction

Fractional calculus originated on September 30, 1695, when Leibniz expressed his idea of derivative in a note to De l’Hospital. De l’Hospital asked about the meaning of $\frac{\mathrm{d}^{n}f(x)}{\mathrm{d}x^{n}}$ when $n=\frac{1}{2}$. But by now, the field of fractional calculus has been revolutionized. Nowadays, this field has become very popular among the scientists and a great number of different forms of fractional operators have been introduced by notable researchers [1,2,3,4,5,6,7].

When it comes to practical applications, the substantial derivatives, introduced by Friedrich et al. [8], have a wide range of utilization. For example, Friedrich et al. found that a fractional substantial derivative which represents important non-local couplings in space and time is involved in generalized Fokker–Planck collision operator. By taking a modified shifted substantial Grunwald formula, Hao et al. [9] found a second-order approximation of fractional substantial derivative. Chen and Deng [10] presented the numerical discretizations and some properties of the fractional substantial operators. Turgeman et al. [11] used the CTRW model [12] and derived forward as well as backward fractional Feynman–Kac equation by replacing the ordinary temporal derivative with substantial derivative.

The selection of a suitable fractional operator depends on the physical system under consideration. As a result, we observe numerous definitions of different fractional operators in the literature. So, it is logical to establish and study the generalized fractional operators, for which the existing ones are particular cases. One such general class, namely fractional operators with analytic kernels, proposed by Fernandez et al. [13] is a useful way of generalizing several types of fractional calculus which have been deeply studied in the past few years. Another general class of fractional operators, sometimes known as $\varPsi$-fractional calculus, is given by fractional operators of a function with respect to another function [5, 14, 15]. Erdelyi [16] discussed fractional integration with respect to a power function $t^{n}$; the same operator was rediscovered by Katugampola [17], and nowadays, it is often called Katugampola fractional calculus which generalizes the Hadamard and Riemann–Liouville fractional calculus. In [18], he presented a more generalized fractional integral operator such that the famous fractional operators, Erdelyi–Kober, Liouville, Katugampola, Hadamard, Riemann–Liouville and Weyl become special cases of it. Agrawal [19] presented some new operators which unified Riesz–Riemann–Liouville, the left and the right fractional Riemann–Liouville, Riesz–Caputo and Caputo derivatives and the fractional Riemann–Liouville integrals. These operators further investigated by Lupa et al. [20] and Odzijewicz et al. [21].

In this paper, we introduce the generalized substantial fractional operators in both Riemann–Liouville and Caputo sense and obtain the relations between the generalized substantial fractional integral and some other famous fractional integrals, namely Riemann–Liouville-type Katugampola, standard Riemann–Liouville, standard substantial and Hadamard fractional integrals. We establish the relations between the generalized substantial and Riemann–Liouville-type Katugampola fractional operators. Proofs of the composition rules for the newly defined generalized operators are also the part of this work. Finally, we prove the well-posedness results for a class of generalized substantial fractional differential equations.

The paper is organized as follows. In Sect. 2, we state definitions and some important properties of substantial and Katugampola fractional operators. In Sect. 3, the generalized substantial fractional operators are introduced and fundamental properties of these operators are analyzed. Sections 4 and 5 are devoted to well-posedness results for a class of generalized substantial fractional differential equations. Section 6 finally makes concluding statements about our main results.

2 Preliminaries

Prior to introducing fractional differential operators, we first give some notations for sake of convenience in further developments.

The notation ${}_\sigma D^m:=\left( \frac{\mathrm{d}}{\mathrm{d}t}+\sigma \right) ^m$ where $\left( \frac{\mathrm{d}}{\mathrm{d}t}+\sigma \right) ^m=\left( D+\sigma \right) \left( D+\sigma \right) \left( D+\sigma \right) \cdots \left( D+\sigma \right)$ appears frequently in the literature [10]. Along with the operator ${}_\sigma D$, in sequel we shall use the operator ${}_\sigma D^{m,\rho }:=\left( \frac{t^{1-\rho }}{\rho }\frac{\mathrm{d}}{\mathrm{d}t}+\sigma \right) ^m$, where $\sigma \in \mathbb {R}$ and $\rho \ne 0$. Also the generalized differential operator defined as $\frac{t^{1-\rho }}{\rho }\frac{\mathrm{d}^m}{\mathrm{d}t^m}$ will be denoted by $D^{m,\rho }$. We define function spaces $\varOmega _{\sigma ,\rho }^m [a,b]:=\left\{ \psi :e^{\sigma t^{\rho }}t^{1-\rho }\psi (t) \in AC^m[a,b] \right\}$ and $\varLambda _{\sigma ,\rho }^{p}[a,b]:=\left\{ \psi :e^{\sigma t^{\rho }}t^{1-\rho }\psi \in L_{p}[a,b] \right\}$ where AC[a, b] is the space of absolutely continuous functions and $L_{p}[a,b]~(1\le p<\infty )$ denotes the space of measurable functions on [a, b]. For simplicity, $\varOmega _{\sigma ,1}$ and $\varOmega _{0,\rho }$ will be denoted by $\varOmega _{\sigma }$ and $\varOmega _{\rho }$, respectively.

2.1 Substantial Fractional Operators

Definition 1

[8, 10] Let $\alpha$ and $\sigma$ be real numbers such that $\alpha > 0$ and $\psi \in \varLambda _{\sigma }^{1}[a,b]$, then substantial fractional integral operator is defined as $\tilde{I}^{\alpha }_{a}\psi (t)=\frac{1}{\varGamma (\alpha )}\int _{a}^{t} (t - s)^{\alpha -1} e^{-\sigma (t-s)} \psi (s) \mathrm{d}s$. Furthermore, for $\psi \in \varOmega _{\sigma }^{m}[a,b]$, $m-1<\alpha \le m$, the Riemann–Liouville-type substantial fractional derivative is defined as $\tilde{D}^{\alpha }_{a}\psi (t)=\tilde{D}^{m}_{a}{}^{}_{}\tilde{I}^{m-\alpha }_{a}\psi (t).$ The Caputo-type substantial fractional derivative is defined as ${}^c\tilde{D}^{\alpha }_{a}\psi (t)=\tilde{I}^{m-\alpha }_{a}\tilde{D}^{m}_{a}\psi (t).$

2.2 Katugampola Fractional Operators

For $\rho \ne 0$, let $I^{1,\rho }_{a}\psi (t)=\int _{a}^{t}\psi (s)d(s^\rho )$, where $d(s^\rho )=\rho s^{\rho -1}\mathrm{d}s$. Then, mth iterate of the integral operator $I^{1,\rho }_{a}$ is given by

$$\begin{aligned} \begin{aligned} I^{m,\rho }_{a}\psi (t)&=\int _{a}^{t}d\left( t_1^\rho \right) \int _{a}^{t_1}d\left( t_2^\rho \right) \int _{a}^{t_2}d\left( t_3^\rho \right) \cdots \int _{a}^{t_{n-1}}\psi (t_{n-1})d\left( t_{n-1}^\rho \right) \\&=\frac{\rho }{\varGamma (m)}\int _{a}^{t}\left( t^\rho -s^\rho \right) ^{m-1}\psi (s)s^{\rho -1}\mathrm{d}s. \end{aligned} \end{aligned}$$

(1)

Replacing the m by real $\alpha >0$ in (1), the Katugampola fractional integral is defined as

Definition 2

[17] For $\rho \ne 0$, $\alpha >0$ and $\psi \in \varLambda _{\rho }^{1}[a,b]$, the Katugampola fractional integral is given by

$$\begin{aligned} I^{\alpha ,\rho }_{a}\psi (t)=\frac{\rho }{\varGamma (\alpha )}\int _{a}^{t}(t^\rho -s^\rho )^{\alpha -1}\psi (s)s^{\rho -1}\mathrm{d}s. \end{aligned}$$

Furthermore, for $m-1<\alpha \le m$ and $\phi \in \varOmega _{\rho }^{m}[a,b]$ the Riemann–Liouville-type Katugampola fractional derivative is defined as $D^{\alpha ,\rho }_{a}\psi (t)=D^{m,\rho }I^{m-\alpha ,\rho }_{a}\psi (t)$ and the Caputo-type Katugampola is defined as ${}^cD_{a}^{\alpha }\psi (t)=I^{m-\alpha ,\rho }_{a}D^{m,\rho }\psi (t)$.

Remark 1

We introduced a slight modification in the definition of Katugampola fractional integral operator. The factor $\rho ^{1-\alpha }$ in original definition is now replaced with $\rho$. This avoids repeated appearance of some factors of $\rho$ in calculations [5, p. 103].

It is to be noted $D^{1,\rho } I^{1,\rho }_{a}\psi (t)=\psi (t)$. A repeated application of this identity leads us to the identity $D^{m,\rho } I^{m,\rho }_{a}\psi (t)=\psi (t)$. Furthermore, $I^{1,\rho } _{a}D^{1,\rho } \psi (t)=\int _a^tD^\rho \psi (s)d(s^\rho )=\int _a^td(\psi (s))=\psi (t)-\psi (a).$ Similarly, $I^{2,\rho } _{a}D^{2,\rho } \psi (t)=\psi (t)-\psi (a)-(t^\rho -a^\rho )D^{1,\rho }\psi (a).$ In general, a repeated application of preceding steps leads to Taylor-type expansion of $\psi$ as

$$\begin{aligned} \psi (t)= & {} \sum _{k=0}^{m-1}\frac{D^{k,\rho }\psi (a)}{k!}(t^\rho -a^{\rho })^k+ I_{a}^{m,\rho } D^{m,\rho } \psi (t). \end{aligned}$$

(2)

The Katugampola fractional differential and integral operators satisfy following properties [22, 23]:

• (P1) For $\psi \in \varLambda _{\rho }^{1}[a,b]$, ${I}_a^{\alpha ,\rho } I_a^{\beta ,\rho }\psi (t) ={I}_a^{\beta ,\rho } I_a^{\alpha ,\rho }\psi (t)= I_a^{\alpha +\beta ,\rho }\psi (t)$.

• (P2) For $\beta \ge \alpha$, and $\psi \in \varLambda _{\rho }^{1}[a,b]$, $D_a^{\alpha ,\rho } I_a^{\beta ,\rho }\psi (t)= I_a^{\beta -\alpha ,\rho }\psi (t)$ and ${}^cD_a^{\alpha ,\rho } I_a^{\beta ,\rho }\psi (t)=I_a^{\beta -\alpha ,\rho }\psi (t)$.

• (P3) For $\alpha >1$ and $m-1< \alpha \le m$ and $\psi \in \varOmega _{\rho }^{m}[a,b]$, we have

  $$\begin{aligned} I_{a}^{\alpha ,\rho } D_a^{\alpha ,\rho } \psi (t)=\psi (t)- \sum _{k=1}^{m-1} \frac{ \lim _{s\rightarrow a^+} D_{a}^{\alpha -k,\rho }\psi (s)}{\varGamma (\alpha -k+1)}(t^\rho -a^{\rho })^{\alpha -1}. \end{aligned}$$

  Specifically, for $0< \alpha < 1$, $I_{a}^{\alpha ,\rho } D_a^{\alpha ,\rho } \psi (t)=\psi (t)- \frac{I^{1-\alpha ,\rho }_{a}\psi (a)}{\varGamma (\alpha )}(t^\rho -a^{\rho })^{\alpha -1}$.

• (P4) For $\psi \in \varOmega _{\rho }^{m}[a,b]$ and $\beta \ge \alpha$,

  $$\begin{aligned} I_{a}^{\alpha ,\rho } {}^c D_a^{\alpha ,\rho } \psi (t)=\psi (t)- \sum _{k=0}^{m-1} \frac{D^{k,\rho }\psi (a)}{k!}(t^\rho -a^{\rho })^k. \end{aligned}$$

Lemma 1

Assume that $\psi \in \varOmega _{\rho }^{m}[a,b]$. Then ${}_\sigma D^{m,\rho } (e^{-\sigma t^\rho }\psi (t))=e^{-\sigma t^\rho }D^{m,\rho }\psi (t)$ and $e^{\sigma t^\rho } {}_\sigma D^{m,\rho } (\psi (t))=D^{m,\rho }(e^{\sigma t^\rho }\psi (t))$.

Proof

We prove this Lemma by induction. For $m=1$, we have

$$\begin{aligned} {}_\sigma D^{1,\rho } (e^{-\sigma t^\rho }\psi (t))=\frac{t^{1-\rho }}{\rho }\frac{\mathrm{d}}{\mathrm{d}t}\left( e^{-\sigma t^\rho }\psi (t)\right) +\sigma e^{-\sigma t^\rho }\psi (t)=e^{-\sigma t^\rho }D^{1,\rho }\psi (t). \end{aligned}$$

Assume the conclusion follows for $m-1$. Then,

$$\begin{aligned} {}_\sigma D^{m,\rho } (e^{-\sigma t^\rho }\psi (t))= & {} {}_{\sigma } D^{1,\rho }{}_{\sigma }D^{m-1,\rho }( e^{-\sigma t^\rho }\psi (t))\\= & {} {}_{\sigma } D^{1,\rho }\left( e^{-\sigma t^\rho }D^{m-1,\rho }\psi (t)\right) =e^{-\sigma t^\rho }D^{m,\rho }\psi (t). \end{aligned}$$

Second identity can be obtained in a similar way. $\square$

3 Generalized Substantial Fractional Integral and Derivatives

Motivated by definitions of substantial fractional operators, here we introduce new definitions for substantial fractional operators by generalizing Katugampola fractional operators. We also establish relation between generalized substantial fractional operators and the Katugampola fractional operators.

For $\rho \ne 0$ and $\sigma \in \mathbb {R}$, define ${}_\sigma I_{a}^{1,\rho }\psi (t)=\int _{a}^{t}\psi (s)e^{-\sigma (t^\rho -s^\rho )}d(s^\rho )$. Then, generalized substantial integral of order m is given by mth iterate of the integral ${}_\sigma I_{a}^{1,\rho }$ as

$$\begin{aligned} \begin{aligned} {}_\sigma I_{a}^{m,\rho }\psi (t)&=\int _{a}^{t}e^{-\sigma (t^\rho -t_1^\rho )}d(t_1^\rho ) \int _{a}^{t_1}e^{-\sigma (t^\rho -t_2^\rho )}d(t_2^\rho )\\&\quad \cdots \int _{a}^{t_{n-1}}e^{-\sigma (t^\rho -t_{n-1}^\rho )}\psi (t_{n-1})d(t_{n-1}^\rho )\\&=\frac{\rho }{\varGamma (m)}\int _{a}^{t}(t^\rho -s^\rho )^{m-1}e^{-\sigma (t^\rho -s^\rho )}\psi (s)s^{\rho -1}\mathrm{d}s. \end{aligned} \end{aligned}$$

(3)

We observe that ${}_\sigma D^{1,\rho }{}_\sigma I^{1,\rho }_{a}\psi (t)=\psi (t)$. A repeated application of this identity leads us to the identity ${}_\sigma D^{m,\rho }{}_\sigma I^{m,\rho }_{a}\psi (t)=\psi (t)$. Thus, for $m>n$, we have ${}_\sigma D^{m-n,\rho }{}_\sigma I^{m-n,\rho }_{a}\psi (t)=\psi (t)$. Application of the operator ${}_\sigma D^{n,\rho }$ to both sides of this identity leads to the identity ${}_\sigma D^{n,\rho }\psi (t)={}_\sigma D^{m,\rho }{}_\sigma I^{m-n,\rho }_{a}\psi (t).$ This relation will lead us to the definition of generalized fractional derivative. Furthermore, ${}_\sigma I^{1,\rho }_{a}{}_\sigma D^{1,\rho } \psi (t) =\int _a^t (\frac{s^{1-\rho }}{\rho }\frac{\mathrm{d}}{\mathrm{d}s}+\sigma )\psi (s)d(s^\rho ) =\psi (t)-\psi (a)e^{-\sigma (t^\rho -a^\rho )}.$ In general, a repeated application of this process leads us to generalized Taylor expansion involving generalized operators

$$\begin{aligned} {}_\sigma I^{m,\rho }_{a}{}_\sigma D^{m,\rho }\psi (t)=\psi (t)- e^{-\sigma (t^\rho - a^\rho )} \sum _{k=1}^{m} \frac{(t^\rho -a^{\rho })^{m-k} }{\varGamma (m-k+1)}\underset{s\rightarrow a^+}{\lim } {}_\sigma D^{m-k,\rho } \psi (s) \end{aligned}$$

provided $\psi \in \varOmega _{\rho }^{m}[a,b]$.

Definition 3

For real numbers $\sigma$, $\rho \ne 0$, $\alpha >0$ and $\psi \in \varLambda ^1_{\sigma ,\rho }[a,b]$, we define generalized substantial integral as

$$\begin{aligned}{}_\sigma I_a^{\alpha ,\rho }\psi (t)=\frac{\rho }{\varGamma (\alpha )}\int _a^t\frac{s^{\rho -1} e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }}\psi (s) \mathrm{d}s. \end{aligned}$$

Furthermore, the Riemann–Liouville-type generalized substantial fractional derivative is defined as ${}_\sigma D_a^{\alpha ,\rho }\psi (t)={}_\sigma D^{m,\rho } {}_\sigma I_a^{m-\alpha , \rho }\psi (t)$ where $m-1<\alpha \le m$.

It is to be noted that for $\sigma \rightarrow 0$, ${}_\sigma I_a^{\alpha ,\rho }\psi (t)\rightarrow I_a^{\alpha ,\rho }\psi (t)$, which is Katugampola fractional integral. Furthermore, for $\sigma =0$ and $\rho \rightarrow 1$, the generalized substantial integral approaches to standard Riemann–Liouville integral and the lower limit $a\rightarrow -\infty$ leads to the Weyl fractional integral. For $\sigma \ne 0$ and $\rho =1$, the generalized substantial integral becomes the standard substantial integral. Finally, for $\sigma =0$ and $\rho \rightarrow 0$, we get the Hadamard fractional integral.

Definition 4

For $m-1<\alpha \le m$, $a<b<\infty$ and $\psi \in \varOmega _{\sigma ,\rho }^m[a,b]$. Then, generalized Caputo-type substantial derivative is defined as

$$\begin{aligned} {}_\sigma ^c{D}_a^{\alpha ,\rho }\psi (t)={}_\sigma {D}_a^{\alpha ,\rho }\left( \psi (t)-\sum _{k=0}^{m-1}\frac{{}_\sigma {D}^{k,\rho }\psi (a)}{k!}(t^\rho -a^\rho )^k e^{-\sigma (t^\rho -a^\rho )}\right) . \end{aligned}$$

Theorem 1

Assume $\alpha >0$, $\sigma >0$, $\rho > 0$ and $\{\psi _k\}_{k=1}^\infty$ is a uniformly convergent sequence of continuous functions on [0, b]. Then $( {}_\sigma I_0^{\alpha ,\rho }\lim _{k\rightarrow \infty } \psi _k)(t)=(\lim _{k\rightarrow \infty }\ {}_\sigma I_0^{\alpha ,\rho } \psi _k)(t).$

Proof

We denote the limit of sequence $\{\psi _k\}_{k=1}^\infty$ by $\psi$. It is well known that $\psi$ is continuous. We then have following estimates

$$\begin{aligned} \begin{aligned}&|{}_\sigma I_0^{\alpha ,\rho }\psi _k(t)- {}_\sigma I_0^{\alpha ,\rho }\psi (t)|\\&\quad \le \frac{\rho }{\varGamma (\alpha )}\int _{0}^{t}s^{\rho -1} (t^\rho - s^\rho )^{\alpha -1} | ( e^{-\sigma (t^\rho -s^\rho )} )(\psi _k(s)-\psi (s))| \mathrm{d}s\\&\quad \le \frac{\rho }{\varGamma (\alpha )}\Vert \psi _k - \psi \Vert _{\infty } \int _{0}^{t} s^{\rho -1} (t^\rho - s^\rho )^{\alpha -1}\mathrm{d}s\\&\quad = \frac{{b^\rho }}{\varGamma (\alpha +1)}\Vert \psi _k - \psi \Vert _{\infty }. \end{aligned} \end{aligned}$$

The conclusion follows, since $\Vert \psi _k - \psi \Vert _{\infty }\rightarrow 0$ as ${k\rightarrow \infty }$ uniformly on [0, b] . $\square$

In the forthcoming results, we shall demonstrate the relationship between Riemann–Liouville-type Katugampola fractional operators and the generalized substantial fractional operators.

Lemma 2

Assumptions $\psi \in \varLambda _{\sigma ,\rho }^1[a,b]$. Then, ${}_{\sigma }I_a^{\alpha ,\rho }\psi (t)=e^{-\sigma t^\rho }I_a^{\alpha ,\rho }(e^{\sigma t^\rho }\psi (t)).$

Theorem 2

Assumptions $\psi \in \varOmega _{\sigma ,\rho }^m[a,b]$. Then, ${}_{\sigma }D_{a}^{\alpha ,\rho }\psi (t)=e^{-\sigma t^\rho }D_{a}^{\alpha ,\rho }(e^{\sigma t^\rho }\psi (t))$.

Proof

By Definition (3) of substantial fractional differential operator, Lemmas 1, 2 and Definition 2, we have

$$\begin{aligned} {}_{\sigma }D_{a}^{\alpha ,\rho }\psi (t)&={}_{\sigma }D^{m,\rho } {}_{\sigma }I_a^{m-\alpha ,\rho }\psi (t) ={}_{\sigma }D^{m,\rho }\left( e^{-\sigma t^\rho }I_a^{m-\alpha ,\rho }(e^{\sigma t^\rho }\psi (t))\right) \\&=e^{-\sigma t^\rho }D^{m,\rho }I_a^{m-\alpha ,\rho }(e^{\sigma t^\rho }\psi (t)) =e^{-\sigma t^\rho }D_a^{\alpha ,\rho }(e^{\sigma t^\rho }\psi (t)). \end{aligned}$$

$\square$

Now, we introduce composition properties of the generalized substantial operators. First, we show that generalized integral satisfies the semi-group property.

Theorem 3

Let $\alpha , \beta > 0$ and $\psi \in \varLambda _{\sigma ,\rho }^1[a,b]$. Then, ${}_\sigma I_a^{\alpha ,\rho }{}_\sigma I_a^{\beta ,\rho }\psi (t)={}_\sigma I_a^{\alpha +\beta ,\rho }\psi (t).$

Proof

Using (P1) and Lemma 2 repeatedly, we have

$$\begin{aligned} \begin{aligned} {}_\sigma I_a^{\alpha ,\rho }({}_\sigma I_a^{\beta ,\rho }\psi (t))&= {}_\sigma I_a^{\alpha ,\rho } ( e^{-\sigma t^\rho } I_a^{\beta ,\rho } (e^{\sigma t^\rho }\psi (t) ) ) \\&=e^{-\sigma t^\rho } I_a^{\alpha +\beta ,\rho } (e^{\sigma t^\rho }\psi (t))={}_\sigma I_a^{\alpha +\beta ,\rho }\psi (t). \end{aligned} \end{aligned}$$

$\square$

Theorem 4

Let $m-1<\alpha \le m$, $\beta \ge \alpha$ and $\psi \in \varLambda _{\sigma ,\rho }^1[a,b]$. Then, ${}_\sigma D_a^{\alpha ,\rho } {}_\sigma I_a^{\beta ,\rho }\psi (t)={}_\sigma I_a^{\beta -\alpha ,\rho }\psi (t).$

The proof of Theorem 4 is same as the proof of Theorem 3. Therefore, we omit it.

Theorem 5

Assume $\alpha >0$, $m-1<\alpha \le m$ and ${}_\sigma I_a^{m-\alpha ,\rho }\psi \in \varOmega _{\sigma ,\rho }^m[a,b]$. Then

$$\begin{aligned} {}_\sigma I_a^{\alpha ,\rho }{}_\sigma D_a^{\alpha ,\rho } \psi (t)=\psi (t)- e^{-\sigma (t^\rho - a^\rho )} \sum _{k=1}^{m} \frac{(t^\rho -a^{\rho })^{\alpha -k} }{\varGamma (\alpha -k+1)}\underset{s\rightarrow a^+}{\lim } {}_\sigma D_{a}^{\alpha -k,\rho } \psi (s). \end{aligned}$$

Specifically, for $0<\alpha <1$ we have

$$\begin{aligned} {}_\sigma I_a^{\alpha ,\rho }{}_\sigma D_a^{\alpha ,\rho } \psi (t)=\psi (t)- e^{-\sigma (t^\rho - a^\rho )} \frac{(t^\rho -a^{\rho })^{\alpha -1} }{\varGamma (\alpha )}\underset{s\rightarrow a^+}{\lim } {}_\sigma I_{a}^{1-\alpha ,\rho } \psi (s). \end{aligned}$$

Proof

Using Leibniz rule, following relation can be established.

$$\begin{aligned}&\left( \frac{t^{1-\rho }}{\rho }\frac{\mathrm{d}}{\mathrm{d}t}+\sigma \right) \int _a^t\frac{s^{\rho -1} e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{-\alpha }}{}_\sigma D_a^{\alpha ,\rho }\psi (s) \mathrm{d}s \nonumber \\&\quad = \alpha \int _a^t\frac{s^{\rho -1} e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }}{}_\sigma D_a^{\alpha ,\rho }\psi (s) \mathrm{d}s. \end{aligned}$$

(4)

By definition of ${}_\sigma I_a^{\alpha ,\rho }$, we have

$$\begin{aligned} {}_\sigma I_a^{\alpha ,\rho }{}_\sigma D_a^{\alpha ,\rho } \psi (t)&=\frac{\rho }{\varGamma (\alpha )}\int _a^t\frac{s^{\rho -1} e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }}{}_\sigma D_a^{\alpha ,\rho }\psi (s) \mathrm{d}s. \end{aligned}$$

(5)

From Eqs. (4) and (5), we get

$$\begin{aligned} {}_\sigma I_a^{\alpha ,\rho }{}_\sigma D_a^{\alpha ,\rho } \psi (t)&=\frac{\rho }{\varGamma (\alpha +1)} \left( \frac{t^{1-\rho }}{\rho }\frac{\mathrm{d}}{\mathrm{d}t}+\sigma \right) \int _a^t\frac{s^{\rho -1} e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{-\alpha }}{}_\sigma D_a^{\alpha ,\rho }\psi (s) \mathrm{d}s. \end{aligned}$$

(6)

From Definition 3 and Eq. (6), we find

$$\begin{aligned} {}_\sigma I_a^{\alpha ,\rho }{}_\sigma D_a^{\alpha ,\rho } \psi (t)&=\frac{\rho }{\varGamma (\alpha +1)} \left( \frac{t^{1-\rho }}{\rho }\frac{\mathrm{d}}{\mathrm{d}t}+\sigma \right) \int _a^t\frac{s^{\rho -1} e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{-\alpha }} \left( \frac{s^{1-\rho }}{\rho }\frac{\mathrm{d}}{\mathrm{d}s}+\sigma \right) \\&\quad {}_\sigma D^{m-1,\rho } {}_\sigma I_a^{m-\alpha ,\rho } \psi (s) \mathrm{d}s. \end{aligned}$$

Applying integration by parts and product rule for classical derivatives, we have

$$\begin{aligned} {}_\sigma I_a^{\alpha ,\rho }{}_\sigma D_a^{\alpha ,\rho } \psi (t)&= - \frac{ e^{-\sigma (t^\rho -a^\rho )}}{\varGamma (\alpha ) (t^\rho -s^\rho )^{1-\alpha }}\underset{s\rightarrow a^+}{\lim } {}_\sigma D^{m-1,\rho } {}_\sigma I_a^{m-\alpha ,\rho } \psi (s) \\&\quad + \frac{\rho }{\varGamma (\alpha )} \left( \frac{t^{1-\rho }}{\rho }\frac{\mathrm{d}}{\mathrm{d}t}+\sigma \right) \int _a^t\frac{s^{\rho -1} e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} \left( \frac{s^{1-\rho }}{\rho }\frac{\mathrm{d}}{\mathrm{d}s}+\sigma \right) \\&\quad {}_\sigma D^{m-2,\rho } {}_\sigma I_a^{m-\alpha ,\rho } \psi (s) \mathrm{d}s. \end{aligned}$$

Continuing in this manner, we get

$$\begin{aligned} {}_\sigma I_a^{\alpha ,\rho }{}_\sigma D_a^{\alpha ,\rho } \psi (t)&=-e^{-\sigma (t^\rho - a^\rho )} \sum _{k=1}^{m} \frac{(t^\rho -a^{\rho })^{\alpha -k} }{\varGamma (\alpha -k+1)}\underset{s\rightarrow a^+}{\lim } {}_\sigma D_{a}^{\alpha -k,\rho } \psi (s) \\&\quad + \frac{\rho }{\varGamma (\alpha -m+1)} \left( \frac{t^{1-\rho }}{\rho }\frac{\mathrm{d}}{\mathrm{d}t}+\sigma \right) \int _a^t\frac{s^{\rho -1} e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{m-\alpha }}\\&\quad {}_\sigma I_a^{m-\alpha ,\rho } \psi (s) \mathrm{d}s, \end{aligned}$$

where

$$\begin{aligned} \frac{\rho }{\varGamma (\alpha -m+1)} \left( \frac{t^{1-\rho }}{\rho }\frac{\mathrm{d}}{\mathrm{d}t}+\sigma \right) \int _a^t\frac{s^{\rho -1} e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{m-\alpha }} {}_\sigma I_a^{m-\alpha ,\rho } \psi (s) \mathrm{d}s&=\psi (t). \end{aligned}$$

Finally, we get the desired result

$$\begin{aligned} {}_\sigma I_a^{\alpha ,\rho }{}_\sigma D_a^{\alpha ,\rho } \psi (t)=\psi (t)- e^{-\sigma (t^\rho - a^\rho )} \sum _{k=1}^{m} \frac{(t^\rho -a^{\rho })^{\alpha -k} }{\varGamma (\alpha -k+1)}\underset{s\rightarrow a^+}{\lim } {}_\sigma D_{a}^{\alpha -k,\rho } \psi (s). \end{aligned}$$

$\square$

Theorem 6

Assume $\psi \in \varOmega _{\sigma ,\rho }^m[a,b]$. Then, generalized Caputo-type substantial derivative can be written as

$$\begin{aligned} {}_\sigma ^c{D}_a^{\alpha ,\rho }\psi (t)= & {} {}_\sigma {I}_a^{m-\alpha ,\rho }{}_\sigma D^{m,\rho }\psi (t) =\frac{\rho }{\varGamma (m-\alpha )}\\&\quad \int _a^t\frac{e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1+\alpha -m}}{}_\sigma D^{m,\rho }(\psi (s)) \mathrm{d}s. \end{aligned}$$

Proof

By Definition 4 and Eq. (3), we have ${}_\sigma ^c{D}_a^{\alpha ,\rho }\psi (t)={}_\sigma {D}_a^{\alpha ,\rho }{}_\sigma I^{m,\rho }{}_\sigma D^{m,\rho }\psi (t)$. An application of Definition 3 and Properties (P1) and (P2) leads us to

$$\begin{aligned} {}_\sigma ^c{D}_a^{\alpha ,\rho }\psi (t)&={}_\sigma {D}^{m,\rho }{}_\sigma {}I_a^{m-\alpha ,\rho }{}_\sigma I_a^{m,\rho }{}_\sigma D^{m,\rho }\psi (t) ={}_\sigma {D}^{m,\rho }{} {}_\sigma {I}_a^{m,\rho } {}_\sigma I_a^{m-\alpha ,\rho }{}_\sigma D^{m,\rho }\psi (t)\\&={}_\sigma I_a^{m-\alpha ,\rho }{}_\sigma D^{m,\rho }\psi (t). \end{aligned}$$

$\square$

Lemma 3

For $\psi \in \varOmega _{\rho }^m[a,b]$, the operator ${}_\sigma ^c{D}_a^{\alpha ,\rho }$ satisfies the relation ${}_\sigma ^c{D}_a^{\alpha ,\rho }\psi (t)=e^{-\sigma t^\rho }{}^cD_{a}^{\alpha ,\rho }(e^{\sigma t^\rho }\psi (t)).$

Proof

By using Lemma 1, Lemma 2 and Theorem 6, we have

$$\begin{aligned} {}_\sigma ^c{D}_a^{\alpha ,\rho }\psi (t)&={}_\sigma I_a^{m-\alpha ,\rho }{}_\sigma D^{m,\rho }\psi (t) =e^{-\sigma t^\rho }{I}_a^{m-\alpha ,\rho }(e^{\sigma t^\rho }{}_\sigma D^{m,\rho }\psi (t))\\&=e^{-\sigma t^\rho }{I}_a^{m-\alpha ,\rho }{}D^{m,\rho }(e^{\sigma t^\rho }\psi (t)) =e^{-\sigma t^\rho }{} ^c{D}_a^{\alpha ,\rho }(e^{\sigma t^\rho }\psi (t)). \end{aligned}$$

$\square$

Theorem 7

Let $m-1<\alpha \le m$, $\beta \ge \alpha$ and $\psi \in \varOmega _{\sigma ,\rho }^m [a,b]$. Then, ${}_\sigma ^c{D}_a^{\alpha ,\rho } {}_\sigma I_a^{\beta ,\rho } \psi (t)={}_\sigma I_a^{\beta -\alpha ,\rho }\psi (t).$

By using Lemma 2 and Lemma 3, the result can easily be proved.

Theorem 8

Assume $m-1<\alpha \le m$ and $\psi \in \varOmega _{\sigma ,\rho }^m[a,b]$. Then

$$\begin{aligned} {}_\sigma {I}_a^{\alpha ,\rho }{}_\sigma ^c{D}_a^{\alpha ,\rho }\psi (t) =\psi (t)-\sum _{k=0}^{m-1}\frac{{} {}_\sigma {D}^{k,\rho }\psi (a)}{k!}e^{-\sigma (t^\rho -a^\rho )}(t^\rho -a^\rho )^k. \end{aligned}$$

Proof

From Lemmas 3 and 2, we have

${}_\sigma {I}_a^{\alpha ,\rho }{}_\sigma ^c{D}_a^{\alpha ,\rho }\psi (t) ={}_\sigma {I}_a^{\alpha ,\rho }\left( e^{-\sigma t^\rho }{}^c{D}_a^{\alpha ,\rho }(e^{\sigma t^\rho }\psi (t))\right) =e^{-\sigma t^\rho }{I}_a^{\alpha ,\rho }({}^c{D}_a^{\alpha ,\rho }(e^{\sigma t^\rho }\psi (t))).$

Now by property (P4), we have

$$\begin{aligned} {}_\sigma {I}_a^{\alpha ,\rho }{}_\sigma ^c{D}_a^{\alpha ,\rho }\psi (t)&= e^{-\sigma t^\rho }\left( e^{\sigma t^\rho }\psi (t)-\sum _{k=0}^{m-1}\frac{ D^{k,\rho }(e^{\sigma t^\rho }\psi (t))|_{t=a}}{k!}(t^\rho -a^\rho )^k\right) \\&=\psi (t)-\sum _{k=0}^{m-1}\frac{{} {}_\sigma D^{k,\rho }\psi (a)}{k!}e^{-\sigma (t^\rho -a^\rho )}(t^\rho -a^\rho )^k. \end{aligned}$$

$\square$

Fig. 1

Fractional integrals ${}_\sigma I_a^{\alpha ,\rho }$ of $\psi (t)=(t^\rho -a^\rho )^{\beta }e^{-\sigma t^\rho }$

Example 1

Consider $\psi (t)=(t^\rho -a^\rho )^{\beta }e^{-\sigma t^\rho }$. Then, from Lemma 2 we have ${}_\sigma I_a^{\alpha ,\rho }\psi (t)=e^{-\sigma t^{\rho }}I_a^{\alpha ,\rho }(t^\rho -a^\rho )^{\beta }$. Now, by Lemma 3 in [23] we have

$$\begin{aligned} {}_\sigma I_a^{\alpha ,\rho }\psi (t)=\frac{\varGamma (\beta +1)e^{-\sigma t^{\rho }}}{\varGamma (\alpha +\beta +1)}(t^\rho -a^\rho )^{\alpha +\beta }. \end{aligned}$$

(7)

Fractional integrals of $\psi (t)$ for different values of $\alpha$, $\beta$, $\sigma$ and $\rho$ are graphically illustrated in Fig. 1. Now we compute Riemann–Liouville substantial derivative of $\psi (t)=(t^\rho -a^\rho )^{\beta }e^{-\sigma t^\rho }$. Note that

$$\begin{aligned} D^{1,\rho }(t^\rho -a^\rho )^\beta = \frac{t^{1-\rho }}{\rho }\frac{\mathrm{d}}{\mathrm{d}t}(t^\rho -a^\rho )^\beta =\beta (t^\rho -a^\rho )^{\beta -1}. \end{aligned}$$

(8)

Therefore, from definition of Riemann–Liouville substantial derivative, Lemma (1) and Eq. (8), we have

$$\begin{aligned} {}_\sigma D_a^{\alpha ,\rho }\psi (t)&={}_\sigma D^{1,\rho }{}_\sigma I_a^{1-\alpha ,\rho }\left[ e^{-\sigma t^{\rho }}(t^\rho -a^\rho )^\beta \right] \\&=\frac{\varGamma (\beta +1)}{\varGamma (\beta -\alpha +2)}{}_\sigma D^{1,\rho }\left[ e^{-\sigma t^{\rho }}(t^\rho -a^\rho )^{\beta -\alpha +1}\right] \\&=\frac{\varGamma (\beta +1)e^{-\sigma t^{\rho }}}{\varGamma (\beta -\alpha +2)}D^{1,\rho }(t^\rho -a^\rho )^{\beta -\alpha +1}\\&=\frac{\varGamma (\beta +1)e^{-\sigma t^{\rho }}}{\varGamma (\beta -\alpha +1)}(t^\rho -a^\rho )^{\beta -\alpha }. \end{aligned}$$

Similarly, Caputo-type substantial derivative of $\psi (t)=(t^\rho -a^\rho )^{\beta }e^{-\sigma t^\rho }$ can be computed as

$$\begin{aligned} {}^c_\sigma D_a^{\alpha ,\rho }\psi (t)&={}_\sigma I_a^{1-\alpha ,\rho }{}_\sigma D^{1,\rho }\left[ e^{-\sigma t^{\rho }}(t^\rho -a^\rho )^\beta \right] \\&\quad ={}_\sigma I_a^{1-\alpha ,\rho }\left[ e^{-\sigma t^{\rho }}D^{1,\rho }((t^\rho -a^\rho )^\beta )\right] \\&={}_\sigma I_a^{1-\alpha ,\rho } e^{-\sigma t^{\rho }}D^{1,\rho }(t^\rho -a^\rho )^\beta =e^{-\sigma t^{\rho }}I_a^{1-\alpha ,\rho } D^{1,\rho }(t^\rho -a^\rho )^\beta \\&=\beta e^{-\sigma t^{\rho }}I_a^{1-\alpha ,\rho } (t^\rho -a^\rho )^{\beta -1} =\frac{\varGamma (\beta +1)e^{-\sigma t^{\rho }}}{\varGamma (\beta -\alpha +1)}(t^\rho -a^\rho )^{\beta -\alpha }. \end{aligned}$$

4 Existence and Uniqueness of Solutions

When it comes to the problem of solving a fractional differential equation, the existence and uniqueness results have their own importance. It is necessary to notice in advance whether there is a solution to a given fractional differential equation. With this in view, here we prove the equivalence between initial value problem (IVP) and Volterra equation. Then, using this equivalence along with Weissinger’s fixed point theorem, we prove the existence and uniqueness of solution for the following IVP

$$\begin{aligned} {}_\sigma ^c{D}_{0}^{\alpha ,\rho }\psi (t)&=f(t,\psi (t)),\ \ \ t>0, \end{aligned}$$

(9)

$$\begin{aligned} {}_{\sigma }D_{}^{k,\rho }\psi (0)&=b_k, \ \ k \in \left\{ 0,1,2,\ldots ,m-1 \right\} , \end{aligned}$$

(10)

where $\sigma >0$, $\rho >0$, $\alpha >0$, $m= \lceil \alpha \rceil$, ${}_\sigma ^c{D}_{0}^{\alpha ,\rho }$ is the generalized Caputo-type substantial fractional derivative and $f: \mathbb {R}^{+} \times \mathbb {R}\rightarrow \mathbb {R}.$

For $K>0$, $h^{*}>0$ and $b_1,\ldots ,b_m \in \mathbb {R}$, define the set

$$\begin{aligned} H:= & {} \left\{ (t,\psi (t)): 0 \le t \le h^{*}, \left| \psi (t)- e^{-\sigma t^\rho } \sum _{k=0}^{m-1}\frac{b_k}{\varGamma (k+1)} t^{\rho k} \right| \le K \right\} . \end{aligned}$$

Following will be assumed while establishing the subsequent results.

• (H1) $f:H \rightarrow \mathbb {R}$ is both continuous and bounded in H;

• (H2) f satisfies the Lipschitz condition with respect to the second variable, i.e., for some constant $L>0$ and for all $(t,\psi (t)),(t,\tilde{\psi }(t)) \in H$, we have

  $$\begin{aligned} |f(t,\psi (t))-f(t,\tilde{\psi }(t))|\le L|\psi (t)-\tilde{\psi }(t)|. \end{aligned}$$

For convenience, we introduce some notations. Let $h:= \min \left\{ h^{*}, \tilde{h}, \left( \frac{\varGamma (\alpha +1)K}{M}\right) ^{\frac{1}{\rho \alpha }} \right\}$ where $M:= \sup _{(x,y )\in H} |f(x,y)|$ and $\tilde{h}$ being a positive real number fulfills the inequality $\tilde{h}< \left( \frac{\varGamma (\alpha +1)}{L} \right) ^{\frac{1}{\rho \alpha }}.$ These notations appear frequently in this section. The main results of this section are the generalizations of existence and uniqueness results presented in [24,25,26].

Theorem 9

Assume that $h>0$ and $f: \mathbb {R}^{+} \times \mathbb {R}\rightarrow \mathbb {R}$ is continuous. Then, $\psi \in C[0,h]$ is the solution of IVP (9)–(10) if and only if $\psi \in C[0,h]$ satisfies the Volterra equation

$$\begin{aligned} \psi (t)=e^{-\sigma t^\rho } \sum _{k=0}^{m-1}\frac{b_k }{\varGamma (k+1)}t^{\rho k}+\frac{\rho }{\varGamma (\alpha )}\int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} f(s,\psi (s)) \mathrm{d}s. \end{aligned}$$

Proof

Let $\psi \in C[0,h]$ be a solution of Volterra equation

$$\begin{aligned} \psi (t)=e^{-\sigma t^\rho } \sum _{k=0}^{m-1}\frac{b_k }{\varGamma (k+1)}t^{\rho k}+{}_{\sigma }I_{0}^{\alpha ,\rho } f(t,\psi (t)). \end{aligned}$$

Apply ${}_\sigma ^c{D}_{0}^{\alpha ,\rho }$ to both sides of the above equation. Using Theorem 3 and Example 1, we get

$$\begin{aligned} {}_\sigma ^c{D}_{0}^{\alpha ,\rho }\psi (t)&= \sum _{k=0}^{m-1}\frac{b_k }{\varGamma (k+1)}{}_\sigma ^c{D}_{0}^{\alpha ,\rho }e^{-\sigma t^\rho } t^{\rho k}+{}_\sigma ^c{D}_{0}^{\alpha ,\rho }{}_{\sigma }I_{0}^{\alpha ,\rho } f(t,\psi (t)) \\&= f(t,\psi (t)). \end{aligned}$$

Now we apply ${}_{\sigma }D_{}^{k,\rho }$ to both sides of Volterra equation, where $0 \le k \le m-1$. Using Theorem 3, Theorem 7 and Example 1, we have

$$\begin{aligned} {}_{\sigma }D_{0}^{k,\rho }\psi (t)&= \sum _{j=0}^{m-1}\frac{b_j }{\varGamma (j+1)} {}_{\sigma }D_{}^{k,\rho } e^{-\sigma t^\rho }t^{\rho j}+ {}_{\sigma }D_{}^{k,\rho } {}_{\sigma }I_{0}^{\alpha ,\rho } f(t,\psi (t))\\&=\sum _{j=0}^{m-1}\frac{b_j }{\varGamma (j+1)} \left( \frac{\varGamma (j+1)}{\varGamma (j-k+1)} e^{-\sigma t^\rho }t^{\rho (j-k)} \right) \\&+\quad {}_{\sigma }D_{}^{k,\rho } {}_{\sigma }I_{}^{k,\rho } {}_{\sigma }I_{0}^{\alpha -k,\rho } f(t,\psi (t))\\&= e^{-\sigma t^\rho } \sum _{j=0}^{m-1}\frac{b_j }{\varGamma (j-k+1)} t^{\rho (j-k)}+ \frac{\rho }{\varGamma (\alpha -k)}\\&\quad \int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha +k}} s^{\rho -1} f(s,\psi (s)) \mathrm{d}s. \end{aligned}$$

Clearly for $j<k$, the summands become identically zero because reciprocal of Gamma function for non-positive integers, vanishes. Furthermore, for $k<j$, the summands vanish if $t=0$. Since $\alpha -k$ is a positive real number, so the integral also vanishes when $t=0$. Thus, we are left with the case $j=k$.

$$\begin{aligned} {}_{\sigma }D_{}^{k,\rho }\psi (0) = \frac{b_k }{\varGamma (k-k+1)} e^{-\sigma t^\rho }t^{\rho (k-k)} \Big |_{t=0} =b_k. \end{aligned}$$

Conversely, assume that $\psi \in C[0,h]$ is the solution of the given IVP. Applying ${}_{\sigma }I_{0}^{\alpha ,\rho }$ to both sides of the fractional differential Eq. (9), using the initial conditions (10) and result of Theorem 8 , we get Volterra equation. $\square$

Theorem 10

Assume that f satisfies (H1) and (H2) . Then, Volterra equation

$$\begin{aligned} \psi (t)=e^{-\sigma t^\rho } \sum _{k=0}^{m-1}\frac{b_k }{\varGamma (k+1)}t^{\rho k}+\frac{\rho }{\varGamma (\alpha )}\int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} f(s,\psi (s)) \mathrm{d}s \end{aligned}$$

possesses a uniquely determined solution $\psi \in [0,h]$.

Proof

Define a set

$$\begin{aligned} B:= \left\{ \psi \in C[0,h] : \sup _{0 \le t \le h} \left| \psi (t) - e^{-\sigma t^\rho } \sum _{k=0}^{m-1}\frac{b_k }{\varGamma (k+1)} t^{\rho k} \right| \le K \right\} \end{aligned}$$

equipped with the norm $||.||_{B}$

$$\begin{aligned} ||\psi ||_{B} := \sup _{0 < t \le h} | \psi (t) |. \end{aligned}$$

It can be seen that $(B,||.||_{B})$ is a Banach space. Define the operator E by

$$\begin{aligned} E\psi (t):=e^{-\sigma t^\rho } \sum _{k=0}^{m-1}\frac{b_k }{\varGamma (k+1)}t^{\rho k}+\frac{\rho }{\varGamma (\alpha )}\int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} f(s,\psi (s)) \mathrm{d}s. \end{aligned}$$

It is easy to check that $E\psi$ is a continuous on interval [0, h] for $\psi \in B$. Moreover,

$$\begin{aligned} \left| E\psi (t)- e^{-\sigma t^\rho } \sum _{k=0}^{m-1}\frac{b_k }{\varGamma (k+1)} t^{\rho k} \right|&=\left| \frac{\rho }{\varGamma (\alpha )}\int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} f(s,\psi (s)) \mathrm{d}s \right| \\&\le \frac{\rho }{\varGamma (\alpha )} M \int _0^t\frac{s^{\rho -1} }{(t^\rho -s^\rho )^{1-\alpha }} \mathrm{d}s \\&= \frac{\rho }{\varGamma (\alpha )} M \frac{t^{\rho \alpha }}{\rho \alpha }= \frac{ M }{\varGamma (\alpha +1)} t^{\rho \alpha }\le K \end{aligned}$$

for $t \in [0,h]$, the last step follows from the definition of h. This means that $E\psi \in B$ for $\psi \in B$, i.e., E is the self-map.

From the definition of operator E and Volterra equation, it follows that fixed points of E are solutions of Volterra equation.

We use Weissinger’s fixed point theorem to prove that the operator E has a unique fixed point. For $\psi _1,\psi _2 \in B$, first we will show the following inequality

$$\begin{aligned} ||E^{j}\psi _1-E^{j}\psi _2||_{B} \le \left( \frac{ L h^{\rho \alpha }}{\varGamma (\alpha +1)} \right) ^{j} ||\psi _1-\psi _2||_{B}. \end{aligned}$$

Clearly, the above inequality is true for the case $j=0$. Assume that it is true for $j=k-1$. For $j=k$, we have

$$\begin{aligned} ||E^{k}\psi _1-E^{k}\psi _2||_{{B}}&=\sup _{0 \le t \le h} \left| E^{k}\psi _1(t)-E^{k}\psi _2(t) \right| \\&= \sup _{0 \le t \le h} \left| E E^{k-1}\psi _1(t)-EE^{k-1}\psi _2(t) \right| \\&= \sup _{0 \le t \le h} \frac{1}{\varGamma (\alpha )}\left| \rho \int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1}\right. \\&\left. \quad \left[ f(s,E^{k-1}\psi _1(s))-f(s,E^{k-1}\psi _2(s))\right] \mathrm{d}s \right| \\&\le \sup _{0 \le t \le h} \frac{ L }{\varGamma (\alpha )} \left\{ \rho \int _0^t\frac{ 1}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1}\mathrm{d}s \right\} \\&\quad ||E^{k-1}\psi _1-E^{k-1}\psi _2||_{{B}} \\&= \frac{ L }{\varGamma (\alpha )} \left\{ \frac{ h^{\rho \alpha }}{\alpha } \right\} ||E^{k-1}\psi _1-E^{k-1}\psi _2||_{{B}} \\&\quad = \left( \frac{ L h^{\rho \alpha }}{\varGamma (\alpha +1)} \right) ||E^{k-1}\psi _1-E^{k-1}\psi _2||_{{B}} \\&=\left( \frac{ L h^{\rho \alpha }}{\varGamma (\alpha +1)} \right) ^{k}||\psi _1-\psi _2||_{{B}}. \end{aligned}$$

Since $h \le \tilde{h}$, we have $\left( \frac{ L h^{\rho \alpha }}{\varGamma (\alpha +1)} \right) < 1.$ Thus, the series $\sum _{j=0}^{\infty } \left( \frac{ L h^{\rho \alpha }}{\varGamma (\alpha +1)} \right) ^{j}$ is convergent. This completes the proof. $\square$

Following is an example for which a general method to determine the analytical solution is not available, but Theorem 9 and Theorem 10 allow us to comment on the existence of its unique solution.

Example 2

Consider the IVP

$$\begin{aligned} {}_1^c{D}_{0}^{0.5,2}\psi (t)&=t e^{-t^2}\frac{(\psi (t))^2}{1+(\psi (t))^2}, \end{aligned}$$

(11)

$$\begin{aligned} \psi (0)&=b_0. \end{aligned}$$

(12)

It can easily be verified that $f(t,\psi (t))= t e^{-t^2}\frac{(\psi (t))^2}{1+(\psi (t))^2}$ is both, continuous and bounded in H. Furthermore, we show that f satisfies the Lipschitz condition

$$\begin{aligned} |f(t,\psi (t))-f(t,\tilde{\psi }(t))|&=\left| t e^{-t^2}\frac{(\psi (t))^2}{1+(\psi (t))^2}-t e^{-t^2}\frac{(\tilde{\psi }(t))^2}{1+(\tilde{\psi }(t))^2}\right| \\&=\left| t e^{-t^2}\right| \left| \frac{(\tilde{\psi }(t))^2-(\psi (t))^2}{(1+(\psi (t))^2)(1+(\tilde{\psi }(t))^2)}\right| . \end{aligned}$$

Since $1+(\psi (t))^2 \ge 1$ and $1+(\tilde{\psi }(t))^2 \ge 1$, so

$$\begin{aligned} |f(t,\psi (t))-f(t,\tilde{\psi }(t))|&\le \left\{ \sup _{0 \le t \le h} \left| t e^{-t^2} (\tilde{\psi }(t)+\psi (t)) \right| \right\} |\tilde{\psi }(t)-\psi (t)| \\&\le h \left\{ |\sup _{0 \le t \le h}|\tilde{\psi }(t)|+\sup _{0 \le t \le h}|\psi (t) | \right\} |\tilde{\psi }(t)-\psi (t)| \\&= h(K_1+K_2)|\psi (t)-\tilde{\psi }(t)|, \end{aligned}$$

where $L:= h(K_1+K_2)$ is the Lipschitz constant. Thus, hypotheses (H1) and (H2) hold. From Theorems 9 and 10, we can deduce that there exists a unique solution of IVP (11)–(12).

5 Continuous Dependence of Solutions on the Given Data

In this section, first we prove a Gronwall-type inequality which is the generalized version of Gronwall-type inequalities presented in [27,28,29]. Undoubtedly, this inequality plays an important role in the qualitative theory of integral and differential equations. Furthermore, we analyze the continuous dependence of solution of a fractional differential equation on the given data.

Theorem 11

Assume that p and q are non-negative integrable functions and g is non-negative and non-decreasing continuous function on [a, b] .

If

$$\begin{aligned} p(t) \le q(t) + {\rho }^{1-\alpha } g(t) \int _a^t\frac{s^{\rho -1} e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }}p(s) \mathrm{d}s,\quad \forall t \in [a,b], \end{aligned}$$

then

$$\begin{aligned}&p(t) \le q(t) + \int _a^t \sum _{k=1}^{\infty } \frac{ {\rho }^{1-k\alpha }[g(t)\varGamma (\alpha )]^{k}}{\varGamma (k\alpha )}\\&\quad e^{-\sigma (t^\rho -s^\rho )}(t^\rho -s^\rho )^{k\alpha - 1} s^{\rho -1} q(s) \mathrm{d}s, \quad \forall \ t \in [a,b]. \end{aligned}$$

Moreover, if q is non-decreasing, then

$$\begin{aligned} p(t) \le q(t) E_\alpha \left[ g(t) \varGamma (\alpha ) \left( \frac{(t^\rho -a^\rho )}{\rho }\right) ^{\alpha }\right] , \quad \forall \ t \in [a,b]. \end{aligned}$$

Proof

Define operator A as

$$\begin{aligned} A \psi (t):= {\rho }^{1-\alpha } g(t) \int _a^t\frac{s^{\rho -1} e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }}\psi (s) \mathrm{d}s. \end{aligned}$$

Then,

$$\begin{aligned} p(t) \le q(t)+ A p(t). \end{aligned}$$

Iterating successively, for $n \in \mathbb {N}$, we obtain

$$\begin{aligned} p(t) \le \sum _{k=0}^{n-1} A^{k} q(t)+ A^{n} p(t). \end{aligned}$$

By mathematical induction, we show that if $\psi$ is non-negative, then

$$\begin{aligned} A^{k} \psi (t) \le {\rho }^{1-k\alpha } \int _a^t \frac{ [g(t)\varGamma (\alpha )]^{k}}{\varGamma (k\alpha )} e^{-\sigma (t^\rho -s^\rho )}(t^\rho -s^\rho )^{k\alpha - 1} s^{\rho -1} \psi (s) \mathrm{d}s. \end{aligned}$$

For $k=1$, the equality holds. Assume that it is true for $k \in \mathbb {N}$. Then,

$$\begin{aligned} A^{k+1} \psi (t)&= A(A^{k} \psi (t)) \\&\le A \left( {\rho }^{1-k\alpha } \int _a^\tau \frac{ [g(\tau )\varGamma (\alpha )]^{k}}{\varGamma (k\alpha )} e^{-\sigma (\tau ^\rho -s^\rho )}(\tau ^\rho -s^\rho )^{k\alpha - 1} s^{\rho -1} \psi (s) \mathrm{d}s \right) \\&= {\rho }^{1-\alpha } g(t) \int _a^t\frac{\tau ^{\rho -1} e^{-\sigma (t^\rho -\tau ^\rho )}}{(t^\rho -\tau ^\rho )^{1-\alpha }}\\&\quad \left( {\rho }^{1-k\alpha } \int _a^\tau \frac{ [g(\tau )\varGamma (\alpha )]^{k}}{\varGamma (k\alpha )} e^{-\sigma (\tau ^\rho -s^\rho )}(\tau ^\rho -s^\rho )^{k\alpha - 1} s^{\rho -1} \psi (s) \mathrm{d}s \right) d\tau . \end{aligned}$$

By assumption, g is non-decreasing, so $g (\tau ) \le g(t)$, $\forall \ \tau \le t$. Thus, we have

$$\begin{aligned}&A^{k+1} \psi (t) \le \frac{ (\varGamma (\alpha ))^{k}}{\varGamma (k\alpha )} {\rho }^{2-(k+1)\alpha } (g(t))^{k+1} \int _a^t \\&\quad \int _a^\tau e^{-\sigma (t^\rho -s^\rho )} (t^\rho -\tau ^\rho )^{\alpha -1} \tau ^{\rho -1} (\tau ^\rho -s^\rho )^{k\alpha - 1} s^{\rho -1} \psi (s) \mathrm{d}s \mathrm{d}\tau . \end{aligned}$$

Using Fubini’s theorem and Dirichlet’s technique, we get

$$\begin{aligned} A^{k+1} \psi (t)&\le \frac{ (\varGamma (\alpha ))^{k}}{\varGamma (k\alpha )} {\rho }^{2-(k+1)\alpha } (g(t))^{k+1} \int _a^t e^{-\sigma (t^\rho -s^\rho )} s^{\rho -1} \psi (s)\\&\quad \int _s^t (t^\rho -\tau ^\rho )^{\alpha -1} \tau ^{\rho -1} (\tau ^\rho -s^\rho )^{k\alpha - 1} \mathrm{d}\tau \mathrm{d}s \\&= \frac{ (\varGamma (\alpha ))^{k}}{\varGamma (k\alpha )} {\rho }^{2-(k+1)\alpha } (g(t))^{k+1}\\&\quad \int _a^t e^{-\sigma (t^\rho -s^\rho )} s^{\rho -1} \psi (s) \left( \frac{\varGamma (\alpha )\varGamma (k\alpha )}{\rho \varGamma (k\alpha +\alpha )}(t^\rho -s^\rho )^{(k+1)\alpha -1} \right) \mathrm{d}s \\&={\rho }^{1-(k+1)\alpha } \int _a^t \frac{ [g(t)\varGamma (\alpha )]^{(k+1)}}{\varGamma ((k+1)\alpha )}\\&\quad e^{-\sigma (t^\rho -s^\rho )}(t^\rho -s^\rho )^{(k+1)\alpha - 1} s^{\rho -1} \psi (s) \mathrm{d}s. \end{aligned}$$

Now we prove that $A^{n}p(t) \rightarrow 0$ as $n \rightarrow \infty$. Since g is continuous on [a, b] , so by Weierstrass theorem, $\exists$ a constant $M > 0$ such that $g (t) \le M, \ \ \forall \ t \in [a,b].$

$$\begin{aligned} \implies A^{n}p(t) \le {\rho }^{1-n\alpha } \int _a^t \frac{ [M\varGamma (\alpha )]^{n}}{\varGamma (n\alpha )} e^{-\sigma (t^\rho -s^\rho )}(t^\rho -s^\rho )^{n\alpha - 1} s^{\rho -1} p(s) \mathrm{d}s. \end{aligned}$$

Consider the series

$$\begin{aligned} \sum _{n=1}^{\infty } \frac{ [M\varGamma (\alpha )]^{n}}{\varGamma (n\alpha )}. \end{aligned}$$

Using the relation

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{\varGamma (n \alpha )(n \alpha )^{\alpha }}{\varGamma (n \alpha +\alpha )} = 1, \end{aligned}$$

and ratio test, we deduce that the series converges and therefore $A^{n}p(t) \rightarrow 0$ as $n \rightarrow \infty$. Thus,

$$\begin{aligned} p(t)&\le \sum _{k=0}^{\infty } A^{k} q(t) \\&\le q(t) + \int _a^t \sum _{k=1}^{\infty } \frac{ {\rho }^{1-k\alpha }[g(t)\varGamma (\alpha )]^{k}}{\varGamma (k\alpha )} e^{-\sigma (t^\rho -s^\rho )}(t^\rho -s^\rho )^{k\alpha - 1} s^{\rho -1} q(s) \mathrm{d}s. \end{aligned}$$

Additionally, if q is non-decreasing, then, $q(s) \le q(t)$, $\forall \ s \in [a,t]$. So,

$$\begin{aligned} p(t)&\le q(t) \left[ 1 + \sum _{k=1}^{\infty } \frac{ {\rho }^{1-k\alpha }[g(t)\varGamma (\alpha )]^{k}}{\varGamma (k\alpha )} \int _a^t e^{-\sigma (t^\rho -s^\rho )}(t^\rho -s^\rho )^{k\alpha - 1} s^{\rho -1} \mathrm{d}s \right] \\&\le q(t) \left[ 1 + \sum _{k=1}^{\infty } \frac{ {\rho }^{1-k\alpha }[g(t)\varGamma (\alpha )]^{k}}{\varGamma (k\alpha )} \int _a^t (t^\rho -s^\rho )^{k\alpha - 1} s^{\rho -1} \mathrm{d}s \right] \\&= q(t) \left[ 1 + \sum _{k=1}^{\infty } \frac{ {\rho }^{-k\alpha }[g(t)\varGamma (\alpha )(t^\rho -a^\rho )^{\alpha }]^{k}}{\varGamma (k\alpha +1)} \right] \\&= q(t) E_\alpha \left[ g(t) \varGamma (\alpha ) \Big (\frac{(t^\rho -a^\rho )}{\rho }\Big )^{\alpha } \right] . \end{aligned}$$

$\square$

Next we look at the dependence of solution of a fractional differential equation on the initial values.

Theorem 12

Assume that $\psi$ is the solution of the IVP (9)–(10) and $\phi$ is the solution of the following IVP

$$\begin{aligned} {}_\sigma ^c{D}_{0}^{\alpha ,\rho }\phi (t)&=f(t,\phi (t)),\ \ \ t>0, \end{aligned}$$

(13)

$$\begin{aligned} {}_{\sigma }D_{}^{k,\rho }\phi (0)&=c_k, \ \ k \in \left\{ 0,1,2,\ldots ,m-1 \right\} . \end{aligned}$$

(14)

Let $\epsilon := \max _{k=0,1,\ldots ,m-1} |b_k-c_k|$. If $\epsilon$ is sufficiently small, then $\exists$ some constant $h>0$ such that $\psi$ and $\phi$ are defined on [0, h] , and

$$\begin{aligned} \sup _{0 \le t \le h} |\psi (t)-\phi (t)|=\mathcal {O} (\epsilon ). \end{aligned}$$

Proof

Let $\psi$ and $\phi$ be defined on $[0,h_1]$ and $[0,h_2]$, respectively. Taking $h=\min \left\{ h_1,h_2\right\}$, then both the functions $\psi$ and $\phi$ are at least defined on interval [0, h] . Defining $\delta (t):= \psi (t)-\phi (t)$, then $\delta$ is the solution of the following IVP

$$\begin{aligned} {}_\sigma ^c{D}_{0}^{\alpha ,\rho }\delta (t)&=f(t,\psi (t))-f(t,\phi (t)),\ \ \ t>0, \end{aligned}$$

(15)

$$\begin{aligned} {}_{\sigma }D_{}^{k,\rho }\delta (0)&=b_k-c_k, \ \ k \in \left\{ 0,1,2,\ldots ,m-1 \right\} . \end{aligned}$$

(16)

The IVP (15)–(16) is equivalent to Volterra equation

$$\begin{aligned} \delta (t)= & {} e^{-\sigma t^\rho } \sum _{k=0}^{m-1}\frac{(b_k-c_k) }{\varGamma (k+1)}t^{\rho k}+\frac{\rho }{\varGamma (\alpha )}\\&\quad \int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} \left( f(s,\psi (s))-f(s,\phi (s))\right) \mathrm{d}s. \end{aligned}$$

Taking absolute of above equation and using triangle inequality and Lipschitz condition on f, we get

$$\begin{aligned} |\delta (t)|&=\left| e^{-\sigma t^\rho } \sum _{k=0}^{m-1}\frac{(b_k-c_k) }{\varGamma (k+1)}t^{\rho k}+\frac{\rho }{\varGamma (\alpha )}\int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1}\right. \\&\left. \quad \left( f(s,\psi (s))-f(s,\phi (s))\right) \mathrm{d}s\right| \\&\le \left| e^{-\sigma t^\rho } \right| \sum _{k=0}^{m-1}\frac{ t^{\rho k} }{\varGamma (k+1)}\left| b_k-c_k\right| +\frac{\rho }{\varGamma (\alpha )}\int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1}\\&\quad \left| f(s,\psi (s))-f(s,\phi (s))\right| \mathrm{d}s \\&\le \sum _{k=0}^{m-1}\frac{ h^{\rho k} }{\varGamma (k+1)}\max _{k=0,1,\ldots ,m-1} \left| b_k-c_k\right| +\frac{\rho L}{\varGamma (\alpha )}\int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1}\\&\quad \left| \psi (s)-\phi (s)\right| \mathrm{d}s \\&= m\epsilon \sum _{k=0}^{m-1}\frac{ h^{\rho k} }{\varGamma (k+1)} +\frac{\rho L}{\varGamma (\alpha )}\int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} |\delta (s)| \mathrm{d}s. \end{aligned}$$

Taking $p(t)=|\delta (t)|$, $q(t)= m\epsilon \sum _{k=0}^{m-1}\frac{ h^{\rho k} }{\varGamma (k+1)}$ and $g(t)=\frac{\rho ^{\alpha } L}{\varGamma (\alpha )}$, and using Theorem 11, we find

$$\begin{aligned} |\delta (t)| \le m\epsilon \sum _{k=0}^{m-1}\frac{ h^{\rho k} }{\varGamma (k+1)} E_{\alpha } ( L t^{\rho \alpha } ) \le m\epsilon \sum _{k=0}^{m-1}\frac{ h^{\rho k} }{\varGamma (k+1)} E_{\alpha } ( L h^{\rho \alpha } ) = \mathcal {O} (\epsilon ), \end{aligned}$$

and this completes the proof.$\square$

Now we discuss an example to verify the statement of Theorem 12.

Example 3

The unique analytical solutions of the following four IVPs

$$\begin{aligned} {}_1^c{D}_{0}^{0.5,0.5}\psi _{i}(t)=0.9\psi _{i}(t), \ \ \psi _{1}(0)=1, \ \ \psi _{2}(0)=1.2, \ \ \psi _{3}(0)=1.4, \ \ \psi _{4}(0)=1.6, \end{aligned}$$

are given by

$$\begin{aligned} \psi _{i}(t)=\psi _{i}(0) e^{-t^{0.5}} E_{0.5}(0.9t^{0.25}), \ \ 0 \le t \le h. \end{aligned}$$

Plots of these solutions are given in Fig. 2. From Fig. 2, we can see that change in solutions is bounded by the change in initial conditions on the closed interval [0, h]. Thus, Example 3 verifies the statement of Theorem 12.

Fig. 2

Graphs of solutions from Example 3

In the next theorem, we analyze the dependence of solution of the fractional differential equation on the force function f.

Theorem 13

Assume that $\psi$ is the solution of the IVP (9)–(10) and $\phi$ is the solution of the following IVP

$$\begin{aligned} {}_\sigma ^c{D}_{0}^{\alpha ,\rho }\phi (t)&=\tilde{f}(t,\phi (t)),\ \ \ t>0, \end{aligned}$$

(17)

$$\begin{aligned} {}_{\sigma }D_{}^{k,\rho }\phi (0)&=b_k, \ \ k \in \left\{ 0,1,2,\ldots ,m-1 \right\} , \end{aligned}$$

(18)

where $\tilde{f}$ satisfies the same conditions as f. Let $\epsilon := \max _{(t,\phi (t)) \in H} |f(t,\phi (t))-\tilde{f}(t,\phi (t))|$. If $\epsilon$ is sufficiently small, then $\exists$ some constant $h>0$ such that $\psi$ and $\phi$ are defined on [0, h] , and

$$\begin{aligned} \sup _{0 \le t \le h} |\psi (t)-\phi (t)|=\mathcal {O} (\epsilon ). \end{aligned}$$

Proof

Let $\psi$ and $\phi$ be defined on $[0,h_1]$ and $[0,h_2]$, respectively. Taking $h=\min \left\{ h_1,h_2\right\}$, then both the functions $\psi$ and $\phi$, are at least defined on interval [0, h] . Defining $\delta (t):= \psi (t)-\phi (t)$, then $\delta$ is the solution of the following IVP

$$\begin{aligned} {}_\sigma ^c{D}_{0}^{\alpha ,\rho }\delta (t)&=f(t,\psi (t))-\tilde{f}(t,\phi (t)),\ \ \ t>0, \end{aligned}$$

(19)

$$\begin{aligned} {}_{\sigma }D_{}^{k,\rho }\delta (0)&=0, \ \ k \in \left\{ 0,1,2,\ldots ,m-1 \right\} . \end{aligned}$$

(20)

The IVP (19)–(20) is equivalent to Volterra equation

$$\begin{aligned} \delta (t)=\frac{\rho }{\varGamma (\alpha )}\int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} \left( f(s,\psi (s))-\tilde{f}(s,\phi (s))\right) \mathrm{d}s. \end{aligned}$$

Taking absolute of above equation and using Lipschitz condition on f, we get

$$\begin{aligned} |\delta (t)|&=\left| \frac{\rho }{\varGamma (\alpha )}\int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} \left[ \left( f(s,\psi (s))-{f}(s,\phi (s))\right) +\right. \right. \\&\left. \left. \quad \left( f(s,\phi (s))-\tilde{f}(s,\phi (s))\right) \right] \mathrm{d}s\right| \\&\le \frac{\rho }{\varGamma (\alpha )} \left\{ \int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} \left| f(s,\psi (s))-{f}(s,\phi (s))\right| \mathrm{d}s \right. \\&\left. \quad + \int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} \left| f(s,\phi (s))-\tilde{f}(s,\phi (s))\right| \mathrm{d}s \right\} \\&\le \frac{\rho L}{\varGamma (\alpha )} \int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} \left| \delta (s)\right| \mathrm{d}s + \frac{\rho }{\varGamma (\alpha )} \int _0^t\frac{ s^{\rho -1}}{(t^\rho -s^\rho )^{1-\alpha }}\\&\quad \max _{(s,\phi (s)) \in H} \left| f(s,\phi (s))-\tilde{f}(s,\phi (s))\right| \mathrm{d}s \\&\le \frac{\rho L}{\varGamma (\alpha )} \int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} |\delta (s)| \mathrm{d}s + \frac{\epsilon }{\varGamma (\alpha +1)}t^{\rho \alpha } \le \frac{\epsilon h^{\rho \alpha } }{\varGamma (\alpha +1)}\\&\quad +\frac{\rho L}{\varGamma (\alpha )} \int _0^t\frac{ e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} |\delta (s)| \mathrm{d}s. \end{aligned}$$

Taking $p(t)=|\delta (t)|$, $q(t)= \frac{\epsilon h^{\rho \alpha } }{\varGamma (\alpha +1)}$ and $g(t)=\frac{\rho ^{\alpha } L}{\varGamma (\alpha )}$ and using Theorem 11, we find

$$\begin{aligned} |\delta (t)| \le \frac{\epsilon h^{\rho \alpha } }{\varGamma (\alpha +1)} E_{\alpha } ( L t^{\rho \alpha } ) \le \frac{\epsilon h^{\rho \alpha } }{\varGamma (\alpha +1)} E_{\alpha } ( L h^{\rho \alpha } ) = \mathcal {O} (\epsilon ). \end{aligned}$$

This completes the proof. $\square$

Finally, we explore the consequences of perturbing the order of the fractional differential equation.

Theorem 14

Assume that $\psi$ is the solution of the IVP (9)–(10) and $\phi$ is the solution of the following IVP

$$\begin{aligned} {}_\sigma ^c{D}_{0}^{\tilde{\alpha },\rho }\phi (t)&=f(t,\phi (t)),\ \ \ t>0, \end{aligned}$$

(21)

$$\begin{aligned} {}_{\sigma }D_{}^{k,\rho }\phi (0)&=b_k, \ \ k \in \left\{ 0,1,2,\ldots ,\tilde{m}-1 \right\} , \end{aligned}$$

(22)

where $\tilde{\alpha } > \alpha$ and $\tilde{m}:= \lceil \tilde{\alpha } \rceil$. Let $\epsilon := \tilde{\alpha }-\alpha$ and

$$\begin{aligned} \tilde{\epsilon } := {\left\{ \begin{array}{ll} 0 &{} \text {if}\quad \tilde{m}=m, \\ \max \left\{ |b_k|: m \le k \le \tilde{m}-1 \right\} &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

If $\epsilon$ and $\tilde{\epsilon }$ are sufficiently small, then $\exists$ some constant $h>0$ such that $\psi$ and $\phi$ are defined on [0, h] , and

$$\begin{aligned} \sup _{0 \le t \le h} |\psi (t)-\phi (t)|=\mathcal {O} (\epsilon ) + \mathcal {O}(\tilde{\epsilon }). \end{aligned}$$

Proof

Let $\psi$ and $\phi$ be defined on $[0,h_1]$ and $[0,h_2]$, respectively. Taking $h=\min \left\{ h_1,h_2\right\}$, then both the functions $\psi$ and $\phi$, are at least defined on interval [0, h] . Defining $\delta (t):= \psi (t)-\phi (t)$, then using Theorem 9

$$\begin{aligned} \delta (t)&=-e^{-\sigma t^\rho } \sum _{k=m}^{\tilde{m}-1}\frac{b_k }{\varGamma (k+1)}t^{\rho k}+\frac{\rho }{\varGamma (\alpha )}\int _0^t\frac{e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} f(s,\psi (s)) \mathrm{d}s\\&\quad -\frac{\rho }{\varGamma (\tilde{\alpha })}\int _0^t\frac{e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\tilde{\alpha }}} s^{\rho -1} f(s,\phi (s)) \mathrm{d}s \\&=-e^{-\sigma t^\rho } \sum _{k=m}^{\tilde{m}-1}\frac{b_k }{\varGamma (k+1)}t^{\rho k}+\frac{\rho }{\varGamma (\alpha )}\int _0^t\frac{e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} \\&\quad \left( f(s,\psi (s))-f(s,\phi (s))\right) \mathrm{d}s\\&\quad + \int _0^t \left( \frac{\rho (t^\rho -s^\rho )^{\alpha -1}}{\varGamma (\alpha )} -\frac{\rho (t^\rho -s^\rho )^{\tilde{\alpha }-1}}{\varGamma (\tilde{\alpha })} \right) e^{-\sigma (t^\rho -s^\rho )} s^{\rho -1} f(s,\phi (s)) \mathrm{d}s. \end{aligned}$$

Taking absolute of above equation and using Lipschitz condition on f, we get

$$\begin{aligned} |\delta (t)|&\le \sum _{k=m}^{\tilde{m}-1}\frac{h^{\rho k} }{\varGamma (k+1)} \Big | b_k \Big | +\frac{\rho L}{\varGamma (\alpha )}\int _0^t\frac{e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} |\delta (s)| \mathrm{d}s\\&\quad + \max _{(x,y) \in H} \Big |f(x,y)\Big | \int _0^t \left| \frac{\rho (t^\rho -s^\rho )^{\alpha -1}}{\varGamma (\alpha )} -\frac{\rho (t^\rho -s^\rho )^{\tilde{\alpha }-1}}{\varGamma (\tilde{\alpha })} \right| s^{\rho -1} \mathrm{d}s \\&\le \mathcal {O} (\tilde{\epsilon }) +\frac{\rho L}{\varGamma (\alpha )}\int _0^t\frac{e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} |\delta (s)| \mathrm{d}s\\&\quad + M \int _0^h \left| \frac{(v)^{\alpha -1}}{\varGamma (\alpha )} -\frac{(v)^{\tilde{\alpha }-1}}{\varGamma (\tilde{\alpha })} \right| dv. \end{aligned}$$

It can be seen that the zero of above integrand is $v_0=\left( \frac{\varGamma (\tilde{\alpha })}{\varGamma ({\alpha })} \right) ^\frac{1}{{\tilde{\alpha }-\alpha }}$. If $h \le v_0$, then absolute value sign can be taken outside the integral. In other case, the interval of integration must be separated at $v_0$, and each integral can be evaluated easily. Thus in any case, we find that the integral is bounded by $\mathcal {O} (\tilde{\alpha }-\alpha )=\mathcal {O} (\epsilon )$. Thus, we have

$$\begin{aligned} |\delta (t)| \le \mathcal {O} (\tilde{\epsilon }) + \mathcal {O} (\epsilon ) +\frac{\rho L}{\varGamma (\alpha )}\int _0^t\frac{e^{-\sigma (t^\rho -s^\rho )}}{(t^\rho -s^\rho )^{1-\alpha }} s^{\rho -1} |\delta (s)| \mathrm{d}s, \end{aligned}$$

and using Theorem 11, the desired result can be obtained.

$\square$

6 Concluding Remarks

In this paper, we have focused on substantial fractional integral and differential operators. We have introduced a new generalized substantial fractional integral. Generalizations of fractional substantial derivatives have also been introduced in both Riemann–Liouville and Caputo sense. Furthermore, we have proved several fundamental properties of these operators. We have also considered a class of generalized substantial fractional differential equations and discussed the existence, uniqueness and continuous dependence of solutions on initial data.