The extended fractional Caputo–Fabrizio derivative of order $0\leq \sigma <1$ on $C_{\mathbb{R}}[0,1]$ and the existence of solutions for two higher-order series-type differential equations

Abstract

We extend the fractional Caputo–Fabrizio derivative of order $0\leq \sigma <1$ on $C_{\mathbb{R}}[0,1]$ and investigate two higher-order series-type fractional differential equations involving the extended derivation. Also, we provide an example to illustrate one of the main results.

1 Introduction

Recently, Caputo and Fabrizio suggested a new fractional derivative [15, 16]), and Losada and Nieto [21] investigated some of its properties. Later, some authors tried to utilized it for solving various equations (see [2–14, 17], and [26]), whereas some researchers studied some singular fractional integro-differential equations [22–25]. As you know, the fractional Caputo–Fabrizio derivative is defined on the space $H^{1}$ (which is not necessarily a Banach space), and because of this reason, a researcher has to investigate approximate solutions for some problems [11, 13]. It seems that Caputo and Fabrizio tried to give a formula for an extension of their definition (see formula (3) in [15]), but they did not use it in their investigation. In 2016, Alqahtani tried to extend the Caputo–Fabrizio derivative by using formula (2.2) in [5]. Again, he did not use it for investigating the problems reported in [5]. In this manuscript, we extend the fractional Caputo–Fabrizio derivative on $C_{\mathbb{R}}[0,1]$. Using it, we discuss some higher-order series-type fractional integro-differential equations.

The properties of the fractional Caputo–Fabrizio derivative were investigated very recently in [7]. Specifically, the Caputo–Fabrizio fractional derivative is discussed in he distributional setting [7]. For more detail about physical interpretation of the Caputo–Fabrizio derivative, the reader can see the new results presented recently in [19]. Specifically, the physical origin of Caputo–Fabrizio derivative is demonstrated in [18]. Besides, very recently the determination of the fractional order (relation to physical characteristics of the process) was investigated in [20].

Having all the mentioned things in mind, in this paper, we extend the fractional Caputo–Fabrizio derivative on $C_{\mathbb{R}}[0,1]$. Using it, we investigate some higher-order series-type fractional integro-differential equations.

Let $b>0$, $\kappa \in H^{1}(0,b)$, and $\sigma \in [0,1]$. Thus, for the function κ, its Caputo–Fabrizio fractional derivative is written as ${}^{CF\hspace{-1pt}}D^{\sigma }\kappa (t)=\frac{ B(\sigma)}{ 1-\sigma } \int_{0}^{t}\exp (\frac{-\sigma }{1-\sigma }(t-s))\kappa^{\prime }(p)\,dp$, where $t\geq 0$, and $B(\sigma)$ denotes a normalization constant obeying $B(0)=B(1)=1$ [1, 15]. The associated fractional integral of order σ for the function κ is defined by ${}^{CF\hspace{-1pt}}I^{\sigma } \kappa (t)=\frac{ 1-\sigma }{ B(\sigma)}\kappa (t) +\frac{\sigma }{ B(\sigma)}\int_{0}^{t} \kappa (s)\,ds$ for $0<\sigma <1$ [1, 21].

If $n\geq 1$ and $\sigma \in [0,1]$, then the fractional derivative ${}^{CF\hspace{-1pt}}D^{\sigma +n}$ of order $n+\sigma$ is defined by ${}^{CF\hspace{-1pt}}D^{ \sigma +n}\kappa:={}^{CF\hspace{-1pt}}D^{\sigma }(D^{n}\kappa (t))$ [21]. Also, we have $\lim_{\sigma \to 0}{}^{CF\hspace{-1pt}}D^{\sigma }\kappa (t)=\kappa (t)- \kappa (0)$, $\lim_{\sigma \to 1}{}^{CF\hspace{-1pt}}D^{\sigma }\kappa (t)= \kappa^{\prime }(t)$, and ${}^{CF\hspace{-1pt}}D^{\sigma } (\lambda \kappa (t)+ \gamma \upsilon (t))=\lambda {}^{CF\hspace{-1pt}}D^{\sigma }\kappa (t)+\gamma {}^{CF\hspace{-1pt}}D ^{\sigma }\upsilon (t)$ for all $\kappa,\upsilon \in H^{1}$ and $\lambda,\gamma \in \mathbb{R}$ [15]. We now present the following important results.

Lemma 1.1

([21])

Let $0<\sigma <1$. Then the unique solution of ${}^{CF\hspace{-1pt}}D^{\sigma }\kappa (p)=\upsilon (p)$ such that $\kappa (0)=c$ is written as $\kappa (p)=c+a_{\sigma }(\upsilon (p)-\upsilon (0))+b_{ \sigma }\int_{0}^{p} \upsilon (s)\,ds$, where $a_{\sigma }=\frac{ 1- \sigma }{B(\sigma)}$ and $b_{\sigma }=\frac{ \sigma }{B(\sigma)}$. Note that $\upsilon (0)=0$.

Lemma 1.2

([27])

Let $t\in \mathbb{R}$ and $0\leq |t|<\infty$. Then $t\prod_{i=1} ^{\infty }(1-\frac{t^{2}}{i^{2} \pi^{2}})=\sin t$, $\prod_{i=1} ^{\infty }(1-\frac{4t^{2}}{(2i-1)^{2} \pi^{2}})=\cos t$ and $e^{t}=\sum_{i=0} ^{\infty } \frac{t^{i}}{i!}$ for $0<|t|<\infty$.

2 Results and discussion

We further show our main results. Let $\kappa \in C_{\mathbb{R}}[0,b]$, $b>0$, and $\sigma \in (0,1)$. We define the expended fractional Caputo–Fabrizio derivative of order σ by

$$\begin{aligned} {}^{CF\hspace{-1pt}}_{{N}} D^{\sigma }\kappa (p) =&\frac{B(\sigma)}{1-\sigma } \bigl( \kappa (p)-\kappa (0)\bigr)\exp \biggl(\frac{-\sigma }{1-\sigma }p\biggr) \\ &{}+\frac{\sigma B(\sigma)}{(1-\sigma)^{2}} \int_{0}^{p} \bigl(\kappa (p)-\kappa (s)\bigr) \exp \biggl(\frac{-\sigma }{1-\sigma }(p-s)\biggr)\,ds. \end{aligned}$$

If $\kappa (0)=0$, then we have ${}^{CF\hspace{-1pt}}_{{N}} D^{\sigma }\kappa (p)=\frac{B( \sigma)}{1-\sigma }\kappa (p) - \frac{\sigma B(\sigma)}{(1-\sigma)^{2}}\int_{0}^{p} \exp (-\frac{ \sigma }{1-\sigma }(p-s))\kappa (s)\,ds$. We recall that

$$\begin{aligned} J^{n}\kappa (p) =& \underbrace{ \int_{0}^{p=p_{n}} \int_{0}^{s=p_{n-1}} \int_{0}^{p_{n-2}} \cdots \int_{0}^{p_{1}} }_{n~\text{times}} \kappa (p_{0})\,dp _{0}\,dp_{1}\cdots d(p_{n-2})\,ds \\ =&\frac{1}{(n-1)!} \int_{0}^{p}\kappa (s) (p-s)^{n-1}\,ds. \end{aligned}$$

Now, let us po define ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma^{[n]}}\kappa (p):= \underbrace{ \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }({}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }({}^{CF\hspace{-1pt}}_{{N}}D ^{\sigma }\cdots ({}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }}_{n~\text{times}}\kappa (p))\cdots ))$ for $n\geq 1$, $a, p\in \mathbb{R}$, and $p>0$. Also, we define $J^{0}\kappa (p)=\int_{0}^{p^{[0]}} \kappa (s)\,ds=\kappa (p)$,

$$\int_{0}^{p^{[n]}} \kappa (s)\,ds=\underbrace{ \int_{0}^{p=p_{n}} \int _{0}^{s=p_{n-1}} \int_{0}^{p_{n-2}} \cdots \int_{0}^{p_{1}} }_{n~\text{times}} \kappa (p_{0})\,dp_{0}\,dp_{1}\cdots d(p_{n-2}) \,ds=J^{n}\kappa (p), $$

and

$$\begin{array}{rcl}{(a_{\sigma }+b_{\sigma }J\kappa (p))}^{[n]}& =& {(a_{\sigma }+b_{\sigma }{\int }_0^p\kappa (s)ds)}^{[n]}\\ & =& \left(\begin{array}{c}n\\ 0\end{array}\right)a_{\sigma }^n{b}_{\sigma }^0{\int }_0^{p^{[0]}}\kappa (s)ds+\left(\begin{array}{c}n\\ 1\end{array}\right)a_{\sigma }^{n-1}{b}_{\sigma }^1{\int }_0^{p^{[1]}}\kappa (s)ds\\ & & +\cdots +\left(\begin{array}{c}n\\ n-1\end{array}\right)a_{\sigma }^1{b}_{\sigma }^{n-1}{\int }_0^{p^{[n-1]}}\kappa (s)ds+\left(\begin{array}{c}n\\ n\end{array}\right)a_{\sigma }^0{b}_{\sigma }^n{\int }_0^{p^{[n]}}\kappa (s)ds\\ & =& \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)a_{\sigma }^{n-i}{b}_{\sigma }^i{\int }_0^{p^{[i]}}\kappa (s)ds\\ & =& \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)a_{\sigma }^{n-i}{b}_{\sigma }^i{J}^i\kappa (p).\end{array}$$

The following result shows that our definition is a generalization of the Caputo–Fabrizio derivative.

Lemma 2.1

Let $\kappa \in H^{1}(0,b)$, $b>0$, and $\sigma \in (0,1)$. Then ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)= {}^{CF\hspace{-1pt}}D^{\sigma }\kappa (t)$. If $\kappa \in C_{\mathbb{R}}[0,b]$, then there exists a sequence $(\kappa_{n} )_{n=1}^{\infty }$ of $H^{1}(0,b)$ such that ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)= \lim_{n\to \infty } {}^{CF\hspace{-1pt}}_{{N}}D ^{\sigma }\kappa_{n}(t)$ and $\lim_{\sigma \to 0} {}^{CF\hspace{-1pt}}_{{N}}D ^{\sigma }\kappa (t)=\kappa (t)-\kappa (0)$.

Proof

Let $\kappa \in H^{1}(0,b)$. Note that

$$\begin{aligned}& {}^{CF\hspace{-1pt}}D^{\sigma }\kappa (t) \\& \quad =\frac{B(\sigma)}{1-\sigma } \int_{0} ^{t}\exp \biggl(-\frac{\sigma }{1-\sigma }(t-p) \biggr)\kappa^{\prime }(p)\,dp \\& \quad =\frac{B(\sigma)}{1-\sigma }\exp \biggl(-\frac{\sigma }{1-\sigma }(t-p)\biggr) \kappa (p)|_{0}^{t}-\frac{B(\sigma)}{1-\sigma } \int_{0}^{t} \frac{ \sigma }{1-\sigma }\exp \biggl(- \frac{\sigma }{1-\sigma }(t-p)\biggr)\kappa (p)\,dp \\& \quad = \frac{B(\sigma)}{1-\sigma }\kappa (t)-\frac{B(\sigma)}{1-\sigma } \exp \biggl(- \frac{\sigma }{1-\sigma }t\biggr)\kappa (0)- \frac{\sigma B(\sigma)}{(1- \sigma)^{2}} \int_{0}^{t} \exp \biggl(-\frac{\sigma }{1-\sigma }(t-p) \biggr) \kappa (p)\,dp \\& \quad =\frac{B(\sigma)}{1-\sigma }\kappa (t)-\frac{B(\sigma)}{1-\sigma } \exp \biggl(- \frac{\sigma }{1-\sigma }t\biggr)\kappa (0)-\frac{\sigma B(\sigma)}{(1- \sigma)^{2}} \int_{0}^{t} \exp \biggl(-\frac{\sigma }{1-\sigma }(t-p) \biggr) \kappa (p)\,dp \\& \qquad {}+\frac{\sigma B(\sigma)}{(1-\sigma)^{2}} \int_{0}^{t} \exp \biggl(-\frac{ \sigma }{1-\sigma }(t-p) \biggr)\kappa (t)\,dp \\& \qquad {}- \frac{\sigma B(\sigma)}{(1- \sigma)^{2}} \int_{0}^{t} \exp \biggl(-\frac{\sigma }{1-\sigma }(t-p) \biggr) \kappa (t)\,dp \\& \quad =\frac{B(\sigma)}{1-\sigma } \bigl(\kappa (t)-\kappa (0)\bigr)\exp \biggl( \frac{- \sigma }{1-\sigma }t\biggr) \\& \qquad {}+\frac{\sigma B(\sigma)}{(1-\sigma)^{2}} \int _{0}^{t} \bigl(\kappa (t)-\kappa (p)\bigr)\exp \biggl(\frac{-\sigma }{1-\sigma }(t-p)\biggr)\,dp. \end{aligned}$$

Now, let $\kappa \in C_{\mathbb{R}}[0,b]$. Choose a sequence of polynomials ${\{\kappa_{n} =P_{n}\}}_{n=1}^{\infty }$ that converges uniformly to κ. Hence ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)= \lim_{n\to \infty } \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa_{n}(t)$. Since $P_{n}\in H^{1}$, we conclude that $\lim_{\sigma \to 0} {}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t) =\lim_{\sigma \to 0} \lim_{n\to \infty } {}^{CF\hspace{-1pt}}D^{\sigma }P_{n}(t)=\lim_{n\to \infty } \lim_{\sigma \to 0} {}^{CF\hspace{-1pt}}D^{\sigma }P_{n}(t)= \lim_{n\to \infty }[P_{n}(t)-P_{n}(0)]= \kappa (t)-\kappa (0)$. □

Note that if $\kappa \in H^{1}(0,b)$, then $\lim_{\sigma \to 1} {}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)=\kappa^{\prime }(t)$. But this may be not true for $\kappa \in C_{\mathbb{R}}[0,b]$.

Lemma 2.2

A solution of the problem ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)= \upsilon (t)$ such that $\kappa (0)=0$ is of the form $\kappa (t)= a _{\sigma } \upsilon (t) +b_{\sigma }\int_{0}^{t} \upsilon (s)\,ds$ for $0<\sigma <1$.

Proof

Note that ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)=\frac{B(\sigma)}{1- \sigma }\kappa (t) - \frac{\sigma B(\sigma)}{(1-\sigma)^{2}}\int _{0}^{t} \exp (-\frac{\sigma }{1-\sigma }(t-s))\kappa (s)\,ds=\upsilon (t)$. Hence $\frac{\sigma B(\sigma)}{(1-\sigma)^{2}}\int_{0}^{t} \exp ( \frac{\sigma }{1-\sigma } s)\kappa (s)\,ds=\exp ( \frac{\sigma }{1-\sigma } t)[\frac{B(\sigma)}{1-\sigma }\kappa (t)-\upsilon (t) ]$. By differentiating both sides we get $\upsilon (t)=\frac{1-\sigma }{ \sigma }[\frac{B(\sigma)}{1-\sigma } \kappa (t)-\upsilon (t)]^{ \prime }$. Now by integrating we obtain $\kappa (t)= a_{\sigma } \upsilon (t) +b_{\sigma }\int_{0}^{t} \upsilon (s)\,ds$. □

It is crucial to check that in the last result the equation ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)=0$ with $\kappa (0)=0$ possesses a unique solution. Using this note and Lemma 2.2, we deduce the next result.

Lemma 2.3

Let $0<\sigma <1$. Then the unique solution of ${}^{CF\hspace{-1pt}}_{{N}}D^{ \sigma }\kappa (t)=\upsilon (t)$ with $\kappa (0)=0$ is written as $\kappa (t)= a_{\sigma } \upsilon (t) +b_{\sigma }\int_{0}^{t} \upsilon (p)\,dp$.

Lemma 2.4

Let $0<\sigma <1$ and $\kappa (0)=0$. Then the unique solution for the problem ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma^{[n]}}\kappa (t)=\upsilon (t)$ is given by $\kappa (t)= (a_{\sigma } +b_{\sigma }J\upsilon (t) )^{[n]}$.

Proof

Applying Lemma 2.3 to ${}^{CF\hspace{-1pt}}_{{N}} D^{\sigma } \kappa (t)= \upsilon (t)$, we conclude that $\kappa (t)= a_{\sigma } \upsilon (t) + b_{\sigma } \int_{0}^{t} \upsilon (p)\,dp$. Using Lemma 2.3 for equation ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma^{[2]}} \kappa (t)=\upsilon (t)$, we get ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)= a_{\sigma } \upsilon (t) +b _{\sigma } \int_{0}^{t} \upsilon (s)\,ds$, and so

$$\begin{aligned} \kappa (t) =&a_{\sigma } \biggl( a_{\sigma } \upsilon (t) +b_{\sigma } \int_{0}^{t} \upsilon (s)\,ds \biggr)+b_{\sigma } \int_{0}^{t} \biggl( a_{ \sigma } \upsilon (t) +b_{\sigma } \int_{0}^{s} \upsilon (r)\,dr \biggr)\,ds \\ =& a_{\sigma }^{2} \upsilon (t)+2a_{\sigma }b_{\sigma } \int_{0}^{t} \upsilon (s)\,ds+b_{\sigma }^{2} \int_{0}^{t} \int_{0}^{s} \upsilon (r)\,dr\,ds = \biggl(a_{\sigma } +b_{\sigma } \int_{0}^{t} \upsilon (s)\,ds \biggr)^{[2]}. \end{aligned}$$

Now, suppose that $\kappa (t)= (a_{\sigma } +b_{\sigma }J\upsilon (t) )^{[n]}$ is the solution of ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma^{[n]}} \kappa (t)=\upsilon (t)$. We prove that $\kappa (t)= (a_{\sigma } +b _{\sigma }J\upsilon (t) )^{[n+1]}$ is the solution of $\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\sigma^{[n+1]}} \kappa (t)=\upsilon (t)$. If ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma^{[n]}} ({}^{CF\hspace{-1pt}}_{{N}}D^{\sigma } \kappa (t))=\upsilon (t)$, then ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma } \kappa (t))= (a_{\sigma } +b _{\sigma }J\upsilon (t) )^{[n ]}$, and so

$$\begin{array}{rcl}\kappa (t)& =& a_{\sigma }{(a_{\sigma }+b_{\sigma }J\upsilon (t))}^{[n]}+b_{\sigma }{\int }_0^t{(a_{\sigma }+b_{\sigma }J\upsilon (s))}^{[n]}ds\\ & =& a_{\sigma }[\left(\begin{array}{c}n\\ 0\end{array}\right)a_{\sigma }^n{b}_{\sigma }^0{\int }_0^{t^{[0]}}\upsilon (s)ds+\left(\begin{array}{c}n\\ 1\end{array}\right)a_{\sigma }^{n-1}{b}_{\sigma }^1{\int }_0^{t^{[1]}}\upsilon (s)ds\\ & & +\cdots +\left(\begin{array}{c}n\\ n-1\end{array}\right)a_{\sigma }^1{b}_{\sigma }^{n-1}{\int }_0^{t^{[n-1]}}\upsilon (s)ds+\left(\begin{array}{c}n\\ n\end{array}\right)a_{\sigma }^0{b}_{\sigma }^n{\int }_0^{t^{[n]}}\upsilon (s)ds]\\ & & +b_{\sigma }[\left(\begin{array}{c}n\\ 0\end{array}\right)a_{\sigma }^n{b}_{\sigma }^0{\int }_0^{t^{[1]}}\upsilon (s)ds+\left(\begin{array}{c}n\\ 1\end{array}\right)a_{\sigma }^{n-1}{b}_{\sigma }^1{\int }_0^{t^{[2]}}\upsilon (s)ds\\ & & +\cdots +\left(\begin{array}{c}n\\ n-1\end{array}\right)a_{\sigma }^1{b}_{\sigma }^{n-1}{\int }_0^{t^{[n]}}\upsilon (s)ds+\left(\begin{array}{c}n\\ n\end{array}\right)a_{\sigma }^0{b}_{\sigma }^n{\int }_0^{t^{[n+1]}}\upsilon (s)ds]\\ & =& \left(\begin{array}{c}n\\ 0\end{array}\right)a_{\sigma }^{n+1}{b}_{\sigma }^0{\int }_0^{t^{[0]}}\upsilon (s)ds+[\left(\begin{array}{c}n\\ 1\end{array}\right)+\left(\begin{array}{c}n\\ 0\end{array}\right)]a_{\sigma }^n{b}_{\sigma }^1{\int }_0^{t^{[1]}}\upsilon (s)ds\\ & & +\cdots +[\left(\begin{array}{c}n\\ n\end{array}\right)+\left(\begin{array}{c}n\\ n-1\end{array}\right)]a_{\sigma }^1{b}_{\sigma }^n{\int }_0^{t^{[n]}}\upsilon (s)ds+\left(\begin{array}{c}n\\ n\end{array}\right)a_{\sigma }^0{b}_{\sigma }^{n+1}{\int }_0^{t^{[n+1]}}\upsilon (s)ds\\ & =& \left(\begin{array}{c}n+1\\ 0\end{array}\right)a_{\sigma }^{n+1}{b}_{\sigma }^0{\int }_0^{t^{[0]}}\upsilon (s)ds+\left(\begin{array}{c}n+1\\ 1\end{array}\right)a_{\sigma }^n{b}_{\sigma }^1{\int }_0^{t^{[1]}}\upsilon (s)ds\\ & & +\cdots +\left(\begin{array}{c}n+1\\ n\end{array}\right)+a_{\sigma }^1{b}_{\sigma }^n{\int }_0^{t^{[n]}}\upsilon (s)ds+\left(\begin{array}{c}n+1\\ n+1\end{array}\right)a_{\sigma }^0{b}_{\sigma }^{n+1}{\int }_0^{t^{[n+1]}}\upsilon (s)ds\\ & =& {(a_{\sigma }+b_{\sigma }J\upsilon (t))}^{[n+1]}.\end{array}$$

 □

By using the details of the proof of the last result, we can easily deduce the following results (see [13]).

Lemma 2.5

Let $\kappa,\upsilon \in C_{\mathbb{R}}[0,1]$. If there exists a real number K such that $|\kappa (t)-\upsilon (t)|\leq K$ for all $t\in [0,1]$, then $|{}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)-{}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\upsilon (t)| \leq \frac{(2-\sigma)B(\sigma) }{(1- \sigma)^{2}}K$ for all $t\in [0,1]$. If $\kappa (0)=\upsilon (0)$, then $|{}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)-{}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\upsilon (t)| \leq \frac{B(\sigma) }{(1-\sigma)^{2}}K$.

This result implies that $|{}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)| \leq \frac{(2-\sigma)B(\sigma) }{(1-\sigma)^{2}}K$ for all $t\in [0,1]$ whenever $\kappa \in C_{\mathbb{R}}[0,1]$ with $|\kappa (t)|\leq K$ for some $K\geq 0$ and all $t\in [0,1]$.

Lemma 2.6

Suppose that $\kappa,\upsilon \in C_{\mathbb{R}}[0,1]$ and there exists a real number K such that $|\kappa (t)-\upsilon (t)|\leq K$ for all $t\in [0,1]$. Then $|{}^{CF\hspace{-1pt}}_{{N}}D^{\sigma^{[n]}}\kappa (t)-{}^{CF\hspace{-1pt}}_{{N}}D^{\sigma^{[n]}}\upsilon (t)| \leq (\frac{ (2-\sigma) B( \sigma)}{(1-\sigma)^{2 }})^{n}K$ for all $t\in [0,1]$. If $\kappa (0)=\upsilon (0)$, then $|{}^{CF\hspace{-1pt}}_{{N}}D^{\sigma^{[n]}} \kappa (t)-{}^{CF\hspace{-1pt}}_{{N}}D^{\sigma^{[n]}}\upsilon (t)| \leq (\frac{ B( \sigma)}{(1-\sigma)^{2 }})^{n}K$ for all $t\in [0,1]$.

In [21], the nonlinear fractional differential problem ${}^{CF\hspace{-1pt}}D^{\sigma }\kappa (t)= f(t,\kappa (t))$ with $0\leq \sigma <1$ was studied. In the proof of the related result (Theorem 1), the self-map $F: \in C_{\mathbb{R}}[0,1]\to C_{\mathbb{R}}[0,1]$ defined by $(F\kappa)(t)=a_{\sigma } f(t,\kappa (t)) + b_{\sigma } \int_{0} ^{t} f(s,\kappa (s))\,ds$ is well defined, and there is no problem. Note that this method of proofs cannot be useful for the problem

$${}^{CF\hspace{-1pt}}D^{\sigma }\kappa (t)= f\bigl(t,\kappa (t), {}^{CF\hspace{-1pt}}D^{ \sigma } \kappa (t)\bigr) $$

because the map F cannot be defined on the space $H^{1}$. Here $\sigma, \beta \in (0,1)$. For finding a new method for solving such problems, we defined a new notion by replacing ${}^{CF\hspace{-1pt}}D^{\sigma } \kappa (t)$ with ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)$. Note that our extended derivative can only be used for order $\sigma \in (0,1)$. First, we investigate the fractional differential problem

$$ {}^{CF\hspace{-1pt}}_{{N}} D^{\sigma }\kappa (t)= f\bigl(t,\kappa (t), g(t) \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}} D^{ \beta }\kappa (t)\bigr) $$

(1)

with $\kappa (0)=0$, where $\sigma, \beta \in (0,1)$.

Theorem 2.7

Let $\sigma, \beta \in (0,1)$, and let $f:[0,1]\times \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a continuous function such that $|f(t,x,y)-f(t,x ^{\prime},y^{\prime} )|\leq \eta (t)(|x-x^{\prime}|+|y-y^{\prime}| )$ for all $t\in [0,1]$ and $x,y, x^{\prime},y^{\prime} \in \mathbb{R}$. Then problem (1) has a unique solution in $H^{1}(0,1)$ whenever $\Delta =\frac{1}{B(\sigma)} [ \eta ^{*}[1+ \frac{MB(\beta)}{(1-\beta)^{2}} ]<1$.

Proof

Consider the map $F: \in C_{\mathbb{R}}[0,1]\to C_{\mathbb{R}}[0,1]$ defined by

$$(F\kappa) (t)=a_{\sigma } f\bigl(t,\kappa (t),g(t) \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \beta } \kappa (t)\bigr) + b_{\sigma } \int_{0}^{t} f\bigl(s,\kappa (s), g(s) \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \beta }\kappa (s)\bigr)\,ds. $$

By Lemma 2.5 we get

$$\begin{aligned}& \bigl\vert f\bigl(t,\kappa (t), g(t) \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \beta }\kappa (t)\bigr)-f\bigl(t, \upsilon (t),g(t) \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \beta }\upsilon (t)\bigr) \bigr\vert \\& \quad \leq \eta^{*}\biggl( \Vert \kappa -\upsilon \Vert + \frac{MB(\beta)}{(1-\beta)^{2}} \Vert \kappa -\upsilon \Vert \biggr)= \eta^{*} \biggl[1+ \frac{MB(\beta)}{(1-\beta)^{2}} \biggr] \Vert \kappa -\upsilon \Vert \end{aligned}$$

for all $\kappa,\upsilon \in C_{\mathbb{R}}[0,1]$ and $t\in [0,1]$. Hence

$$\begin{aligned} \bigl\vert (F\kappa) (t)-(F\upsilon) (t) \bigr\vert \leq& a_{\sigma } \eta^{*}\biggl[1+ \frac{MB( \beta)}{(1-\beta)^{2}} \biggr] \Vert \kappa -\upsilon \Vert +b_{\sigma } \int_{0} ^{t} \eta^{*}\biggl[1+ \frac{MB(\beta)}{(1-\beta)^{2}} \biggr] \Vert \kappa -\upsilon \Vert \,ds \\ \leq& [ a_{\sigma } + b_{\sigma }] [ \eta^{*}\biggl[1+ \frac{MB(\beta)}{(1- \beta)^{2}} \biggr] \Vert \kappa -\upsilon \Vert \\ =&\frac{1}{B(\sigma)} [ \eta^{*}\biggl[1+ \frac{MB(\beta)}{(1-\beta)^{2}} \biggr] \Vert \kappa -\upsilon \Vert \end{aligned}$$

for all $\kappa,\upsilon \in C_{\mathbb{R}}[0,1]$. Since $\Delta <1$, F is a contraction. By the Banach contraction principle F has a unique fixed point, which is the unique solution for problem (1). □

Let h and g be bounded functions on $[0,1]$ with $M_{1}=\sup_{t \in I}|h(t)| <\infty$ and $M_{2}=\sup_{t\in I}|g(t)| <\infty$. Now, we investigate the fractional higher-order series-type differential problem

$$ {}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)= \sum_{j=0}^{\infty } \frac{ {}^{CF\hspace{-1pt}}_{{N}}D^{ \varrho^{[j]}}f(t,\kappa (t),(\phi \kappa)(t), h(t) \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\nu } \kappa (t),g(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \delta } \kappa (t)) }{2^{j}} $$

(2)

with boundary condition $\kappa (0)=0$, where $\sigma, \nu,\varrho,\delta \in (0,1)$. Note that the functions h and g may be discontinuous. Since the left side of equation (2) is continuous, so is the right side as problem (2) should be a well defined equation (check our example). For this reason, we add the continuity of the function f to the assumptions of the next two results in order equations (2) and (3) to be well defined. Consider the Banach space $C_{\mathbb{R}}[0,1]$ endowed with the norm $\|\kappa \|= \sup_{t \in I}|\kappa (t)|$.

Theorem 2.8

Let $f:[0,1]\times \mathbb{R}^{4}\rightarrow \mathbb{R}$ be a continuous function such that

$$\bigl\vert f(t,x,y,w,v)-f_{1}\bigl(t,x^{\prime},y^{\prime},w^{\prime},v^{\prime} \bigr) \bigr\vert \leq \xi_{1} \bigl\vert x -x^{\prime} \bigr\vert + \xi_{2} \bigl\vert y -y^{\prime} \bigr\vert + \xi_{3} \bigl\vert w-w^{\prime} \bigr\vert + \xi_{4} \bigl\vert v -v^{\prime} \bigr\vert $$

for some nonnegative real numbers $\xi_{1}$, $\xi_{2}$, $\xi_{3}$, $\xi_{4}$ and all $x,y,w,v,x^{\prime},y^{\prime},w^{\prime},v^{\prime} \in \mathbb{R}$ and $t \in [0,1]$. If $\Delta = \frac{1}{ B(\sigma)} \sum_{j=0} ^{ \infty } \frac{1}{2^{j}}(\frac{B(\varrho)}{ (1-\varrho)^{2 }})^{j}( \xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac{M_{1}B(\nu)}{(1-\nu)^{2}}+\xi_{4} \frac{M_{2}B(\delta)}{(1- \delta)^{2}}) <1$, then problem (2) has a unique solution.

Proof

Define the map $F:C_{\mathbb{R}}[0,1]\to C_{\mathbb{R}}[0,1]$ by

$$\begin{aligned} (F\kappa) (t) =&a_{\sigma } \sum_{j=0} ^{\infty }\frac{ {}^{CF\hspace{-1pt}}_{{N}}D^{ \varrho^{[j]}}f(t,\kappa (t),(\phi \kappa)(t), h(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\nu } \kappa (t),g(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \delta }\kappa (t))}{2^{j}} \\ &{}+ b_{\sigma } \int_{0}^{t} \sum_{j=0} ^{\infty } \frac{ {}^{CF\hspace{-1pt}}_{{N}}D^{ \varrho^{[j]}}f(s,\kappa (s),(\phi \kappa)(s), h(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\nu } \kappa (s),g(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \delta }\kappa (s))}{2^{j}}\,ds, \end{aligned}$$

where $a_{\sigma }$ and $b_{\sigma }$ are introduced in Lemma 1.1. By Lemmas 2.5 and 2.6 we get

$$\begin{aligned}& \Bigg|\Bigg[ \sum_{j=0} ^{\infty } \frac{1}{2^{j}} \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \varrho^{[j]}}f\bigl(t,\kappa (t),(\phi \kappa) (t), h(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D ^{\nu } \kappa (t),g(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \delta } \kappa (t)\bigr) \\& \qquad {} -\sum_{j=0} ^{\infty } \frac{1}{2^{j}} \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \varrho ^{[j]}}f\bigl(t,\upsilon (t),(\phi \upsilon) (t), h(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \nu } \upsilon (t),g(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \delta }\upsilon (t)\bigr) \Bigg| \\& \quad \leq \sum_{j=0} ^{\infty } \frac{1}{2^{j}}\biggl(\frac{B(\varrho)}{ (1- \varrho)^{2 }}\biggr)^{j}\biggl( \xi_{1} \Vert \kappa -\upsilon \Vert + \xi_{2} \gamma _{0} \Vert \kappa -\upsilon \Vert \\& \qquad {} +\xi_{3} \frac{M_{1}B(\nu)}{(1-\nu)^{2}} \Vert \kappa -\upsilon \Vert +\xi_{4} \frac{M_{2}B(\delta)}{(1-\delta)^{2}} \Vert \kappa -\upsilon \Vert \biggr) \\& \qquad \leq \sum_{j=0} ^{\infty } \frac{1}{2^{j}}\biggl(\frac{B(\varrho)}{ (1- \varrho)^{2 }}\biggr)^{j}\biggl( \xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac{M _{1}B(\nu)}{(1-\nu)^{2}} +\xi_{4} \frac{M_{2}B(\delta)}{(1-\delta)^{2}} \biggr) \Vert \kappa - \upsilon \Vert \end{aligned}$$

for all $\kappa,\upsilon \in C_{\mathbb{R}}[0,1]$ and $t\in [0,1]$. Hence

$$\begin{aligned}& \bigl\vert (F\kappa) (t)-(F\upsilon) (t) \bigr\vert \\& \quad \leq a_{\sigma } \Biggl\vert \sum_{j=0} ^{\infty }\frac{1}{2^{j}} \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \varrho^{[j]}}f\bigl(t, \kappa (t),(\phi \kappa) (t), h(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\nu } \kappa (t),g(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \delta }\kappa (t)\bigr) \\& \qquad {} -\sum_{j=0} ^{\infty } \frac{1}{2^{j}} \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \varrho ^{[j]}}f\bigl(t,\upsilon (t),(\phi \upsilon) (t), h(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \nu } \upsilon (t),g(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \delta }\upsilon (t)\bigr) \Biggr\vert \\& \qquad {} +b_{\sigma } \Biggl[ \int_{0}^{t} \Biggl\vert \sum _{j=0} ^{\infty } \frac{1}{2^{j}} \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \varrho^{[j]}}f\bigl(s,\kappa (s),( \phi \kappa) (s),h(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\nu } \kappa (s), g(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \delta } \kappa (s)\bigr) \\& \qquad {} - \sum_{j=0} ^{\infty } \frac{1}{2^{j}} \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \varrho^{[j]}}f\bigl(s,\upsilon (s),(\phi \upsilon) (s), h(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\nu } \upsilon (s),g(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \delta }\upsilon (s)\bigr) \Biggr\vert \,ds \Biggr] \\& \quad \leq \Biggl(a_{\sigma } \sum_{j=0} ^{\infty } \frac{1}{2^{j}}\biggl(\frac{B( \varrho)}{ (1-\varrho)^{2 }}\biggr)^{j} \biggl(\xi_{1} + \xi_{2} \gamma_{0} +\xi _{3} \frac{M_{1}B(\nu)}{(1-\nu)^{2}}+\xi_{4} \frac{M_{2}B(\delta)}{(1- \delta)^{2}}\biggr) ] \\& \qquad {} + b_{\sigma } \int_{0}^{t} \sum_{j=0} ^{\infty } \frac{1}{2^{j}}\biggl(\frac{B( \varrho)}{ (1-\varrho)^{2 }}\biggr)^{j} \biggl(\xi_{1} + \xi_{2} \gamma_{0} +\xi _{3} \frac{M_{1}B(\nu)}{(1-\nu)^{2}}+\xi_{4} \frac{M_{2}B(\delta)}{(1- \delta)^{2}}\biggr) \,ds \Biggr) \\& \quad \leq [ a_{\sigma } + b_{\sigma }] \Biggl[ \sum _{j=0} ^{\infty } \frac{1}{2^{j}}\biggl( \frac{B(\varrho)}{ (1-\varrho)^{2 }}\biggr)^{j}\biggl(\xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac{M_{1}B(\nu)}{(1-\nu)^{2}} + \xi _{4} \frac{M_{2}B(\delta)}{(1-\delta)^{2}}\biggr) \Biggr] \Vert \kappa -\upsilon \Vert \end{aligned}$$

for all $\kappa,\upsilon \in C_{\mathbb{R}}[0,1]$ and $t\in [0,1]$. This implies that

$$\Vert F\kappa -F\upsilon \Vert \leq \frac{1}{ B(\sigma)} \sum _{j=0} ^{ \infty } \frac{1}{2^{j}}\biggl( \frac{B(\varrho)}{ (1-\varrho)^{2 }}\biggr)^{j}\biggl( \xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac{M_{1}B(\nu)}{(1-\nu)^{2}}+ \xi_{4} \frac{M_{2}B(\delta)}{(1- \delta)^{2}}\biggr) \Vert \kappa -\upsilon \Vert $$

for all $\kappa,\upsilon \in C_{\mathbb{R}}[0,1]$. Note that $\Delta <1$, so that F is a contraction. The Banach contraction principle implies that F has a unique fixed point, which is the unique solution for (2). □

Note that we used the boundary conditions $\kappa (0)=0$ for obtaining the key inequality

$$\begin{aligned}& \bigl\vert (F\kappa) (t)-(F\upsilon) (t) \bigr\vert \\& \quad \leq \frac{1}{ B(\sigma)} \sum_{j=0} ^{\infty } \frac{1}{2^{j}}\biggl(\frac{B(\varrho)}{ (1-\varrho)^{2 }}\biggr)^{j} \biggl( \xi_{1} + \xi_{2} \gamma_{0} + \xi_{3} \frac{M_{1}B(\nu)}{(1-\nu)^{2}}+\xi_{4} \frac{M_{2}B(\delta)}{(1- \delta)^{2}} \biggr) \Vert \kappa -\upsilon \Vert . \end{aligned}$$

Let k, s, h, and g be bounded functions on $[0,1]$ with $M_{1}= \sup_{t\in I}|k(t)|<\infty$, $M_{2}=\sup_{t\in I}|s(t)|<\infty$, $M_{3}= \sup_{t\in I}|h(t)| <\infty$, and $M_{4}=\sup_{t\in I}|g(t)| <\infty$. Now, we investigate the fractional higher-order series-type differential problem

$$\begin{aligned}& \frac{{}^{CF\hspace{-1pt}}_{{N}}D^{\sigma^{[n]}}\kappa (t)-[\lambda k(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\beta^{[m]}}\kappa (t)+{\mu }s(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\rho^{[p]}} \kappa (t)]}{\sum_{j=0} ^{\infty }\frac{1}{j!}\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \theta^{[j]}}f(t,\kappa (t),(\phi \kappa)(t), h(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D ^{\nu^{[q]}} \kappa (t),g(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \delta^{[r]}}\kappa (t)) } \\& \quad =\prod_{i=1} ^{\infty } \biggl(1- \frac{[\sum_{j=0} ^{\infty } \frac{1}{j!}\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \theta^{[j]}}f(t,\kappa (t),(\phi \kappa)(t), h(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\nu^{[q]}} \kappa (t),g(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \delta^{[r]}}\kappa (t)) ]^{2}}{i^{2} \pi^{2}} \biggr) \\& \qquad {} +\prod_{i=1} ^{\infty } \biggl(1- \frac{4[\sum_{j=0} ^{\infty } \frac{1}{j!}\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \theta^{[j]}}f(t,\kappa (t),(\phi \kappa)(t), h(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\nu^{[q]}} \kappa (t),g(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \delta^{[r]}}\kappa (t)) ]^{2}}{(2i-1)^{2} \pi^{2}} \biggr) \end{aligned}$$

(3)

such that $\kappa (0)=0$, where $\lambda,\mu \geq 0$, $\sigma, \beta,\rho,\theta,\nu,\delta,\in (0,1)$, and n, m, p, q, and r are natural numbers.

Theorem 2.9

Assume that $f:[0,1]\times \mathbb{R}^{4}\rightarrow \mathbb{R}$ is a continuous function such that

$$\begin{aligned}& \bigl\vert f(t,x,y,w,z)-f\bigl(t,x^{\prime},y^{\prime},w^{\prime},z^{\prime} \bigr) \bigr\vert \\& \quad \leq \xi_{1} \bigl( \bigl\vert x-x^{\prime} \bigr\vert +\xi _{2} \bigl\vert y-y^{\prime} \bigr\vert + \xi_{3} \bigl\vert w-w^{\prime} \bigr\vert + \xi_{4} \bigl\vert z-z^{\prime} \bigr\vert \bigr) \end{aligned}$$

for some nonnegative real numbers $\xi_{1}$, $\xi_{2}$, $\xi_{3}$, $\xi_{4}$ and all $x,y,w,z,x^{\prime},y^{\prime},w^{\prime},z^{\prime} \in \mathbb{R}$ and $t \in [0,1]$. If $\Delta =(\frac{1}{B(\sigma)})^{n}([ \lambda \frac{M_{1}B(\beta)}{(1- \beta)^{2m}}+\mu \frac{ M_{2}B(\rho)}{(1-\rho)^{2p}} ]+ [ e^{\frac{B( \theta)}{(1-\theta)^{2}}}(\xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac{ M_{3}B(\nu)}{(1-\nu)^{2q}} +\xi_{4} \frac{M_{4}B(\delta)}{(1- \delta)^{2r}})])<1$, then problem (3) has a unique solution.

Proof

Define $(G\kappa)(s)=\sum_{j=0} ^{\infty }\frac{1}{j!}\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \theta^{[j]}}f(s,\kappa (s),(\phi \kappa)(s), h(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\nu^{[q]}} \kappa (s),g(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \delta^{[r]}} \kappa (s))$ for all $\kappa \in C_{\mathbb{R}}[0,1]$ and $s\in [0,1]$. Consider the map $F:C_{\mathbb{R}}[0,1] \to C_{ \mathbb{R}}[0,1]$ defined by

$$\begin{aligned} F(\kappa) =& \Biggl(a_{\sigma } +b_{\sigma } \int_{0}^{t} \Biggl[ (G\kappa) (s) \prod _{i=1} ^{\infty }\biggl(1-\frac{[(G\kappa)(s)]^{2}}{i^{2} \pi^{2}}\biggr)+(G \kappa) (t)\prod_{i=1} ^{\infty }\biggl(1- \frac{4[(G\kappa)(t)]^{2}}{(2i-1)^{2} \pi^{2}}\biggr) \\ &{}+\bigl(\lambda k(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\beta^{[m]}}\kappa (s)+{\mu }s(s) \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\rho^{[p]}}\kappa (s)\bigr) \Biggr]\,ds \Biggr)^{[n]}. \end{aligned}$$

By Lemma 1.2 we have

$$\begin{aligned} {}^{CF\hspace{-1pt}}_{{N}}D^{\sigma^{[n]}} \kappa (t) =&(G\kappa) (t)\prod _{i=1} ^{\infty } \biggl(1-\frac{[(G\kappa)(t)]^{2}}{i^{2} \pi^{2}} \biggr)+(G \kappa) (t)\prod_{i=1} ^{\infty } \biggl(1- \frac{4[(G\kappa)(t)]^{2}}{(2i-1)^{2} \pi^{2}} \biggr) \\ &{} +\bigl(\lambda k(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\beta^{[m]}}\kappa (t)+{\mu }s(t) \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\rho^{[p]}}\kappa (t)\bigr) \\ =&\sin (G\kappa) (t)+\cos (G\kappa) (t)+\bigl(\lambda k(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D ^{\beta^{[m]}}\kappa (t)+{\mu }s(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\rho^{[p]}} \kappa (t)\bigr) \\ =&2^{\frac{1}{2} }\sin \biggl((G\kappa) (t)+\frac{\pi }{4} \biggr) +\bigl( \lambda k(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\beta^{[m]}}\kappa (t)+{\mu }s(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \rho^{[p]}}\kappa (t)\bigr) \end{aligned}$$

for all $\kappa \in C_{\mathbb{R}}[0,1]$ and $s\in [0,1]$. Hence

$$\begin{aligned}& \bigl\vert (F\kappa) (t)-(F\upsilon) (t) \bigr\vert \\& \quad \leq \biggl(a_{\sigma }+b_{\sigma } \int _{0} ^{t }\biggl[2^{\frac{1}{2} } \biggl\vert \sin \biggl((G\kappa) (s)+\frac{\pi }{4}\biggr)- \sin \biggl((G\upsilon) (s)+ \frac{\pi }{4}\biggr) \biggr\vert \\& \qquad {} + \bigl\vert \bigl(\lambda k(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\beta^{[m]}} \kappa (s)+{\mu }s(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\rho^{[p]}}\kappa (s)\bigr)-\bigl( \lambda k(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D ^{\beta^{[m]}}\upsilon (s) \\& \qquad {}+{\mu }s(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\rho^{[p]}} \upsilon (s)\bigr) \bigr\vert \biggr]\,ds \biggr)^{[n]} \\& \quad \leq \biggl(a_{\sigma }+ \int_{0} ^{t }2^{\frac{1}{2} } \bigl\vert (G \kappa) (s)-(G \upsilon) (s) \bigr\vert \biggr) \\& \qquad {} + \bigl\vert \bigl(\lambda k(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\beta^{[m]}} \kappa (s)+{\mu }s( s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\rho^{[p]}} \kappa (s)\bigr)-\bigl( \lambda k(s)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D ^{\beta^{[m]}}\upsilon (s) \\& \qquad {}+{\mu }s(t)\hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{\rho^{[p]}} \upsilon (s)\bigr) \bigr\vert \,ds )^{[n]} \\& \quad \leq \biggl(a_{\sigma }+ \int_{0} ^{t }2^{\frac{1}{2} } \bigl\vert (G \kappa) (s)-(G \upsilon) (s) \bigr\vert \biggr) \\& \qquad {} + \lambda \bigl\vert k(s) \bigr\vert \bigl\vert {}^{CF\hspace{-1pt}}_{{N}}D^{\beta^{[m]}}\bigl(\kappa (s)-\upsilon (s)\bigr) \bigr\vert + {\mu } \bigl\vert s( s) \bigr\vert \bigl\vert {}^{CF\hspace{-1pt}}_{{N}}D^{\rho^{[p]}} \bigl(\kappa (s)-\upsilon (s)\bigr) \bigr\vert \,ds )^{[n]} \end{aligned}$$

for all $\kappa,\upsilon \in C_{\mathbb{R}}[0,1]$ and $t\in [0,1]$. Also, by Lemma 2.6 we get

$$\bigl\vert (G\kappa) (s)-(G\upsilon) (s) \bigr\vert \leq \sum _{i=0} ^{\infty } \frac{1}{i!}\biggl( \frac{B(\theta)}{ (1-\theta)^{2 }}\biggr)^{i}\biggl[\xi_{1} + \xi _{2} \gamma_{0} +\xi_{3}\frac{ M_{3}B(\nu)}{(1-\nu)^{2q}} + \xi_{4} \frac{ M_{4}B(\delta)}{(1-\delta)^{2r}}\biggr] \Vert \kappa -\upsilon \Vert $$

for all $\kappa \in C_{\mathbb{R}}[0,1]$ and $s\in [0,1]$. Thus we get

$$\begin{aligned}& \Vert F\kappa -F\upsilon \Vert \\& \quad \leq (a_{\sigma }+b_{\sigma })^{n}\biggl[ \lambda \frac{M _{1}B(\beta)}{(1-\beta)^{2m}}+\mu \frac{ M_{2}B(\rho)}{(1-\rho)^{2p}} \biggr] \\& \qquad {} +(a_{\sigma }+b_{\sigma })^{n}\biggl[ e^{\frac{B(\theta)}{(1-\theta)^{2}}}\biggl( \xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac{ M_{3}B(\nu)}{(1-\nu)^{2q}} +\xi_{4} \frac{M_{4}B(\delta)}{(1- \delta)^{2r}}\biggr)\biggr] \Vert \kappa -\upsilon \Vert \\& \quad =\biggl(\frac{1}{B(\sigma)}\biggr)^{n}\biggl(\biggl[ \lambda \frac{M_{1}B(\beta)}{(1-\beta)^{2m}}+\mu \frac{ M_{2}B(\rho)}{(1-\rho)^{2p}} \biggr] \\& \qquad {}+ \biggl[ e^{\frac{B( \theta)}{(1-\theta)^{2}}}\biggl( \xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac{ M_{3}B(\nu)}{(1-\nu)^{2q}} +\xi_{4} \frac{M_{4}B(\delta)}{(1- \delta)^{2r}}\biggr)\biggr]\biggr) \Vert \kappa -\upsilon \Vert \end{aligned}$$

for all $\kappa,\upsilon \in C_{\mathbb{R}}[0,1]$.

Since $\Delta <1$, F is a contraction. By the Banach contraction principle, F possesses a unique fixed point, which is the unique solution for (3). □

Now, we explicitly show an example to illustrate our aim.

Example 2.1

Consider $\gamma:[0,1] \times [0,1]\to [0,\infty)$ defined by $\gamma (t,s)=\frac{e^{2t-s}}{e}$. Note that $\gamma_{0}\leq e$. Put $\sigma =\frac{1}{4}$, $\nu =\frac{1}{4}$, $\delta =\frac{1}{4}$, $\varrho =\frac{1}{8}$, $\xi_{1}=\frac{2}{41}$, $\xi_{2}=\frac{1}{48}$, $\xi_{3}=\frac{1}{e^{2}}$, and $\xi_{4}=\frac{1}{2e^{2}}$. Let $B(\sigma)=1$ for $\sigma \in (0,1)$, $h(t)=1$ for $x \in Q\cap [0,1]$ and $h(t)=0$ for $x \in Q^{c}\cap [0,1]$, $g(t)=0$ for $x \in Q\cap [0,1]$ and $g(t)=2$ for $x \in Q^{c}\cap [0,1]$. Then, $M_{1}=\sup_{t\in [0,1]}|h(t)|=1$ and $M_{2}=\sup_{t\in [0,1]}|g(t)|=2$. We further discuss the fractional problem

$$\begin{aligned} {}^{CF\hspace{-1pt}}D^{\frac{1}{4}}\kappa (t) =&\sum_{j=0} ^{\infty }\frac{1}{2^{j}} \hspace{1pt}{}^{CF\hspace{-1pt}}_{{N}}D^{ \varrho^{[j]}}\biggl( t+ \frac{2}{41}\kappa (t)+ \frac{1}{48} \int_{0}^{t} \frac{e^{2t-s}}{e}\kappa (s)\,ds \\ &{}+\frac{1}{e^{2}}h(t){}^{CF\hspace{-1pt}}D^{\frac{1}{4}} \kappa (t) + \frac{1}{2e^{2}}g(t){}^{CF\hspace{-1pt}}D ^{\frac{1}{4}}\kappa (t)\biggr))) \end{aligned}$$

(4)

such that $\kappa (0)=0$. Consider $f(t,x,y,w,v)= t+\frac{2}{41}x+ \frac{1}{48}y+\frac{1}{e^{2}}w+\frac{1}{2e^{2}}v$ for all $t\in I$ and $x,y,w, v \in \mathbb{R}$. Note that $\Delta = \frac{1}{ B(\alpha)} \sum_{j=0} ^{\infty } \frac{1}{2^{j}}(\frac{B(\varrho)}{ (1-\varrho)^{2 }})^{j}(\xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac{M_{1}B( \nu)}{(1-\nu)^{2}}+\xi_{4} \frac{M_{2}B(\delta)}{(1-\delta)^{2}}) <0.3<1$. By Theorem 2.8 we conclude that problem (4) has a unique solution.

3 Conclusion

The nonlinear fractional differential problem ${}^{CF\hspace{-1pt}}D^{\sigma } \kappa (t)= f(t,\kappa (t))$ with $0\leq \sigma <1$ has been studied in some works. The method used in the proofs cannot be used for the problem ${}^{CF\hspace{-1pt}}D^{\sigma }\kappa (t)= f(t,\kappa (t), {}^{CF\hspace{-1pt}}D ^{ \sigma }\kappa (t))$ for technical reasons. For finding a new method for solving such problems, we define a new extended fractional derivative ${}^{CF\hspace{-1pt}}_{{N}}D^{\sigma }\kappa (t)$ replacing ${}^{CF\hspace{-1pt}}D^{\sigma }\kappa (t)$, and we study two higher-order series-type fractional differential equations involving the extended derivative. We emphasize that our extended derivative can be used only for order $\sigma \in (0,1)$.