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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.

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 [214, 17], and [26]), whereas some researchers studied some singular fractional integro-differential equations [2225]. 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 \(0<|t|<\infty \).

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

( a σ + b σ J κ ( p ) ) [ n ] = ( a σ + b σ 0 p κ ( s ) d s ) [ n ] = ( n 0 ) a σ n b σ 0 0 p [ 0 ] κ ( s ) d s + ( n 1 ) a σ n 1 b σ 1 0 p [ 1 ] κ ( s ) d s + + ( n n 1 ) a σ 1 b σ n 1 0 p [ n 1 ] κ ( s ) d s + ( n n ) a σ 0 b σ n 0 p [ n ] κ ( s ) d s = i = 0 n ( n i ) a σ n i b σ i 0 p [ i ] κ ( s ) d s = i = 0 n ( n i ) a σ n i b σ i J i κ ( p ) .

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

κ ( t ) = a σ ( a σ + b σ J υ ( t ) ) [ n ] + b σ 0 t ( a σ + b σ J υ ( s ) ) [ n ] d s = a σ [ ( n 0 ) a σ n b σ 0 0 t [ 0 ] υ ( s ) d s + ( n 1 ) a σ n 1 b σ 1 0 t [ 1 ] υ ( s ) d s + + ( n n 1 ) a σ 1 b σ n 1 0 t [ n 1 ] υ ( s ) d s + ( n n ) a σ 0 b σ n 0 t [ n ] υ ( s ) d s ] + b σ [ ( n 0 ) a σ n b σ 0 0 t [ 1 ] υ ( s ) d s + ( n 1 ) a σ n 1 b σ 1 0 t [ 2 ] υ ( s ) d s + + ( n n 1 ) a σ 1 b σ n 1 0 t [ n ] υ ( s ) d s + ( n n ) a σ 0 b σ n 0 t [ n + 1 ] υ ( s ) d s ] = ( n 0 ) a σ n + 1 b σ 0 0 t [ 0 ] υ ( s ) d s + [ ( n 1 ) + ( n 0 ) ] a σ n b σ 1 0 t [ 1 ] υ ( s ) d s + + [ ( n n ) + ( n n 1 ) ] a σ 1 b σ n 0 t [ n ] υ ( s ) d s + ( n n ) a σ 0 b σ n + 1 0 t [ n + 1 ] υ ( s ) d s = ( n + 1 0 ) a σ n + 1 b σ 0 0 t [ 0 ] υ ( s ) d s + ( n + 1 1 ) a σ n b σ 1 0 t [ 1 ] υ ( s ) d s + + ( n + 1 n ) + a σ 1 b σ n 0 t [ n ] υ ( s ) d s + ( n + 1 n + 1 ) a σ 0 b σ n + 1 0 t [ n + 1 ] υ ( s ) d s = ( a σ + b σ J υ ( t ) ) [ n + 1 ] .

 □

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.

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 \(\sigma \in (0,1)\).

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Acknowledgements

The second and third authors were supported by Azarbaijan Shahid Madani University. The authors express their gratitude to the referees for their helpful suggestions, which improved the final version of this paper.

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Correspondence to Dumitru Baleanu.

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Baleanu, D., Mousalou, A. & Rezapour, S. 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. Adv Differ Equ 2018, 255 (2018). https://doi.org/10.1186/s13662-018-1696-6

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MSC

  • 34A08
  • 34A99

Keywords

  • The extended Caputo–Fabrizio derivative of order \(0\leq \sigma <1\)
  • Higher-order fractional differential equation
  • Series-type equation