Open Access

On the existence of solutions for some infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential equations

Boundary Value Problems20172017:145

https://doi.org/10.1186/s13661-017-0867-9

Received: 1 April 2017

Accepted: 4 September 2017

Published: 5 October 2017

Abstract

By mixing the idea of 2-arrays, continued fractions, and Caputo-Fabrizio fractional derivative, we introduce a new operator entitled the infinite coefficient-symmetric Caputo-Fabrizio fractional derivative. We investigate the approximate solutions for two infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential problems. Finally, we analyze two examples to confirm our main results.

Keywords

continued fractioncontraction mapfractional integro-differential equationinfinite Caputo-Fabrizio fractional derivative

MSC

30B7034A08

1 Introduction

Fractional calculus has many real-world applications in various fields of science and engineering [110]. During the recent years, the researchers started to think how to enlarge the range of fractional calculus by constructing operators with different nonlocal kernels. For example, a new nonlocal derivative without singular kernel was introduced in [11]. After that, this new fractional operator was utilized to get more information from solving different fractional differential equations corresponding to complex phenomena (the reader can see, for example, [1120], and the references therein). Let use consider \(b>0\) and \(x\in H^{1}(0,b)\) together with \(\alpha\in(0,1)\). For a function x, Caputo and Fabrizio defined its fractional derivative (CF) of order α as \({}^{\mathrm{CF}}C^{\alpha}x(p)=\frac{(2-\alpha)M(\alpha)}{2(1-\alpha)}\int _{0}^{p}\exp (\frac{-\alpha}{1-\alpha}(p-w))x^{\prime}(w)\,dw\), where \(t\geq0\), and \(M(\alpha)\) is such that \(M(0)=M(1)=1\) [11]. The corresponding fractional integral of order α for the function x is \({}^{\mathrm{CF}}I^{\alpha} x(p)=\frac{2(1-\alpha)}{(2-\alpha)M(\alpha)}x(p) +\frac{2\alpha}{(2-\alpha)M(\alpha)}\int_{0}^{p} x(w) \,dw\) whenever \(0<\alpha<1\) [21]. Also, the values of the function M were found as \(M(\alpha)=\frac{2}{2-\alpha}\) for all \(0\leq\alpha\leq1\) [21]. Taking into account the results mentioned, for a given function x, its fractional CF of order α becomes \({}^{\mathrm{CF}}C^{\alpha}x(p)=\frac{1}{1-\alpha}\int_{0}^{p}\exp(-\frac {\alpha}{1-\alpha}(p-w))x^{\prime}(w)\,dw\) for \(t\geq0\) and \(0<\alpha<1\) [21]. In this way a new type of fractional calculus was established. The aim of the manuscript is to propose a new operator named the infinite coefficient-symmetric Caputo-Fabrizio fractional derivative and to study some its properties.

2 Basic tools and new fractional operators

We further introduce some basic notation.

Lemma 2.1

[21]

Let us consider the equation \({}^{\mathrm{CF}}C^{\alpha}x(t)=y(t)\) such that \(x(0)=c\) and \(0<\alpha<1\). The solutions of this equation has the form \(x(p)=c+a_{\alpha}(y(p)-y(0))+b_{\alpha}\int_{0}^{p} y(z)\,dz\), where \(a_{\alpha}=\frac{2(1-\alpha)}{(2-\alpha)M(\alpha)}=1-\alpha\) and \(b_{\alpha}=\frac{2\alpha}{(2-\alpha)M(\alpha)}=\alpha\).

Let \(\varepsilon> 0\). We consider a metric space \((Z,d_{1})\), a selfmap G on Z, and a mapping \(\alpha : Z\times Z \to[ 0 , \infty) \). As a result, we say that G is α-admissible whenever \(\alpha(t,s) \geq1\) implies \(\alpha (Gt , Gs )\geq1\) [22]. An element \(z_{0}\in Z \) is called an ε-fixed point of G if \(d(G z_{0},z_{0}) \leq\varepsilon \). We say that G possess the approximate fixed point property if G possesses an ε-fixed point for all \(\varepsilon> 0\) [22]. Denote by \(\mathcal{R}\) the set of all continuous mappings \(j : [0,\infty)^{5} \to[0,\infty)\) satisfying \(j(1,1,1,2,0)= j(1,1,1,0,2):=l \in(0,1)\), \(j(\mu t_{1},\mu t_{2},\mu t_{3},\mu t_{4},\mu t_{5}) \leq\mu j(t_{1}, t_{2},t_{3},t_{4},t_{5})\) for all \((t_{1},t_{2},t_{3},t_{4},t_{5}) \in[0,\infty)^{5} \) and \(\mu\geq0 \) and also \(j( t_{1},t_{2},t_{3},0,t_{4}) \leq j( s_{1},s_{2},s_{3},0,s_{4})\) and \(j(t_{1},t_{2},t_{3},t_{4},0)\leq j(s_{1},s_{2},s_{3},s_{4},0)\) whenever \(t_{1},\dots,t_{4},s_{1},\dots,s_{4} \in[0,\infty)\) with \(t_{k}< s_{k} \) for \(k=1,2,3,4\) [22]. Next, we recall that G is called a generalized α-contractive mapping if there exists \(j \in\mathcal{R}\) such that \(\alpha(t,s)d_{1}(Gt,Gs)\leq j(d_{1}(t_{1},s_{1}),d_{1}(t_{1},Gt_{1}),d_{1}(s_{1},Gs_{1}),d_{1}(t_{1},Gs_{1}), d_{1}(s_{1},Gt_{1}))\) for all \(t_{1},s_{1} \in Z\) [22]. We need the following key result.

Theorem 2.2

[22]

Suppose that there exists \(t_{0}\in Z\) such that \(\alpha(t_{0},Gt_{0}) \geq1\). Then G possesses an approximate fixed point, where \((Z,d)\) is a metric space, \(\alpha: Z\times Z \to [0,\infty)\) denotes a mapping, and G represents a generalized α-contractive and α-admissible selfmap on Z.

Let \(\{L_{i,2^{i}}\}_{i\geq1}\) be a sequence of operators on a set. For reduction and approximation in large and infinite potential-driven flow networks, there is a method of using 2-arrays and continued fractions (see [23] and [24]). In fact, it is sufficient to arrange the operators \(\{L_{i,2^{i}}\}_{i\geq1}\) symmetrically on a 2-array, and by using a continued fraction we make a new operator \(L_{N}\) from the operators \(L_{i,2^{i}}\), where N is a natural number (see [23] and [24]). First, we arrange the operators \(L_{i,2^{i}}\) on a 2-array (tree) as in Figure 1 (see [23]).
Figure 1

An N generation tree network composed of the operators \(\pmb{L_{i,2^{i}}}\) .

Now, using a finite continued fraction, consider the new operator \(L_{N}\) defined by
$$ L_{N}=\frac{1}{\frac{1}{L_{11}+\frac{1}{\frac{1}{L_{21}+\cdots+\frac {1}{\frac{1}{L_{N1}}+\frac{1}{L_{N2}}}}+\frac{1}{L_{22}+\cdots+\frac {1}{\frac{1}{L_{N3}}+\frac{1}{L_{N4}}}}}}+\frac{1}{L_{12}+\frac{1}{\frac {1}{L_{23}+\cdots+\frac{1}{\frac{1}{L_{N2^{N-3}}}+\frac {1}{L_{N2^{N-2}}}}}+\frac{1}{L_{24}+\cdots+\frac{1}{\frac {1}{L_{N2^{N-1}}}+\frac{1}{L_{N2^{N}}}}}}}}. $$
Here, we replace symmetrically the operators \(L_{ij}\) with \({}^{\mathrm{CF}}C^{\alpha}\) for j odd (the upper branch) and \({}^{\mathrm{CF}}C^{\beta}\) for j even (the lower branch) as in Figure 2.
Figure 2

A symmetric generation tree network composed of the operators \(\pmb{{}^{\mathrm{CF}}C^{\alpha}}\) and \(\pmb{{}^{\mathrm{CF}}C^{\beta}}\) .

Put
$${}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{1}=\frac{1}{\frac {1}{{}^{\mathrm{CF}}C^{\alpha} }+\frac{1}{{}^{\mathrm{CF}}C^{\beta} }},\quad \quad {}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{2}=\frac{1}{\frac {1}{{}^{\mathrm{CF}}C^{\alpha}+\frac{1}{\frac{1}{{}^{\mathrm{CF}}C^{\alpha}}+\frac {1}{{}^{\mathrm{CF}}C^{\beta}}}}+\frac{1}{{}^{\mathrm{CF}}C^{\beta}+\frac{1}{\frac {1}{{}^{\mathrm{CF}}C^{\alpha}}+\frac{1}{{}^{\mathrm{CF}}C^{\beta}}}}} $$
and
$$\begin{aligned}& {}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{3} \\& \quad= \frac{1}{\frac{1}{{}^{\mathrm{CF}}C^{\alpha}+\frac{1}{\frac {1}{{}^{\mathrm{CF}}C^{\alpha}+\frac{1}{\frac{1}{{}^{\mathrm{CF}}C^{\alpha} }+\frac{1}{{}^{\mathrm{CF}}C^{\beta} }}}+\frac{1}{{}^{\mathrm{CF}}C^{\beta}+\frac{1}{\frac{1}{{}^{\mathrm{CF}}C^{\alpha} }+\frac{1}{{}^{\mathrm{CF}}C^{\beta} }}}}}+\frac{1}{{}^{\mathrm{CF}}C^{\beta}+\frac{1}{\frac{1}{{}^{\mathrm{CF}}C^{\alpha }+\frac{1}{\frac{1}{{}^{\mathrm{CF}}C^{\alpha} }+\frac{1}{{}^{\mathrm{CF}}C^{\beta} }}}+\frac{1}{{}^{\mathrm{CF}}C^{\beta}+\frac{1}{\frac{1}{{}^{\mathrm{CF}}C^{\alpha} }+\frac{1}{{}^{\mathrm{CF}}C^{\beta} }}}}}}. \end{aligned}$$
Continuing this process, we can define the new operator \({}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{N}\). Now, we define the infinite symmetric CF fractional derivative by \({}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{\infty}=\lim_{N \to \infty} {}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{N}\). A simple calculation shows that \({}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{\infty} =({}^{\mathrm{CF}}C^{\alpha} {}^{\mathrm{CF}}C^{\beta})^{\frac{1}{2}}\). Similarly, we can define the infinite symmetric CF fractional integral \({}^{\mathrm{CF}}\mathbb{I}^{(\alpha,\beta)}_{\infty}\) by
$$ {}^{\mathrm{CF}}\mathbb{I}^{(\alpha,\beta)}_{\infty}= \frac{1}{\frac {1}{{}^{\mathrm{CF}}I^{\alpha}+\frac{1}{\frac{1}{{}^{\mathrm{CF}}I^{\alpha}+\cdots }+\frac{1}{{}^{\mathrm{CF}}I^{\beta}+\cdots}}}+\frac{1}{{}^{\mathrm{CF}}I^{\beta}+\frac {1}{\frac{1}{{}^{\mathrm{CF}}I^{\alpha}+\cdots}+\frac{1}{{}^{\mathrm{CF}}I^{\beta }+\cdots}}}}. $$
Let \(\mu\geq0\), \(\mu\neq2\). Putting \(\mu^{i-1} {}^{\mathrm{CF}}C^{\alpha}\) on the upper branch and \(\mu^{i-1} {}^{\mathrm{CF}}C^{\beta}\) on the lower branch in the ith stage as in Figure 3, we can make the infinite coefficient-symmetric CF fractional derivative as a generalization for last case.
Figure 3

A coefficient-symmetric generation tree composed of the operators \(\pmb{{}^{\mathrm{CF}}C^{\alpha}}\) and \(\pmb{{}^{\mathrm{CF}}C^{\beta}}\) .

In fact, we define
$${}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{(\mu,\infty)}=\frac{1}{\frac {1}{{}^{\mathrm{CF}}C^{\alpha}+\frac{1}{\frac{1}{\mu {}^{\mathrm{CF}}C^{\alpha}+\cdots}+\frac{1}{\mu {}^{\mathrm{CF}}C^{\beta}+\cdots}}}+\frac{1}{{}^{\mathrm{CF}}C^{\beta}+\frac{1}{\frac {1}{\mu {}^{\mathrm{CF}}C^{\alpha}+\cdots}+\frac{1}{\mu{}^{\mathrm{CF}}C^{\beta}+\cdots}}}}, $$
and so
$${}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\alpha)}_{(\mu,\infty)}=\frac{1}{\frac {1}{{}^{\mathrm{CF}}C^{\alpha}+ \mu{}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\alpha)}_{(\mu,\infty)}}+\frac {1}{{}^{\mathrm{CF}}C^{\alpha}+ \mu{}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\alpha)}_{(\mu,\infty)}}}. $$
This implies that
$$ (*)\quad\quad {}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{(\mu,\infty)} = \frac{1}{2-\mu} {}^{\mathrm{CF}}C^{\alpha}. $$

3 Results

To show our results, we recall below two lemmas [15] under the assumption that \(x,y\in H^{1}(0,1)\).

Lemma 3.1

[15]

If there exists a real number \(K_{1}\) such that \(\vert x(p)-y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}C^{\alpha}x(p)-{{}^{\mathrm{CF}}}C^{\alpha}y(p) \vert \leq \frac{2-\alpha}{(1-\alpha)^{2}}K_{1}\) for all \(p \in[0,1]\).

Lemma 3.2

[15]

Assume that \(x(0)=y(0)\) and there exists a real number \(K_{1}\) such that \(\vert x(p)-y(p) \vert \leq K_{1}\) for \(p\in[0,1]\). Then \(\vert {}^{\mathrm{CF}}C^{\alpha}x(p)-{{}^{\mathrm{CF}}}C^{\alpha}y(p) \vert \leq\frac{1}{(1-\alpha)^{2}}K_{1}\) for all \(p\in[0,1]\).

Let \(x,y\in C_{\mathbb{R}}[0,1]\).

Lemma 3.3

[15]

If there is \(K_{1}\geq0\) such that \(\vert x(p)-y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}I^{\alpha}x(p)-{}^{\mathrm{CF}}I^{\alpha}y(p) \vert \leq K_{1}\) for \(p\in[0,1]\).

Now we are ready to show our main results. Using Lemmas 3.1 and 3.2, we obtain the next key results.

Lemma 3.4

Let \(x,y\in H^{1}\). If there exists a real number \(K_{1}\) such that \(\vert x(p)-y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{C}_{\infty}^{(\alpha,\alpha)}y(p) \vert \leq\frac{2-\alpha}{(1-\alpha)^{2}}K_{1}\) for all \(p\in[0,1]\).

Lemma 3.5

Let \(x,y\in H^{1}\) with \(x(0)=y(0)\) and \(K_{1}\in\mathbb{R}\). If \(\vert x(p)-y(p) \vert \leq K_{1}\) for \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{C}_{\infty}^{(\alpha,\alpha)}y(p) \vert \leq\frac{1}{(1-\alpha)^{2}}K_{1}\) for all \(p\in[0,1]\).

Using Lemmas 3.4 and 3.5 and (*), we get the following results.

Lemma 3.6

Let \(x,y\in H^{1}\). If there exists a real number \(K_{1}\) such that \(\vert x(p)-y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha)}y(p) \vert \leq\frac{(2-\alpha)}{(2-\mu)(1-\alpha)^{2}} K_{1}\) for all \(p\in [0,1]\).

Lemma 3.7

Let \(x,y\in H^{1}\) with \(x(0)=y(0)\) and \(K_{1}\in\mathbb{R}\). If \(\vert x(p)-y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha)}y(p) \vert \leq\frac{1}{(2-\mu)(1-\alpha)^{2}} K_{1}\) for all \(p\in[0,1]\).

Lemma 3.8

Let \(x,y\in C_{\mathbb{R}}[0,1]\). Let \(K_{1}\) be a real number such that \(\vert x(p)-y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}\mathbb{I}_{\infty}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{I}_{\infty}^{(\alpha,\alpha)}y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\).

Using Lemma 2.1, we can prove the next key result.

Lemma 3.9

Let \(\alpha\in(0,1)\) and \(c\in\mathbb{R}\). The unique solution of the problem
$$ {}^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\alpha,\alpha)}x(p)=y(p) $$
with boundary condition \(x(0)=c\) is given by \(x(p)=c+a_{\alpha}(y(p)-y(0))+b_{\alpha}\int_{0}^{t} y (s)\,ds\).

Also, using Lemma 2.1 and (*), we can prove the next key result.

Lemma 3.10

Let \(\alpha\in(0,1)\) and \(c\in\mathbb{R}\). The unique solution of the problem
$$ {}^{\mathrm{CF}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha)}x(p)=y(p) $$
with boundary condition \(x(0)=c\) is given by
$$ x(p)=c+a_{\alpha}(2-\mu) \bigl(y(p)-y(0) \bigr)+b_{\alpha}(2-\mu) \int_{0}^{p} y (s)\,ds. $$
Let \(I=[0,1]\), and let \(\gamma,\lambda:[0,1] \times[0,1]\to [0,\infty)\) be two continuous maps such that \(\sup_{p\in I} \vert \int_{0}^{p} \lambda(p,r) \,dr \vert <\infty\) and \(\sup_{p\in I} \vert \int_{0}^{p} \gamma(p,r) \,dr \vert <\infty\). We introduce the following maps ϕ and φ defined by \((\phi u)(p)= \int_{0}^{p} \gamma(p,r)u(r)\,dr \) and \((\varphi u)(p)= \int_{0}^{p} \lambda(p,r)u(r)\,dr \), respectively. Let us consider \(\gamma_{0}=\sup \vert \int_{0}^{p} \gamma(p,r) \,dr \vert \) and \(\lambda_{0}=\sup \vert \int_{0}^{p} \lambda(p,r) \,dr \vert \), respectively. Let \(\eta(p)\in L^{\infty}(I)\) with \(\eta^{\ast}=\sup_{p\in I} \vert \eta(p) \vert \). We further are going to investigate the infinite CF fractional integro-differential problem, namely
$$ \begin{aligned}[b] {}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\alpha )}_{\infty}u_{1}^{\prime}(r)&= \mu{ \bigl(} ^{\mathrm{CF}}\mathbb{C}^{(\beta,\beta )}_{\infty}u_{1}^{\prime}(r)+^{\mathrm{CF}} \mathbb{C}^{(\gamma,\gamma)}_{\infty}u_{1}^{\prime}(r) { \bigr)} \\ &\quad{}+f^{\prime} \bigl(r,u_{1}^{\prime}(r), \bigl(\phi u_{1}^{\prime} \bigr) (r), \bigl(\varphi u_{1}^{\prime} \bigr) (r), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta )}_{\infty} u_{1}^{\prime}(r),^{\mathrm{CF}}\mathbb{C}^{ (\delta,\delta)}_{\infty}u_{1}^{\prime}(r) \bigr) \end{aligned} $$
(1)
with \(u_{1}^{\prime}(0)=0\). Here \(\alpha,\beta,\gamma,\theta ,\delta\in (0,1)\), and \(\mu\geq0\).

Theorem 3.11

Let \(f^{\prime}:[0,1]\times\mathbb{R}^{5}\rightarrow\mathbb{R}\) be a continuous function satisfying
$$\begin{aligned}& \bigl\vert f^{\prime}(r,x_{1},y_{1},w_{1},u_{1},u_{2})-f^{\prime} \bigl(r,x_{1}^{\prime},y_{1}^{\prime},w_{1}^{\prime},v_{1},v_{2} \bigr) \bigr\vert \\& \quad \leq\eta(r) \bigl( \bigl\vert x_{1}-x_{1}^{\prime} \bigr\vert + \bigl\vert y_{1}-y_{1}^{\prime} \bigr\vert + \bigl\vert w_{1}-w_{1}^{\prime} \bigr\vert + \vert u_{1}-v_{1} \vert + \vert u_{2}-v_{2} \vert \bigr) \end{aligned}$$
for all \(r\in I\) and \(x_{1},y_{1},w_{1},x_{1}^{\prime},y_{1}^{\prime },w_{1}^{\prime},u_{1}, u_{2},v_{1},v_{2} \in\mathbb{R}\). If \(\Delta= [\eta^{*}(2+\gamma _{0} + \lambda_{0} + \frac{1}{(1-{\delta})^{2}})+{ \mu}( \frac{1}{(1-{\gamma})^{2} }+\frac{1}{(1-{\beta})^{2}})]<1\), then problem (1) possesses an approximate solution.

Proof

Let \(H^{1}\) be equipped with \(d(u_{1}^{\prime},v_{1}^{\prime })= \Vert u_{1}^{\prime}-v_{1}^{\prime} \Vert \), where \(\Vert u_{1}^{\prime} \Vert =\sup_{t\in I} \vert u_{1}^{\prime}(t) \vert \). Now, consider the selfmap \(F:H^{1}\to H^{1}\) defined by
$$\begin{aligned} \bigl(Fu_{1}^{\prime} \bigr) (r) &=a_{ \alpha} \bigl[ \mu{ \bigl(} ^{\mathrm{CF}}\mathbb{C}^{(\beta,\beta)}_{\infty }u_{1}^{\prime}(r)+^{\mathrm{CF}} \mathbb{C}^{(\gamma,\gamma)}_{\infty}u_{1}^{\prime}(r) { \bigr)} \\ &\quad{} {}+f^{\prime} \bigl(r,u_{1}^{\prime}(r), \bigl(\phi u_{1}^{\prime} \bigr) (r), \bigl(\varphi u_{1}^{\prime} \bigr) (r), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} u_{1}^{\prime}(r),^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta )}_{\infty} u_{1}^{\prime}(r) \bigr) \bigr] \\ &\quad {}+b_{\alpha} \int_{0}^{r} \bigl[ \mu{ \bigl(} ^{\mathrm{CF}} \mathbb{C}^{(\beta,\beta)}_{\infty} u_{1}^{\prime}(s)+^{\mathrm{CF}} \mathbb{C}^{(\gamma,\gamma)}_{\infty}u_{1}^{\prime}(s) { \bigr)} \\ &\quad {}+ f^{\prime} \bigl(s,u_{1}^{\prime}(s), \bigl(\phi u_{1}^{\prime} \bigr) (s), \bigl(\varphi u_{1}^{\prime} \bigr) (s), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta )}_{\infty} u_{1}^{\prime}(r),^{\mathrm{CF}}\mathbb{C}^{(\delta ,\delta)}_{\infty} u_{1}^{\prime}(s) \bigr) \bigr]\,ds \end{aligned}$$
for all \(r\in I\) and \(u_{1}^{\prime},v_{1}^{\prime}\in H^{1}\), where \(a_{\alpha}\) and \(b_{\alpha}\) have the meaning given in Lemma 3.9. Now, utilizing Lemmas 3.5 and 3.8, we get
$$\begin{aligned} & \bigl\vert \bigl(Fu_{1}^{\prime} \bigr) (r) - \bigl(F v_{1}^{\prime} \bigr) (r) \bigr\vert \\ &\quad\leq a_{\alpha}\bigl( \mu \bigl\vert \bigl( ^{\mathrm{CF}} \mathbb{C}^{(\beta,\beta )}_{\infty}u_{1}^{\prime}(r)+ ^{\mathrm{CF}}\mathbb{C}^{(\gamma,\gamma)}_{\infty}u_{1}^{\prime}(r) \bigr)- \bigl( ^{\mathrm{CF}}\mathbb{C}^{(\beta,\beta )}_{\infty}v_{1}^{\prime}(r)+^{\mathrm{CF}} \mathbb{C}^{(\gamma,\gamma)}_{\infty}v_{1}^{\prime}(r) \bigr) \bigr\vert \\ &\quad\quad {}+ \bigl\vert f^{\prime} \bigl(r,u_{1}^{\prime}(r), \bigl(\phi u_{1}^{\prime} \bigr) (r), \bigl(\varphi u_{1}^{\prime} \bigr) (r), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} u_{1}^{\prime }(r),^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta)}_{\infty} u_{1}^{\prime}(r) \bigr) \\ &\quad\quad {}-f^{\prime} \bigl(r,v_{1}^{\prime}(t), \bigl( \phi v_{1}^{\prime} \bigr) (r), \bigl(\varphi v_{1}^{\prime} \bigr) (r), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} v_{1}^{\prime}(r), ^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta )}_{\infty} v_{1}^{\prime}(r) \bigr) \bigr\vert \bigr) \\ &\quad\quad {}+b_{\alpha} \int_{0}^{r} \bigl[ \mu \bigl\vert \bigl( ^{\mathrm{CF}}\mathbb{C}^{(\beta,\beta)}_{\infty}u_{1}^{\prime }(s)+^{\mathrm{CF}} \mathbb{C}^{(\gamma,\gamma)}_{\infty}u_{1}^{\prime}(s) \bigr)- \bigl( ^{\mathrm{CF}}\mathbb{C}^{(\beta,\beta)}_{\infty }v_{1}^{\prime}(s)+^{\mathrm{CF}} \mathbb{C}^{(\gamma,\gamma)}_{\infty}v_{1}^{\prime}(s) \bigr) \bigr\vert \\ &\quad\quad {}+ \bigl\vert f^{\prime} \bigl(s,u_{1}^{\prime}(r), \bigl( \phi u_{1}^{\prime} \bigr) (s), \bigl(\varphi u_{1}^{\prime} \bigr) (s), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} u_{1}^{\prime}(s),^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta )}_{\infty} u_{1}^{\prime}(s) \bigr) \\ &\quad\quad {}-f^{\prime} \bigl(s,v_{1}^{\prime}(s), \bigl( \phi v_{1}^{\prime} \bigr) (s), \bigl(\varphi v_{1}^{\prime} \bigr) (s), ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} v_{1}^{\prime}(s), ^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta )}_{\infty} v_{1}^{\prime}(s) \bigr) \bigr\vert \bigr] \,ds \\ &\quad \leq a_{ \alpha} \mu \bigl[ \bigl\vert ^{\mathrm{CF}} \mathbb{C}^{(\beta,\beta)}_{\infty} \bigl(u_{1}^{\prime}(r)-v_{1}^{\prime}(r) \bigr) \bigr\vert \\ &\quad\quad {}+ \bigl\vert ^{\mathrm{CF}}\mathbb{C}^{(\gamma,\gamma)}_{\infty} \bigl(u_{1}^{\prime}(r)-v_{1}^{\prime}(r) \bigr) \bigr\vert \bigr]+a_{ \alpha} \bigl\vert \eta(r) \bigr\vert \bigl[ \bigl\vert u_{1}^{\prime}(r)-v_{1}^{\prime}(r) \bigr\vert + \bigl\vert \bigl(\phi u_{1}^{\prime} \bigr) (r)- \bigl( \phi v_{1}^{\prime} \bigr) (r) \bigr\vert \\ &\quad\quad {}+ \bigl\vert \bigl(\varphi u_{1}^{\prime} \bigr) (r)- \bigl( \varphi v_{1}^{\prime} \bigr) (r) \bigr\vert + \bigl\vert ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} u_{1}^{\prime}(r)- ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} v_{1}^{\prime}(r) \bigr\vert \\ &\quad\quad{} + \bigl\vert ^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta)} _{\infty} u_{1}^{\prime}(r)-^{\mathrm{CF}} \mathbb{C}^{(\delta,\delta)}_{\infty} v_{1}^{\prime}(r) \bigr\vert \bigr] \\ &\quad\quad {}+b_{\alpha} \int_{0}^{r} \bigl[ \mu \bigl( \bigl\vert ^{\mathrm{CF}}\mathbb{C}^{(\beta,\beta)}_{\infty} \bigl(u_{1}^{\prime}(s)-v_{1}^{\prime}(s) \bigr) \bigr\vert + \bigl\vert ^{\mathrm{CF}}\mathbb{C}^{(\gamma ,\gamma)}_{\infty} \bigl(u_{1}^{\prime}(s)-v_{1}^{\prime}(s) \bigr) \bigr\vert \bigr) \\ &\quad\quad{} + \bigl\vert \eta(s) \bigr\vert \bigl( \bigl\vert u_{1}^{\prime}(s)-v_{1}^{\prime}(s) \bigr\vert \\ &\quad\quad {}+ \bigl\vert \bigl(\phi u_{1}^{\prime} \bigr) (s)- \bigl( \phi v_{1}^{\prime} \bigr) (s) \bigr\vert + \bigl\vert \bigl( \varphi u_{1}^{\prime} \bigr) (s)- \bigl(\varphi v_{1}^{\prime} \bigr) (s) \bigr\vert + \bigl\vert ^{\mathrm{CF}}\mathbb{I}^{(\theta,\theta)}_{\infty} u_{1}^{\prime}(s) - ^{\mathrm{CF}}I^{(\theta,\theta)}_{\infty} v_{1}^{\prime}(s) \bigr\vert \\ &\quad\quad {}+ \bigl\vert ^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta)}_{\infty} u_{1}^{\prime}(s)-^{\mathrm{CF}}\mathbb{C}^{(\delta,\delta )}_{\infty} v_{1}^{\prime}(s) \bigr\vert \bigr) \bigr] \,ds \\ &\quad \leq \biggl[\eta^{*} \biggl(2+\gamma_{0} + \lambda _{0} + \frac{1}{(1-{\delta})^{2}} \biggr)+{ \mu}\biggl( \frac{1}{(1-{\gamma})^{2}}+ \frac{1}{(1-{\beta})^{2}}\biggr) \biggr] [a_{\alpha}+b_{\alpha}] \bigl\Vert u_{1}^{\prime}-v_{1}^{\prime} \bigr\Vert \end{aligned}$$
for all \(r\in I\) and \(u_{1}^{\prime},v_{1}^{\prime}\in H^{1}\). Hence,
$$ \bigl\Vert Fu_{1}^{\prime}-Fv_{1}^{\prime} \bigr\Vert \leq \biggl[\eta^{*} \biggl(2+\gamma_{0} + \lambda_{0} + \frac{1}{(1-{\delta})^{2}} \biggr)+{ \mu} \biggl( \frac{1}{(1-{\gamma})^{2} }+\frac{1}{(1-{\beta})^{2}} \biggr) \biggr] \bigl\Vert u_{1}^{\prime}-v_{1}^{\prime} \bigr\Vert $$
for all \(u_{1}^{\prime},v_{1}^{\prime}\in H^{1}\). Consider the mappings \(j:[0,\infty)^{5} \to[0,\infty) \) and \(\alpha :H^{1}\times H^{1}\to[0,\infty)\) defined by \(j(t_{1},t_{2},t_{3},t_{4},t_{5})= \Delta t_{1}\) and \(\alpha(t,s) =1\) for all \(t,s\in H^{1}\). We can check that \(j \in\mathcal{R}\) and F is a generalized α-contraction. From Theorem 2.2 we conclude that F possesses an approximate fixed point, which is an approximate solution for problem (1). □
Let c be a real number, and k, s, and q bounded functions on \(I=[0,1]\) with \(M_{1}=\sup_{p\in I} \vert k(p) \vert <\infty\), \(M_{2}=\sup_{p\in I} \vert s(p) \vert <\infty\), and \(M_{3}=\sup_{t\in I} \vert q(p) \vert <\infty\). We investigate the infinite coefficient-symmetric CF fractional integro-differential problem
$$ \begin{aligned}[b] ^{\mathrm{CF}}\mathbb{C}_{(\mu,\infty)}^{(\alpha ,\alpha)}x(p)&= \lambda k(p)^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\delta,\delta )}x(p) +{\rho }s(p)^{\mathrm{CF}}\mathbb{I}_{\infty}^{(\theta,\theta)}x(p) \\ &\quad{}+ \int_{0}^{p} f \bigl(w,x(w),(\varphi x) (w),q(w)^{\mathrm{CF}}\mathbb{C}_{(m,\infty)}^{(\gamma, \gamma )}x(w) \bigr)\,dw \end{aligned} $$
(2)
with \(x(0)=c\), where \(\lambda,\rho\geq0\) and \(\alpha, \gamma ,\delta,\theta\in(0,1)\).

Theorem 3.12

Let \(\xi_{1},\xi_{2},\xi_{3}\geq0\), and let \(f:[0,1]\times\mathbb {R}^{3}\rightarrow\mathbb{R}\) be a bounded integrable function satisfying \(\vert f(p,x_{1},y_{1},w_{1})-f(p,x_{1}^{\prime },y_{1}^{\prime},w_{1}^{\prime}) \vert \leq \xi_{1} \vert x_{1} -x_{1}^{\prime} \vert +\xi_{2} \vert y_{1} -y_{1}^{\prime} \vert +\xi_{3} \vert w_{1}-w_{1}^{\prime} \vert \) for all \(p \in I\) and \(x_{1},y_{1},w_{1},v_{1},x_{1}^{\prime },y_{1}^{\prime},w_{1}^{\prime}\in\mathbb{R}\). If \(\Delta= \vert 2-\mu \vert [\lambda\frac{M_{1}}{(1-\delta )^{2}}+\rho M_{2}+ \xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac {M_{3}}{(1-\gamma)^{2} \vert 2-m \vert }]<1\), then problem (2) admits an approximate solution.

Proof

Let \(H^{1}\) be equipped with \(d(x ,y )= \Vert x -y \Vert \), where \(\Vert x \Vert =\sup_{t\in I} \vert x(t) \vert \). Now, consider the selfmap \(\mathcal{F}:H^{1}\to H^{1}\) defined by
$$\begin{aligned} (\mathcal{F}x) (p)&=x(0)+(2-\mu)a_{\alpha} \biggl[ (\lambda k(p)^{\mathrm{CF}} \mathbb{C}_{\infty}^{(\delta,\delta)}x(p)+{ \rho}s(p)^{\mathrm{CF}} \mathbb{I}_{\infty}^{(\theta,\theta)}x(p) \\ &\quad {}+ \int_{0}^{p} f \bigl(w,x(w),(\varphi x) (w),q(w)^{\mathrm{CF}}\mathbb{C}_{(m,\infty)}^{ (\gamma, \gamma )}x(w) \bigr)\,dw \biggr] \\ &\quad{}+ b_{\alpha}(2-\mu) \int_{0}^{p} \biggl[ \lambda k(w)^{\mathrm{CF}} \mathbb{C}_{\infty }^{(\delta,\delta)}x(w)+{\rho}s(w)^{\mathrm{CF}} \mathbb{I}_{\infty }^{(\theta,\theta)}x(w)\,dw \\ &\quad {}+ \int_{0}^{w}f \bigl(r,x(r),(\varphi x) (r),q(r)^{\mathrm{CF}}\mathbb{C}_{(m,\infty)}^{ (\gamma, \gamma )}x(r) \bigr)\,dr \biggr] \,dw \end{aligned}$$
for all \(p\in I\) and \(x,y\in H^{1}\), where \(a_{\alpha}\) and \(b_{\alpha}\) are given in Lemma 3.10. As a result, utilizing Lemmas 3.5, 3.7, and 3.8, we get
$$\begin{aligned} & \biggl\vert \biggl[\lambda k(p)^{\mathrm{CF}}\mathbb{C}_{\infty }^{(\delta,\delta)}x(p)+{ \rho}s(p)^{\mathrm{CF}}\mathbb{I}_{\infty}^{(\theta,\theta)}x(p)+ \int_{0}^{p} f \bigl(w,x(w),(\varphi x) (w), q(w)^{\mathrm{CF}}\mathbb{C}_{(m,\infty)}^{(\gamma, \gamma)}x(w) \bigr)\,dw \biggr] \\ &\quad\quad{} - \biggl[\lambda k(p)^{\mathrm{CF}}\mathbb{C}_{\infty }^{(\delta,\delta)}y(p)+{ \rho} s(w)^{\mathrm{CF}}\mathbb{I}_{\infty}^{(\theta,\theta)}y(p) \\ &\quad\quad{} + \int _{0}^{p} f \bigl(w,y(w),(\varphi y) (w),q(w)^{\mathrm{CF}}\mathbb {C}_{(m,\infty)}^{(\gamma, \gamma)}y(w) \bigr)\,dw \biggr] \biggr\vert \\ &\quad \leq \biggl[\lambda\frac{M_{1}}{(1-\delta)^{2}}+\rho M_{2} \biggr] \Vert x-y \Vert + \xi_{1} \Vert x-y \Vert + \xi_{2} \gamma _{0} \Vert x-y \Vert +\xi_{3} \frac{M_{3}}{(1-\gamma)^{2} \vert 2-m \vert } \Vert x-y \Vert \\ &\quad\leq \biggl[\lambda\frac{M_{1}}{(1-\delta)^{2}}+\rho M_{2}+ \xi _{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac{M_{3}}{(1-\gamma)^{2} \vert 2-m \vert } \biggr] \Vert x-y \Vert \end{aligned}$$
for all \(p\in I\) and \(x,y\in H^{1}\). As a result, we get
$$\begin{aligned} & \bigl\vert (\mathcal{F}x) (p)-(\mathcal{F}x) (p) \bigr\vert \\ &\quad \leq a_{\alpha} \vert 2-\mu \vert \biggl[\lambda\frac{M_{1}}{(1-\delta )^{2}}+\rho M_{2}+ \xi_{1} + \xi_{2} \gamma_{0} + \xi_{3} \frac{M_{3}}{(1-\gamma)^{2} \vert 2-m \vert } \biggr] \Vert x-y \Vert \\ &\quad \quad{}+b_{\alpha} \vert 2-\mu \vert \int _{0}^{p} \biggl[\lambda\frac{M_{1}}{(1-\delta)^{2}}+\rho M_{2}+ \xi_{1} + \xi_{2} \gamma_{0} + \xi_{3} \frac{M_{3}}{(1-\gamma)^{2} \vert 2-m \vert } \biggr] \Vert x-y \Vert \,ds \\ &\quad \leq \vert 2-\mu \vert \biggl[\lambda\frac{M_{1}}{(1-\delta )^{2}}+\rho M_{2}+ \xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac{M_{3}}{(1-\gamma)^{2} \vert 2-m \vert } \biggr] \Vert x-y \Vert \end{aligned}$$
for all \(p \in I\) and \(x,y \in H^{1}\). Now we consider the mappings \(j:[0,\infty)^{5} \to[0,\infty)\) and \(\alpha:H^{1}\times H^{1}\to[0,\infty)\) defined by \(\alpha(t,s) = 1\) and \(j(t_{1}, t_{2},t_{3},t_{4},t_{5})=\frac{\Delta}{3}(t_{1} +2t_{2})\). We can check that \(j \in\mathcal{R}\) and \(\mathcal{F}\) is a generalized α-contraction. With the help of Theorem 2.2, we conclude that \(\mathcal{F}\) possesses an approximate fixed point, which represents an approximate solution for the investigated problem (2). □

The next step is to study two applications to describe the reported results.

Example 1

Let us define \(\eta\in L^{\infty}([0,1])\) and \(\gamma ,\lambda:[0,1]\times[0,1]\to[0,\infty)\) by \(\eta(p)=\frac{\pi}{e^{(p+12)}}\), \(\gamma(p,s)=e^{ p-s}\) and \(\lambda(p,s)= \ln(5^{\sin(\pi p -s)})\). Then, we have \(\eta^{*}=\frac{\pi}{e^{12}}\), \(\gamma_{0}\leq e\), and \(\lambda_{0}\leq\ln5\). Let us consider \(\alpha=\frac{1}{5}\), \(\mu= \frac{1}{20}\), \(\beta =\frac{1}{4}\), \(\gamma=\frac{1}{2}\), \(\theta=\frac{3}{4}\), and \(\delta=\frac{3}{5}\). Consider the problem
$$ \begin{aligned}[b] ^{\mathrm{CF}}\mathbb{C}^{(\frac{1}{5},\frac {1}{5})}_{\infty}u_{1}^{\prime}(p)&= \frac{1}{20} { \bigl(}^{\mathrm{CF}}\mathbb{C}_{\infty} ^{(\frac{1}{4},\frac{1}{4})} u_{1}^{\prime}(p)+ ^{\mathrm{CF}} \mathbb{C}_{\infty} ^{(\frac{1}{2},\frac{1}{2})}u_{1}^{\prime}(p) { \bigr)} \\ &\quad{} +e^{-\pi(t+12)} \biggl[p+u_{1}^{\prime}(p)+ \int_{0}^{p} e^{ p-s}u_{1}^{\prime}(s) \,ds \\ &\quad{}+ \int_{0}^{p} \ln \bigl(5^{\sin(\pi p -s)} \bigr)u_{1}^{\prime}(s)\,ds+^{\mathrm{CF}}\mathbb{I}_{\infty}^{(\frac {3}{4},\frac{3}{4})} u_{1}^{\prime}(p)+^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\frac{3}{5}, \frac{3}{5})}u_{1}^{\prime}(p) \biggr] \end{aligned} $$
(3)
with \(u_{1}^{\prime}(0)=0\). Considering \(f(p,x,y,w,u_{1},u_{2})= e^{-\pi(p+12)}(p+x+y+w+u_{1}+u_{2})\), we note that \(\Delta= [\eta^{*}(2+\gamma_{0} + \lambda_{0} + \frac{1}{(1-{\delta })^{2}})+{ \mu}( \frac{1}{(1-{\gamma})^{2}(1-{\beta})^{2}} )]<0/4447<1\). Now, by Theorem 3.11 problem (3) admits an approximate solution.

Example 2

Consider the function \(\lambda:[0,1] \times[0,1]\to[0,\infty)\) by \(\lambda(p,s )=\frac{e^{2p-s}}{e}\). Thus, \({\lambda_{0}\leq e}\). Let us consider \(\mu=3\), \(m=\frac{1}{2}\), \(\alpha=\frac{1}{4}\), \(\delta=\frac{1}{4}\), \(\theta=\frac{1}{2}\), \(\gamma=\frac{1}{2}\), \(\lambda= \frac{1}{200}\), \(\rho=\frac{1}{122}\), \(\xi_{1}=\frac{1}{320}\), \(\xi_{2}=\frac{1}{40}\), and \(\xi_{3}=\frac {1}{119}\). Let \(k(t)=\frac{2-p}{p+1}\), \(s(p)=\sin p\) and \(q(p)=\tan^{-1}(p)\). Then, \(M_{1}=\sup_{p\in[0,1]} \vert k(p) \vert =2\), \(M_{2}=\sup_{t\in[0,1]} \vert s(p) \vert =1\), and \(M_{3}=\sup_{t\in[0,1]} \vert q(p) \vert =\frac{\pi}{2} \). As a next step, we consider the problem
$$ \begin{aligned}[b] ^{\mathrm{CF}}\mathbb{C}_{(\mu,\infty)}^{(\frac {1}{4},\frac{1}{4})}x(p) &= \frac{1}{200}k(p)^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\frac{1}{4}, \frac{1}{4})}x(p)+ \frac{1}{122}s(p)^{\mathrm{CF}}\mathbb{I}_{\infty}^{(\frac {1}{2},\frac{1}{2})}x(p) \\ &\quad{}+ \int_{0}^{p} \biggl[\frac{2}{56}s+ \frac{1}{320}x(s)+ \frac{1}{40} \int_{0}^{s}\frac{e^{2s-r}}{e}x(r) \,dr \\ &\quad{} + \frac{1}{119}\tan^{-1}(s) ^{\mathrm{CF}}\mathbb{C}_{(m,\infty )}^{(\frac{1}{2},\frac{1}{2})}x(s) \biggr]\,ds \end{aligned} $$
(4)
with \(x(0)=0\). Considering \(f(p,x_{1},y_{1},w_{1})= \frac{2}{56}p+\xi _{1}x_{1}+\xi_{2}y_{1}+\xi_{3}w_{1}\) for all \(p\in I\) and \(x_{1},y_{1},w_{1},v\in\mathbb{R}\), we note that
$$ \Delta= \vert 2-\mu \vert \biggl[\lambda\frac{M_{1}}{(1-\delta )^{2}}+\rho M_{2}+ \xi_{1} + \xi_{2} \gamma_{0} + \xi_{3} \frac{M_{3}}{(1-\gamma)^{2} \vert 2-m \vert } \biggr]< 0.111< 1. $$
Now, by Theorem 3.12, problem (4) admits an approximate solution.

4 Conclusion

Fractional derivatives with nonsingular kernels started to be utilized from both theoretical and applied viewpoints. Particularly, the fractional Caputo-Fabrizio derivative was applied to models possessing memory effect of exponential type. Therefore, new generalizations of this operator should be investigated and applied to the dynamics of real-world problems. In this manuscript, we suggested a new operator called the infinite coefficient-symmetric CF fractional derivative. Besides, its properties were investigated, and two examples clearly show the advantages of the newly introduced concept.

Declarations

Acknowledgements

The second and third authors were supported by Azarbaijan Shahid Madani University.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Cankaya University
(2)
Institute of Space Sciences
(3)
Department of Mathematics, Azarbaijan Shahid Madani University

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