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

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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.

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.

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
figure1

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
figure2

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
figure3

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

$$ (*)\quad\quad {}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{(\mu,\infty)} = \frac{1}{2-\mu} {}^{\mathrm{CF}}C^{\alpha}. $$

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.

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.

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Acknowledgements

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

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Correspondence to Dumitru Baleanu.

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MSC

  • 30B70
  • 34A08

Keywords

  • continued fraction
  • contraction map
  • fractional integro-differential equation
  • infinite Caputo-Fabrizio fractional derivative