# Ill-posedness for the two component Degasperis-Procesi equation in critical
Besov space

Jinlu Li1, Min Li2,111E-mail<EMAIL_ADDRESS>limin@jxufe.edu.cn(Corresponding author<EMAIL_ADDRESS>and Weipeng Zhu3
1 School of Mathematics and Computer Sciences, Gannan Normal University,
Ganzhou 341000, China
2 Department of mathematics, Jiangxi University Of Finance and Economics,
Nanchang 330032, China
3 School of Mathematics and Big Data, Foshan University, Foshan, Guangdong
528000, China

Abstract: In this paper, we study the Cauchy problem for the two component
Degasperis-Procesi equation in critical Besov space
$B^{1}_{\infty,1}(\mathbb{R})$. By presenting a new construction of initial
data, we proved the norm inflation of the corresponding solutions in
$B^{1}_{\infty,1}(\mathbb{R})$ and hence ill-posedness. This is quite
different from the local well-posedness result [27] for the Degasperis–Procesi
equation in critical Besov space $B^{1}_{\infty,1}(\mathbb{R})$ due to the
coupled structure of density function.

Keywords: two component Degasperis-Procesi equation; Ill-posedness; Critical
Besov space.

MSC (2010): 35Q53, 37K10.

##  1 Introduction

In this paper, we consider the Cauchy problem for the following two component
Degasperis-Procesi equation [2, 24]

$\displaystyle\begin{cases}\partial_{t}m+3mu_{x}+m_{x}u+k_{3}\rho\rho_{x}=0,&\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R},\\\
\partial_{t}\rho+k_{2}u\rho_{x}+(k_{1}+k_{2})u_{x}\rho=0,&\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R},\\\
m=u-u_{xx},&\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R},\\\
u(0,x)=u_{0}(x),\quad\rho(0,x)=\rho_{0}(x),&\quad x\in\mathbb{R},\end{cases}$
(1.1)

here $(k_{1},k_{2},k_{3})=(1,1,c)$ or $(c,1,0)$ and $c$ is an arbitrary
constant. System (1.1) first appeared in [2] as the Hamiltonian extension of
the Degasperis–Procesi equation. It is worth mentioning that, in the case
$(k_{1},k_{2},k_{3})=(c,1,0),$ system (1.1) is no more coupled and the
equation on $\rho$ becomes linear. Therefore, we only consider the first case
$(k_{1},k_{2},k_{3})=(1,1,c).$ For simplicity, we might as well assume that
$c=1$, which makes system (1.1) equivalent to the following quasi-linear
evolution equation of hyperbolic type:

$\displaystyle\begin{cases}\partial_{t}u+uu_{x}=-\partial_{x}(1-\partial^{2}_{x})^{-1}(\frac{3}{2}u^{2}+\frac{1}{2}\rho^{2}),&\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R},\\\
\partial_{t}\rho+u\rho_{x}=-2u_{x}\rho,&\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R},\\\
u(0,x)=u_{0}(x),\quad\rho(0,x)=\rho_{0}(x),&\quad x\in\mathbb{R}.\end{cases}$
(1.2)

In particular, for $\rho_{0}\equiv 0,$ system (1.2) reduces to the
Degasperis–Procesi (DP) equation[22]. The DP equation can be regarded as a
model for nonlinear shallow water dynamics and its asymptotic accuracy is the
same as for the well known Camassa-Holm equation[3]. The DP equation was also
proved formally integrable by constructing a Lax pair [13] and the direct and
inverse scattering approach to pursue it can be seen in [23]. Moreover, they
also presented [13] that the DP equation has a bi-Hamiltonian structure and an
infinite number of conservation laws, and admits travelling wave and exact
peakon solutions which are analogous to the Camassa–Holm peakons[4, 6]. The
Cauchy problem of the DP equation is locally well-posed in certain Sobolev and
Besov spaces [5, 7, 8], also the non-uniform dependence and ill-posedness
problem have recently been studied in [31, 28]. It has global strong solutions
[16, 8], the finite-time blow-up solutions [10, 11] and global weak solutions
[12, 10, 9]. Different form the Camassa-Holm equation, the DP equation has not
only peakon solutions [13], periodic peakon solutions [9], but also shock
peakons[15] and the periodic shock waves [11]. Both Camassa-Holm and DP
equation admit cusped travelling waves [17, 4]. This type of weak solutions
are known to arise as solutions for the governing equations for waves in a
channel or along a sloped beach [18, 19], and also as solutions for the
governing equations for equatorial ocean waves [20, 21].

Recently, a lot of literature was devoted to studying the well-posedness
(especially the ill-posedness) problem of the Camassa-Holm type equations (CH,
DP and Novikov equation etc.) in the critical Besov spaces. For example, Guo
et al. [14] prove norm inflation and hence ill-posedness for the Camassa-Holm
type equations in the critical Sobolev space $H^{\frac{3}{2}}$ and even in the
Besov space $B^{1+\frac{1}{p}}_{p,r}(\mathbb{R})$ with
$p\in[1,\infty],r\in(1,\infty],$ which implies
$B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ is the critical Besov space for these
Camassa-Holm type equations. Li et al. [29, 30] demonstrated the non-
continuity and then sharp ill-posedness of the Camassa-Holm type equations in
$B^{\sigma}_{p,\infty}(\mathbb{R})$ with
$\sigma>\max\\{\frac{3}{2},1+\frac{1}{p}\\}$. Later, it has been proved the
local well-posedness for the Cauchy problem of the Camassa-Holm type equations
in $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $p\in[1,\infty)$ by adopt the
compactness argument and Lagrangian coordinate transformation[25]. In the
remaining case $p=\infty$, Guo et al. [26] prove the ill-posedness for the
Camassa-Holm equation in $B^{1}_{\infty,1}.$

Different from the CH equation, the authors in [27] have proved the well-
posedness for the Cauchy problem of the DP equation in critical Besov space
$B^{1}_{\infty,1}.$ For the two component DP equation(1.2), the local well-
posedness problem in certain Sobolev and Besov spaces has been studied by Yan
et al.in [24], here we mainly focus on the well-posedness problem in critical
Besov space $B^{1}_{\infty,1}(\mathbb{R})$. As the structure is changed from
the DP equation due to the coupled density function $\rho$, here we have a
different result for the two component DP equation (1.2) in this paper. Now,
let us state our main theorem of this paper.

######  Theorem 1.1.

For any $n\in\mathbb{Z}^{+}$ large enough there exists $(u_{0},\rho_{0})$ with

$\|u_{0}\|_{B^{1}_{\infty,1}}+\|\rho_{0}\|_{B^{0}_{\infty,1}}\leq\frac{1}{\log\log
n},$

such that if we denote by $(u,\rho)\in\mathcal{C}([0,1];H^{3}\times H^{2})$,
the solution of the two component Degasperis-Procesi with initial data
$(u_{0},\rho_{0})$, then

$\|u(t_{0})\|_{B^{1}_{\infty,1}}+\|\rho(t_{0})\|_{B^{0}_{\infty,1}}\geq{\log\log
n},$

with $t_{0}\in(0,\frac{1}{\log n}]$.

######  Remark 1.1.

This implies the ill-posedness of the two component DP equation (1.2) in Besov
space $B^{1}_{\infty,1}(\mathbb{R})$. In fact, by Theorem 1.1 we have
construct the norm inflation in space $B^{1}_{\infty,1}(\mathbb{R})$, thus the
data-to-solution map is not continuous at origin in this space, in the sense
of Hadamard this implies the ill-posedness.

##  2 Littlewood-Paley analysis

Next, we will recall some facts about the Littlewood-Paley decomposition, the
nonhomogeneous Besov spaces and their some useful properties (see [1] for more
details).

Let $\mathcal{B}:=\\{\xi\in\mathbb{R}:|\xi|\leq 4/3\\}$ and
$\mathcal{C}:=\\{\xi\in\mathbb{R}:3/4\leq|\xi|\leq 8/3\\}.$ Choose a radial,
non-negative, smooth function $\chi:\mathbb{R}\mapsto[0,1]$ such that it is
supported in $\mathcal{B}$ and $\chi\equiv 1$ for $|\xi|\leq 3/4$. Setting
$\varphi(\xi):=\chi(\xi/2)-\chi(\xi)$, then we deduce that $\varphi$ is
supported in $\mathcal{C}$. Moreover,

$\displaystyle\chi(\xi)+\sum_{j\geq 0}\varphi(2^{-j}\xi)=1\quad\mbox{ for any
}\xi\in\mathbb{R}.$

We should emphasize that the fact $\varphi(\xi)\equiv 1$ for $4/3\leq|\xi|\leq
3/2$ will be used in the sequel.

For every $u\in\mathcal{S^{\prime}}(\mathbb{R})$, the inhomogeneous dyadic
blocks ${\Delta}_{j}$ are defined as follows

${\Delta_{j}u=}\begin{cases}0,&if\quad j\leq-2;\\\
\chi(D)u=\mathcal{F}^{-1}(\chi\mathcal{F}u),&if\quad j=-1;\\\
\varphi(2^{-j}D)u=\mathcal{F}^{-1}\big{(}\varphi(2^{-j}\cdot)\mathcal{F}u\big{)},&if\quad
j\geq 0.\end{cases}$

In the inhomogeneous case, the following Littlewood-Paley decomposition makes
sense

$u=\sum_{j\geq-1}{\Delta}_{j}u\quad\text{for
any}\;u\in\mathcal{S^{\prime}}(\mathbb{R}).$

######  Definition 2.1 (see [1]).

Let $s\in\mathbb{R}$ and $(p,r)\in[1,\infty]^{2}$. The nonhomogeneous Besov
space $B^{s}_{p,r}(\mathbb{R})$ is defined by

$\displaystyle
B^{s}_{p,r}(\mathbb{R}):=\Big{\\{}f\in\mathcal{S}^{\prime}(\mathbb{R}):\;\|f\|_{B^{s}_{p,r}(\mathbb{R})}<\infty\Big{\\}},$

where

$\displaystyle\|f\|_{B^{s}_{p,r}(\mathbb{R})}=$
$\displaystyle\left(\sum_{j\geq-1}2^{sjr}\|\Delta_{j}f\|^{r}_{L^{p}(\mathbb{R})}\right)^{\frac{1}{r}},$
if $1\leq r<\infty$, $\displaystyle\|f\|_{B^{s}_{p,r}(\mathbb{R})}=$
$\displaystyle\sup_{j\geq-1}2^{sj}\|\Delta_{j}f\|_{L^{p}(\mathbb{R})},$ if
$r=\infty$.

The following Bernstein’s inequalities will be used in the sequel.

######  Lemma 2.1 (see Lemma 2.1 in [1]).

Let $\mathcal{B}$ be a Ball and $\mathcal{C}$ be an annulus. There exist
constants $C>0$ such that for all $k\in\mathbb{N}\cup\\{0\\}$, any positive
real number $\lambda$ and any function $f\in L^{p}$ with $1\leq p\leq
q\leq\infty$, we have

$\displaystyle{\rm{supp}}\hat{f}\subset\lambda\mathcal{B}\;\Rightarrow\;\|\partial_{x}^{k}f\|_{L^{q}}\leq
C^{k+1}\lambda^{k+(\frac{1}{p}-\frac{1}{q})}\|f\|_{L^{p}},$
$\displaystyle{\rm{supp}}\hat{f}\subset\lambda\mathcal{C}\;\Rightarrow\;C^{-k-1}\lambda^{k}\|f\|_{L^{p}}\leq\|\partial_{x}^{k}f\|_{L^{p}}\leq
C^{k+1}\lambda^{k}\|f\|_{L^{p}}.$

######  Lemma 2.2 (see Lemma 2.100 in [1]).

Let $1\leq r\leq\infty$, $1\leq p\leq p_{1}\leq\infty$ and
$\frac{1}{p_{2}}=\frac{1}{p}-\frac{1}{p_{1}}$. There exists a constant $C$
depending continuously on $p,p_{1}$, such that

$\left\|\left(2^{j}\left\|[\Delta_{j},v\partial_{x}]f\right\|_{L^{p}}\right)_{j}\right\|_{\ell^{r}}\leq
C\left(\|\partial_{x}v\|_{L^{\infty}}\|f\|_{B_{p,r}^{1}}+\|\partial_{x}f\|_{L^{p_{2}}}\|\partial_{x}v\|_{B_{p_{1},r}^{0}}\right).$

##  3 Proof of Theorem 1.1

For localization, we first introduce the following bump function in the
frequency space. let $\widehat{\theta}\in\mathcal{C}^{\infty}_{0}(\mathbb{R})$
be a smooth and even function with values in $[0,1]$ which satisfies

$\displaystyle\widehat{\theta}(\xi)=\begin{cases}1,\quad\mathrm{if}\
|\xi|\leq\frac{1}{200},\\\ 0,\quad\mathrm{if}\
|\xi|\geq\frac{1}{100}.\end{cases}$

Then, shift the frequency of the above function by letting
$\phi(x)=\theta(x)\sin(\frac{17}{24}x)$. For the convenience of notation,
hereafter

$\displaystyle n\in
16\mathbb{N}^{+}=\\{16,32,48,\cdots\\},~{}~{}~{}~{}\mathbb{N}(n)=\\{k\in\mathbb{N}^{+}:\frac{n}{4}\leq
k\leq\frac{n}{2}\\}.$

and define the norm $B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)$ by

$\displaystyle\left\|u\right\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}=\sum_{j\in\mathbb{N}(n)}\left\|\Delta_{j}u\right\|_{L^{\infty}}.$

Now, we construct the initial data $(u_{0},\rho_{0})$ as following:

$\displaystyle u_{0}(x)=0,\qquad\rho_{0}(x)$
$\displaystyle=n^{-\frac{1}{2}}\log
n\cdot\sin(\frac{17}{12}2^{n}x)\cdot\sum_{\ell\in\mathbb{N}(n)}\phi(2^{\ell}(x-2^{2n+\ell})).$

###  3.1 Estimation of initial data

First of all, we give the estimates for $\rho_{0}$.

######  Lemma 3.1.

There exists a positive constant $C$ independent of $n$ such that

$\displaystyle\|\rho_{0}\|_{L^{\infty}}\leq Cn^{-\frac{1}{2}}\log n,$
$\displaystyle\|\rho_{0}\|_{B^{0}_{\infty,1}}\leq
C\|\rho_{0}\|_{L^{\infty}}\leq Cn^{-\frac{1}{2}}\log n.$

###### Proof.

By the definition of $\rho_{0}$ and using convolution expressions, it’s easy
to verify that the support set of $\rho_{0}$ satisfies

$\displaystyle\mathrm{supp}\
\widehat{\rho_{0}}\subset\left\\{\xi\in\mathbb{R}:\
\frac{17}{12}2^{n}-\frac{17}{12}2^{\ell}-\frac{1}{2}\leq|\xi|\leq\frac{17}{12}2^{n}+\frac{17}{12}2^{\ell}+\frac{1}{2}\right\\}.$
(3.3)

Since $\phi$ and $\check{\chi}$ are Schwartz functions, we have

$\displaystyle|\phi(x)|+|\check{\chi}(x)|\leq C(1+|x|)^{-M},\qquad M\geq 100.$
(3.4)

then, by a direct computation, we have

$\displaystyle\|\rho_{0}\|_{L^{\infty}}$ $\displaystyle\leq
Cn^{-\frac{1}{2}}\log
n\left\|\sum_{\ell\in\mathbb{N}(n)}\phi(2^{\ell}(x-2^{2n+\ell}))\right\|_{L^{\infty}}$
$\displaystyle\leq Cn^{-\frac{1}{2}}\log
n\left\|\sum_{\ell\in\mathbb{N}(n)}\frac{1}{(1+|2^{\ell}x-2^{2n+2\ell}|)^{M}}\right\|_{L^{\infty}}\leq
Cn^{-\frac{1}{2}}\log n.$

Due to the fact of (3.3) and the definition of $\Delta_{j}$, it’s easy to
check that $\Delta_{j}\rho_{0}=\rho_{0}$ if $j=n$ and $\Delta_{j}u_{0}=0$ if
$j\neq n$, we deduce that

$\displaystyle\|\rho_{0}\|_{B^{0}_{\infty,1}}$ $\displaystyle\leq
C\|\rho_{0}\|_{L^{\infty}}\leq Cn^{-\frac{1}{2}}\log n.$

This completes the proof of Lemma 3.1. ∎

The following lower bound estimate of the squared term is crucial to our
proof.

######  Lemma 3.2.

There exists a positive constant $c$ independent of $n$ such that

$\displaystyle\left\|(\rho_{0})^{2}\right\|_{B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)}\geq
c\log^{2}n,\qquad n\gg 1.$

###### Proof.

It follows that for any $j\in\mathbb{N}(n)$,

$\displaystyle\Delta_{j}[(\rho_{0})^{2}]=\frac{1}{2}n^{-1}\log^{2}n\cdot\Delta_{j}U,\qquad
U:=\left(\sum_{\ell\in\mathbb{N}(n)}\phi(2^{\ell}(x-2^{2n+\ell}))\right)^{2}.$
(3.5)

We rewrite $U$ as

$\displaystyle
U=\sum_{\ell\in\mathbb{N}(n)}\phi^{2}(2^{\ell}(x-2^{2n+\ell}))+\sum_{\ell,m\in\mathbb{N}(n)\setminus\\{\ell=m\\}}\phi(2^{\ell}(x-2^{2n+\ell}))\phi(2^{m}(x-2^{2n+m})):=U_{1}+U_{2}.$

Then, we have $\Delta_{j}U=\Delta_{j}U_{1}+\Delta_{j}U_{2}$. For
$\Delta_{j}U_{2}$, we have

$\displaystyle\begin{split}||\Delta_{j}U_{2}||_{L^{\infty}}&\leq
C||U_{2}||_{L^{\infty}}\\\ &\leq
C\sum_{\ell,m\in\mathbb{N}(n)\setminus\\{\ell=m\\}}||\phi(2^{\ell}(x-2^{2n+\ell}))\phi(2^{m}(x-2^{2n+m}))||_{L^{\infty}}\\\
&\leq
C\sum_{\ell,m\in\mathbb{N}(n)\setminus\\{\ell=m\\}}||\frac{1}{(1+2^{\ell}|x-2^{2n+\ell}|)^{M}}\cdot\frac{1}{(1+2^{m}|x-2^{2n+m}|)^{M}}||_{L^{\infty}}\\\
&\leq C\sum_{\ell,m\in\mathbb{N}(n)\setminus\\{\ell=m\\}}2^{-2nM}\leq
Cn^{2}2^{-2nM}.\end{split}$ (3.6)

By direct computations, for large enough $n$, we have

$\displaystyle\Delta_{j}U_{1}=\Delta_{j}\phi^{2}(2^{j}(x-2^{2n+j}))+\Delta_{j}\sum_{j\neq\ell\in\mathbb{N}(n)}\phi^{2}(2^{\ell}(x-2^{2n+\ell}))=:\Delta_{j}U_{1,1}+\Delta_{j}U_{1,2}.$

Let us introduce the set $\mathbf{B}_{j}$ defined by
$\mathbf{B}_{j}=\\{x:2^{j}|x-2^{2n+j}|\leq 1\\}$.

We can show that

$\displaystyle||\Delta_{j}U_{1}||_{L^{\infty}(\mathbf{B}_{j})}\geq||\Delta_{j}U_{1,1}||_{L^{\infty}(\mathbf{B}_{j})}-||\Delta_{j}U_{1,2}||_{L^{\infty}(\mathbf{B}_{j})}$

Notice that

$\displaystyle\phi^{2}(x)=\frac{1}{2}\theta^{2}(x)-\frac{1}{2}\theta^{2}(x)\cos(\frac{17}{12}x):=\Phi_{1}(x)+\Phi_{2}(x),$

then we have

$\displaystyle\Delta_{j}U_{1,1}=\Delta_{j}\Phi_{2}(2^{j}(x-2^{2n+j}))=\Phi_{2}(2^{j}(x-2^{2n+j})).$

This implies

$\displaystyle||\Delta_{j}U_{1,1}||_{L^{\infty}(\mathbf{B}_{j})}\geq\frac{1}{2}\theta^{2}(0):=c.$
(3.7)

By (3.4), we have

$\displaystyle\|\Delta_{j}U_{1,2}\|_{L^{\infty}(\mathbf{B}_{j})}$
$\displaystyle\leq
C\sum_{j\neq\ell\in\mathbb{N}(n)}\left\|2^{j}\int_{\mathbb{R}}\check{\chi}(2^{j}(x-y))\phi^{2}(2^{\ell}(y-2^{2n+\ell}))\mathrm{d}y\right\|_{L^{\infty}(\mathbf{B}_{j})}$
$\displaystyle\leq
C\sum_{j\neq\ell\in\mathbb{N}(n)}\left\|2^{j}\int_{\mathbb{R}}\frac{1}{(1+2^{j}|x-y|)^{M}}\frac{1}{(1+2^{\ell}|y-2^{2n+\ell}|)^{2M}}\mathrm{d}y\right\|_{L^{\infty}(\mathbf{B}_{j})}.$

Dividing the integral region in terms of $y$ into the following two parts to
estimate:

$\displaystyle\mathbf{A}_{1}$ $\displaystyle:=\left\\{y:\;|y-2^{j+2n}|\leq
2^{2n}\right\\},$ $\displaystyle\mathbf{A}_{2}$
$\displaystyle:=\left\\{y:\;|y-2^{j+2n}|\geq 2^{2n}\right\\}.$

For $x\in\mathbf{B}_{j}$ and $y\in\mathbf{A}_{1}$, it is easy to check that

$\displaystyle\left|y-2^{\ell+2n}\right|$
$\displaystyle=\left|y-2^{j+2n}+2^{j+2n}-2^{\ell+2n}\right|$
$\displaystyle\geq\left|2^{\ell+2n}-2^{j+2n}\right|-\left|y-2^{j+2n}\right|\geq
2^{2n}.$

For $x\in\mathbf{B}_{j}$ and $y\in\mathbf{A}_{2}$, it is easy to check that

$\displaystyle|x-y|$ $\displaystyle=\left|x-2^{j+2n}+2^{j+2n}-y\right|$
$\displaystyle\geq\left|y-2^{j+2n}\right|-\left|x-2^{j+2n}\right|\geq
2^{2n}-2^{-j}\geq 2^{2n-1}.$

Then, we have

$\displaystyle\left\|2^{j}\int_{\mathbb{R}}\frac{1}{(1+2^{j}|x-y|)^{M}}\frac{1}{(1+2^{\ell}|y-2^{2n+\ell}|)^{2M}}\mathrm{d}y\right\|_{L^{\infty}(\mathbf{B}_{j})}\leq
C2^{j}2^{-2nM}\leq C2^{\frac{n}{2}}2^{-nM},$

which implies

$\displaystyle\|\Delta_{j}U_{1,2}\|_{L^{\infty}(\mathbf{B}_{j})}$
$\displaystyle\leq C2^{-n}$ (3.8)

Combining (3.6)-(3.8), we deduce that for $n\gg 1$,

$\displaystyle\left\|\Delta_{j}[(\rho_{0})^{2}]\right\|_{L^{\infty}}$
$\displaystyle\geq\left\|\Delta_{j}[(\rho_{0})^{2}]\right\|_{L^{\infty}(\mathbf{B}_{j})}$
$\displaystyle\geq\left\|\Delta_{j}U_{1,1}\right\|_{L^{\infty}(\mathbf{B}_{j})}-\left\|\Delta_{j}U_{1,2}\right\|_{L^{\infty}(\mathbf{B}_{j})}-\left\|\Delta_{j}U_{2}\right\|_{L^{\infty}}$
$\displaystyle\geq cn^{-1}\log^{2}n.$

Therefore, by the definition of the Besov norm, we have

$\displaystyle\left\|(\rho_{0})^{2}\right\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}=\sum_{j\in\mathbb{N}(n)}\left\|\Delta_{j}[(\rho_{0})^{2}]\right\|_{L^{\infty}}\geq
c\log^{2}n.$

This completes the proof of lemma 3.2. ∎

###  3.2 Norm Inflation

Firstly, we define the Lagrangian flow-map $\psi$ associated to $u$ by solve
the following ODE:

$\left\\{\begin{array}[]{ll}\frac{\mathrm{d}}{\mathrm{d}t}\psi(t,x)=u(t,\psi(t,x)),\\\\[4.30554pt]
\psi(0,x)=x.\end{array}\right.$ (3.9)

Considering the equation

$\left\\{\begin{array}[]{ll}\partial_{t}v+u\partial_{x}v=P,&t\in[0,T),\quad
x\in\mathbb{R},\\\\[4.30554pt] v(0,x)=x,x\in\mathbb{R}.\end{array}\right.$
(3.10)

Taking $\Delta_{j}$ on (3.10), we get

$\partial_{t}(\Delta_{j}v)+u\partial_{x}\Delta_{j}v=[u,\Delta_{j}]\partial_{x}v+\Delta_{j}P:=R_{j}+\Delta_{j}P,$

Due to (3.9), then

$\frac{\mathrm{d}}{\mathrm{d}t}\big{(}(\Delta_{j}v)\circ\psi\big{)}=R_{j}\circ\psi+\Delta_{j}P\circ\psi,$

which is equivalent to the integral form

$\displaystyle(\Delta_{j}v)\circ\psi$
$\displaystyle=\Delta_{j}v_{0}+\int^{t}_{0}R_{j}\circ\psi\mathrm{d}\tau+\int^{t}_{0}\Delta_{j}P\circ\psi\mathrm{d}\tau.$
(3.11)

Now we can go back to the proof of the Theorem 1.1. For $n\gg 1$, it is easy
to check that $(u,\rho)\in\mathcal{C}([0,1];H^{3}\times H^{2})$ which satisfy
for $t\in[0,1]$

$\displaystyle\|u(t)\|_{W^{1,\infty}}+||\rho(t)||_{L^{\infty}}\leq(\|u_{0}\|_{W^{1,\infty}}+||\rho_{0}||_{L^{\infty}})\exp\left(C\int_{0}^{t}\|\partial_{x}u(\tau),\rho(\tau)\|_{L^{\infty}}\mathrm{d}\tau\right).$

For $n\gg 1$, we have for $t\in[0,1]$

$\displaystyle\|u\|_{W^{1,\infty}}+\|\rho\|_{L^{\infty}}\leq
C\|\rho_{0}\|_{L^{\infty}}\leq Cn^{-\frac{1}{2}}\log n.$ (3.12)

Moreover, we also have $t\in[0,1]$

$\displaystyle\|u\|_{L^{\infty}}$ $\displaystyle\leq
C\int^{t}_{0}(\|u\|^{2}_{W^{1,\infty}}+||\rho||^{2}_{L^{\infty}})\mathrm{d}\tau\leq
Cn^{-1}\log^{2}n.$ (3.13)

To prove Theorem 1.1, it suffices to show that there exists
$t_{0}\in(0,\frac{1}{\log n}]$ such that

$\displaystyle\|u(t_{0},\cdot)\|_{B^{1}_{\infty,1}}\geq\log\log n.$ (3.14)

We prove (3.14) by contraction. If (3.14) were not true, then

$\displaystyle\sup_{t\in(0,\frac{1}{\log
n}]}\|u(t,\cdot)\|_{B^{1}_{\infty,1}}<\log\log n.$ (3.15)

We divide the proof into two steps.

Step 1: Lower bounds for $(\Delta_{j}u)\circ\psi$

Now we consider the equation along the Lagrangian flow-map associated to $u$.
Utilizing (3.11) to (1.2) yields

$\displaystyle(\Delta_{j}u)\circ\psi$
$\displaystyle=\Delta_{j}u_{0}+\int^{t}_{0}R^{1}_{j}\circ\psi\mathrm{d}\tau+\int^{t}_{0}\Delta_{j}F\circ\psi\mathrm{d}\tau+\int^{t}_{0}\big{(}\Delta_{j}E\circ\psi-\Delta_{j}E_{0}\big{)}\mathrm{d}\tau+t\Delta_{j}E_{0},$

where

$\displaystyle R^{1}_{j}=[u,\Delta_{j}]\partial_{x}u,\quad
F=-\frac{3}{2}\partial_{x}(1-\partial^{2}_{x})^{-1}(u^{2}),\quad
E=-\frac{1}{2}\partial_{x}(1-\partial^{2}_{x})^{-1}(\rho^{2}).$

Due to Lemma 3.2, we deduce

$\displaystyle\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E_{0}\|_{L^{\infty}}\approx\sum_{j\in\mathbb{N}(n)}\|\Delta_{j}\partial_{x}E_{0}\|_{L^{\infty}}\geq
c\sum_{j\in\mathbb{N}(n)}\|\Delta_{j}(\rho_{0})^{2}\|_{L^{\infty}}\geq
c\log^{2}n.$ (3.16)

Notice that, for fixed $t$ the Lagrangian flow-map $\psi(t,\cdot)$ is a
diffeomorphism of $\mathbb{R}$, then we have for $t\in(0,\frac{1}{\log n}]$,

$\displaystyle\|f(t,\psi(t,x))\|_{L^{\infty}}=\|f(t,\cdot)\|_{L^{\infty}}.$

Then, using (3.12), (3.15) and the commutator estimate from Lemma 2.2, we have

$\displaystyle\sum_{j\geq-1}2^{j}\|R^{1}_{j}\circ\psi\|_{L^{\infty}}$
$\displaystyle\leq C\sum_{j\geq-1}2^{j}\|R^{1}_{j}\|_{L^{\infty}}\leq
C\|\partial_{x}u\|_{L^{\infty}}\|u\|_{B^{1}_{\infty,1}}\leq
Cn^{-\frac{1}{2}}\log n\cdot\log\log n.$ (3.17)

Also, by (3.13), we have

$\displaystyle 2^{j}\|\Delta_{j}F\circ\psi\|_{L^{\infty}}$ $\displaystyle\leq
C2^{j}\|\Delta_{j}F\|_{L^{\infty}}\leq C\|u\|^{2}_{L^{\infty}}\leq
Cn^{-2}\log^{4}n,$

which implies

$\displaystyle\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}F\circ\psi\|_{L^{p}}\leq
Cn^{-1}\log^{4}n.$ (3.18)

Combining (3.16)-(3.18) and using Lemmas 3.1-3.2 yields

$\displaystyle\sum_{j\in\mathbb{N}(n)}2^{j}\|(\Delta_{j}u)\circ\psi\|_{L^{\infty}}$
$\displaystyle\geq
t\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E_{0}\|_{L^{\infty}}-\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E\circ\psi-\Delta_{j}E_{0}\|_{L^{\infty}}-Cn^{-1}\log^{4}n$
$\displaystyle\geq
ct\log^{2}n-\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E\circ\psi-\Delta_{j}E_{0}\|_{L^{\infty}}-Cn^{-1}\log^{4}n.$

Step 2: Upper bounds for $\Delta_{j}E\circ\psi-\Delta_{j}E_{0}$

By easy computations, we can see that

$\displaystyle\partial_{t}E+u\partial_{x}E$
$\displaystyle=-\partial_{x}(1-\partial^{2}_{x})^{-1}(\partial_{t}\rho\rho)-\frac{1}{2}u\partial^{2}_{x}(1-\partial^{2}_{x})^{-1}(\rho^{2})$
(3.19)
$\displaystyle=-\partial_{x}(1-\partial^{2}_{x})^{-1}(-\frac{1}{2}\partial_{x}(u\rho^{2})-\frac{3}{2}u_{x}\rho^{2})-\frac{1}{2}u(1-\partial^{2}_{x})^{-1}(\rho^{2})+\frac{1}{2}u\rho^{2}$
$\displaystyle=\frac{1}{2}\partial^{2}_{x}(1-\partial^{2}_{x})^{-1}(u\rho^{2})+\frac{3}{2}\partial_{x}(1-\partial^{2}_{x})^{-1}(u_{x}\rho^{2})-\frac{1}{2}u(1-\partial^{2}_{x})^{-1}(\rho^{2})+\frac{1}{2}u\rho^{2}$
$\displaystyle=\frac{1}{2}(1-\partial^{2}_{x})^{-1}(u\rho^{2})-\frac{1}{2}u\rho^{2}+\frac{3}{2}\partial_{x}(1-\partial^{2}_{x})^{-1}(u_{x}\rho^{2})-\frac{1}{2}u(1-\partial^{2}_{x})^{-1}(\rho^{2})+\frac{1}{2}u\rho^{2}$
$\displaystyle=:G,$

where

$\displaystyle G=$
$\displaystyle-\frac{1}{2}u(1-\partial^{2}_{x})^{-1}(\rho^{2})-\frac{1}{2}(1-\partial^{2}_{x})^{-1}\left(-u\rho^{2}-\partial_{x}(3u_{x}\rho^{2})\right).$

Utilizing (3.11) to (3.19) yields

$\displaystyle\Delta_{j}E\circ\psi-\Delta_{j}E_{0}=\int^{t}_{0}[u,\Delta_{j}]\partial_{x}E\circ\psi\mathrm{d}\tau+\int^{t}_{0}\Delta_{j}G\circ\psi\mathrm{d}\tau.$

Using the commutator estimate from Lemma 2.2, one has

$\displaystyle 2^{j}\|[u,\Delta_{j}]\partial_{x}E\|_{L^{\infty}}\leq
C(\|\partial_{x}u\|_{L^{\infty}}\|E\|_{B^{1}_{\infty,\infty}}+\|\partial_{x}E\|_{L^{\infty}}\|u\|_{B^{1}_{\infty,\infty}})\leq
C(\|u\|_{W^{1,\infty}}+\|\rho\|_{L^{\infty}})^{3}$

and

$\displaystyle 2^{j}\|\Delta_{j}G\|_{L^{\infty}}\leq
C(\|u\|_{W^{1,\infty}}+\|\rho\|_{L^{\infty}})^{3}.$

Then, we have from (3.12)

$\displaystyle 2^{j}\|\Delta_{j}E\circ\psi-\Delta_{j}E_{0}\|_{L^{\infty}}\leq
C(\|u\|_{W^{1,\infty}}+\|\rho\|_{L^{\infty}})^{3}\leq
Cn^{-\frac{3}{2}}\log^{3}n,$

which leads to

$\displaystyle\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E\circ\psi-\Delta_{j}E_{0}\|_{L^{p}}\leq
Cn^{-\frac{1}{2}}\log^{3}n.$

Combining Step 1 and Step 2, then for $t=\frac{1}{\log n}$, we obtain for
$n\gg 1$

$\displaystyle\|u(t)\|_{B^{1}_{\infty,1}}$
$\displaystyle\geq\|u(t)\|_{B^{1}_{\infty,1}(\mathbb{N}(n))}\geq
C\sum_{j\in\mathbb{N}(n)}2^{j}\|(\Delta_{j}u)\circ\psi\|_{L^{\infty}}$
$\displaystyle\geq ct\log^{2}n-Cn^{-\frac{1}{2}}\log^{3}n\geq\log\log n,$

which contradicts the hypothesis (3.15).

In conclusion, we obtain the norm inflation and hence the ill-posedness of the
two component Degasperis-Procesi equation. Thus, Theorem 1.1 is proved.

## Acknowledgments

J. Li is supported by the National Natural Science Foundation of China
(11801090 and 12161004) and Jiangxi Provincial Natural Science Foundation,
China (20212BAB211004). M. Li was supported by Educational Commission Science
Programm of Jiangxi Province (No. GJJ190284) and Natural Science Foundation of
Jiangxi Province (No. 20212BAB211011). W. Zhu is supported by the Guangdong
Basic and Applied Basic Research Foundation, China (2021A1515111018).

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# The Duality Theory of Fractional Calculus
and a New Fractional Calculus of Variations Involving Left Operators Only

Delfim F. M. Torres
(ORCID: 0000-0001-8641-2505)  Center for Research and Development in
Mathematics and Applications (CIDMA), Department of Mathematics, University of
Aveiro, 3810-193 Aveiro, Portugal<EMAIL_ADDRESS>
(Date: Please cite this paper as follows: D. F. M. Torres, The duality theory
of fractional calculus and a new fractional calculus of variations involving
left operators only, Mediterr. J. Math. 21 (2024), no. 3, Paper No. 106, 16
pp. (https://doi.org/10.1007/s00009-024-02652-x).)

###### Abstract.

Through duality it is possible to transform left fractional operators into
right fractional operators and vice versa. In contrast to existing literature,
we establish integration by parts formulas that exclusively involve either
left or right operators. The emergence of these novel fractional integration
by parts formulas inspires the introduction of a new calculus of variations,
where only one type of fractional derivative (left or right) is present. This
applies to both the problem formulation and the corresponding necessary
optimality conditions. As a practical application, we present a new Lagrangian
that relies solely on left-hand side fractional derivatives. The fractional
variational principle derived from this Lagrangian leads us to the equation of
motion for a dissipative/damped system.

###### Key words and phrases:

Duality, fractional calculus, integration by parts, fractional calculus of
variations, Euler–Lagrange equations, dissipative systems.

###### 2020 Mathematics Subject Classification:

34A08, 49K05, 49S05

##  1\. Introduction

Fractional differentiation means “differentiation of arbitrary order”. Its
origin goes back more than 325 years, when in 1695 L’Hopital asked Leibniz the
meaning of $\frac{d^{n}x}{dt^{n}}$ for $n=1/2$. After that, many famous
mathematicians, like Fourier, Abel, Liouville, and Riemann, among others,
contributed to the development of Fractional Calculus [16].

In 1931, Bauer proved that it is impossible to use a variational principle to
derive a linear dissipative equation of motion with constant coefficients [5].
Bauer’s result expresses the well-known belief that there is no direct method
of applying variational principles to nonconservative systems, which are
characterized by friction or other dissipative processes. It turns out that
fractional derivatives provide an elegant solution to the problem. Indeed, the
proof of Bauer relies, implicitly, on the assumption that derivatives are of
integer order.

The Fractional Calculus of Variations (FCoV) was born in 1996–1997 with the
work of Riewe [14, 15], precisely with the aim to obtain a Lagrangian for a
simple dissipative system with a damping force proportional to the velocity.
Riewe proposed to represent the dissipative effects with a Lagrangian
dependent on left and right fractional Riemann–Liouville derivatives of order
$1/2$, showing that in such case one can obtain the equations of motion of a
dissipative linear system with constant coefficients by a (fractional)
variational principle. Riewe’s idea is very simple and natural: if the
Lagrangian contains a term proportional to
$\left(\frac{d^{n}x}{dt^{n}}\right)^{2}$, then the respective Euler–Lagrange
equation has a corresponding term proportional to $\frac{d^{2n}x}{dt^{2n}}$.
Therefore, a damping force of the form $c\frac{dx}{dt}$ should follow from a
Lagrangian containing a term proportional to
$\left(\frac{d^{1/2}x}{dt^{1/2}}\right)^{2}$. The FCoV has therefore
significant importance in physics and engineering, as a means to circumvent
Bauer’s result [9]. This explains the fact why FCoV is under strong
development. For those interested on the subject, we refer to the books [1, 2,
10, 11] and the survey papers [3, 12].

The Lagrangian proposed by Riewe marked the beginning of several discussions
in the literature, in particular among physicists, due to the presence of
right fractional derivatives and other related issues. For a good and recent
account about the fractional Lagrangian proposed by Riewe and other proposals
by different mathematicians, we refer the reader to [8]. Here we remark that,
to the best of our knowledge, all such fractional Lagrangians involve always a
right derivative, which causes some physical concerns of non-causality. From a
strictly mathematical point of view, however, the right operators appear
naturally in the FCoV due to the central role of integration by parts in the
proof of the Euler–Lagrange necessary optimality conditions. Indeed, under
fractional integration by parts, a left fractional operator is transformed
into a right fractional operator and vice versa (see Section 2), so even if
the Lagrangian only includes left operators, then the right operators will
appear in the Euler–Lagrange equation. To circumvent the undesirable
phenomenon of appearance of right-fractional operators either on the
Lagrangian or the Euler–Lagrange equation, here we develop the theory of
duality for fractional calculus (see Section 3.1), as formulated by Caputo and
Torres in 2015 [7], with early contributions discussed in [16].

The duality theory gives a way to express left fractional operators in terms
of right fractional operators and the other way around. In simple terms, the
duality theory shows that the right fractional operators of a function are the
dual of the left operators of the dual function or, equivalently, the left
fractional derivative/integral of a function is the dual of the right
fractional derivative/integral of the dual function. Here we further develop
and use such duality theory to derive new fractional integration by parts
formulas involving only left (or right) fractional operators (see Section
3.2). With the help of the new fractional integration by parts formulas, as
well as results relating fractional derivatives with classical ones (see
Section 3.3), we provide a new perspective to the FCoV by proposing a new FCoV
where both Lagrangians and respective Euler–Lagrange equations involve only
one type of operators, e.g., both variational problem and respective necessary
optimality conditions with left-hand-side operators only (see Section 3.4). As
an application of the obtained results, we go back to the original motivation
behind the seminal papers of Riewe and provide a new Lagrangian that
circumvents the Bauer theorem, for which the respective Euler–Lagrange
equation coincide with the equation of motion of a dissipative system (see
Section 4). In contrast with the example of Riewe, and others available in the
literature, our quadratic Lagrangian for the linear friction problem is a real
valued Lagrangian involving let-hand side fractional derivatives only. This
provides a better physical meaning to the proposed Lagrangian, in contrast
with the ones available in the literature. We end up with a conclusion,
summarizing the main contributions of the paper (see Section 5).

##  2\. Preliminaries

We recall some well known definitions and results from fractional calculus,
fixing also our notations. Here we only give the notions and results that will
be useful in the sequel. For more details we refer to the book [11], where all
such definitions and results are found, and references therein.

######  Definition 2.1 (Fractional integrals of order $\alpha$).

Let $\varphi\in L^{1}([a,b],\mathbb{R})$. The integrals

(2.1)
$\left(\mathcal{I}^{\alpha}_{a+}\varphi\right)(t)=\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\varphi(\tau)(t-\tau)^{\alpha-1}d\tau,\quad
t>a,$

and

(2.2)
$\left(\mathcal{I}^{\alpha}_{b-}\varphi\right)(t)=\frac{1}{\Gamma(\alpha)}\int_{t}^{b}\varphi(\tau)(\tau-t)^{\alpha-1}d\tau,\quad
t<b,$

where $\alpha>0$ and $\Gamma(\cdot)$ is the Gamma function, are called,
respectively, the left and the right fractional integrals of order $\alpha$.
Additionally, we define

$\mathcal{I}^{0}_{a+}\varphi=\mathcal{I}^{0}_{b-}\varphi=\varphi,$

that is, for $\alpha=0$ (2.1) and (2.2) are the identity operator.

######  Proposition 2.2 (Integration by parts for fractional integrals).

Let $\alpha>0$ and $1/p+1/q\leq 1+\alpha$, $p\geq 1$, $q\geq 1$, with $p\neq
1$ and $q\neq 1$ in the case $1/p+1/q=1+\alpha$. If $\varphi\in
L^{p}([a,b],\mathbb{R})$ and $\psi\in L^{q}([a,b],\mathbb{R})$, then the
following equality holds:

(2.3)
$\int_{a}^{b}\varphi(\tau)\left(\mathcal{I}^{\alpha}_{a+}\psi\right)(\tau)d\tau=\int_{a}^{b}\left(\mathcal{I}^{\alpha}_{b-}\varphi\right)(\tau)\psi(\tau)d\tau.$

######  Proposition 2.3.

If $f\in L^{p}([a,b],\mathbb{R})$, $1\leq p\leq\infty$, $\alpha>0$ and
$\beta>0$, then

(2.4)
$\left(\mathcal{I}^{\alpha}_{a+}\left(\mathcal{I}^{\beta}_{a+}f\right)\right)(t)=\left(\mathcal{I}^{\alpha+\beta}_{a+}f\right)(t)$

almost everywhere on $[a,b]$.

######  Definition 2.4 (Left/right Riemann–Liouville fractional derivatives of
order $\alpha$).

Let $0<\alpha<1$ and $\varphi\in W^{1,1}([a,b],\mathbb{R})$. The left
Riemann–Liouville derivative of order $\alpha$ is defined by

(2.5)
$\left(\mathcal{D}^{\alpha}_{a+}\varphi\right)(t)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int_{a}^{t}\varphi(\tau)(t-\tau)^{-\alpha}d\tau,$

while the right Riemann–Liouville derivative of order $\alpha$ is defined by

(2.6)
$\left(\mathcal{D}^{\alpha}_{b-}\varphi\right)(t)=-\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int_{t}^{b}\varphi(\tau)(\tau-t)^{-\alpha}d\tau.$

For $\alpha=0$ we have
$\mathcal{D}^{0}_{a+}\varphi=\mathcal{D}^{0}_{b-}\varphi=\varphi$; while for
$\alpha=1$ one has
$\left(\mathcal{D}^{1}_{a+}\varphi\right)(t)=\frac{d}{dt}\varphi(t)$ and
$\left(\mathcal{D}^{1}_{b-}\varphi\right)(t)=-\frac{d}{dt}\varphi(t)$.

######  Remark 2.5.

Let $\alpha\in[0,1]$. For functions $\varphi\in W^{1,1}([a,b],\mathbb{R})$,
each of the following expressions hold:

(2.7)
$\left(\mathcal{D}^{\alpha}_{a+}\varphi\right)(t)=\frac{d}{dt}\left[\left(\mathcal{I}^{1-\alpha}_{a+}\varphi\right)(t)\right]$

and

(2.8)
$\left(\mathcal{D}^{\alpha}_{b-}\varphi\right)(t)=-\frac{d}{dt}\left[\left(\mathcal{I}^{1-\alpha}_{b-}\varphi\right)(t)\right].$

The following properties can be found, for example, in Propositions 2.4 and
2.5 of [11]. They are valid under appropriate space of functions
$\mathcal{I}^{\alpha}_{a+}\left(L^{p}(a,b)\right)$ and
$\mathcal{I}^{\alpha}_{b-}\left(L^{p}(a,b)\right)$, $1\leq p\leq\infty$,
defined respectively by

$\mathcal{I}^{\alpha}_{a+}\left(L^{p}(a,b)\right):=\left\\{\varphi:\varphi(t)=\left(\mathcal{I}^{\alpha}_{a+}\psi\right)(t),\psi\in
L^{p}([a,b],\mathbb{R})\right\\}$

and

$\mathcal{I}^{\alpha}_{b-}\left(L^{p}(a,b)\right):=\left\\{\varphi:\varphi(t)=\left(\mathcal{I}^{\alpha}_{b-}\psi\right)(t),\psi\in
L^{p}([a,b],\mathbb{R})\right\\}.$

######  Proposition 2.6.

If $\alpha>0$, then

(2.9)
$\left(\mathcal{D}^{\alpha}_{a+}\left(\mathcal{I}^{\alpha}_{a+}\varphi\right)\right)(t)=\varphi(t)$

for any $\varphi\in L^{1}([a,b],\mathbb{R})$, while

(2.10)
$\left(\mathcal{I}^{\alpha}_{a+}\left(\mathcal{D}^{\alpha}_{a+}\varphi\right)\right)(t)=\varphi(t)$

is satisfied for $\varphi\in\mathcal{I}^{\alpha}_{a+}\left(L^{1}(a,b)\right)$.

######  Proposition 2.7 (Integration by parts for fractional derivatives).

Let $0<\alpha<1$. If
$\varphi\in\mathcal{I}^{\alpha}_{b-}\left(L^{p}(a,b)\right)$ and
$\psi\in\mathcal{I}^{\alpha}_{a+}\left(L^{q}(a,b)\right)$ with $1/p+1/q\leq
1+\alpha$, then the following equality holds:

(2.11)
$\int_{a}^{b}\varphi(\tau)\left(\mathcal{D}^{\alpha}_{a+}\psi\right)(\tau)d\tau=\int_{a}^{b}\left(\mathcal{D}^{\alpha}_{b-}\varphi\right)(\tau)\psi(\tau)d\tau.$

##  3\. Main Results

In what follows the $\alpha$’s are always real numbers between $0<\alpha<1$.

###  3.1. A duality theory for the FCoV

Here we extend the duality of [7] to the context of the calculus of
variations.

######  Definition 3.1 (Dual function and the dual operator $*$).

Given $f:[a,b]\rightarrow\mathbb{R}$, we define the dual function
$f^{*}:[a,b]\rightarrow\mathbb{R}$ (and the dual operator $*$) by

(3.1) $f^{*}(t)=f(b-t+a)$

for all $t\in[a,b]$.

The following properties are straightforward but important.

######  Proposition 3.2.

For any function $f:[a,b]\rightarrow\mathbb{R}$, let
$f^{**}:=\left(f^{*}\right)^{*}$. Then,

(3.2) $f^{**}=f.$

If $c_{1},c_{2}\in\mathbb{R}$ and $f_{1},f_{2}:[a,b]\rightarrow\mathbb{R}$,
then

(3.3) $\left(f_{1}\cdot f_{2}\right)^{*}=f_{1}^{*}\cdot f_{2}^{*};$ (3.4)
$\left(c_{1}f_{1}+c_{2}f_{2}\right)^{*}=c_{1}f_{1}^{*}+c_{2}f_{2}^{*}.$

###### Proof.

The proof is a trivial exercise. ∎

######  Remark 3.3.

An immediate consequence of the linearity given by (3.4) is that the sign
change of the dual is the dual of the sign change. This property will be
useful for us to establish a relation between left and right fractional
derivatives. Precisely, for any function $f:[a,b]\rightarrow\mathbb{R}$ we
have

(3.5) $\left(-f\right)^{*}=-f^{*}.$

######  Proposition 3.4.

Under integration, the following relation holds:

(3.6) $\int_{a}^{b}f^{*}(t)\cdot g(t)dt=\int_{a}^{b}f(t)\cdot g^{*}(t)dt.$

###### Proof.

From Definition 3.1 of duality one has

$\int_{a}^{b}f^{*}(t)\cdot g(t)dt=\int_{a}^{b}f(b-t+a)\cdot g(t)dt$

and doing the change of variables $s=b-t+a$ we have

$\int_{a}^{b}f^{*}(t)\cdot g(t)dt=-\int_{b}^{a}f(s)\cdot
g(b-s+a)ds=\int_{a}^{b}f(s)\cdot g^{*}(s)ds.$

The proof is complete. ∎

The following important lemma asserts that the left fractional integral of the
dual is the dual of the right fractional integral (of the same order). See
equation (2.19) in the classical book of Samko–Kilbas–Marichev [16] and also
[4].

######  Lemma 3.5.

Let $\varphi\in L^{1}([a,b],\mathbb{R})$. The following relation holds:

(3.7)
$\mathcal{I}^{\alpha}_{a+}f^{*}=\left(\mathcal{I}^{\alpha}_{b-}f\right)^{*}.$

As a corollary of Lemma 3.5, it follows that the right fractional integral of
the dual is the dual of the left fractional integral.

######  Corollary 3.6.

If $f\in L^{1}([a,b],\mathbb{R})$, then

(3.8)
$\mathcal{I}^{\alpha}_{b-}f^{*}=\left(\mathcal{I}^{\alpha}_{a+}f\right)^{*}.$

An equivalent way to look to the duality of the fractional integral operators
consists to say that the left fractional integral is the dual of the right
fractional integral of the dual; and the right fractional integral is the dual
of the left fractional integral of the dual.

######  Corollary 3.7.

If $\varphi\in L^{1}([a,b],\mathbb{R})$, then

(3.9)
$\mathcal{I}^{\alpha}_{a+}f=\left(\mathcal{I}^{\alpha}_{b-}f^{*}\right)^{*}$

and

(3.10)
$\mathcal{I}^{\alpha}_{b-}f=\left(\mathcal{I}^{\alpha}_{a+}f^{*}\right)^{*}.$

Roughly speaking, now we show that, up to a minus signal, the duality
relations proved for the fractional integrals also hold true for fractional
derivatives.

Lemma 3.8 asserts that the left fractional derivative of the dual is the sign
change of the dual of the right fractional derivative (of the same order).

######  Lemma 3.8.

Let $f\in W^{1,1}([a,b],\mathbb{R})$. The following relation holds:

(3.11)
$\mathcal{D}^{\alpha}_{a+}f^{*}=-\left(\mathcal{D}^{\alpha}_{b-}f\right)^{*}.$

###### Proof.

From relations (2.7) and (3.7), we can write that

(3.12)
$\left(\mathcal{D}^{\alpha}_{a+}f^{*}\right)(t)=\frac{d}{dt}\left[\left(\mathcal{I}^{1-\alpha}_{a+}f^{*}\right)(t)\right]=\frac{d}{dt}\left[\left(\mathcal{I}^{1-\alpha}_{b-}f\right)^{*}(t)\right].$

Using now (3.1) and relation (2.8) between the right fractional derivative of
order $\alpha$ and the right fractional integral of order $1-\alpha$, we
obtain from (3.12) that

$\begin{split}\left(\mathcal{D}^{\alpha}_{a+}f^{*}\right)(t)&=\frac{d}{dt}\left[\left(\mathcal{I}^{1-\alpha}_{b-}f\right)(b-t+a)\right]\\\
&=-\left(\mathcal{D}^{\alpha}_{b-}f\right)(b-t+a)=-\left(\mathcal{D}^{\alpha}_{b-}f\right)^{*}(t),\end{split}$

which completes the proof. ∎

As a corollary, we can also say that the right fractional derivative of the
dual is the sign change of the dual of the left fractional derivative.

######  Corollary 3.9.

If $f\in W^{1,1}([a,b],\mathbb{R})$, then

(3.13)
$\mathcal{D}^{\alpha}_{b-}f^{*}=-\left(\mathcal{D}^{\alpha}_{a+}f\right)^{*}.$

###### Proof.

Let $g=f^{*}$, which by Proposition 3.2 is equivalent to $f=g^{*}$. Using
Lemma 3.8 for $g$, we can write from (3.11) that

$\left(\mathcal{D}^{\alpha}_{b-}g\right)^{*}(t)=-\left(\mathcal{D}^{\alpha}_{a+}g^{*}\right)(t)$

for all $t\in[a,b]$, that is,

(3.14)
$\left(\mathcal{D}^{\alpha}_{b-}f^{*}\right)^{*}(t)=-\left(\mathcal{D}^{\alpha}_{a+}f\right)(t),\quad
t\in[a,b].$

Applying the dual operator $*$ both sides of (3.14) gives, by Proposition 3.2
and (3.5), that

$\left(\mathcal{D}^{\alpha}_{b-}f^{*}\right)(t)=-\left(\mathcal{D}^{\alpha}_{a+}f\right)^{*}(t)$

and the result is proved. ∎

Recalling Remark 3.3, together with the linearity of the fractional operators,
allow us to look to Lemma 3.8 as saying that the left fractional derivative of
the dual is the dual of the right fractional derivative of the sign change or,
in other words, the right fractional derivative is the sign change of the dual
of the left fractional derivative of the dual.

######  Corollary 3.10.

Let $f\in W^{1,1}([a,b],\mathbb{R})$. The following relations hold:

(3.15)
$\mathcal{D}^{\alpha}_{a+}f^{*}=\left(\mathcal{D}^{\alpha}_{b-}(-f)\right)^{*}$

and

(3.16)
$\mathcal{D}^{\alpha}_{b-}f=-\left(\mathcal{D}^{\alpha}_{a+}f^{*}\right)^{*}.$

###### Proof.

Using (3.11) and (3.5) we can write that

$\mathcal{D}^{\alpha}_{a+}f^{*}=-\left(\mathcal{D}^{\alpha}_{b-}f\right)^{*}=\left(-\mathcal{D}^{\alpha}_{b-}f\right)^{*}$

and equality (3.15) follows by the linearity of operator
$\mathcal{D}^{\alpha}_{b-}$. Applying the dual operator $*$ to both sides of
(3.15), already proved, gives

$\mathcal{D}^{\alpha}_{b-}(-f)=\left(\mathcal{D}^{\alpha}_{a+}f^{*}\right)^{*},$

which is equivalent, by the linearity of $\mathcal{D}^{\alpha}_{b-}$, to

$-\mathcal{D}^{\alpha}_{b-}f=\left(\mathcal{D}^{\alpha}_{a+}f^{*}\right)^{*}.$

The result is proved. ∎

Similarly, Remark 3.3 together with the linearity of the fractional
derivatives give us a new look to the duality of Corollary 3.9: the right
fractional derivative of the dual is the dual of the left fractional
derivative of the sign change or, equivalently, the left fractional derivative
is the sign change of the dual of the right fractional derivative of the dual.

######  Corollary 3.11.

If $f\in W^{1,1}([a,b],\mathbb{R})$, then

(3.17)
$\mathcal{D}^{\alpha}_{b-}f^{*}=\left(\mathcal{D}^{\alpha}_{a+}(-f)\right)^{*}$

and

(3.18)
$\mathcal{D}^{\alpha}_{a+}f=-\left(\mathcal{D}^{\alpha}_{b-}f^{*}\right)^{*}.$

###### Proof.

We prove Corollary 3.11 as a corollary of Corollary 3.10. Let $g=f^{*}$, that
is, $f=g^{*}$. From relation (3.16) applied to $g$ we know that

$\mathcal{D}^{\alpha}_{b-}g=-\left(\mathcal{D}^{\alpha}_{a+}g^{*}\right)^{*}\Leftrightarrow\mathcal{D}^{\alpha}_{b-}f^{*}=-\left(\mathcal{D}^{\alpha}_{a+}f\right)^{*},$

which, by (3.5) and the linearity of $\mathcal{D}^{\alpha}_{a+}$, is
equivalent to

$\mathcal{D}^{\alpha}_{b-}f^{*}=\left(-\mathcal{D}^{\alpha}_{a+}f\right)^{*}\Leftrightarrow\mathcal{D}^{\alpha}_{b-}f^{*}=\left(\mathcal{D}^{\alpha}_{a+}(-f)\right)^{*}.$

We have just proved (3.17). From (3.15) applied to $g$, we also know that

$\mathcal{D}^{\alpha}_{a+}g^{*}=\left(\mathcal{D}^{\alpha}_{b-}(-g)\right)^{*}\Leftrightarrow\mathcal{D}^{\alpha}_{a+}f=\left(\mathcal{D}^{\alpha}_{b-}(-f^{*})\right)^{*},$

which by the linearity of the operator $\mathcal{D}^{\alpha}_{b-}$ and (3.5)
is equivalent to

$\mathcal{D}^{\alpha}_{a+}f=\left(-\mathcal{D}^{\alpha}_{b-}f^{*}\right)^{*}\Leftrightarrow\mathcal{D}^{\alpha}_{a+}f=-\left(\mathcal{D}^{\alpha}_{b-}f^{*}\right)^{*}.$

The result is proved. ∎

###  3.2. New fractional formulas of integration by parts

Now we are in a good position to prove some important results: formulas of
integration by parts, for both fractional integrals and fractional
derivatives, that involve only left operators or only right operators. This
contrasts with the results available in the literature (cf. Propositions 2.2
and 2.7) for which a left operator on the left-hand side is converted into a
right operator on the right-hand side — see formulas (2.3) and (2.11).

As we shall see in Section 3.4, our new results allow us to develop a new
fractional calculus of variations (FCoV) more suitable for mechanics.

######  Theorem 3.12 (Integration by parts involving left fractional integrals
only).

Let $1/p+1/q\leq 1+\alpha$, $p\geq 1$, $q\geq 1$, with $p\neq 1$ and $q\neq 1$
in the case $1/p+1/q=1+\alpha$. If $f\in L^{p}([a,b],\mathbb{R})$ and $g\in
L^{q}([a,b],\mathbb{R})$, then the following equality holds:

(3.19)
$\int_{a}^{b}f(\tau)\cdot\left(\mathcal{I}^{\alpha}_{a+}g\right)^{*}(\tau)d\tau=\int_{a}^{b}\left(\mathcal{I}^{\alpha}_{a+}f\right)(\tau)\cdot
g^{*}(\tau)d\tau.$

###### Proof.

From (3.8) and (2.3) we can write that

$\begin{split}\int_{a}^{b}f(\tau)\left(\mathcal{I}^{\alpha}_{a+}g\right)^{*}(\tau)d\tau&=\int_{a}^{b}f(\tau)\cdot\left(\mathcal{I}^{\alpha}_{b-}g^{*}\right)(\tau)d\tau\\\
&=\int_{a}^{b}\left(\mathcal{I}^{\alpha}_{a+}f\right)(\tau)\cdot
g^{*}(\tau)d\tau,\end{split}$

which proves the intended relation. ∎

Similarly to Theorem 3.12, one can prove an integration by parts formula
involving right fractional integrals only.

######  Theorem 3.13 (Integration by parts involving right fractional
integrals only).

Let $1/p+1/q\leq 1+\alpha$, $p\geq 1$, $q\geq 1$, with $p\neq 1$ and $q\neq 1$
in the case $1/p+1/q=1+\alpha$. If $f\in L^{p}([a,b],\mathbb{R})$ and
$g^{*}\in L^{q}([a,b],\mathbb{R})$, then the following equality holds:

(3.20)
$\int_{a}^{b}f(\tau)\cdot\left(\mathcal{I}^{\alpha}_{b-}g\right)^{*}(\tau)d\tau=\int_{a}^{b}\left(\mathcal{I}^{\alpha}_{b-}f\right)(\tau)\cdot
g^{*}(\tau)d\tau.$

###### Proof.

From (3.7) and (2.3) it follows that

$\begin{split}\int_{a}^{b}f(\tau)\left(\mathcal{I}^{\alpha}_{b-}g\right)^{*}(\tau)d\tau&=\int_{a}^{b}f(\tau)\cdot\left(\mathcal{I}^{\alpha}_{a+}g^{*}\right)(\tau)d\tau\\\
&=\int_{a}^{b}\left(\mathcal{I}^{\alpha}_{b-}f\right)(\tau)\cdot
g^{*}(\tau)d\tau\end{split}$

and the result is proved. ∎

We now give a proper formula of integration by parts for the FCoV that
involves left fractional derivatives only.

######  Theorem 3.14 (Integration by parts involving left fractional
derivatives only).

If $f\in\mathcal{I}^{\alpha}_{a+}\left(L^{p}(a,b)\right)$ and
$g^{*}\in\mathcal{I}^{\alpha}_{b-}\left(L^{q}(a,b)\right)$ with $1/p+1/q\leq
1+\alpha$, then the following equality holds:

(3.21)
$\int_{a}^{b}f(t)\cdot\left(\mathcal{D}^{\alpha}_{a+}g\right)^{*}(t)dt=-\int_{a}^{b}\left(\mathcal{D}^{\alpha}_{a+}f\right)(t)\cdot
g^{*}(t)dt.$

###### Proof.

From (3.13) and (2.11) we can write that

$\begin{split}\int_{a}^{b}f(t)\left(\mathcal{D}^{\alpha}_{a+}g\right)^{*}(t)dt&=-\int_{a}^{b}f(t)\cdot\left(\mathcal{D}^{\alpha}_{b-}g^{*}\right)(t)dt\\\
&=-\int_{a}^{b}\left(\mathcal{D}^{\alpha}_{a+}f\right)(t)\cdot
g^{*}(t)dt,\end{split}$

which proves the intended relation. ∎

In classical mechanics it makes sense to consider causal operators. For this
reason, we shall develop our new FCoV based on left operators only. For
completeness, however, we also provide here a similar relation to (3.21)
involving right fractional derivatives only.

######  Theorem 3.15 (Integration by parts involving right fractional
derivatives only).

If $f\in\mathcal{I}^{\alpha}_{b-}\left(L^{p}(a,b)\right)$ and
$g^{*}\in\mathcal{I}^{\alpha}_{a+}\left(L^{q}(a,b)\right)$ with $1/p+1/q\leq
1+\alpha$, then the following equality holds:

(3.22)
$\int_{a}^{b}f(t)\cdot\left(\mathcal{D}^{\alpha}_{b-}g\right)^{*}(t)dt=-\int_{a}^{b}\left(\mathcal{D}^{\alpha}_{b-}f\right)(t)\cdot
g^{*}(t)dt.$

###### Proof.

Using Lemma 3.8 and Proposition 2.7, we obtain that

$\begin{split}\int_{a}^{b}f(t)\cdot\left(\mathcal{D}^{\alpha}_{b-}g\right)^{*}(t)dt&\stackrel{{\scriptstyle\eqref{eq:16}}}{{=}}-\int_{a}^{b}f(t)\cdot\left(\mathcal{D}^{\alpha}_{a+}g^{*}\right)(t)dt\\\
&\stackrel{{\scriptstyle\eqref{eq:12}}}{{=}}-\int_{a}^{b}\left(\mathcal{D}^{\alpha}_{b-}f\right)(t)\cdot
g^{*}(t)dt.\end{split}$

The result is proved. ∎

###  3.3. From fractional to classical derivatives

Before formulating our fundamental problem of the fractional calculus of
variations and proving its Euler–Lagrange necessary optimality condition, we
prove here several auxiliary results that relate expressions involving
fractional derivatives into expressions involving classical derivatives only.
The results are here formulated and proved for left derivatives but similar
formulas for right derivatives can also be obtained.

######  Proposition 3.16.

Let $\alpha_{1},\alpha_{2}\in[0,1]$ with $\alpha_{1}+\alpha_{2}=1$. If $h\in
C^{1}([a,b];\mathbb{R})$, then

(3.23)
$\left(\mathcal{D}^{\alpha_{1}}_{a+}\left(\mathcal{D}^{\alpha_{2}}_{a+}h\right)\right)(t)=h^{\prime}(t)$

for all $t\in[a,b]$.

###### Proof.

Let $\psi=\mathcal{D}^{\alpha_{2}}_{a+}h$. Then,

$\begin{split}\left(\mathcal{D}^{\alpha_{1}}_{a+}\psi\right)(t)&\stackrel{{\scriptstyle\eqref{eq:05b}}}{{=}}\frac{d}{dt}\left[\left(\mathcal{I}^{1-\alpha_{1}}_{a+}\psi\right)(t)\right]=\frac{d}{dt}\left[\left(\mathcal{I}^{\alpha_{2}}_{a+}\mathcal{D}^{\alpha_{2}}_{a+}h\right)(t)\right]\\\
&\stackrel{{\scriptstyle\eqref{eq:08}}}{{=}}\frac{d}{dt}\left[h(t)\right]=h^{\prime}(t).\end{split}$

The proof is complete. ∎

######  Theorem 3.17.

Let $\alpha_{1},\alpha_{2}\in[0,1]$ with $\alpha_{1}+\alpha_{2}=1$. If $f\in
C^{1}([a,b];\mathbb{R})$, then

(3.24)
$\int_{a}^{b}\left(\mathcal{D}^{\alpha_{1}}_{a+}f\right)(t)\cdot\left(\mathcal{D}^{\alpha_{2}}_{a+}g\right)^{*}(t)dt=-\int_{a}^{b}f^{\prime}(t)\cdot
g^{*}(t)dt.$

###### Proof.

Using Theorem 3.14 and Proposition 3.16 we obtain:

(3.25)
$\begin{split}\int_{a}^{b}\left(\mathcal{D}^{\alpha_{1}}_{a+}f\right)(t)\cdot\left(\mathcal{D}^{\alpha_{2}}_{a+}g\right)^{*}(t)dt&\stackrel{{\scriptstyle\eqref{eq:20}}}{{=}}-\int_{a}^{b}\left(\mathcal{D}^{\alpha_{2}}_{a+}\left(\mathcal{D}^{\alpha_{1}}_{a+}f\right)\right)(t)\cdot
g^{*}(t)dt\\\
&\stackrel{{\scriptstyle\eqref{eq:22}}}{{=}}-\int_{a}^{b}f^{\prime}(t)\cdot
g^{*}(t)dt.\end{split}$

The result is proved. ∎

######  Remark 3.18.

As a consequence of Proposition 3.4, the left-hand side of (3.24) can be
written as

$\int_{a}^{b}\left(\mathcal{D}^{\alpha_{1}}_{a+}f\right)^{*}(t)\cdot\left(\mathcal{D}^{\alpha_{2}}_{a+}g\right)(t)dt$

while the right-hand side of (3.24) can be written as

$-\int_{a}^{b}\left(f^{\prime}\right)^{*}(t)\cdot g(t)dt.$

######  Corollary 3.19.

Let $\alpha_{1},\alpha_{2}\in[0,1]$ with $\alpha_{1}+\alpha_{2}=1$. If $f\in
C^{1}([a,b];\mathbb{R})$ and $h\in L^{1}([a,b],\mathbb{R})$, then

(3.26)
$\int_{a}^{b}\left(\mathcal{D}^{\alpha_{1}}_{a+}f\right)(t)\cdot\left(\mathcal{D}^{\alpha_{2}}_{b-}h\right)(t)dt=\int_{a}^{b}f^{\prime}(t)\cdot
h(t)dt.$

###### Proof.

Using (3.24) with $g=h^{*}\Leftrightarrow h=g^{*}$ we obtain that

(3.27)
$\int_{a}^{b}\left(\mathcal{D}^{\alpha_{1}}_{a+}f\right)(t)\cdot\left(\mathcal{D}^{\alpha_{2}}_{a+}h^{*}\right)^{*}(t)dt=-\int_{a}^{b}f^{\prime}(t)\cdot
h(t)dt.$

From (3.15) and Proposition 3.2 we know that

$\left(\mathcal{D}^{\alpha_{2}}_{a+}h^{*}\right)^{*}=\mathcal{D}^{\alpha_{2}}_{b-}(-h)=-\mathcal{D}^{\alpha_{2}}_{b-}h$

and (3.27) can be rewritten as

$-\int_{a}^{b}\left(\mathcal{D}^{\alpha_{1}}_{a+}f\right)(t)\cdot\left(\mathcal{D}^{\alpha_{2}}_{b-}h\right)(t)dt=-\int_{a}^{b}f^{\prime}(t)\cdot
h(t)dt,$

which proves the result. ∎

######  Proposition 3.20.

Let $\alpha_{1},\alpha_{2}\in[0,1]$ with $\alpha_{1}+\alpha_{2}=1$. If
$f^{*}\in C^{1}([a,b],\mathbb{R})$, then

(3.28)
$\mathcal{D}^{\alpha_{1}}_{a+}\left(\mathcal{D}^{\alpha_{2}}_{b-}f\right)^{*}=-\left(f^{*}\right)^{\prime}.$

###### Proof.

Let $\psi=\left(\mathcal{D}^{\alpha_{2}}_{b-}f\right)^{*}$. Then,

$\begin{split}\left(\mathcal{D}^{\alpha_{1}}_{a+}\left(\mathcal{D}^{\alpha_{2}}_{b-}f\right)^{*}\right)(t)&=\left(\mathcal{D}^{\alpha_{1}}_{a+}\psi\right)(t)\\\
&\stackrel{{\scriptstyle\eqref{eq:05b}}}{{=}}\frac{d}{dt}\left[\left(\mathcal{I}^{\alpha_{2}}_{a+}\psi\right)(t)\right]\\\
&=\frac{d}{dt}\left[\left(\mathcal{I}^{\alpha_{2}}_{a+}\left(\mathcal{D}^{\alpha_{2}}_{b-}f\right)^{*}\right)(t)\right]\\\
&\stackrel{{\scriptstyle\eqref{eq:16}}}{{=}}\frac{d}{dt}\left[\left(\mathcal{I}^{\alpha_{2}}_{a+}\left(-\mathcal{D}^{\alpha_{2}}_{a+}f^{*}\right)\right)(t)\right]\\\
&\stackrel{{\scriptstyle\eqref{eq:08}}}{{=}}-\frac{d}{dt}\left[f^{*}(t)\right]=-\left(f^{*}\right)^{\prime}(t),\end{split}$

and the proof is complete. ∎

###  3.4. A new FCoV involving left fractional operators only

Here we propose a different approach to the FCoV so that both variational
problems and their necessary optimality conditions involve left fractional
derivatives only. This contrasts completely with the results found in the
literature.

Let $0<\alpha<1$ and let us use the standard notation of mechanics for the
derivative with respect to time: $\dot{x}(t)=x^{\prime}(t)$. Consider the
following problem: find a function $x\in C^{1}([a,b];\mathbb{R})$ that gives
an extremum (minimum or maximum) to the integral functional $\mathcal{J}$,

(3.29)
$\mathcal{J}[x]=\int_{a}^{b}L\left(t,x(t),\dot{x}(t),\left(\mathcal{D}^{\alpha_{1}}_{a+}x\right)(t),\left(\mathcal{D}^{\alpha_{2}}_{a+}x\right)^{*}(t)\right)dt,$

when subject to given boundary conditions

(3.30) $x(a)=x_{a},\quad x(b)=x_{b}.$

We assume that the Lagrangian $L\in
C^{2}([a,b]\times\mathbb{R}^{4};\mathbb{R})$ and that $\partial_{4}L$ and
$\partial_{5}L$ (the partial derivatives of $L(\cdot,\cdot,\cdot,\cdot,\cdot)$
with respect to its 4th and 5th arguments, respectively) have continuous
Riemann–Liouville fractional derivatives.

######  Definition 3.21.

A function $x\in C^{1}([a,b];\mathbb{R})$ that satisfies the given boundary
conditions (3.30) is said to be an admissible trajectory for problem
(3.29)–(3.30).

Trivially, for the particular case when the Lagrangian $L$ only depends on
$t$, $x$, and $\dot{x}$, then problem (3.29)–(3.30) reduces to the classical
fundamental problem of the calculus of variations [13]. We now prove an
extension of the classical Euler–Lagrange equation: when $L=L(t,x,\dot{x})$ in
(3.29), then our necessary optimality condition (3.31) reduces to the
classical Euler–Lagrange equation:

$\partial_{2}L-\frac{d}{dt}\left(\partial_{3}L\right)=0.$

######  Theorem 3.22 (The fractional Euler–Lagrange equation).

If $x$ is an extremizer (minimizer or maximizer) to problem (3.29)–(3.30),
then $x$ satisfies the Euler–Lagrange equation

(3.31)
$\partial_{2}L-\frac{d}{dt}\left(\partial_{3}L\right)-\left(\mathcal{D}^{\alpha_{1}}_{a+}\left(\partial_{4}L\right)^{*}\right)^{*}-\left(\mathcal{D}^{\alpha_{2}}_{a+}\left(\partial_{5}L\right)\right)^{*}=0.$

###### Proof.

Suppose that $x$ is a solution of (3.29)–(3.30). Note that
$\hat{x}_{\epsilon}(t)=x(t)+\epsilon h(t)$ is admissible for our problem for
any $h\in C^{1}([a,b];\mathbb{R})$ with $h(a)=h(b)=0$ and for any
$\epsilon\in\mathbb{R}$. Clearly, for $\epsilon=0$ one gets the solution of
the problem: $\hat{x}_{0}=x$. Let us define the real function
$J:\mathbb{R}\rightarrow\mathbb{R}$ by

$J(\epsilon)=\mathcal{J}[\hat{x}_{\epsilon}]=\int_{a}^{b}L\left(t,\hat{x}_{\epsilon}(t),\dot{\hat{x}}_{\epsilon}(t),\left(\mathcal{D}^{\alpha_{1}}_{a+}\hat{x}_{\epsilon}\right)(t),\left(\mathcal{D}^{\alpha_{2}}_{a+}\hat{x}_{\epsilon}\right)^{*}(t)\right)dt.$

Since $\epsilon=0$ is an extremizer for function $J$, the classical Fermat’s
theorem asserts that a necessary optimality condition for our problem is given
by $J^{\prime}(0)=0$. Using the linearity of the involved operators, a direct
computation tells us that

(3.32)
$\begin{split}0&=J^{\prime}(0)=\left.\frac{d}{d\epsilon}J(\epsilon)\right|_{\epsilon=0}\\\
&=\int_{a}^{b}\left[\partial_{2}L\cdot
h(t)+\partial_{3}L\cdot\dot{h}(t)+\partial_{4}L\cdot\left(\mathcal{D}^{\alpha_{1}}_{a+}h\right)(t)+\partial_{5}L\cdot\left(\mathcal{D}^{\alpha_{2}}_{a+}h\right)^{*}(t)\right]dt,\end{split}$

where we omit the arguments
$\left(t,x(t),\dot{x}(t),\left(\mathcal{D}^{\alpha_{1}}_{a+}x\right)(t),\left(\mathcal{D}^{\alpha_{2}}_{a+}x\right)^{*}(t)\right)$
of $\partial_{i}L$, for $i=2,\ldots,5$. Using the classical formula of
integration by parts, and having in mind that $h(a)=h(b)=0$, it follows that

(3.33)
$\int_{a}^{b}\partial_{3}L\cdot\dot{h}(t)\,dt=-\int_{a}^{b}\frac{d}{dt}\left(\partial_{3}L\right)\cdot
h(t)\,dt.$

On the other hand, our Proposition 3.2 and Theorem 3.14 allow us to write that

(3.34)
$\begin{split}\int_{a}^{b}\partial_{4}L\cdot\left(\mathcal{D}^{\alpha_{1}}_{a+}h\right)(t)\,dt&\stackrel{{\scriptstyle\eqref{eq:13b}}}{{=}}\int_{a}^{b}\left(\left(\partial_{4}L\right)^{*}\right)^{*}\cdot\left(\mathcal{D}^{\alpha_{1}}_{a+}h\right)(t)\,dt\\\
&\stackrel{{\scriptstyle\eqref{eq:20}}}{{=}}-\int_{a}^{b}\left(\mathcal{D}^{\alpha_{1}}_{a+}\left(\partial_{4}L\right)^{*}\right)^{*}\cdot
h(t)\,dt,\end{split}$

while from Theorem 3.14 and Proposition 3.4 we obtain that

(3.35)
$\begin{split}\int_{a}^{b}\partial_{5}L\cdot\left(\mathcal{D}^{\alpha_{2}}_{a+}h\right)^{*}(t)\,dt&\stackrel{{\scriptstyle\eqref{eq:20}}}{{=}}-\int_{a}^{b}\mathcal{D}^{\alpha_{2}}_{a+}\left(\partial_{5}L\right)\cdot
h^{*}(t)\,dt\\\
&\stackrel{{\scriptstyle\eqref{eq:prop:04}}}{{=}}-\int_{a}^{b}\left(\mathcal{D}^{\alpha_{2}}_{a+}\left(\partial_{5}L\right)\right)^{*}\cdot
h(t)\,dt.\end{split}$

Substituting (3.33), (3.34) and (3.35) into (3.32), we conclude that

(3.36)
$\int_{a}^{b}\left[\partial_{2}L-\frac{d}{dt}\left(\partial_{3}L\right)-\left(\mathcal{D}^{\alpha_{1}}_{a+}\left(\partial_{4}L\right)^{*}\right)^{*}-\left(\mathcal{D}^{\alpha_{2}}_{a+}\left(\partial_{5}L\right)\right)^{*}\right]\cdot
h(t)\,dt=0.$

We obtain the intended necessary optimality condition (3.31) applying the
fundamental lemma of the calculus of variations to (3.36). ∎

As an application of our new proposed calculus of variations, in Section 4 we
give the first example of the literature of a fractional mechanical Lagrangian
$L$ involving left fractional derivatives only, for which the respective
Euler–Lagrange equation coincides with the motion of a dissipative/damped
system.

##  4\. A nonconservative/dissipative equation of motion

Let $\alpha_{1},\alpha_{2}\in[0,1]$ with $\alpha_{1}+\alpha_{2}=1$ and the
Lagrangian $L$ be given by

(4.1)
$L\left(t,x,\dot{x},\mathcal{D}^{\alpha_{1}}_{a+}x,\left(\mathcal{D}^{\alpha_{2}}_{a+}x\right)^{*}\right)=\frac{1}{2}m\dot{x}^{2}-U(x)+\frac{1}{2}c\left(\mathcal{D}^{\alpha_{1}}_{a+}x\right)\cdot\left(\mathcal{D}^{\alpha_{2}}_{a+}x\right)^{*}.$

In this case, one has

$\begin{split}\partial_{2}L&=-U^{\prime}(x),\\\ \partial_{3}L&=m\dot{x},\\\
\partial_{4}L&=\frac{1}{2}c\left(\mathcal{D}^{\alpha_{2}}_{a+}x\right)^{*}\Rightarrow\left(\partial_{4}L\right)^{*}=\frac{1}{2}c\left(\mathcal{D}^{\alpha_{2}}_{a+}x\right),\\\
\partial_{5}L&=\frac{1}{2}c\left(\mathcal{D}^{\alpha_{1}}_{a+}x\right),\end{split}$

and thus the Euler–Lagrange equation (3.31) takes the form

(4.2)
$-U^{\prime}(x)-\frac{d}{dt}\left(m\dot{x}\right)-\frac{1}{2}c\left(\mathcal{D}^{\alpha_{1}}_{a+}\left(\mathcal{D}^{\alpha_{2}}_{a+}x\right)\right)^{*}-\frac{1}{2}c\left(\mathcal{D}^{\alpha_{2}}_{a+}\left(\mathcal{D}^{\alpha_{1}}_{a+}x\right)\right)^{*}=0.$

It follows from Proposition 3.16 that (4.2) is equivalent to

(4.3) $U^{\prime}(x)+c(\dot{x})^{*}+m\ddot{x}=0,$

that is, we have obtained an equation of motion (4.3) with a nonconservative
Rayleigh term [6].

##  5\. Conclusion

In 1996, Riewe has proved that a calculus of variations can be formulated to
include derivatives of fractional (non-integer) order [14]. By doing that,
Riewe has shown possible to define Lagrangians, involving both left and right
fractional derivatives, that lead directly to equations of motion with
nonconservative forces, such as friction, circumventing Bauer’s corollary that
“The equations of motion of a dissipative linear dynamical system with
constant coefficients are not given by a variational principle” [5]. Here we
continued the development of the fractional-derivative calculus of variations
by providing a completely new perspective to the subject. Our main
contributions are: (i) new formulas of integration by parts that involve left
fractional operators only; (ii) a new fractional calculus of variations where
both the Lagrangian and respective Euler–Lagrange equation involve left
fractional derivatives only; (iii) a new example of a Lagrangian for which the
respective Euler–Lagrange equation coincide with the equation of motion of a
dissipative system. Such contributions are radically different from other
results found in the vast literature on the subject. Indeed, to the best of
our knowledge, (i) all available formulas of fractional integration by parts
involve both left and right fractional operators; (ii) all available forms of
the fractional calculus of variations for which the Lagrangian involves left
fractional derivatives result in Euler–Lagrange equations involving a right
fractional derivative; (iii) all available examples of fractional Lagrangians
giving rise to equations of motion of a dissipative system involve both left
and right fractional derivatives.

In this paper we have restricted ourselves to ideas and to the central result
of any calculus of variations: the celebrated Euler–Lagrange equation, which
is a first-order necessary optimality condition. Of course our results can be
extended in many different ways and directions. We trust that the new
perspective introduced here marks the beginning of a fruitful road for
fractional mechanics, the fractional calculus of variations and fractional
optimal control.

## Statements and Declarations

### Funding

This research was supported by The Portuguese Foundation for Science and
Technology (FCT) and the Center for Research and Development in Mathematics
and Applications (CIDMA), under Grant Agreement No UIDB/04106/2020
(https://doi.org/10.54499/UIDB/04106/2020).

### Availability of data and materials

Not applicable.

### Competing interests

The author has no competing interests to declare.

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