\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 106, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2013/106\hfil Existence of solutions]
{Existence of solutions to boundary value problems arising
from the fractional \\ advection dispersion equation}
\author[L. Kong \hfil EJDE-2013/106\hfilneg]
{Lingju Kong} % in alphabetical order
\address{Lingju Kong \newline
Department of Mathematics,
the University of Tennessee at Chattanooga,
Chattanooga, TN 37403, USA}
\email{Lingju-Kong@utc.edu}
\thanks{Submitted December 17, 2012. Published April 24, 2013.}
\subjclass[2000]{34B15, 34A08}
\keywords{Three solutions; fractional boundary value problem;
\hfill\break\indent variational methods}
\begin{abstract}
We study the existence of multiple solutions to the
boundary value problem
\begin{gather*}
\frac{d}{dt}\Big(\frac12{}_0D_t^{-\beta}(u'(t))+\frac12{}_tD_T^{-\beta}(u'(t))
\Big)+\lambda \nabla F(t,u(t))=0,\quad t\in [0,T],\\
u(0)=u(T)=0,
\end{gather*}
where $T>0$, $\lambda>0$ is a parameter, $0\leq\beta<1$, ${}_0D_t^{-\beta}$
and ${}_tD_T^{-\beta}$ are, respectively, the left and right Riemann-Liouville
fractional integrals of order $\beta$,
$F: [0,T]\times\mathbb{R}^N\to\mathbb{R}$ is a given function.
Our interest in the above system arises from studying the steady fractional
advection dispersion equation.
By applying variational methods, we obtain sufficient conditions
under which the above equation has at least three solutions.
Our results are new even for the special case when $\beta=0$.
Examples are provided to illustrate the applicability of our results.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}\label{sec1}
In recent years, the subject of fractional calculus has gained considerable
popularity and importance due mainly to its applications in numerous
seemingly diverse and widespread fields of science and engineering.
The monographs \cite{kst, m, p} are excellent sources for
the theory and applications of fractional calculus.
In this article, we study the existence of three solutions to
fractional boundary value problems (BVPs) of the form
\begin{equation}\label{1.1}
\begin{gathered}
\frac{d}{dt}\Big(\frac12{}_0D_t^{-\beta}(u'(t))+\frac12{}_tD_T^{-\beta}(u'(t))
\Big)+\lambda \nabla F(t,u(t))=0,\quad t\in[0,T],\\
u(0)=u(T)=0,
\end{gathered}
\end{equation}
where $T>0$, $\lambda>0$ is a parameter, $0\leq\beta<1$, ${}_0D_t^{-\beta}$
and ${}_tD_T^{-\beta}$ are the left and right
Riemann-Liouville fractional integrals of order $\beta$, respectively,
$N\geq 1$ is an integer,
$F: [0,T]\times\mathbb{R}^N\to\mathbb{R}$ is a given function
such that $F(t,\mathbf{x})$ is measurable in $t$ for each
$\mathbf{x}=(x_1,\dots,x_N)\in\mathbb{R}^N$ and
continuously differentiable in $\mathbf{x}$ for a.e. $t\in [0,T]$,
$F(t,0,\dots,0)\equiv 0$ on $[0,T]$, and
$\nabla F(t,\mathbf{x})=(\partial F/\partial x_1,\dots,\partial F/\partial x_N)$
is the gradient of $F$ at $\mathbf{x}$.
By a solution of \eqref{1.1}, we mean an absolutely continuous
function $u: [0,T]\to\mathbb{R}^N$ such that $u(t)$
satisfies both equation for a.e. $t\in [0,T]$ and
the boundary conditions in \eqref{1.1}.
We notice that when $\beta=0$, problem \eqref{1.1} has the form
\begin{equation}\label{1.2}
\begin{gathered}
u''(t)+\lambda \nabla F(t,u(t))=0,\quad t\in [0,T],\\
u(0)=u(T)=0,
\end{gathered}
\end{equation}
which has been extensively studied.
The equation in \eqref{1.1} is motivated by the steady fractional advection
dispersion equation studied in \cite{er},
\begin{equation}\label{1.3}
-D\, a\, (p\, {}_0D_t^{-\beta}+q\, {}_tD_T^{-\beta})Du+b(t)Du+c(t)u=f,
\end{equation}
where $D$ represents a single spatial derivative, $0\leq p, q\leq 1$
satisfying $ p+q = 1$, $a>0$ is a constant, and $b$, $c$, $f$ are functions
satisfying some suitable conditions.
The interest in \eqref{1.3} arises from its application as a model for physical
phenomena exhibiting anomalous diffusion; i.e., diffusion not accurately
modeled by the usual advection dispersion equation.
Anomalous diffusion has been used in modeling turbulent flow \cite{clz, swk},
and chaotic dynamics of classical conservative systems \cite{zsw}.
The reader may find more background information and applications
on \eqref{1.3} in \cite{bwm, er}.
\begin{remark}\label{r1.1} \rm
When $N=1$, problem \eqref{1.1} reduces to the scalar BVP
\begin{equation}\label{1.4}
\begin{gathered}
\frac{d}{dt}\Big(\frac12{}_0D_t^{-\beta}(u'(t))+\frac12{}_tD_T^{-\beta}(u'(t))
\Big)+\lambda f(t,u(t))=0,\quad t\in[0,T],\\
u(0)=u(T)=0,
\end{gathered}
\end{equation}
where $f: [0,T]\times\mathbb{R}\to\mathbb{R}$ is such that $f(t,x)$
is measurable in $t$ for each
$x\in \mathbb{R}$ and continuous in $x$ for a.e. $t\in [0,T]$.
It is clear that the equation in \eqref{1.4}
is of the form of \eqref{1.3} with $D=d/dt$, $a=1$,
$p=q=1/2$, $b(t)=c(t)=0$, and $f=\lambda f(t,u)$.
We also notice that since \eqref{1.3} is the steady fractional advection
dispersion equation, it has no dependence on the time variable
and it just depends on the space variable $t$ (here, the notation $t$
stands for the space variable in \eqref{1.3}).
Since the space we studied is one dimensional and has the form of
an interval, say $[0,T]$, the boundary conditions in the space reduce
to the conditions at the two endpoints $t=0$ and $t=T$ of the interval.
In our system, we study the Dirichlet type boundary conditions.
\end{remark}
In recent years, the existence of solutions of various BVPs of fractional
differential equations is under strong research. For a small sample of
the work on this subject,
we refer the reader to \cite{aos, bl, en, g, GKK, GKY, jz, zshz}.
We remark that most existing results on fractional BVPs were obtained
by using various fixed point theorems
and that few results were established by using variational methods.
This is because that it is often very difficult to establish a suitable
space and variational functional for fractional BVPs.
As pointed out in \cite{er, jz}, these difficulties are mainly caused by
the following properties of fractional differential operators:
(i) fractional differential operators are not local operators, and
(ii) the adjoint of a fractional differential operator is not the negative
of itself.
Recently, in \cite{er, jz} suitable fractional derivative
spaces and variational
structures for the system \eqref{1.1} were developed.
Moreover, the existence of at least one solution for the system
\eqref{1.1} was established in \cite{jz} by using the minimax methods
in critical point theory.
Our goal in this paper is to obtain some sufficient conditions to
guarantee that the system \eqref{1.1} has at least three solutions.
Our analysis is mainly based on a recent three critical
points theorem appeared in \cite{ab, bc}, see Lemma \ref{l4.1} below.
This lemma and its various variations have been frequently used to obtain
multiplicity theorems for nonlinear problems of
variational nature. See, for example, \cite{ab,bb1,bb,bc,R1,R} and
the references therein.
The rest of this article is organized as follows.
Section \ref{sec2} contains some preliminaries on fractional calculus,
Section \ref{sec3} contains the main results of this paper and
two illustrative examples, and the
proofs of the main results are presented in Section \ref{sec4}.
\section{Preliminaries on fractional calculus}\label{sec2}
To make this paper self-contained, in this section, we recall some basic
definitions and properties of the fractional calculus.
The presentation here and more information on fractional calculus can be
found in, for example, \cite{kst, p}.
\begin{definition}\label{d2.1}\rm
Let $f$ be a function defined on $[a,b]$ and $\gamma>0$.
The left and right Riemann-Liouville
fractional integrals of order $\gamma$ for the function $f$,
denoted respectively by ${}_aD_t^{-\gamma}$ and ${}_tD_b^{-\gamma}$,
are defined by
\begin{gather*}
_aD_t^{-\gamma}f(t)=\frac{1}{\Gamma(\gamma)}\int_a^t(t-s)^{\gamma-1}f(s)ds,\quad t\in [a,b],
\\
_tD_b^{-\gamma}f(t)=\frac{1}{\Gamma(\gamma)}\int_t^b(s-t)^{\gamma-1}f(s)ds,\quad t\in [a,b],
\end{gather*}
provided the right-hand sides are pointwise defined on $[a,b]$, where
$\Gamma>0$ is the gamma function.
\end{definition}
\begin{remark}\label{r2.1} \rm
When $\gamma=n\in\mathbb{N}$, $_aD_t^{-\gamma}f(t)$ and $_tD_b^{-\gamma}f(t)$ coincide
with the $n$th integrals of the form
\begin{gather*}
_aD_t^{-n}f(t)=\frac{1}{(n-1)!}\int_a^t(t-s)^{n-1}f(s)ds,\quad t\in [a,b],
\\
_tD_b^{-n}f(t)=\frac{1}{(n-1)!}\int_t^b(s-t)^{n-1}f(s)ds,\quad t\in [a,b].
\end{gather*}
\end{remark}
\begin{definition}\label{d2.2} \rm
Let $f$ be a function defined on $[a,b]$ and $\gamma>0$.
The left and right Riemann-Liouville fractional derivatives of order $\gamma$
for the function $f$,
denoted respectively by $_aD_t^{\gamma}$ and $_tD_b^{\gamma}$, are defined by
\begin{equation*}
_aD_t^{\gamma}f(t)=\frac{d^n}{dt^n}{}_aD_t^{\gamma-n}f(t)=\frac{1}{\Gamma(n-\gamma)}
\frac{d^n}{dt^n}\Big(\int_a^t(t-s)^{n-\gamma-1}f(s)ds\Big),
\end{equation*}
and
\begin{equation*}
_tD_b^{\gamma}f(t)=(-1)^n\frac{d^n}{dt^n}{}_tD_b^{\gamma-n}f(t)
=\frac{1}{\Gamma(n-\gamma)}(-1)^n\frac{d^n}{dt^n}
\Big(\int_t^b(s-t)^{n-\gamma-1}f(s)ds\Big),
\end{equation*}
where $t\in [a,b]$, $n-1\leq\gamma0$. For $(t,x,y)\in [0,T]\times\mathbb{R}^2$, let $F(t,x,y)=tG(x,y)$,
where $G: \mathbb{R}^2\to\mathbb{R}$ satisfies that $G(-x,-y)=G(x,y)$, and that
for $x\in [0,\infty)$ and $y\in\mathbb{R}$,
\begin{equation}\label{3.24}
G(x,y)=\begin{cases}
x^3+|y|^3, & 0\leq x\leq 1,\; 0\leq |y|\leq 1,\\
x^3+2|y|^{3/2}-1, & 0\leq x\leq 1,\; |y|>1,\\
2x^{3/2}+|y|^3-1, & x> 1,\; 0\leq |y|\leq 1,\\
2x^{3/2}+2|y|^{3/2}-2, & x> 1,\; |y|>1.
\end{cases}
\end{equation}
It is easy to verify that $F : [0,T]\times\mathbb{R}^2\to\mathbb{R}$
is measurable in $t$ for
$(x,y)\in\mathbb{R}^2$ and continuously differentiable in $x$
and $y$ for $t\in [0,T]$, and $F(t,0,0)\equiv 0$ on $[0,T]$.
Let $0\leq\beta <1$, $\alpha=1-\beta/2\in (1/2,1]$, $\rho_{\alpha}$
be defined by \eqref{3.1}, and $u(t)=(u_1(t),u_2(t))$.
We claim that for each
$$
\lambda\in \Big(\frac{\rho_{\alpha}(1+\cos^2(\pi\alpha))}{T^2|\cos(\pi\alpha)|}, \infty\Big),
$$
the system
\begin{equation}\label{3.25}
\begin{gathered}
\frac{d}{dt}\Big(\frac12{}_0D_t^{-\beta}(u'(t))
+\frac12{}_tD_T^{-\beta}(u'(t))\Big)+\lambda \nabla F(t,u(t))=0,\quad
t\in [0,T],\\
u(0)=u(T)=0,
\end{gathered}
\end{equation}
has at least three solutions.
In fact, system \eqref{3.25} is a special case of system \eqref{1.1} with $N=2$.
For $01$, in view of \eqref{3.24}, we have
\begin{gather}\label{3.26}
\frac{\int_0^TF(t,c,c)dt}{c^2}=\frac{2c^3\int_0^Ttdt}{c^2}=T^2c,\\
\label{3.27}
\frac{\int_0^TF(t,p,p)dt}{p^2}=\frac{(4p^{3/2}-2)\int_0^Ttdt}{p^2}
=\frac{T^2(2p^{3/2}-1)}{p^2}.
\end{gather}
Choose $d=1$. Then,
\begin{equation}\label{3.38}
\int_{T/4}^{3T/4}F(t,d,d)dt=2\int_{T/4}^{3T/4}tdt=\frac{1}{2}T^2.
\end{equation}
By \eqref{3.26}--\eqref{3.38}, we see that there exist $01$ such that \eqref{3.12}, \eqref{3.15}, and \eqref{3.16}
hold for any $0p^*$.
Moreover, \eqref{3.13} and \eqref{3.14} hold for any $c,p>0$.
Finally, note from \eqref{3.17} and \eqref{3.18} that
\begin{gather*}
\underline{\lambda}_2=\frac{\rho_{\alpha}(1+\cos^2(\pi\alpha))}{T^2|\cos(\pi\alpha)|},\\
\overline{\lambda}_2\to\infty\quad \text{as}\ c\to 0^+\ \text{and}\ p\to\infty.
\end{gather*}
Then, the claim follows from Corollary \ref{c3.2}.
\end{example}
\begin{example}\label{e3.2}\rm
Let $ F: \mathbb{R}^2\to\mathbb{R}$ satisfies that $F(-x,-y)=F(x,y)$,
and that for $x\in [0,\infty)$ and $y\in\mathbb{R}$,
\begin{equation}\label{3.29}
F(x,y)=\begin{cases}
x^3, & 0\leq x\leq 1,\; 0\leq |y|\leq 1,\\
x^3+2|y|^{3/2}-3|y|+1, & 0\leq x\leq 1,\; |y|>1,\\
2x^{3/2}-1, & x> 1,\; 0\leq |y|\leq 1,\\
2x^{3/2}+2|y|^{3/2}-3|y|, & x> 1,\; |y|>1.
\end{cases}
\end{equation}
It is easy to verify that $F : \mathbb{R}^2\to\mathbb{R}$ is continuously
differentiable in $x$ and $y$ and $F(0,0)=0$.
Let $T>0$ and $u(t)=(u_1(t),u_2(t))$. We claim that for each
$\lambda\in (32/T^2, \infty)$, the system
\begin{equation}\label{3.30}
\begin{gathered}
u''(t)+\lambda \nabla F(u(t))=0,\quad t\in [0,T],\\
u(0)=u(T)=0
\end{gathered}
\end{equation}
has at least three solutions.
In fact, the system \eqref{3.30} is a special case of the
system \eqref{1.2} with $N=2$.
For $01$, from \eqref{3.29}, we have
\begin{gather}\label{3.31}
\frac{F(c,c)}{c^2}=\frac{c^3}{c^2}=c,\\
\label{3.32}
\frac{F(p,p)}{p^2}=\frac{4p^{3/2}-3p}{p^2}=\frac{4p^{1/2}-3}{p}.
\end{gather}
Choose $d=1$. Then
\begin{equation}\label{3.33}
\frac{F(d,d)}{32Nd^2}=\frac{1}{64}\quad \text{and}\quad
\frac{F(d,d)}{64Nd^2}=\frac{1}{128}.
\end{equation}
By \eqref{3.31}--\eqref{3.33}, we see that there exist $01$ such that \eqref{3.19} and \eqref{3.22} hold for
any $0p^*$.
Moreover, \eqref{3.13} and \eqref{3.14} hold for any $c,p>0$.
Finally, note from \eqref{3.23} that
\begin{equation*}
\underline{\lambda}_3=\frac{32}{T^2}\quad \text{and}\quad
\overline{\lambda}_3\to\infty\quad \text{as}\ c\to 0^+\ \text{and}\ p\to\infty
\end{equation*}
Then, the claim follows from Corollary \ref{c3.3} and Remark \ref{r3.1}.
\end{example}
\begin{remark}\label{r3.3} \rm
As noted in Remark \ref{r3.2}, one of the three solutions in the conclusions
of the above examples may be trivial.
\end{remark}
\section{Proofs of the main results}\label{sec4}
Let $X$ be nonempty set and $\Phi, \tilde{\Psi}: X\to\mathbb{R}$ be two functionals.
For $r, r_1, r_2, r_3\in\mathbb{R}$ with $r_1<\sup_X\Phi$,
$r_2>\inf_X\Phi$, $r_2>r_1$, and $r_3>0$, we define
\begin{gather}\label{4.1}
\varphi(r):=\inf_{u\in\Phi^{-1}(-\infty,r)}\frac{\big(\sup_{u\in \Phi^{-1}(-\infty,r)}
\tilde{\Psi}(u)\big)-\tilde{\Psi}(u)}{r-\Phi(u)}, \\
\label{4.2}
\beta(r_1,r_2):=\inf_{u\in\Phi^{-1}(-\infty,r_1)}\sup_{v\in\Phi^{-1}[r_1,r_2)}
\frac{\tilde{\Psi}(v)-\tilde{\Psi}(u)}{\Phi(v)-\Phi(u)}, \\
\label{4.3}
\gamma(r_2,r_3):=\frac{\sup_{u\in\Phi^{-1}(-\infty,r_2+r_3)}\tilde{\Psi}(u)}{r_3},\\
\label{4.4}
\alpha(r_1, r_2, r_3):=\max\left\{\varphi(r_1), \varphi(r_2), \gamma(r_2, r_3)\right\}.
\end{gather}
The following lemma is fundamental in our proofs. The reader may refer
to \cite[Theorem 5.2]{ab} or \cite[Theorem 3.3]{bc} for its proof.
\begin{lemma}\label{l4.1}
Let $X$ be a reflexive real Banach space, $\Phi: X\to\mathbb{R}$ be a convex, coercive,
and continuously G\^{a}teaux differentiable functional whose G\^{a}teaux
derivative admits a continuous inverse on $X^*$,
$\tilde{\Psi}: X\to\mathbb{R}$ be a continuously G\^{a}teaux differentiable
functional whose G\^{a}teaux derivative is compact, such that
\begin{itemize}
\item[(a)] $\inf_X\Phi=\Phi(0)=\tilde{\Psi}(0)=0$,
\item[(b)] for every $u_1$, $u_2$ satisfying $\tilde{\Psi}(u_1)\geq 0$ and
$\tilde{\Psi}(u_2)\geq 0$, one has
\begin{equation*}
\inf_{t\in [0,1]}\tilde{\Psi}\left(tu_1+(1-t)u_2\right)\geq 0.
\end{equation*}
\end{itemize}
Assume further that there exist three positive constants $r_1$, $r_2$,
and $r_3$, with $r_10$.
For any $u\in E^{\alpha}$, from the first inequality in \eqref{4.16}, we see that
$\|u\|_{\alpha}^2\leq{2\Phi(u)}/{|\cos(\pi\alpha)|}$.
Then, by \eqref{4.7} and \eqref{4.8}, we have
\begin{equation*}
\|u\|_{\infty}^2\leq\frac{T^{2\alpha-1}}{\Gamma^2(\alpha)(2\alpha-1)}\|u\|_{\alpha}^2
\leq \frac{2T^{2\alpha-1}\Phi(u)}{\Gamma^2(\alpha)(2\alpha-1)|\cos(\pi\alpha)|}.
\end{equation*}
Thus, by \eqref{4.20} and \eqref{4.21}, we have the following implications
\begin{equation}\label{4.22}
\begin{gathered}
\Phi(u)\frac{\Gamma^2(\alpha)\cos^2(\pi\alpha)(2\alpha-1)}{T^{2\alpha-1}\rho_{\alpha}d^2}
\int_{T/4}^{3T/4}F(t,\mathbf{d})dt
-\frac{\cos^2(\pi\alpha)}{c^2}\int_0^TF(t,\mathbf{c})dt \\
&>\frac{\Gamma^2(\alpha)\cos^2(\pi\alpha)(2\alpha-1)}{T^{2\alpha-1}
\rho_{\alpha}d^2}\int_{T/4}^{3T/4}F(t,\mathbf{d})dt \\
&\quad -\frac{\Gamma^2(\alpha)\cos^4(\pi\alpha)(2\alpha-1)}{T^{2\alpha-1}
\rho_{\alpha}d^2(1+\cos^2(\pi\alpha))}
\int_{T/4}^{3T/4}F(t,\mathbf{c})dt \\
&= \frac{\Gamma^2(\alpha)\cos^2(\pi\alpha)(2\alpha-1)}{T^{2\alpha-1}
\rho_{\alpha}d^2(1+\cos^2(\pi\alpha))}
\int_{T/4}^{3T/4}F(t,\mathbf{d})dt.
\end{aligned}
\end{equation}
By \eqref{3.15} and \eqref{4.28}--\eqref{4.30}, we see that
\eqref{3.7}--\eqref{3.8} hold.
From \eqref{3.10}, \eqref{3.11}, \eqref{3.17}, \eqref{3.18},
and \eqref{4.30}, we
have $\underline{\lambda}<\underline{\lambda}_2$ and $\overline{\lambda}=\overline{\lambda}_2$.
Therefore, the conclusion now follows from Theorem \ref{t3.1}.
\end{proof}
\begin{proof}[Proof of Corollary \ref{c3.3}]
When $\alpha=1$, from \eqref{3.1}, we have $\rho_{\alpha}=8N/T$.
Under the assumptions of Corollary \ref{c3.3}, it is easy to see that
all the conditions of Corollary \ref{c3.2} hold for $\alpha=1$.
Note that system \eqref{1.2} is a special case of system \eqref{1.1}
with $\alpha=1$.
The conclusion then follows directly from Corollary \ref{c3.2}.
\end{proof}
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