DRAFT 6

# Parabolic frequency monotonicity on the conformal Ricci flow

Abimbola Abolarinwa  Department of Mathematics, University of Lagos, Akoka,
Lagos State, Nigeria<EMAIL_ADDRESS>and  Shahroud Azami  Department
of Pure Mathematics, Faculty of Science, Imam Khomeini International
University, Qazvin, Iran<EMAIL_ADDRESS>
(Date: April 6, 2023)

###### Abstract.

This paper is devoted to the investigation of the monotonicity of parabolic
frequency functional under conformal Ricci flow defined on a closed Riemannian
manifold of constant scalar curvature and dimension not less than $3$.
Parabolic frequency functional for solutions of certain linear heat equation
coupled with conformal pressure is defined and its monotonicity under the
conformal Ricci flow is proved by applying Bakry-Émery Ricci curvature bounds.
Some consequences of the monotonicity are also presented.

###### Key words and phrases:

Frequency functional; Conformal Ricci flow; Drifting Laplacian;
Monotonicity;weighted measure

###### 2010 Mathematics Subject Classification:

53C21, 53E20, 35K65,58J35

###### Contents

  1. 1 Introduction and main results
    1. 1.1 Frequency functionals
    2. 1.2 Conformal Ricci flow
    3. 1.3 Main results
  2. 2 Notation and Preliminaries
  3. 3 Proof of main theorem and its applications
  4. 4 General parabolic equations

##  1\. Introduction and main results

Frequency functional (elliptic and parabolic) monotonicity has been a hot
topic in analysis of partial differential equations and differential geometry
since the classical paper [1] owing to its several applications. On the other
hand, geometric flows of time dependent Riemmanian metrics such as the Ricci
flow and its various extensions and modification have been widely studied
since the seminal paper [14] for their numerous topological, geometric and
physical applications. For better and improved topological and geometric
results, conformal Ricci flow has been introduced in a recent work [11] as a
modification of the Ricci flow. The principal aim of the present paper is to
extend monotonicity of parabolic frequency functional for linear heat equation
to the setting of compact (without boundary) Riemannian maniofld evolving by
the conformal Ricci flow, and then investigate what geometric condition(s)
is/are required for such monotonicity and as well as its possible
applications. The precise definitions, history, applications of, and relevant
literature on frequency functional and conformal Ricci flow are discussed in
what follows.

###  1.1. Frequency functionals

Let $q$ be a fixed point in $\mathbb{R}^{n}$ and $\mathcal{B}(q,r)$ be a ball
of radius $r>0$ centred at $q$. Almgren [1] introduced the frequency
functional (which is known in literature as elliptic frequency functional) for
a harmonic function $v(x)$ on $\mathbb{R}^{n}$, i.e. ,
$\Delta_{\mathbb{R}^{n}}v(x)=0\ \ \text{on}\ \mathbb{R}^{n}$, as follows

(1.1) $\displaystyle E_{F}(r)=\frac{r\int_{\mathcal{B}(q,r)}|\nabla
v(x)|^{2}dx}{\int_{\partial\mathcal{B}(q,r)}|v(x)|^{2}dS},$

where $\partial\mathcal{B}(q,r)$ and $dS$ are respectively the boundary of
$\mathcal{B}(q,r)$ and the induced $(n-1)$-dimensional Hausdorff measure on
$\partial\mathcal{B}(q,r)$. Here, $\nabla$ and $\Delta_{\mathbb{R}^{n}}$ are
gradient and Laplace operators on $\mathbb{R}^{n}$, respectively. The rate of
growth of harmonic function $v$ near the fixed point $q$ is determined by the
functional in (1.1). Furthermore, Almgren [1] proved that $E_{F}(r)$ is
monotone nondecreasing with respect to $r$, the consequence of which led to
the study of the local regularity of harmonic functions and minimal surfaces.

Since the work of Almgren [1], the monotonicity of elliptic functional
$E_{F}(r)$ have been successfully applied in the analysis of more general
elliptic and parabolic partial differential equations, and there have been
also considerable generalization to Riemannian manifolds. We mention but a few
literature: Garofalo and Lin [12, 13] investigated the unique continuation
properties for elliptic operators by using the monotonicity of frequency
functional on Riemannian manifolds. The authors in [16, 20, 21] applied
monotonicity of the frequency functional to estimate the size of nodal and
critical sets of solutions to elliptic and parabolic equations. Colding and
Minicozzi in [9] applied frequency monotonicity to prove finite dimensionality
of the space of polynomial growth of harmonic functions on manifolds with
nonnegative Ricci curvature and Euclidean volume growth, while in [10] they
extended the result to the case of static manifold using drifting Laplacian.
The counterpart of $E_{F}(r)$ for solutions to the heat equation on
$\mathbb{R}^{n}$ is called parabolic frequency functional, which was first
introduced by Poon [26] to the study of the unique continuation of solutions
to parabolic equations on $\mathbb{R}^{n}$. Consider a smooth solution
$u=u(x,t)$ to the heat equation

(1.2) $\displaystyle\partial_{t}u-\Delta_{M}u=0\ \ \ \text{in}\ M\times[0,T],$

where $(M,g)$ is a complete Riemannian manifold, $\Delta_{M}$ is the Laplace-
Beltrami operator on $M$ and $T>0$. The Parabolic frequency for $u$ is defined
by [25] (see also [26])

(1.3) $\displaystyle P_{F}(t)=t\cdot\frac{\int_{M}|\nabla
u|^{2}(x,T-t)H(x,y;t)d\mu(g)}{\int_{M}u^{2}(x,T-t)H(x,y;t)d\mu(g)},$

where $H(x,y;t)$ is the fundamental solution to the heat equation (1.2), $y$
being a referenced point (not so important) in $M$, and $d\mu(g)$ is the
volume form with respect to the Riemannian metric $g$. Restricting $M$ to
possessing nonnegative sectional (or bisectional for holomorphic function)
curvature and parallel Ricci curvature, Poon [26] and Ni [25] proved that
$P_{F}(t)$ is monotone nondecreasing by using Hamilton’s matrix Harnack
estimate [15]. Recently, these results have been generalized to more general
Riemannian manifolds by Li and Wang [27].

Let $\tau(t)$ be the backward time, $\kappa(t)$ be the time-dependent function
and $d\nu$ be the weighted measure. For a solution $u(t)$ of the heat
equation, Baldauf and Kim [5] defined the parabolic frequency as follows

$U(t)=-\frac{\tau(t)||\nabla
u||_{L^{2}(d\nu)}^{2}}{||u||_{L^{2}(d\nu)}^{2}}e^{-\int\frac{1-\kappa(t)}{\tau(t)}dt}.$

In the above definition, the exponential term involving time-dependent
function $\kappa$ serves as a correction term which depends on the geometry of
the flow, analogous to the error term involving $r$ in the elliptic case. The
authors [5] proved that parabolic frequency $U(t)$ for the solution of heat
equation is monotone increasing along the Ricci flow with bounded Bakry-Émery
Ricci curvature. See also the recent preprints [3, 4, 6, 18] for related
results under the Ricci-Bourguignon, Ricci-harmonic and mean curvature flows.
Motivated by the above cited works, we study monotonicity of a well defined
parabolic frequency function (see ((1.11)) below) for a form of linear heat
equation defined in ((1.10)) along the conformal Ricci flow. This study is
more interesting since conformal Ricci flow performs better than (and even
complementary to) Ricci flow in searching for certain geometric features, and
has wider applications in conformal geometry of constant scalar curvature.

###  1.2. Conformal Ricci flow

The conformally modified Ricci flow was introduced by Fischer in [11] and
named conformal Ricci flow as a result of the role played by conformal
geometry in the derivation of its equations. Precisely, let $(M,g_{0})$ be a
smooth $n$-dimensional $(n\geq 3)$ closed connected manifold together with
Riemannian metric $g_{0}$ of constant scalar curvature $R_{0}$. The conformal
Ricci flow is defined by a one-parameter family of metric $g(t)$ satisfying
the following parabolic system

(1.4) $\begin{cases}\displaystyle\frac{\partial g(t)}{\partial
t}+2\left(Ric(t)-\frac{R_{0}}{n}g(t)\right)=-2p(t)g(t),\ &(x,t)\in
M\times(0,T),\\\ \quad R_{g(t)}=R_{0},\ &(x,t)\in M\times[0,T),\end{cases}$

together with the initial condition $g(0)=g_{0}$ and a family of function
$p(t)$, $t\in[0,T)$, where $Ric(t)$ and $R_{g(t)}$ are the Ricci tensor and
scalar curvature of the evolving metric $g(t)$, respectively. By the
constraint equation $R_{g(t)}=R_{0}$ in system (1.4), the flow is known to
preserve constant scalar curvature of the evolving metric. Indeed, this
accounts for naming the function, $p=p(t)$, conformal pressure, since it
serves as time-dependent Lagrange multiplier and makes the term $-p(t)g(t)$
acting as the constraint force necessary to preserve the scalar curvature
constraint. Consequently, $p(t)$ is known to solve a time-dependent elliptic
partial differential equation under the flow

(1.5) $(n-1)\Delta p+R_{0}p=-\left|Ric-\frac{R_{0}}{n}g\right|^{2}\hskip
28.45274pt\text{in}\ \ M\times[0,T).$

Considering the role of the conformal pressure, the function $p(t)$ is
expected to be zero at an equilibrium point and strictly positive otherwise.
Hence, the equilibrium points of the conformal Ricci flow are characterised by
Einstein metrics, and the term $(Ric-\frac{R_{0}}{n}g)$ can then be regarded
as a measure of deviation from an equilibrium point.

Since the volume of a Riemannian manifold $(M,g)$ is a positive real number
and the scalar curvature is a real-valued function on $M$, the constraint on
$R_{g(t)}$ is considerably more drastic than the volume constraint of the
Ricci flow [14, 15]. The Ricci flow in general does not preserve the property
of constant scalar curvature. Thus, the configuration space of the conformal
Ricci flow equations is considerably smaller than that of the Ricci flow.
Obviously, conformal Ricci flow will perform better in searching for certain
geometric features since working on a smaller configuration space is more
advantageous than working on a larger configuration. More concrete
similarities and differences between conformal Ricci flow and the classical
Ricci flow, as well as some possible applications of conformal Ricci flow to
$3$-manifold geometry and conformal geometry are highlighted in [11].
Fischer’s paper [11] has presented a proof of the short-time existence and
uniqueness of the conformal Ricci flow on closed manifolds with negative
constant scalar curvature $R_{0}<0$. In that same paper, he also observed that
Yamabe constant is strictly increasing along the flow on negative Yamabe type
closed manifolds. For detail discussion on Yamabe problem, interested readers
can consult the book by Aubin [2]. The following references [7, 8, 17, 19, 22,
23, 24] can be found for further studies on conformal Ricci flow.

###  1.3. Main results

Denote the partial derivative of any time-dependent quantity by $\partial_{t}$
(i.e., $\partial_{t}u(x,t)=u_{t}(x,t)$). Consider a one-parameter family of
metrics $g=g(t)$, $t\in[0,T)$, $T>0$, on an $(m+1)$-dimensional closed
manifold with initial metric $g(0)$ having constant scalar curvature $-m(m+1)$
which is preserved under the flow as $R_{g(t)}\equiv-m(m+1)$. Referring to
(1.4) and (1.5), one sees that $(M^{m+1},g(t),p(t))$ evolves by conformal
Ricci flow given in the following system

(1.9)
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\partial_{t}g=-2(Ric(t)+(m+p(t))g(t))&\text{on}\
\ M^{m+1}\times[0,T),\\\
\displaystyle(-\Delta_{g(t)}+(m+1))p(t)=\frac{1}{m}|Ric+mg|^{2}&\text{on}\ \
M^{m+1}\times[0,T),\\\ g(0)=g_{0}.\end{array}\right.$

Denote the Laplace-Betrami operator with respect to $g(t)$ by $\Delta_{g(t)}$.
For a smooth function $v$ and a time-dependent nonnegative scalar non-
dynamical field (conformal pressure) $p(t)$ satisfying the second equation in
system (1.9), we consider the following linear heat equation

(1.10) $v_{t}(x,t)-\Delta_{g(t)}v(x,t)=\bar{p}(t)v(x,t),\ \ \ v(0,x)=v_{0},$

where $\displaystyle\bar{p}(t)=\max_{x\in M}p(t)$, on $M\times[0,T)$. Note
that the conformal pressure $p(t)$ is finite [11, 24] (see also Lemma 3.1
below), and $\bar{p}(t)$ can be thought of as its upper bound. Thus,
$\bar{p}(t)$ is viewed as material-constant (or space-constant) pressure
function. Following [18] we define the parabolic frequency functional $Q(t)$
for the solutions of heat equation (1.10) along the conformal Ricci flow as
follows

(1.11)
$Q(t)=\frac{h(t)\int_{M}|\nabla_{g(t)}v|_{g(t)}^{2}dV_{g(t)}}{\int_{M}v^{2}dV_{g(t)}}e^{-\int_{t_{0}}^{t}\left(2p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds}\
$

where $h$ and $k$ are smooth functions with respect to time-variable
$t\in[t_{0},t]\subset[0,T)$, and $dV_{g(t)}$ is the weighted measure on
$(M,g(t))$ (see the appropriate definition of $dV_{g(t)}$ in ((2.4)) below).
The involvement of a finite time-dependent scalar function $p$ in the
exponential term of the functional is natural: First, to reflect the coupling
of the solution of the heat equation with the conformal pressure, and
secondly, to show the conformal nature of $Q(t)$ in preserving the scalar
curvature constraint. (See [11] for complete description of the pressure field
$p(t)$).

In this paper we prove monotonicity of (1.11) along the flow (1.9) coupled
with the heat equation (1.10) and then obtain our main result. Let
$\mathscr{R}ic_{f}$ and $\mathscr{L}_{f}$ be Bakry-Émery Ricci curvature
tensor and drifting Laplacian, respectively.

######  Theorem 1.1.

Let $(M^{m+1},g(t),p(t)),\,\,t\in[0,T)$ be a solution to the conformal Ricci
flow (1.9) with $\mathscr{R}ic_{f}\leq(\frac{k(t)}{2h(t)}+\frac{R_{g}}{m+1})g$
and $R_{g}=-m(m+1)$.

  1. (i)

If $h(t)$ is positive (i.e., $h(t)>0$), then the parabolic frequency $Q(t)$ is
monotone nonincreasing along the conformal Ricci flow.

  2. (ii)

If $h(t)$ is negative (i.e., $h(t)<0$), then the parabolic frequency $Q(t)$ is
monotone nondecreasing along the conformal Ricci flow.

Furthermore, $Q^{\prime}(t)=0$ only if $v$ is an eigenfunction of
$\mathscr{L}_{f}$ satisfying $-\mathscr{L}_{f}v=c(t)v$.

As an application we have the following corollaries.

######  Corollary 1.2.

(Backward uniqueness). Assuming the hypotheses of Theorem 1.1 hold. If
$h(t)<0$ and $v(\cdot,b)=0$, then $v(\cdot,t)=0$ for any
$t\in[a,b)\subset(0,T)$, $a<b$.

######  Corollary 1.3.

(Eigenvalue monotonicity). Define the first nonzero eigenvalue of
$(M^{m+1},g(t),p(t))$ with respect to the drifting Laplacian on weighted
measure $dV_{g(t)}$ by

$\displaystyle\lambda(g(t)):=\inf\frac{\int_{M}-\langle\mathscr{L}_{f}u,u\rangle
dV_{g(t)}}{\int_{M}u^{2}dV_{g(t)}}:\ u\in W^{1,2}(M^{m+1})\setminus\\{0\\}\
\int_{M}udV_{g(t)}=0,$

where $u(t)$ solves the linear heat equation (1.10). Suppose
$(M^{m+1},g(t),p(t)),\,\,t\in[0,T)$ solves (1.9) with
$\mathscr{R}ic_{f}\leq(\frac{k(t)}{2h(t)}+\frac{R_{g}}{m+1})g$ and
$R_{g}=-m(m+1)$. Then for any $t\in[t_{0},t_{1}]\subset(0,T)$

  1. (i)

If $h(t)>0$ then $h(t)\lambda(g(t))$ is a monotone decreasing function.

  2. (ii)

If $h(t)<0$ then $h(t)\lambda(g(t))$ is a monotone increasing function.

######  Remark 1.4.

The Bakry-Émery Ricci condition
$\mathscr{R}ic_{f}\leq(\frac{k(t)}{2h(t)}+\frac{R_{g}}{m+1})g$ is equivalent
to $\mathscr{R}ic_{f}\leq(\frac{k(t)}{2h(t)}-m)g$ by the assumption that
$R_{g}=-m(m+1)$ is preserved by the conformal Ricci flow. Note that the
equation $\mathscr{R}ic_{f}-\frac{1}{m+1}R_{g}=0$ is the quasi-Einstein
equation since $R_{g}$ is constant, which in this can be compared with Ricci
solitons.

######  Remark 1.5.

The above result (Theorem 1.1) is expected to have further applications in the
setting of homogeneous $3$-manifold geometry and conformal geometry of
dimension greater than $3$ in the spirit of Fischer [11].

Lastly, we will consider some more general parabolic equation coupled with
nondynamic conformal pressure $p(t)$

(1.12) $\displaystyle|(\partial_{t}-\Delta)u|\leq\bar{p}(t)(|u|+|\nabla u|)$

along the conformal Ricci flow. The frequency $Q(t)$ for $u$ needs not to be
monotone but its derivative will be suitably bounded yielding backward
uniqueness of solution when $h(t)>0$ (see Theorem 4.1).

The outline of the rest part of this paper is as follows: The next section
(Section 2 is devoted to other notation and some preliminaries. The proof of
main result and its applications are discussed in Section 3. The last section
(Section 4) is basically on the proof of backward uniqueness of solution to
(1.12).

##  2\. Notation and Preliminaries

Other notation that will be required in the sequel is presented first. Let the
dimension of the underlying manifold $M$ be a number $m+1$ not less than $3$.
Let $t$ be the abstract time parameter in the half-closed region $[0,T),T>0$.
The scalar curvature, Ricci curvature and volume element of $(M,g(t))$ are
respectively denoted by $R=R_{g(t)}$, $Ric=Ric(g(t))$ and $d\mu=d\mu_{g(t)}$.
In local coordinates $(x^{1},\cdots,x^{m+1})$,

$d\mu_{g(t)}:=\sqrt{\text{det}g_{ij}(t)}dx^{1}\wedge\cdots\wedge dx^{m+1}.$

Let $\nabla=\nabla_{g(t)}$ and $\Delta=\Delta_{g(t)}$ be the Levi-Civita
connection and the Laplace-Beltrami operator with respect to $g(t)$. Denote
$|\cdot|_{g(t)}=g(t)(\cdot,\cdot)^{\frac{1}{2}}$ called $g(t)$-metric norm,
e.g., $|\nabla u|^{2}_{g(t)}=\langle\nabla u,\nabla u\rangle_{g(t)}$, where
$\langle\cdot,\cdot\rangle_{g(t)}$ is the inner product with respect to metric
$g(t)$.

The time-dependent drifting Laplacian or $f$-Laplacian (also called weighted
Laplacian) for a smooth function $f$ on $M$ is denoted by

$\mathscr{L}_{f}(\cdot):=\mathscr{L}_{g(t),f}(\cdot):=e^{f}\text{div}(e^{-f}\nabla(\cdot))=\Delta(\cdot)-\langle\nabla
f,\nabla(\cdot)\rangle.$

The weighted form of the Ricci curvature tensor is the so called Bakry-Émery
curvature tensor

$\mathscr{R}ic_{f}(t):=Ric(t)+\text{Hess}f,$

where $\text{Hess}f$ is the Hessian of function $f$. Note that having a
condition of the form $\mathscr{R}ic_{f}=\kappa g$ is saying that $(M,g(t))$
is a Ricci soliton, which is a special solution to the Ricci flow [15] and
very useful in the singularity analysis of the Ricci flow. These notations
with and without subscript $g(t)$ are used interchangeably without resulting
to any confusion. It is well know that $\mathscr{L}_{f}$ and
$\mathscr{R}ic_{f}$ are related via the weighted (or drifting) Bochner formula
for an atleast $C^{3}$-function $h$

(2.1) $\frac{1}{2}\mathscr{L}_{f}(|\nabla h|^{2})=|\text{Hess}\
h|^{2}+\langle\nabla h,\nabla\mathscr{L}_{f}h\rangle+\mathscr{R}ic_{f}(\nabla
h,\nabla h).$

Along the flow (1.4) on a closed manifold of dimension $\geq 3$, we let
$\tau(t)=T-t$ be the backward time and define the following conjugate heat
equation on $(M^{m+1},g(t),p(t))$ as

(2.2) $\displaystyle\partial_{t}H(t)=-\Delta_{g(t)}H(t)+(m+1)p(t)H(t)$

with the fundamental solution

$\displaystyle H(t)=(4\pi\tau(t))^{-\frac{m+1}{2}}e^{-f(t)}.$

One can then show that $f(t)$ satisfies the conjugate heat equation

(2.3)
$\displaystyle\partial_{t}f(t)=-\Delta_{g(t)}f(t)+|\nabla_{g(t)}f(t)|^{2}-(m+1)\left(p(t)-\frac{1}{2\tau(t)}\right).$

Define the weighted volume form as

$\displaystyle
dV_{g(t)}=H(t)d\mu_{g(t)}=(4\pi\tau(t))^{-\frac{m+1}{2}}e^{-f(t)}d\mu_{g(t)}$

satisfying $\int_{M}dV_{g(t)}=1$. Recall that $d\mu_{g(t)}$ evolves under
conformal Ricci flow (1.9) by the formula (see [11, 19])

(2.4) $\displaystyle\partial_{t}(d\mu_{g(t)})=-(m+1)p(t)d\mu_{g(t)}.$

Therefore one can compute that

(2.5)
$\displaystyle\partial_{t}(dV_{g(t)})=(H_{t}-(m+1)p(t))d\mu_{g(t)}=-\frac{\Delta_{g(t)}H(t)}{H(t)}dV_{g(t)}.$

For a smooth function $v:M\times[t_{0},t_{1}]\to\mathbb{R}$ with
$v(\cdot,t),\partial_{t}v(\cdot,t)\in W_{0}^{2,2}(dV)$ for any
$t\in[t_{0},t_{1}]\subset[0,T)$, we define quantities $I(t)$ and $E(t)$ as
follows

(2.6) $\displaystyle I(t)=\int_{M}v^{2}dV_{g(t)},$ (2.7) $\displaystyle
E(t)=-h(t)\int_{M}\langle v,\mathscr{L}_{f}v\rangle
dV_{g(t)}=h(t)\int_{M}|\nabla v|_{g(t)}^{2}dV_{g(t)}.$

To this end, referring to definitions in (2.6) and (2.7) and reverting to
(1.11), $Q(t)$ is thus written as

(2.8) $\displaystyle
Q(t)=\frac{E(t)}{I(t)}e^{-\int_{t_{0}}^{t}\left(2p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds},$

where $h(t)$ and $k(t)$ are both smooth functions with respect to time-
variable $t\in[0,T)$.

##  3\. Proof of main theorem and its applications

At first, we state some important results that will be required in the proof
of the main theorem. Here and in what follows, by the virtues of (1.9) and
(1.10), the triple $(g(t),p(t),v(t))$, $t\in[0,T)$ solves the following system

(3.5)
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\partial_{t}g=-2(Ric(t)+(m+p(t))g(t)),\\\
\displaystyle(-\Delta+(m+1))p(t)=\frac{1}{m}|Ric+mg|^{2},\\\
\displaystyle\partial_{t}v-\Delta_{g(t)}v=\bar{p}(t)v,\\\
\displaystyle(g(0),v(0))=(g_{0},v_{0})\end{array}\right.$

on closed connected manifold of dimension $m+1\geq 3$ with constant scalar
curvature. Since the conformal pressure $p(t)$ is a non-dynamical field, no
initial value of $\bar{p}(t)=\max_{x\in M}p(t)$ is required.

######  Lemma 3.1.

(Finiteness of $p(t)$). Let $(g(t),p(t))$, $t\in[0,T)$ be a smooth solution to
the conformal Ricci flow on a closed manifold with $R_{0}=-m(m+1)$ satisfying
$|Ric|(x,t)\leq K(t)$ for all $(x,t)\in M^{m+1}\times[0,T)$. Then the
conformal pressure $p(t)$ satisfies $0\leq p(t)\leq K^{2}(t)$, for all
$t\in[0,T)$, that is, $p(t)$ is finite under the flow $g(t)$.

###### Proof.

We know that $p(t)$ solves the elliptic equation (i.e., the second equation)
in system (3.5) on $M^{m+1}\times[0,T)$. So by the strong maximum principle,
$p(t)\geq 0$ for all $(x,t)\in M^{m+1}\times[0,T)$. The following conclusion
can be reached: either (a) $p(t)=0$ and $Ric+mg=0$ or (b) $p(t)>0$ and
$Ric+mg\neq 0$ (see [11, Proposition 3.3] for detail).

Suppose $(x_{0},t)$ is the maxmum point, that is $p(x_{0},t):=\max_{x\in
M}p(x,t)$, we have $\nabla p(x_{0},t)=0$ and $\Delta p(x_{0},t)\leq 0$. Hence

$\displaystyle
m(m+1)p(x_{0},t)\leq|Ric+mg|^{2}(x_{0},t)=|Ric|^{2}(x_{0},t)-m^{2}(m+1)\leq
K^{2}(t).$

Hence, $0\leq p(x,t)\leq K^{2}(t)$ for all $(x,t)\in M^{m+1}\times[0,T)$.

∎

######  Lemma 3.2.

Let $(g(t),p(t),v(t))$, $t\in[0,T)$ solves the system (3.5). The following
identities hold:

(3.6) $\displaystyle\partial_{t}\left(|\nabla
v|^{2}_{g(t)}\right)=2(Ric+mg(t))(\nabla v,\nabla v)+2(p(t)+\bar{p}(t))|\nabla
v|^{2}_{g(t)}+2\langle\nabla v,\nabla\Delta v\rangle,$ (3.7)
$\displaystyle\left(\partial_{t}-\Delta_{g(t)}\right)|\nabla
v|^{2}_{g(t)}=2(p(t)+\bar{p}(t))|\nabla v|^{2}_{g(t)}+2mg(t)(\nabla v,\nabla
v)-2|\text{Hess}\ v|_{g(t)},$

where $\bar{p}(t):=\max_{x\in M}p(t)$.

###### Proof.

Following the standard computation under geometric flow we have

$\displaystyle\partial_{t}(|\nabla v|^{2})=-[\partial_{t}g](\nabla v,\nabla
v)+2\langle\nabla v,\nabla\partial_{t}v\rangle.$

Reverting to system (3.5), we have values for the quantities
$-[\partial_{t}g]$ and $\partial_{t}v$, which when substituted into the last
expression together with the fact that $\nabla\bar{p}(t)=0$ yields (3.6).
Combining (3.5) with the classical Bochner formula proves (3.7).

∎

######  Lemma 3.3.

For all $u,v\in W^{1,2}_{0}(dV_{g(t)})$, the drifting Laplacian
$\mathscr{L}_{f(t)}$ satisfies integration by parts formula, i.e.,

$\displaystyle\int_{M}u\mathscr{L}_{f(t)}vdV_{g(t)}=-\int_{M}\langle\nabla
u,\nabla v\rangle_{g(t)}dV_{g(t)},$

and it is self-adjoint with respect to the weighted measure $dV_{g(t)}$, i.e.,

$\displaystyle\int_{M}u\mathscr{L}_{f(t)}vdV_{g(t)}=\int_{M}(\mathscr{L}_{f(t)}u)vdV_{g(t)}.$

###### Proof.

Recall that $\mathscr{L}_{f(t)}v:=e^{f(t)}\text{div}(e^{-f(t)}\nabla v)$ and
$dV_{g(t)}:=(4\pi\tau(t))^{-\frac{m+1}{2}}e^{-f(t)}d\mu_{g(t)}$. Direct
computation using classical integration by parts gives

$\displaystyle\int_{M}u\mathscr{L}_{f}v\ dV$
$\displaystyle=(4\pi\tau)^{-\frac{m+1}{2}}\int_{M}ue^{f}\text{div}(e^{-f}\nabla
v)e^{-f}d\mu$
$\displaystyle=-(4\pi\tau)^{-\frac{m+1}{2}}\int_{M}e^{-f}\langle\nabla
u,\nabla v\rangle d\mu:=-\int_{M}\langle\nabla u,\nabla v\rangle dV$
$\displaystyle=(4\pi\tau)^{-\frac{m+1}{2}}\int_{M}\text{div}(e^{-f}\nabla
u)vd\mu$
$\displaystyle=(4\pi\tau)^{-\frac{m+1}{2}}\int_{M}e^{f}\text{div}(e^{-f}\nabla
u)ve^{-f}d\mu:=\int_{M}(\mathscr{L}_{f}u)v\ dV.$

That is,

$\displaystyle\int_{M}u\mathscr{L}_{f}v\ dV=-\int_{M}\langle\nabla u,\nabla
v\rangle dV=\int_{M}(\mathscr{L}_{f}u)v\ dV.$

∎

######  Lemma 3.4.

(Drifting Reilly formula). For any $v\in W^{2,2}_{0}(dV_{g(t)})$, then

(3.8) $\displaystyle\int_{M}|\rm{Hess}\
v|^{2}_{g(t)}=\int_{M}\left[(\mathscr{L}_{f(t)}v)^{2}-\mathscr{R}ic_{f}(\nabla
v,\nabla v)\right]dV_{g(t)}.$

###### Proof.

Applying the drifting Bochner formula (2.1), integration by parts and self-
adjoint properties of $\mathscr{L}_{f}$ with respect to the weighted measure
$dV$, we have

$\displaystyle 0=\frac{1}{2}\int_{M}\mathscr{L}_{f}(|\nabla v|^{2})\
dV=\int_{M}\left[|\text{Hess}\
v|^{2}-(\mathscr{L}_{f}v)^{2}+\mathscr{R}ic_{f}(\nabla v,\nabla v)\right]dV$

which is the required formula. ∎

### Proof of Theorem 1.1

We only give the proof of Part (i) since the proof of Part (ii) is similar.
The procedure is to compute derivatives of $I(t)$ and $E(t)$, and then use
Cauchy-Schwarz inequality to show that $Q^{\prime}(t)$ is nonpositive.

###### Proof.

Suppose $v$ solves the heat equation in the system (3.5), i.e., $v_{t}=\Delta
v+\bar{p}(t)v$. In consideration of evolution of the weighted measure in
(2.5), integration by parts formula and the identity $\Delta v^{2}=2v\Delta
v+2|\nabla v|^{2}$, we first compute the derivative of $I(t)$ (recall that
$I(t)$ is defined in (2.6)) as follows

$\displaystyle I^{\prime}(t)$
$\displaystyle=\int_{M}\left(2vv_{t}-v^{2}\frac{\Delta H}{H}\right)dV$
$\displaystyle=\int_{M}\left(2v(v_{t}-\Delta v)-2|\nabla v|^{2}\right)dV$
$\displaystyle=2\bar{p}(t)\int_{M}v^{2}dv-2\int_{M}|\nabla
v|^{2}dV=2\bar{p}I(t)-\frac{2}{h(t)}E(t).$

Similarly, we can compute derivative of the energy $E(t)$ (recall that $E(t)$
is defined in (2.7)) as follows

(3.9) $\displaystyle E^{\prime}(t)=h^{\prime}(t)\int_{M}|\nabla
v|^{2}dV+h(t)\int_{M}(\partial_{t}-\Delta)|\nabla v|^{2}dV.$

Applying (3.7) of Lemma 3.2 into (3.9), we obtain

$\displaystyle E^{\prime}(t)\leq(h^{\prime}(t)+4h(t)\bar{p}(t))\int_{M}|\nabla
v|^{2}dV+2h(t)\int_{M}\left[mg(t)(\nabla v,\nabla v)-|\text{Hess}\
v|^{2}\right]dV$

due to the condition $h(t)>0$ and the fact that $p(t)\leq\bar{p}(t)$.
Combining the last equation with drifting Reilly formula of Lemma 3.4 we get

$\displaystyle E^{\prime}(t)\leq(h^{\prime}(t)$
$\displaystyle+4h(t)\bar{p}(t))\int_{M}|\nabla v|^{2}dV$
$\displaystyle-2h(t)\int_{M}\left|\mathscr{L}_{f}v|^{2}-(\mathscr{R}ic_{f}+mg(t))(\nabla
v,\nabla v)\right]dV,$

which leads to the following inequality

$\displaystyle
E^{\prime}(t)\leq\left[4\bar{p}(t)+\frac{k(t)+h^{\prime}(t)}{h(t)}\right]E(t)-2h(t)\int_{M}|\mathscr{L}_{f}v|^{2}dV$

by invoking the Bakry-Émery curvature bound
$\mathscr{R}ic_{f}\leq(\frac{k(t)}{2h(t)}+\frac{R_{g}}{m+1})g$. Proceeding
from here, we can express the frequency function $Q(t)$ in terms of rescaled
quantities $\widetilde{I}(t)$ and $\widetilde{E}(t)$ as follows

$\displaystyle Q(t)=\frac{\widetilde{E}(t)}{\widetilde{I}(t)}$

such that

$\widetilde{I}(t)=I(t)e^{-\int_{t_{0}}^{t}2p(s)ds}$

and

$\displaystyle\widetilde{E}(t)=E(t)e^{-\int_{t_{0}}^{t}\left(4p(s)+\frac{k(s)+h^{\prime}(s)}{h(s)}\
\right)ds}$

with their respective derivatives computed as follows

$\widetilde{I}^{\prime}(t)\geq-\frac{2}{h(t)}E(t)e^{-\int_{t_{0}}^{t}2p(s)ds}$

and

$\displaystyle\widetilde{E}^{\prime}(t)\leq-2h(t)e^{-\int_{t_{0}}^{t}\left(4p(s)+\frac{k(s)+h^{\prime}(s)}{h(s)}\right)ds}\int_{M}(\mathscr{L}_{f}v)^{2}dV.$

Now using the bound for $\widetilde{E}(t)$ will allow the derivative of the
frequency functional $Q(t)$ to be bounded. Suppose $h(t)>0$ and denote by
$\Gamma(t)$ the following integral

$\Gamma(t):=\int_{t_{0}}^{t}\left(4p(s)+\frac{k(s)+h^{\prime}(s)}{h(s)}\right)ds.$
$\displaystyle\widetilde{I}^{2}(t)Q^{\prime}(t)$
$\displaystyle=\widetilde{I}(t)\widetilde{E}^{\prime}(t)-\widetilde{I}^{\prime}(t)\widetilde{E}(t)$
$\displaystyle\leq
e^{-\Gamma(t)}\left[-2h(t)I(t)\int_{M}(\mathscr{L}_{f}v)^{2}dV+\frac{2}{h(t)}E^{2}(t)\right]$
$\displaystyle=-2h(t)e^{-\Gamma(t)}\left[I(t)\int_{M}(\mathscr{L}_{f}v)^{2}dV-\left(\frac{1}{h(t)}E(t)\right)^{2}\right]$
$\displaystyle=-2h(t)e^{-\Gamma(t)}\left[\left(\int_{M}v^{2}dV\right)\left(\int_{M}(\mathscr{L}_{f}v)^{2}dV\right)-\left(\int_{M}|\nabla
v|^{2}dV\right)^{2}\right]$ $\displaystyle\leq 0.$

The first inequality is due to the bound for $\widetilde{E}(t)$ whilst the
last inequality is due to integration by parts formula and Cauchy-Schwarz
inequality. That is,

$\displaystyle\int_{M}|\nabla v|^{2}dV$ $\displaystyle=-\int_{M}\langle
v,\mathscr{L}_{f}v\rangle dV\leq\int_{M}|v||\mathscr{L}_{f}v|dV$
$\displaystyle\leq\left(\int_{M}|v|^{2}dV\right)^{\frac{1}{2}}\left(\int_{M}|\mathscr{L}_{f}v|^{2}dV\right)^{\frac{1}{2}}$

which implies

(3.10)
$\displaystyle\left(\int_{M}|v|^{2}dV\right)\left(\int_{M}|\mathscr{L}_{f}v|^{2}dV\right)-\left(\int_{M}|\nabla
v|^{2}dV\right)^{2}\geq 0.$

Hence, $Q(t)$ is a nonincreasing function along the conformal Ricci flow if
$h(t)>0$. The proof for the case of $h(t)<0$ follows suit.

Moreover, suppose $Q^{\prime}(t)=0$, the equality in Cauchy schwarz inequality
(3.10) implies $-\mathscr{L}_{f}v=c(t)v$, where
$c(t)=\frac{1}{h(t)}Q(t)e^{\int_{t_{0}}^{t}\left(2p(s)+^{\prime}\frac{h^{\prime}(s)+k(s)}{h(s}\right)ds}$.

∎

### Proof of Corollary 1.2

Recall that $I^{\prime}(t)=-2\bar{p}(t)I(t)-\frac{2}{h(t)}E(t)$. Therefore

$\displaystyle\frac{d}{dt}\left(\log I(t)\right)=\frac{I^{\prime}(t)}{I(t)}$
$\displaystyle=2\bar{p}(t)-\frac{2}{h(t)}\frac{E(t)}{I(t)}$
$\displaystyle=2\bar{p}(t)-\frac{2}{h(t)}Q(t)e^{\int_{t_{0}}^{t}\left(2p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds}.$

Integrating both sides of the last equation on $[a,b]\subset[t_{0},t_{1})$ and
using the monotonicity of $Q$ in Theorem 1.1, we have

$\displaystyle\log I(b)-\log I(a)$ $\displaystyle\geq
2\int_{a}^{b}\bar{p}(t)dt-2\int_{a}^{b}\frac{Q(t)}{h(t)}e^{\int_{t_{0}}^{t}\left(2p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds}dt$
$\displaystyle\geq
2\int_{a}^{b}\bar{p}(t)dt-2Q(a)\int_{a}^{b}\frac{1}{h(t)}e^{\int_{t_{0}}^{t}\left(2p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds}dt.$

Exponentiating yields (since $p(t)$ and $h(t)$ are finite)

$\displaystyle\frac{I(b)}{I(a)}\geq\exp\left\\{2\int_{a}^{b}\bar{p}(t)dt-2Q(a)\int_{a}^{b}\frac{1}{h(t)}e^{\int_{t_{0}}^{t}\left(2p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds}dt\right\\}.$

Therefore, if $v(\cdot,b)=0$, then $I(b)=0$ and the last inequality implies
$I(a)=0$. Since $a$ is arbitrary we conclude that $I(t)=0$ for any
$t\in[a,b]\subset(0,T)$. Hence $v(\cdot,t)=0$ for any $t\in[a,b]\subset(0,T)$.

∎

### Proof of Corollary 1.3

Given the time interval $[t_{0},t]\subset(0,T)$. Let $(g(t),p(t),v(t))$
$t\in[0,T)$ solves (3.5), that is, $v(t)$ solves $v_{t}=\Delta v+\bar{p}(t)v$.
By the hypothesis, $v(\cdot,t_{0})$ is the eigenfunction of
$-\mathscr{L}_{f(t_{0})}$ corresponding to the first eigenvalue
$\lambda(t_{0}):=\lambda(g(t_{0}))$, thus;
$-\mathscr{L}_{f(t_{0})}v(\cdot,t_{0})=\lambda(t_{0})v(\cdot,t_{0})$. Then we
have the frequency functional $Q(t)$ for $v(t)$ by (2.8) as follows

$\displaystyle Q(t):=\frac{-h(t)\int_{M}\langle v,\mathscr{L}_{f}v\rangle
dV_{g(t)}}{\int_{M}v^{2}dV}e^{-\int_{t_{0}}^{t}\left(2p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds}.$

Based on Theorem 1.1, we know that $Q(t)$ is nonincreasing for $h(t)>0$ and
nondecreasing for $h(t)<0$. To this end, we refer to the definition of
$\lambda(t):=\lambda(g(t))$ (see the statement of the corollary) and have for
any $h(t)>0$ and any $t\in[t_{0},t_{1}]\subset(0,T)$ that

$\displaystyle h(t_{0})\lambda(t_{0})=Q(t_{0})\geq Q(t)\geq
h(t)\lambda(t)e^{-\int_{t_{0}}^{t}\left(2p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds}.$

As a consequence of this, it is clear that $h(t)\lambda(t)$ is monotone
decreasing for $h(t)>0$. The second part of the corollary can be proved in a
similar way.

∎

##  4\. General parabolic equations

In this section we consider some more general parabolic equation coupled with
non-dynamic conformal pressure $p(t)$ along the conformal Ricci flow. Here the
parabolic frequency $Q(t)$ in relation to solution $v$ of

(4.1) $\displaystyle|(\partial_{t}-\Delta)v|\leq\bar{p}(t)(|v|+|\nabla v|),$

where $\displaystyle\bar{p}(t)=\max_{x\in M}p(t)$, along the conformal Ricci
flow needs to be monotone but its derivative can be suitably bounded to imply
backward uniqueness of solution when $h(t)>0$. The main theorem of this
section is the following:

######  Theorem 4.1.

Let $v:M^{m+1}\times[t_{0},t_{1}]\to\mathbb{R}$, $t_{0}<t_{1}$ satisfy (4.1)
along the conformal Ricci flow $(M^{m+1},g(t),p(t))$, $t\in[t_{0},t_{1}]$ with
$\mathscr{R}ic_{f}\leq(\frac{k(t)}{2h(t)}+\frac{R_{g}}{m+1})g$ and
$R_{g}=-m(m+1)$. If $v(\cdot,t_{1})=0$, then $v(\cdot,t)\equiv 0$ for all
$t\in[t_{0},t_{1}]$.

As in Section 2, define $\tau(t)=T-t$ to be the backward time and the
conjugate fundamental solution at some point $(x,t)$ to be

$\displaystyle H(t)=(4\pi\tau(t))^{-\frac{m+1}{2}}e^{-f(t)}$

on $(M^{m+1},g(t),p(t))$ so that the weighted measure remains $dV=Hd\mu$.

For a smooth function $v:M\times[t_{0},t_{1}]\to\mathbb{R}$ with
$v(\cdot,t),\partial_{t}v(\cdot,t)\in W_{0}^{2,2}(dV)$ and for any
$t\in[t_{0},t]\subset[0,T)$, we define the quantities $I(t)$, $E(t)$ and the
parabolic functional $Q(t)$ as in (2.6)–(2.8).

######  Proposition 4.2.

Let $v:M^{m+1}\times[t_{0},t_{1}]\to\mathbb{R}$ satisfy (4.1) along the
conformal Ricci flow $(M^{m+1},g(t),p(t))$, $t\in[t_{0},t_{1}]$ with
$\mathscr{R}ic_{f}\leq(\frac{k(t)}{2h(t)}+\frac{R_{g}}{m+1})g$ and
$R_{g}=-m(m+1)$. Then

(4.2) $\displaystyle(\log
I(t))^{\prime}\geq-3\bar{p}(t)-\frac{\bar{p}(t)+2}{h(t)}Q(t)e^{\int_{t_{0}}^{t}\left(2p(s)+^{\prime}\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds},$
(4.3) $\displaystyle Q^{\prime}(t)\leq\bar{p}^{2}(t)\left[Q+h(t_{0})\right]$

and

(4.4) $\displaystyle\bar{p}^{2}(t)\geq[\log(Q+h(t_{0})]^{\prime}.$

###### Proof.

By direct differentiation and integration by parts formula we have

$\displaystyle I^{\prime}(t)$
$\displaystyle=\int_{M}\left(2vv_{t}-v^{2}\frac{\Delta
H}{H}\right)dV=\int_{M}\left(2v(v_{t}-\Delta v)-2|\nabla v|^{2}\right)dV$
$\displaystyle\geq-2\bar{p}(t)\int_{M}|v|(|v|+|\nabla v|)dV-2\int_{M}|\nabla
v|^{2}dV$ $\displaystyle=-2\bar{p}I(t)-2\bar{p}(t)\int_{M}|v||\nabla
v|dV-\frac{2}{h(t)}E(t)$
$\displaystyle\geq-3\bar{p}(t)I(t)-\bar{p}(t)\int_{M}|\nabla
v|^{2}dV-\frac{2}{h(t)}E(t)$
$\displaystyle=-3\bar{p}(t)I(t)-\frac{\bar{p}(t)+2}{h(t)}E(t)$
$\displaystyle=-3\bar{p}(t)I(t)-\frac{\bar{p}(t)+2}{h(t)}I(t)Q(t)e^{\int_{t_{0}}^{t}\left(2p(s)+^{\prime}\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds},$

where we have used (4.1) to get the first inequality, and an elementary
inequality of the form $2rs\leq r^{2}+s^{2}$ to get the second inequality.
This prove (4.2).

Applying integration by parts formula again we have

$\displaystyle I^{\prime}(t)$
$\displaystyle=\int_{M}\left(2vv_{t}-v^{2}\frac{\Delta H}{H}\right)dV$
$\displaystyle=\int_{M}\left(2vv_{t}-2v\Delta v-2|\nabla v|^{2}\right)dV$
$\displaystyle=2\int_{M}v\left(\partial_{t}-\Delta+\mathscr{L}_{f}\right)vdV$
$\displaystyle=2\int_{M}v\left[\mathscr{L}_{f}+\frac{1}{2}(\partial_{t}-\Delta)\right]vdV+\int_{M}v(\partial_{t}-\Delta)vdV.$

Letting

$\displaystyle\widetilde{I}(t)=I(t)e^{-\int_{t_{0}}^{t}2p(s)ds}.$

Then

(4.5) $\displaystyle\widetilde{I}^{\prime}(t)$
$\displaystyle=\Bigg{\\{}2\int_{M}v\left[\mathscr{L}_{f}+\frac{1}{2}(\partial_{t}-\Delta)\right]vdV+\int_{M}v(\partial_{t}-\Delta)vdV$
$\displaystyle\hskip
56.9055pt-2p(t)I(t)\Bigg{\\}}e^{-\int_{t_{0}}^{t}2p(s)ds}.$

Note that we can also write $E(t)$ as follows

$\displaystyle E(t)$ $\displaystyle=-h(t)\int_{M}v\mathscr{L}_{f}vdV$ (4.6)
$\displaystyle=-h(t)\int_{M}v\left[\mathscr{L}_{f}+\frac{1}{2}(\partial_{t}-\Delta)\right]vdV+\frac{h(t)}{2}\int_{M}v(\partial_{t}-\Delta)vdV.$

Along the conformal Ricci flow, and with the aid of the classical Bochner
formula, we have

$\displaystyle(\partial_{t}-\Delta)|\nabla v|^{2}$
$\displaystyle=2mg(t)(\nabla v,\nabla v)+2p(t)|\nabla v|^{2}$ (4.7)
$\displaystyle\hskip 28.45274pt-2|\text{Hess}\ v|^{2}+2\langle\nabla
v,\nabla(\partial_{t}-\Delta)v\rangle.$

Computing the derivative of $E(t)$ using (4) we have

$\displaystyle E^{\prime}(t)$ $\displaystyle=h^{\prime}(t)\int_{M}|\nabla
v|^{2}dV+h(t)\int_{M}(\partial_{t}-\Delta)|\nabla v|^{2}dV$
$\displaystyle=h(t)\left[\frac{h^{\prime}(t)}{h(t)}\int_{M}|\nabla
v|^{2}dV+\int_{M}(\partial_{t}-\Delta)|\nabla v|^{2}dV\right]$
$\displaystyle=h(t)\int_{M}\Big{[}\frac{h^{\prime}(t)+2h(t)p(t)}{h(t)}|\nabla
v|^{2}+2mg(t)(\nabla v,\nabla v)$ $\displaystyle\hskip
56.9055pt-2|\text{Hess}\ v|^{2}+2\langle\nabla
v,\nabla(\partial_{t}-\Delta)v\rangle\Big{]}dV.$

Applying integration by parts formula and drifting Reilly formula (Lemma 3.4
gives

$\displaystyle E^{\prime}(t)$
$\displaystyle=h(t)\int_{M}\Big{[}\frac{h^{\prime}(t)+2h(t)p(t)}{h(t)}|\nabla
v|^{2}-2|\mathscr{L}_{f}v|^{2}+2(\mathscr{R}ic_{f}+mg)(\nabla v,\nabla v)$
$\displaystyle\hskip
56.9055pt-2(\mathscr{L}_{f}v)(\partial_{t}-\Delta)v\Big{]}dV.$

Applying the Bakry-Émery Ricci curvature bound
$\mathscr{R}ic_{f}\leq(\frac{k(t)}{2h(t)}+\frac{R_{g}}{m+1})g$ with
$R_{g}=-m(m+1)$ gives

$\displaystyle E^{\prime}(t)$
$\displaystyle\leq-2h(t)\int_{M}\left[|\mathscr{L}_{f}v|^{2}+(\mathscr{L}_{f}v)(\partial_{t}-\Delta)v\right]dV$
$\displaystyle\hskip 28.45274pt+(k(t)+h^{\prime}(t)+2h(t)p(t))\int_{M}|\nabla
v|^{2}dV$
$\displaystyle=-2h(t)\int_{M}\left[\left|\left(\mathscr{L}_{f}+\frac{1}{2}\Big{(}\partial_{t}-\Delta\Big{)}\right)v\right|^{2}-\frac{1}{4}\Big{|}\Big{(}\partial_{t}-\Delta\Big{)}v\Big{|}^{2}\right]dV$
$\displaystyle\hskip
28.45274pt+\left[2p(t)+\frac{h^{\prime}(t)+k(t)}{h(t)}\right]E(t),$

where we have used the completing the square method. Also letting

(4.8)
$\displaystyle\widetilde{E}(t)=E(t)e^{-\int_{t_{0}}^{t}\left(4p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds}.$

Then

(4.9) $\displaystyle\widetilde{E}^{\prime}(t)$
$\displaystyle\leq\Bigg{\\{}-2h(t)\int_{M}\left[\left|\left(\mathscr{L}_{f}+\frac{1}{2}\Big{(}\partial_{t}-\Delta\Big{)}\right)v\right|^{2}-\frac{1}{4}\Big{|}\Big{(}\partial_{t}-\Delta\Big{)}v\Big{|}^{2}\right]dV$
$\displaystyle\hskip
28.45274pt-2p(t)E(t)\Bigg{\\}}e^{-\int_{t_{0}}^{t}\left(4p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds}.$

Combining (4), (4) and (4.8) we get

$\displaystyle\widetilde{I}^{\prime}(t)\widetilde{E}(t)$
$\displaystyle=\Bigg{\\{}-2h(t)\left(\int_{M}v\left[\mathscr{L}_{f}+\frac{1}{2}(\partial_{t}-\Delta)\right]vdV\right)^{2}$
(4.10)
$\displaystyle+\frac{h(t)}{2}\left(\int_{M}v((\partial_{t}-\Delta)vdV\right)^{2}-2p(t)I(t)E(t)\Bigg{\\}}e^{-\int_{t_{0}}^{t}\left(6p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds}.$

Now, using the bound for $\widetilde{E}^{\prime}(t)$ (i.e., (4.9)), (4),
Cauchy-Schwarz inequality and elementary inequality of the form $(r+s)^{2}\leq
2(r^{2}+s^{2})$, with $\Gamma_{2}(t)$ denoting

$\Gamma_{2}(t):=\int_{t_{0}}^{t}\left(6p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds$

we obtain

$\displaystyle\widetilde{I}^{2}(t)Q^{\prime}(t)=\widetilde{I}(t)\widetilde{E}^{\prime}(t)-\widetilde{I}^{\prime}(t)\widetilde{E}(t)$
$\displaystyle\leq
e^{-\Gamma_{2}(t)}\Bigg{\\{}-2h(t)I(t)\int_{M}\left[\left|\left(\mathscr{L}_{f}+\frac{1}{2}\Big{(}\partial_{t}-\Delta\Big{)}\right)v\right|^{2}-\frac{1}{4}\Big{|}\Big{(}\partial_{t}-\Delta\Big{)}v\Big{|}^{2}\right]dV$
$\displaystyle\hskip
28.45274pt+2h(t)\left(\int_{M}v\left[\mathscr{L}_{f}+\frac{1}{2}(\partial_{t}-\Delta)\right]vdV\right)^{2}-\frac{h(t)}{2}\left(\int_{M}v((\partial_{t}-\Delta)vdV\right)^{2}\Bigg{\\}}$
$\displaystyle=-2h(t)e^{-\Gamma_{2}(t)}\Bigg{\\{}I(t)\left(\int_{M}\left|\left(\mathscr{L}_{f}+\frac{1}{2}\Big{(}\partial_{t}-\Delta\Big{)}\right)v\right|^{2}\right)-\left(\int_{M}v\left[\mathscr{L}_{f}+\frac{1}{2}(\partial_{t}-\Delta)\right]vdV\right)^{2}$
$\displaystyle\hskip
28.45274pt+\frac{1}{4}\left(\int_{M}v((\partial_{t}-\Delta)vdV\right)^{2}-\frac{1}{4}I(t)\int_{M}\Big{|}\Big{(}\partial_{t}-\Delta\Big{)}v\Big{|}^{2}dV\Bigg{\\}}$
$\displaystyle\leq\ \
\frac{h(t)}{2}I(t)\left(\int_{M}\Big{|}\Big{(}\partial_{t}-\Delta\Big{)}v\Big{|}^{2}dV\right)e^{-\Gamma_{2}(t)}$
$\displaystyle\leq\ \
\frac{h(t)}{2}\bar{p}^{2}(t)I(t)\left(\int_{M}\Big{(}|v|+|\nabla
v|\Big{)}^{2}dV\right)e^{-\Gamma_{2}(t)}$ $\displaystyle\leq\ \
\bar{p}^{2}(t)I(t)\Big{(}h(t)I(t)+E(t)\Big{)}e^{-\Gamma_{2}(t)}.$

Using the condition $h(t)>0$ we have

$\displaystyle Q^{\prime}(t)$
$\displaystyle\leq\bar{p}^{2}(t)\left(h(t)+\frac{E(t)}{I(t)}\right)e^{\int_{t_{0}}^{t}\left(2p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds}$
$\displaystyle=\bar{p}^{2}(t)\left(Q(t)+h(t)e^{\int_{t_{0}}^{t}\left(2p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}\right)ds}\right)$
$\displaystyle\leq\bar{p}^{2}(t)\Big{(}Q(t)+h(t_{0})\Big{)}$

which proves (4.3) and (4.4) of the proposition.

∎

######  Corollary 4.3.

Let $v:M^{m+1}\times[t_{0},t_{1}]\to\mathbb{R}$ satisfy (4.1) along the
conformal Ricci flow $(M^{m+1},g(t),p(t))$, $t\in[t_{0},t_{1}]\subset[0,T)$.
Then

$\displaystyle I(t_{1})\geq$ $\displaystyle
I(t_{0})\exp\Bigg{\\{}-3(t_{1}-t_{0})\sup_{t\in[t_{0},t_{1}]}\max_{x\in
M}p(t)-\Bigg{[}\left(2+\sup_{t\in[t_{0},t_{1}]}\max_{x\in M}p(t)\right)$
(4.11)
$\displaystyle\times\Big{(}Q(t_{0})+h(t_{0})\Big{)}\left(e^{\int_{t_{0}}^{t_{1}}p^{2}(t)dt}\right)\Bigg{]}\int_{t_{0}}^{t_{1}}\frac{1}{h(t)}e^{-\int_{t_{0}}^{t_{1}}2p(s)+\frac{h^{\prime}(s)+k(s)}{h(s)}ds}dt\Bigg{\\}}.$

Moreover, if $v(\cdot,t_{1})=0$, then $v(\cdot,t)\equiv 0$ for all
$t\in[t_{0},t_{1}]$.

This corollary is indeed a restatement of Theorem 4.1.

###### Proof.

Integrating (4.2) on the interval $[t_{0},t_{1}]$ gives

$\displaystyle\log I(t_{1})-$ $\displaystyle\log I(t_{0})$
$\displaystyle\geq-3\int_{t_{0}}^{t_{1}}\bar{p}(t)dt-\int_{t_{0}}^{t_{1}}\frac{\bar{p}(t)+2}{h(t)}Q(t)e^{-\int_{t_{0}}^{t}\left(2p(s)+^{\prime}\frac{h^{\prime}(s)+k(s)}{h(s}\right)ds}dt$
(4.12) $\displaystyle\geq-3(t_{1}-t_{0})\sup_{t\in[t_{0},t_{1}]}\bar{p}(t)$
$\displaystyle\hskip
28.45274pt-\left(2+\sup_{t\in[t_{0},t_{1}]}\bar{p}(t)\right)\int_{t_{0}}^{t_{1}}\frac{Q(t)}{h(t)}e^{-\int_{t_{0}}^{t}\left(2p(s)+^{\prime}\frac{h^{\prime}(s)+k(s)}{h(s}\right)ds}dt.$

Integrating (4.4) we get

$\displaystyle\log[Q(t)+h(t_{0})]\leq\log[Q(t_{0})+h(t_{0})]+\int_{t_{0}}^{t_{1}}\bar{p}^{2}(t)dt.$

Therefore, $Q(t)$ is bounded by

(4.13) $\displaystyle
Q(t)\leq(Q(t_{0})+h(t_{0}))e^{\int_{t_{0}}^{t_{1}}\bar{p}^{2}(t)dt}-h(t_{0}).$

Inserting this bound (4.13) into (4), and then (4.3) is obtained by
exponentiation.

The conclusion in the second part of the corollary follows immediately from
the first part. Note that the fact that $p(t)\geq 0$ is finite and the
assumption that $h(t)>0$ (uniformly) for all $t\in[t_{0},t_{1}]\subset[0,T)$
makes the integral appearing in (4.3) finite.

∎

## Conflict of interest

The authors declare that there is no conflict of interest.

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# Existence and uniqueness of maximal elements for preference relations:
Variational approach

O. Bueno Universidad del Pacífico. Lima, Perú. Email: {o.buenotangoa,
<EMAIL_ADDRESS>J. Cotrina22footnotemark: 2 Y. García
33footnotemark: 3

###### Abstract

In this work, we reformulate the problem of existence of maximal elements for
preference relations as a variational inequality problem in the sense of
Stampacchia. Similarly, we establish the uniqueness of maximal elements using
a variational inequality problem in the sense of Minty. In both of these
approaches, we use the normal cone operator to find existence and uniqueness
results, under mild assumptions. In addition, we provide an algorithm for
finding such maximal elements, which is inspired by the steepest descent
method for minimization. Under certain conditions, we prove that the sequence
generated by this algorithm converges to a maximal element.

Keywords: Maximal elements, Stampacchia variational inequality, Minty
variational inequality

MSC (2010): 91B16, 49J40, 49J53

##  1 Introduction

The theory of preference relations is one of the main tools in the study of
consumer demand. A preference relation is described by means of a binary
relation, which is traditionally derived from a utility function when the
relation satisfies some properties (see for instance [7]). In that sense,
finding a maximal element is equivalent to solving the maximization problem of
the associated utility function. However, it is well-known that there are
preference relations that are not derived from a utility function, for
instance, the lexicographic ordering [10]. We can find in the literature a lot
of works concerning the existence of maximal elements for preference
relations, which are not necessarily transitive nor complete, see for instance
[5, 11, 12, 13, 16, 18] and the references therein. It is interesting to note
that many of these results are consequences of Browder’s theorem [4, Theorem
1].

On the other hand, variational inequalities play an important role in the
study of optimization problems, Nash games, saddle problems, among others, see
[9]. Recently, Milasi _et al._ [12] reformulated the competitive economic
equilibrium problem governed by preference relations as a quasi-variational
inequality problem. Later on, Milasi and Scopelliti [13] reformulated the
problem of finding maximal elements for preference relations as a suitable
variational inequality problem. However, we discovered some inconsistencies in
these works, see Remarks 3.3 and 3.10.

In this work, we aim to study the existence and uniqueness of maximal elements
by considering the Stampacchia and Minty variational inequality. Very
recently, Donato and Villanacci [8] also addressed the existence of maximal
elements in a similar way. However, our existence result is not a consequence
of their results, see Remark 3.13. In addition, we present an algorithm to
obtain these elements, whenever the preference relation is defined on the
whole space. This algorithm is inspired by the classical steepest descent
method for minimization of quasiconvex functions [14], but instead of the
gradient, we use the so-called _Plastria-like normal cone_ of the strict upper
contour.

In Section 2, we present definitions and notations of preference relations.
Section 3 deals with the existence and uniqueness of maximal elements by
considering the variational reformulation. Finally, in Section 4 we provide an
optimization algorithm for preference relations that satisfy certain
conditions.

##  2 Preliminaries

Consider $X$ a non-empty set and $\succeq$ a binary relation on $X$. For each
$x\in X$ we consider the sets

$U(x):=\\{y\in X\>:\>y\succeq x\\},\quad L(x):=\\{y\in X\>:\>x\succeq y\\},$
$U^{s}(x):=\\{y\in X\>:\>y\succ x\\}\text{ and }L^{s}(x):=\\{y\in X\>:\>x\succ
y\\},$

where $\succ$ is the asymmetric part of $\succeq$, that is, $x\succ y$ means
$x\succeq y$ but not $y\succeq x$. The sets $U(x)$ and $U^{s}(x)$ (resp.
$L(x)$ and $L^{s}(x)$) are called the _upper_ and _strictly upper_ (resp.
_lower_) _contour set_.

We recall that a binary relation $\succeq$ [10] is said to be:

  * •

_Complete_ if, for any $x,y\in X$, either $x\succeq y$ or $y\succeq x$.

  * •

_Transitive_ if, for any $x,y,z\in X$, the following implication holds

$(x\succeq y~{}\wedge~{}y\succeq z)\Rightarrow x\succeq z.$

  * •

_Reflexive_ if, for any $x\in X$, we have $x\succeq x$.

  * •

_Rational_ if, it is complete and transitive.

######  Definition 2.1.

Let $m\in\mathbb{N}$. A relation $\succeq$ on $X$ has the $m$-_FIP_ if, for
any $x_{1},x_{2},\ldots,x_{m}\in X$, there exists $x\in X$ such that $x\succeq
x_{i}$, for all $i=1,2,\ldots,m$. If $\succeq$ has the $m$-FIP, for all
$m\in\mathbb{N}$, then it has the _finite intersection property_.

A few remarks are needed.

######  Remark 2.2.

  1. 1.

The relation $\succeq$ has the finite intersection property if, and only if,
the family of sets $\\{U(x)\\}_{x\in X}$ has the finite intersection property.

  2. 2.

If $\succeq$ is complete, then it has the $2$-FIP. The converse is not true in
general, see Example 2.5.

######  Proposition 2.3.

Let $X$ be a non-empty set and $\succeq$ be a relation on $X$. If $\succeq$ is
transitive and has the $2$-FIP, then $\succeq$ has the finite intersection
property.

###### Proof.

We prove this by induction. For $m=1,2$ it follows from the $2$-FIP of
$\succeq$.

Assume now that the family of sets $\\{U(x)\\}_{x\in X}$ satisfies the finite
intersection property for $n-1$ elements of $X$ and consider
$x_{1},\dots,x_{n}\in X$. For $x_{1},x_{2},\dots,x_{n-1}$ there exists $x\in
X$ such that $x\succeq x_{i}$, for all $i\in\\{1,2,\dots,n-1\\}$. Thus, again
by the $2$-FIP of $\succeq$ there exists $z\in X$ such that $z\succeq x_{n}$
and $z\succeq x$. By transitivity of $\succeq$, it follows that $z\succeq
x_{i}$ for all $i\in\\{1,2,\dots,n\\}$. ∎∎

In particular, since completeness implies the 2-FIP, we obtain the following
corollary.

######  Corollary 2.4.

Let $\succeq$ be a rational relation on a non-empty set $X$. Then $\succeq$
has the finite intersection property.

We now present some examples. Example 2.5 shows that neither completeness nor
transitivity of $\succeq$ are necessary to guarantee the finite intersection
property. On the other hand, Example 2.6 shows that completeness, 2-FIP or
transitivity, separately, are not sufficient to obtain the finite intersection
property.

######  Example 2.5.

Consider $X=[0,4]$ and define the relation $\succeq$ on $X$ as

$x\succeq y\text{ if, and only if, }\dfrac{y}{2}+2\leq x\leq 4\text{ and
}(x,y)\neq(7/2,2).$

Clearly $4\in\displaystyle\bigcap_{x\in X}U(x)$, hence $\succeq$ has the
finite intersection property. However, it is straightforward to verify that
$\succeq$ is neither reflexive, nor complete. It is not transitive either,
because $7/2\succeq 3$ and $3\succeq 2$ but $7/2\not\succeq 2$.

######  Example 2.6.

Consider $X=[0,1]$ and, for $j\in\\{a,b\\}$, define the relation $\succeq_{j}$
as

$x\succeq_{j}y\text{ if, and only if, }(x,y)\text{ belongs to the graph in
Figure~{}\ref{FIP-transitivity}, item~{}$j$}.$

Denote with $U_{j}(x)$, $j\in\\{a,b\\}$, the respective upper contour set of
$\succeq_{j}$.

$1$$1$$1/2$$1/2$ | $1$$1$$1/2$$1/2$
---|---
a) Complete relation | b) Transitive relation
Figure 1:

Note that $\succeq_{a}$ is complete (hence, it has the 2-FIP). However it is
not transitive, since $0\succeq_{a}1\succeq_{a}1/2$ and $0\not\succeq_{a}1/2$,
and it does not have the finite intersection property, because

$U_{a}(0)\cap U_{a}(1/2)\cap
U_{a}(1)=\\{0,1/2\\}\cap(]0,1/2]\cup\\{1\\})\cap([0,1]\setminus\\{1/2\\})=\emptyset.$

On the other hand, it is straightforward to verify that $\succeq_{b}$ is
transitive. Moreover, since $U_{b}(0)\cap U_{b}(1)=\emptyset$, $\succeq_{b}$
does not have the 2-FIP, hence it is not complete nor it has the finite
intersection property.

We will now recall some definitions dealing with convexity, which will play an
important role later. Let $X$ be a convex set in a vector space, the relation
$\succeq$ is called _convex_ (resp. _convex s_), if $U(x)$ (resp. $U^{s}(x)$)
is convex, for all $x\in X$. These two notions coincide whenever the relation
$\succeq$ is rational, as it was established without proof in [12, Proposition
2.2, part $(i)$]. We present a proof of this fact, to make our work self-
contained.

######  Proposition 2.7.

Assume that $X$ is convex and the relation $\succeq$ is rational. Then
$\succeq$ is convex if, and only if, it convexs.

###### Proof.

If $U^{s}(x)=\emptyset$, there is nothing to prove. We now consider
$U^{s}(x)\neq\emptyset$. Let $y,z\in U^{s}(x)$ and $t\in[0,1]$. Since
$\succeq$ is complete, without loss of generality we can assume $y\succeq z$.
Thus, $y,z\in U(z)$ which is a convex set. This implies $ty+(1-t)z\succeq z$
and by transitivity we have $ty+(1-t)z\succ x$.

Reciprocally, consider $y,z\in U(x)$ and $t\in[0,1]$. If there exists
$t_{0}\in]0,1[$ such that $t_{0}y+(1-t_{0})z\notin U(x)$ then $x\succ
t_{0}y+(1-t_{0})z$, due to $\succeq$ being complete. Now, by transitivity of
$\succeq$, we have that $y,z\in U^{s}(t_{0}y+(1-t_{0})z)$. However, since
$U^{s}(t_{0}y+(1-t_{0})z)$ is convex we get a contradiction. ∎∎

The following examples show that the previous result is not true in general
when we drop the rationality.

######  Example 2.8.

Consider $X=[0,1]$ and $\succeq$ defined as

$x\succeq y\text{ if and only if }x=0\text{ or }y=x.$

It is easy to see that $U(x)=\\{0,x\\}$, for any $x\in X$. Thus, $\succeq$ is
not convex. However,

$U^{s}(x)=\begin{cases}\emptyset,&x=0,\\\
\\{0\\},&\text{otherwise},\end{cases}$

which is a convex set, for all $x\in X$. Hence, it is convexs.

######  Example 2.9.

Consider $X=[0,1]$ and $\succeq$ defined as

$x\succeq y\text{ if, and only if, }y=0\text{ or
}\begin{cases}x=y,&y\in]0,1]\setminus\\{1/2\\},\\\ x=0,&y=1/2.\end{cases}$

It is not difficult to see that

$U(x)=\begin{cases}[0,1],&x=0,\\\ \\{x\\},&x\in]0,1]\setminus\\{0,1/2\\},\\\
\\{0\\},&x=1/2,\end{cases}\text{ and
}U^{s}(x)=\begin{cases}\emptyset,&x\in]0,1],\\\
]0,1/2[\cup]1/2,1],&x=0.\end{cases}$

Thus, $\succeq$ is convex but it is not convexs.

We finish this section with some topological definitions for binary relations
and correspondences. Given $X$ a topological space, a relation $\succeq$ on
$X$ is said to be _upper_ (resp. _lower_) _semicontinuous_ , if the set
$L^{s}(x)$ (resp. $U^{s}(x)$) is open, for all $x\in X$. Moreover, $\succeq$
is _continuous_ , if it is upper and lower semicontinuous. Clearly, if
$\succeq$ is complete; then it is upper (resp. lower) semicontinuous if, and
only if, $U(x)$ (resp. $L(x)$) is closed, for all $x\in X$.

Let $V,W$ be nonempty sets. A _correspondence_ $T:V\rightrightarrows W$ is an
application from $V$ into $\mathcal{P}(W)$, that is, for $v\in V$,
$T(v)\subset W$. We can see that the upper contour $U$ and the strict upper
contour $U^{s}$ as examples of correspondences from $X$ to itself.

Recall that a correspondence $T:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{m}$
is

  * •

_lower hemicontinuous_ at $x_{0}\in\mathbb{R}^{n}$ when, for any sequence
$(x_{k})_{k\in\mathbb{N}}$ converging to $x_{0}$ and any element $y_{0}$ of
$T(x_{0})$, there exists a sequence $(y_{k})_{k\in\mathbb{N}}$ converging to
$y_{0}$ such that $y_{k}\in T(x_{k})$, for any $k\in\mathbb{N}$.

  * •

_upper hemicontinuous_ at $x_{0}\in\mathbb{R}^{n}$ when, for any neighbourhood
$\mathcal{W}$ of $T(x_{0})$, there exists a neighbourhood $\mathcal{V}$ of
$x_{0}$ such that $T(x)\subset\mathcal{W}$, for all $x\in\mathcal{V}$;

  * •

_lower_ (respectively _upper_) _hemicontinuous_ when it is lower (resp. upper)
hemicontinuous at every $x_{0}\in\mathbb{R}^{n}$;

  * •

_closed_ , if the set
$\operatorname{graph}(T):=\\{(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{m}\>:\>y\in
T(x)\\}$ is a closed subset of $\mathbb{R}^{n}\times\mathbb{R}^{n}$.

It is well known that if $T(\mathbb{R}^{n})$ is bounded and $T$ is closed,
then it is upper hemicontinuous. We also say that $T$ has _open fibres_ when
the set $T^{-1}(y):=\\{x\in\mathbb{R}^{n}\>:\>y\in T(x)\\}$ is open, for all
$y\in\mathbb{R}^{m}$. It is easy to verify that if $T$ has open fibres, then
it is lower hemicontinuous. In particular, if the binary relation $\succeq$ on
$\mathbb{R}^{n}$ is upper (resp. lower) semicontinuous then
$U^{s}:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ (resp. $L^{s}$) has open
fibres, hence it is lower (resp. upper) hemicontinuous.

##  3 A Variational Approach for Maximal Elements

This section is devoted to studying the existence and uniqueness of maximal
elements for binary relations. First, we will introduce a normal cone
correspondence, related to the strict upper contours of a binary relation, and
establish some of its properties. Next, we will deal with the variational
reformulation and the existence of maximal elements. Finally, using the Minty
variational inequality, we will study the uniqueness of such maximal elements.

Let $\succeq$ be a binary relation on $\mathbb{R}^{n}$ with asymmetric part
$\succ$, let $X$ be a non-empty subset of $\mathbb{R}^{n}$ and let $\hat{x}\in
X$. Then

  * •

$\hat{x}$ is a _maximum_ of $X$, if $\hat{x}\succeq y$, for all $y\in X$;

  * •

$\hat{x}$ is a _maximal element_ of $X$, if there is not $y\in X$ such that
$y\succ\hat{x}$.

We denote by $\mathscr{M}_{\succeq}(X)$ and $\mathscr{ME}_{\succeq}(X)$ the
set of maxima and maximal elements of $X$, respectively. It is not difficult
to see that $\mathscr{M}_{\succeq}(X)\subset\mathscr{ME}_{\succeq}(X)$, and
the equality holds under completeness of $\succeq$. Furthermore, it is clear
that

$\mathscr{M}_{\succeq}(X)=\bigcap_{x\in X}U(x)\cap X\text{ and
}\mathscr{ME}_{\succeq}(X)=\\{x\in X\>:\>U^{s}(x)\cap X=\emptyset\\}.$

###  3.1 Variational Inequalities and Normal Cones

Consider $X$ a subset of $\mathbb{R}^{n}$ and
$T:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ a set-valued map. A vector
$\hat{x}\in X$ is said to be a solution of the

  * •

_Stampacchia variational inequality problem_ if there exists $\hat{x}^{*}\in
T(\hat{x})$, such that

$\langle\hat{x}^{*},y-\hat{x}\rangle\geq 0,\quad\forall\,y\in X.$

  * •

_Minty variational inequality problem_ , if

$\langle y^{*},\hat{x}-y\rangle\leq 0,\quad\forall\,y\in X,\,\forall\,y^{*}\in
T(y).$

The solution sets of the Stampacchia and Minty variational inequality problems
will be denoted by $\operatorname{SVIP}(T,X)$ and $\operatorname{MVIP}(T,X)$,
respectively.

Given a subset $A$ of $\mathbb{R}^{n}$ and $x\in\mathbb{R}^{n}$, the _normal
cone_ of $A$ at $x$ is the set

$\mathscr{N}_{A}(x):=\begin{cases}\\{x^{*}\in\mathbb{R}^{n}\>:\>\langle
x^{*},y-x\rangle\leq 0,~{}\forall y\in A\\},&A\neq\emptyset,\\\
\mathbb{R}^{n},&A=\emptyset.\end{cases}$

It is usual in the literature to consider the above definition whenever $A$
convex and closed, and $x\in A$. However, we will not consider such conditions
in this work. See Figure 2 for a geometric interpretation. Note that
$\mathscr{N}_{A}(x)$ is always non-empty, because $0\in\mathscr{N}_{A}(x)$.
Moreover, it is a closed convex cone of $\mathbb{R}^{n}$.

$A$$x$$\mathscr{N}_{A}(x)$ Figure 2: The set $\mathscr{N}_{A}(x)$

Let $\succeq$ be a binary relation on $\mathbb{R}^{n}$. The _normal cone
correspondence_ $N:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$, associated
to $\succeq$, is defined as

$6N(x):=\mathscr{N}_{U^{s}(x)}(x),$ (1)

for all $x\in\mathbb{R}^{n}$.

######  Lemma 3.1.

Let $\succeq$ be a complete relation the following equivalence holds:

$x^{*}\in N(x)~{}\Leftrightarrow~{}\left(\forall y\in\mathbb{R}^{n},~{}\langle
x^{*},y-x\rangle>0~{}\Rightarrow~{}x\succeq y\right).$

In the following example, we explicitly calculate the normal cone
correspondence for a certain preference relation.

######  Example 3.2.

Consider the relation $\succeq$ on $\mathbb{R}$ defined as

$x\succeq y\text{ if, and only if, }y=x\text{ or }y=1.$

It is clear that

$U^{s}(x)=\begin{cases}\emptyset,&x\neq 1,\\\
\mathbb{R}\setminus\\{1\\},&x=1,\end{cases}\text{ and
}N(x)=\begin{cases}\mathbb{R},&x\neq 1,\\\ \\{0\\},&x=1.\end{cases}$

######  Remark 3.3.

Recently, Milasi _et al._ [12], considered $\succeq$ to be a convexs and non-
satiated relation on $X\subset\mathbb{R}^{n}$ and defined
$M_{1}:X\rightrightarrows\mathbb{R}^{n}$ as
$M_{1}(x):=\mathscr{N}_{U^{s}(x)}(x)$. Clearly, if $X=\mathbb{R}^{n}$ then
$M_{1}$ coincides with $N$.

Now, Proposition 2.4 in [12] establishes that for all $x^{*}\in
M_{1}(x)\setminus\\{0\\}$

$\langle x^{*},y-x\rangle<0,\text{ for all }y\in U^{s}(x),$

provided that $\succeq$ is lower semicontinuous, convexs and with no maximal
elements (this property is called _non-satiated_ in [12]). However, this
result as it is stated is not true. Consider for instance
$X=\mathbb{R}\times\\{0\\}\subset\mathbb{R}^{2}$ and the relation $\succeq$ on
$X$ defined as

$(x,0)\succeq(y,0)\text{ if, and only if, }x\geq y.$

It is not difficult to see that $\succeq$ is continuous, convexs and without
maximal elements on $X$. Moreover, for each $(x,0)\in X$ we have

$U^{s}(x,0)=]x,+\infty[\times\\{0\\}\text{ and
}M_{1}(x,0)=\\{(x^{*},y^{*})\in\mathbb{R}^{2}\>:\>x^{*}\leq 0\\}.$

Since $(0,1)\in M_{1}(x,0)\setminus\\{(0,0)\\}$ we obtain that
$\langle(0,1),(y,0)-(x,0)\rangle=0$, for all $(y,0)\in U^{s}(x,0)$.

The following result can be found as Proposition 24 in [8]. However, our
definition of $N$ allows to drop the condition of $U^{s}(x)$ being non-empty.

######  Lemma 3.4.

Let $\succeq$ be a binary relation on $\mathbb{R}^{n}$ and
$x\in\mathbb{R}^{n}$. If $U^{s}(x)$ is a convex set, then
$N(x)\setminus\\{0\\}\neq\emptyset$.

The following result is a generalization of Proposition 2.3 in [12].

######  Proposition 3.5.

Let $\succeq$ be a relation on $\mathbb{R}^{n}$. If $\succeq$ is upper
semicontinuous, then the map $N$ is closed.

###### Proof.

Lower semicontinuity of $U^{s}$ is consequence of the upper semicontinuity of
$\succeq$. Hence the proposition follows from Proposition 25 in [8]. ∎∎

Given a relation $\succeq$ on $\mathbb{R}^{n}$ we define

$N^{*}(x):=\\{x^{*}\in\mathbb{R}^{n}\>:\>\langle
x^{*},y-x\rangle<0,\,\forall\,dy\in U^{s}(x)\\},$

whenever $x\notin\mathscr{ME}_{\succeq}(\mathbb{R}^{n})$, and
$N^{*}(x):=\mathbb{R}^{n}$, otherwise. It is important to note that $N^{*}(x)$
is a cone and $N^{*}(x)\subset N(x)$, for all $x\in X$.

######  Proposition 3.6.

Let $\succeq$ be a rational relation on $\mathbb{R}^{n}$. If
$N^{*}(x)\neq\emptyset$ for all $x\in\mathbb{R}^{n}$; then $\succeq$ is
convex.

###### Proof.

Let $x\in\mathbb{R}^{n}$. If $U(x)=\emptyset$, there is nothing to prove. Now
assume that $U(x)\neq\emptyset$, and take any $y\notin U(x)$. Choosing some
$y^{*}\in N^{*}(y)$, define the set

$H^{-}(y)=\\{z\in\mathbb{R}^{n}\>:\>\langle y^{*},z-y\rangle<0\\}.$

Clearly, $H^{-}(y)$ is convex and $y\notin H^{-}(y)$. Since $y$ was taken
arbitrarily, we have the inclusion

$\bigcap_{y\notin U(x)}H^{-}(y)\subset U(x).$

On the other hand, for $y\notin U(x)$, since $\succeq$ is complete, $x\succ
y$. We now claim that $U(x)\subset H^{-}(y)$. Indeed, for all $z\in U(x)$ we
have $z\succeq x\succ y$, hence $z\succ y$, by transitivity of $\succeq$. Thus
$\langle y^{*},z-y\rangle<0$, because $y^{*}\in N^{*}(y)$, which implies $z\in
H^{-}(y)$. Therefore

$U(x)=\bigcap_{y\notin U(x)}H^{-}(y).$

The proposition follows. ∎∎

The following proposition shows the relation between the operators $N$ and
$N^{*}$.

######  Proposition 3.7.

Let $\succeq$ be a lower semicontinuous relation on $\mathbb{R}^{n}$. Then

  1. 1.

$N(x)=N^{*}(x)\cup\\{0\\}$, for all $x\in\mathbb{R}^{n}$.

  2. 2.

If, in addition, $\succeq$ is convexs, then $N^{*}$ is non-empty valued.

###### Proof.

  1. 1.

The lower semicontinuity of $\succeq$ implies that $U^{s}(x)$ is open. Since
$N(x)$ is the polar cone of $U^{s}(x)\setminus\\{x\\}$, this item now follows
using [17, Exercise 6.22].

  2. 2.

It follows from item 1. and Lemma 3.4. ∎

∎

The following example shows that it is not possible to drop the lower
semicontinuity in the first part of the previous result.

######  Example 3.8.

Consider the relation $\succeq$ on $\mathbb{R}^{2}$ defined as

$(x,y)\succeq(a,b)\text{ if, and only if, }x\geq a\text{ and }y=b=0.$

It is not difficult to verify that

$U^{s}(x,y)=\begin{cases}\emptyset,&y\neq 0,\\\
]x,+\infty[\times\\{0\\},&y=0.\end{cases}$

Thus, the relation $\succeq$ is not lower semicontinuous. Additionally, we can
see that

$N(x,y)=\begin{cases}\mathbb{R}^{2},&y\neq 0,\\\
\\{(x^{*},y^{*})\in\mathbb{R}^{2}\>:\>x^{*}\leq 0\\},&y=0,\end{cases}$

and

$N^{*}(x,y)=\begin{cases}\mathbb{R}^{2},&y\neq 0,\\\
\\{(x^{*},y^{*})\in\mathbb{R}^{2}\>:\>x^{*}<0\\},&y=0.\end{cases}$

Thus, $N(x,0)\neq N^{*}(x,0)\cup\\{(0,0)\\}$.

###  3.2 Reformulation and Existence Results

The following correspondence will allow us to reformulate the problem of
finding maximal elements as a Stampacchia variational inequality problem.
Consider the correspondence $T:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$
defined as

$T(x):=\operatorname*{conv}(N(x)\cap S[0,1]),$ (2)

where $N$ is the correspondence associated to $\succeq$, defined as in (1),
and $S[0,1]$ denotes the unit sphere of $\mathbb{R}^{n}$. For instance, if
$\succeq$ and $N$ are defined as in Example 3.2, then the correspondence $T$
associated to $N$ satisfies $T(x)=[-1,1]$, when $x\neq 1$, and
$T(1)=\emptyset$.

The following proposition establishes a relation between $\succeq$ and $T$.

######  Proposition 3.9.

The following implications hold:

  1. 1.

If $\succeq$ is upper semicontinuous on $\mathbb{R}^{n}$, then $T$ is closed.
In particular $T$ is upper hemicontinuous.

  2. 2.

If $\succeq$ is convexs and lower semicontinuous on $\mathbb{R}^{n}$, then $T$
is non-empty valued.

###### Proof.

  1. 1.

Consider the correspondence $R:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$
such that
$\operatorname{graph}(R)=\operatorname{graph}(N)\cap\big{(}\mathbb{R}^{n}\times
S[0,1]\big{)}$. By Proposition 3.5, we deduce that $R$ is closed. Moreover, it
is upper semicontinuous due to the closed graph theorem. Since
$T(x)=\operatorname*{conv}(R(x))$, for all $x\in X$, the result follows from
Theorem 17.35 in [1].

  2. 2.

It follows from Proposition 3.7, item 2. ∎

∎

######  Remark 3.10.

Milasi and Scopelliti [13] defined the map
$M_{2}:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ as

$M_{2}(x):=\begin{cases}\mathscr{N}_{U^{s}(x)}(x),&x\in X,\\\
\emptyset,&x\notin X.\end{cases}$

The authors also introduced a correspondence similar to $T$ given in (2). More
precisely, they defined the map $G:X\rightrightarrows\mathbb{R}^{n}$ as

$G(x):=\begin{cases}\operatorname*{conv}(M_{2}(x)\cap
S[0,1]),&U^{s}(x)\neq\emptyset,\\\
\overline{B}(0,1),&U^{s}(x)=\emptyset,\end{cases}$

where $\overline{B}(0,1)$ is closed unit ball of $\mathbb{R}^{n}$. It is clear
that $T$ and $G$ coincide when $X=\mathbb{R}^{n}$.

Now, part b) of Theorem 3 in [13] establishes that any solution of the
variational inequality problem (in the sense of Stampacchia) associated to $G$
and $K\subset X$ is a maximal element of $\succeq$ on $K$. However, this is
not true as it is stated. Consider for instance the relation $\succeq$ given
in Remark 3.3, and $K=[0,1]\times\\{0\\}\subset X$. The map
$G:X\rightrightarrows\mathbb{R}^{2}$ is given by

$G(x,0)=\\{(x^{*},y^{*})\in\mathbb{R}^{2}\>:\>x^{*}\leq 0\text{ and
}(x^{*})^{2}+(y^{*})^{2}\leq 1\\}.$

Since $(0,0)\in G(x,0)$ for all $x\in\mathbb{R}$, we deduce that the solution
set of the variational inequality problem, associated to $G$ and $K$,
coincides with $K$. However, the unique maximal element of $K$ is $(1,0)$.

Motivated by the previous remark we establish the following result.

######  Proposition 3.11.

Assume that $X$ is a non-empty subset of $\mathbb{R}^{n}$ and let $\succeq$ be
lower semicontinuous relation on $\mathbb{R}^{n}$. Then
$\operatorname{SVIP}(T,X)\subset\mathscr{ME}_{\succeq}(X)$.

###### Proof.

Let $\hat{x}$ be an element of $\operatorname{SVIP}(T,X)$. There exists
$\hat{x}^{*}\in T(\hat{x})$ satisfying

$\langle\hat{x}^{*},y-\hat{x}\rangle\geq 0,\text{ for all }y\in X.$ (3)

By definition of $T$, there are $x_{1}^{*},x_{2}^{*},\dots,x_{m}^{*}\in
N(\hat{x})\cap S[0,1]$ and $t_{1},t_{2},\dots,t_{m}\in[0,1]$ such that
$\hat{x}^{*}=\sum_{i=1}^{m}t_{i}x_{i}^{*}$ and $\sum_{i=1}^{m}t_{i}=1$. Assume
that $\hat{x}$ is not a maximal element of $\succeq$ on $X$. Then there exists
$y\in X$ such that $y\succ\hat{x}$, i.e. $y\in U^{s}(\hat{x})$. Using
inequality (3), we deduce $\langle x_{i}^{*},y-\hat{x}\rangle\geq 0$ for some
$i$. On the other hand, since $x_{i}^{*}\in N(\hat{x})\setminus\\{0\\}$ and
Proposition 3.7, item 1., $\langle x_{i}^{*},y-\hat{x}\rangle<0$, a
contradiction. ∎∎

We now are able to present the main result of this subsection.

######  Theorem 3.12.

Let $X$ be a convex, compact and non-empty subset of $\mathbb{R}^{n}$, and let
$\succeq$ be a binary relation on $\mathbb{R}^{n}$. If $\succeq$ is continuous
and convexs; then there exists at least a maximal element for $\succeq$ on
$X$.

###### Proof.

From Proposition 3.9, the correspondence $T$ is upper hemicontinuous with
convex, compact and non-empty values. By Theorem 9.9 in [2], there exists
$\hat{x}\in\operatorname{SVIP}(T,X)$. Therefore, the result follows from
Proposition 3.11. ∎∎

######  Remark 3.13.

It is important to note that in Proposition 3.11 we do not require convexitys
of the relation, contrary to Theorem 28 in [8]. On the other hand, Theorem
3.12 is not a consequence of Theorem 33 in [8]. Indeed, consider the relation
$\succeq$ on $\mathbb{R}$, defined as

$x\succeq y\mbox{ if, and only if, }x=y=0.$

Clearly, $\succeq$ is continuous and convexs. Moreover, $T(x)=[-1,1]$, for all
$x\in\mathbb{R}$. Hence, for any compact, convex and non-empty subset $X$ of
$\mathbb{R}$, by Theorem 3.12, $\mathscr{ME}_{\succeq}(X)\neq\emptyset$.
However, we cannot apply Theorem 33 in [8], as $\succeq$ allows for empty
strict upper contours.

###  3.3 On the Uniqueness of Maximal Elements

In this subsection we will show a result concerning the uniqueness of maximal
elements. Before this, we need some previous results. The first result is
about necessary conditions, while the second provides sufficient conditions.
Both are related to the Minty variational inequality problem.

######  Lemma 3.14.

Let $X$ be a subset of $\mathbb{R}^{n}$ and $\succeq$ be a complete relation
on $\mathbb{R}^{n}$. If $\hat{x}\in\mathscr{ME}_{\succeq}(X)$, then

$\langle y^{*},\hat{x}-y\rangle\leq 0,~{}\text{for all }y\in
X\setminus\mathscr{ME}_{\succeq}(X)\text{ and all }y^{*}\in N(y).$ (4)

###### Proof.

Since $\succeq$ is complete, for any $y\in
X\setminus\mathscr{ME}_{\succeq}(X)$, we have $\hat{x}\in U^{s}(y)$. The
result follows from the definition of $N(y)$. ∎∎

The following example shows that we cannot drop the completeness of Lemma
3.14.

######  Example 3.15.

Consider $\succeq$ on $\mathbb{R}$ defined as

$x\succeq y\text{ if, and only if, }(x,y)=(0,0)\vee(x\geq y~{}\wedge~{}x\neq
0),$

and let $X=\mathbb{R}$. It is clear that $U^{s}(x)=]x,+\infty[$ for all $x\neq
0$, and $U^{s}(0)=\emptyset$, hence
$\mathscr{ME}_{\succeq}(\mathbb{R})=\\{0\\}$. Moreover, we can see that
$N(x)=]-\infty,0]$ for all $x\neq 0$, and $N(0)=\mathbb{R}$. However,
$\hat{x}=0$ does not satisfy inequality (4).

######  Proposition 3.16.

Let $\succeq$ be a binary relation on $\mathbb{R}^{n}$. If the sets
$\mathscr{ME}_{\succeq}(\mathbb{R}^{n})$ and
$\operatorname{MVIP}(N,\mathbb{R}^{n})$ are both non-empty, then
$\mathscr{ME}_{\succeq}(\mathbb{R}^{n})=\operatorname{MVIP}(N,\mathbb{R}^{n})=\\{\hat{x}\\}$,
for some $\hat{x}\in\mathbb{R}^{n}$.

###### Proof.

Let $\hat{x}\in\operatorname{MVIP}(N,\mathbb{R}^{n})$ and
$y\in\mathscr{ME}_{\succeq}(\mathbb{R}^{n})$ be arbitrary. Since $y$ is
maximal, $U^{s}(y)=\emptyset$ and $N(y)=\mathbb{R}^{n}$. If $\hat{x}\neq y$
then, there exists $y^{*}\in N(y)=\mathbb{R}^{n}$ such that $\langle
y^{*},\hat{x}-y\rangle>0$, a contradiction with
$\hat{x}\in\operatorname{MVIP}(N,\mathbb{R}^{n})$. Hence $\hat{x}=y$, for all
$\hat{x}\in\operatorname{MVIP}(N,\mathbb{R}^{n})$ and all
$y\in\mathscr{ME}_{\succeq}(\mathbb{R}^{n})$. This implies the proposition. ∎∎

We now present the main result of this subsection.

######  Theorem 3.17.

Let $\succeq$ be a complete relation on $\mathbb{R}^{n}$. If
$\mathscr{ME}_{\succeq}(\mathbb{R}^{n})\neq\emptyset$, then
$\mathscr{ME}_{\succeq}(\mathbb{R}^{n})$ is a singleton if, and only if,
$\mathscr{ME}_{\succeq}(\mathbb{R}^{n})=\operatorname{MVIP}(N,\mathbb{R}^{n})$.

###### Proof.

First assume that $\mathscr{ME}_{\succeq}(\mathbb{R}^{n})=\\{\hat{x}\\}$, and
take $y\in\mathbb{R}^{n}$ and $y^{*}\in N(y)$. If $y\neq\hat{x}$, Lemma 3.14
implies $\langle y^{*},\hat{x}-y\rangle\leq 0$, and, when $y=\hat{x}$,
trivially $\langle y^{*},\hat{x}-y\rangle=0$. Hence
$\hat{x}\in\operatorname{MVIP}(N,\mathbb{R}^{n})$ and, by Proposition 3.16,
$\mathscr{ME}_{\succeq}(\mathbb{R}^{n})=\operatorname{MVIP}(N,\mathbb{R}^{n})=\\{\hat{x}\\}$
. The converse implication is a direct consequence of Proposition 3.16. ∎∎

######  Remark 3.18.

  1. 1.

Proposition 3.16 implies that if either $\mathscr{ME}_{\succeq}(\mathbb{R})$
or $\operatorname{MVIP}(N,\mathbb{R})$ have at least two elements, then the
other one must be empty.

  2. 2.

Example 3.15 also implies that the completeness of $\succeq$ cannot be dropped
in Theorem 3.17. Indeed, for $\succeq$ as in Example 3.15,
$\mathscr{ME}_{\succeq}(\mathbb{R})=\\{0\\}$ but in this case
$\operatorname{MVIP}(N,\mathbb{R})=\emptyset$.

  3. 3.

Existence of maximal elements can be guaranteed by assuming that $\succeq$ is
complete, has the finite intersection property, $\bigcap_{x\in
X}U(x)=\bigcap_{x\in X}\overline{U(x)}$ and $U(x)$ is bounded for some $x$

##  4 An Algorithm to Find Maximal Elements

This section is divided in two parts: the first one deals with the definition
and properties of a normal cone operator inspired by the Plastria
subdifferential [15]. The second subsection contains an algorithm for finding
maximal elements of a binary relation, along with convergence results.

###  4.1 The Plastria-like Normal Cone

Given a relation $\succeq$ on $\mathbb{R}^{n}$ and a function
$f:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ we define

$N_{f}(x):=\begin{cases}\\{x^{*}\in\mathbb{R}^{n}\>:\>\langle
x^{*},y-x\rangle\leq f(x,y),\,\forall\,y\in
U^{s}(x)\\},&x\notin\mathscr{ME}_{\succeq}(\mathbb{R}^{n}),\\\
\mathbb{R}^{n},&\text{otherwise}.\end{cases}$

We will call $N_{f}(x)$ as the _Plastria-like normal cone_ associated to the
binary relation $\succeq$ with respect to the function $f$. Clearly, if $f=0$
then the Plastria-like normal cone $N_{f}$ reduces to the classical normal
cone defined in (1). It is also clear that $N_{f}(x)$ is closed and convex,
for all $x\in\mathbb{R}^{n}$. In general, $N_{f}(x)$ may not be a cone, as it
is shown by the following example.

######  Example 4.1.

Consider $\succeq$ on $\mathbb{R}$ as in Example 3.15 and functions
$f_{1},f_{2}:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ defined as

$f_{1}(x,y)=y^{2}-x^{2}\text{ and }f_{2}(x,y)=y-x.$

It is not difficult to show that the Plastria-like normal cones of $\succeq$,
with respect to $f_{1}$ and $f_{2}$, respectively, are

$N_{f_{1}}(x)=\begin{cases}]-\infty,2x],&x\neq 0,\\\
\mathbb{R},&x=0,\end{cases}\text{ and
}N_{f_{2}}(x)=\begin{cases}]-\infty,1],&x\neq 0,\\\
\mathbb{R},&x=0.\end{cases}$

The following proposition extends Proposition 25 in [8].

######  Proposition 4.2.

Let $\succeq$ be an upper semicontinuous relation on $\mathbb{R}^{n}$ and
$f:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ be a function. If $f$ is
upper semicontinuous then the Plastria-like normal cone $N_{f}$ is closed.

###### Proof.

Let $(x_{k},x_{k}^{*})$ be a sequence converging to $(x,x^{*})$ such that
$x_{k}\in\mathbb{R}^{n}$ and $x_{k}^{*}\in N_{f}(x_{k})$, for all
$k\in\mathbb{N}$. If $U^{s}(x)=\emptyset$, there is nothing to prove. Now, we
consider that $U^{s}(x)\neq\emptyset$ and take any $y\in U^{s}(x)$. Since
$U^{s}$ is a lower hemicontinuous correspondence, there exists a sequence
$(y_{k})$ converging to $y$ such that $y_{k}\in U^{s}(x_{k})$. Thus,

$\langle x^{*}_{k},y_{k}-x_{k}\rangle\leq f(x_{k},y_{k}).$

Letting $k$ tend to $\infty$, we obtain $\langle x^{*},y-x\rangle\leq f(x,y)$.
Since $y$ was arbitrary, we conclude that $x^{*}\in N_{f}(x)$. Therefore,
$N_{f}$ is closed. ∎∎

Note that Proposition 4.2 coincides with Proposition 3.5, when $f=0$. Also,
this proposition holds even if there is no connection between $\succeq$ and
$f$. In the case when $f$ and $\succeq$ are related, we obtain some other
properties, listed in the following propositions.

######  Proposition 4.3.

Let $\succeq$ be a relation on $\mathbb{R}^{n}$ and
$f:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ be a function such that
$f(x,y)<0$ if, and only if, $y\in U^{s}(x)$. The following hold:

  1. 1.

$0\in N_{f}(x)$ if, and only if, $x\in\mathscr{ME}_{\succeq}(\mathbb{R}^{n})$.

  2. 2.

If $f$ is upper semicontinuous with respect to its first variable, then
$\succeq$ is upper semicontinuous.

  3. 3.

If $f$ is upper semicontinuous with respect to its first variable, and
$(x_{0},0)\in\overline{\operatorname{graph}(N_{f})}$, then $0\in
N_{f}(x_{0})$.

###### Proof.

  1. 1.

Follows directly from the definition of $N_{f}$.

  2. 2.

It is enough to note that upper semicontinuity of $f(\cdot,y)$ implies that
the set

$L^{s}(y)=\\{x\>:\>y\succ x\\}=\\{x\>:\>f(x,y)<0\\}$

is open.

  3. 3.

If $x_{0}\in\mathscr{ME}_{\succeq}(\mathbb{R}^{n})$ there is nothing to prove.
Otherwise, there exists $\hat{x}\in U^{s}(x_{0})$, which implies
$f(x_{0},\hat{x})<0$ and $x_{0}\in L^{s}(\hat{x})$. Let
$(x_{k},x_{k}^{*})\in\operatorname{graph}(N_{f})$, such that $x_{k}\to x_{0}$
and $x_{k}^{*}\to 0$. Now, item 2 implies that $L^{s}(\hat{x})$ is open,
therefore $x_{k}\in L^{s}(\hat{x})$, for $k$ large enough. Since $x_{k}^{*}\in
N_{f}(x_{k})$, we deduce

$\langle x_{k}^{*},\hat{x}-x_{k}\rangle\leq f(x_{k},\hat{x}).$

Taking the limit when $k\to\infty$, we obtain $0\leq f(x_{0},\hat{x})<0$, a
contradiction. ∎

∎

######  Proposition 4.4.

Let $\succeq$ be a relation on $\mathbb{R}^{n}$ and
$f:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ be a function such that
the following hold:

  1. 1.

for all $x,y\in\mathbb{R}^{n}$, if $f(x,z)>f(y,z)$, for some
$z\in\mathbb{R}^{n}$, then $x\succ y$;

  2. 2.

there exists $L>0$ such that $|f(x,y)|\leq L\|x-y\|$, for all
$x,y\in\mathbb{R}^{n}$.

Then, for every $u^{*}\in N^{*}(x_{0})$,
$x_{0}^{*}:=\dfrac{Lu^{*}}{\|u^{*}\|}\in N_{f}(x_{0})$.

###### Proof.

If $x_{0}\in\mathscr{ME}_{\succeq}(\mathbb{R}^{n})$, there is nothing to
prove. Otherwise, $U^{s}(x_{0})\neq\emptyset$. Take $u^{*}\in N^{*}(x_{0})$,
that is,

$\langle u^{*},x-x_{0}\rangle<0,\text{ for all }x\in U^{s}(x_{0}).$

Now we define $x_{0}^{*}=\dfrac{L}{\|u^{*}\|}u^{*}$ and consider the
hyperplane $H=\\{z\in\mathbb{R}^{n}\>:\>\langle u^{*},z-x_{0}\rangle=0\\}$.
For each $x\in U^{s}(x_{0})$ we consider $x^{\prime}\in H$ as the projection
of $x$ onto $H$. It is not difficult to verify that

$\langle\frac{u^{*}}{\|u^{*}\|},x-x_{0}\rangle=-\|x-x^{\prime}\|.$

Since $x^{\prime}\notin U^{s}(x_{0})$, assumptions 1. and 2. imply
$f(x_{0},x)\geq f(x^{\prime},x)\geq-L\|x-x^{\prime}\|$. Hence

$f(x_{0},x)\geq\langle x_{0}^{*},x-x_{0}\rangle,$

and, therefore, $x_{0}^{*}\in N_{f}(x_{0})$. The proof is complete. ∎∎

######  Remark 4.5.

In the particular case when $\succeq$ is represented by a utility function
$u:\mathbb{R}^{n}\to\mathbb{R}$ (in the sense that $x\succeq y$ if, and only
if, $u(x)\geq u(y)$), we can define the function
$f_{u}:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$,
$f_{u}(x,y):=u(x)-u(y)$. Using $f_{u}$, the Plastria-like normal cone
$N_{f_{u}}(x)$ coincides with the _Plastria lower subdifferential_ of $u$ at
$x$ [15].

On the other hand, Proposition 4.3, item 1., extends Theorem 3.1 in [15] and
item 3. extends Proposition 7 in [14].

Note that, using $f_{u}$ as above, we readily obtain condition 1. in
Proposition 4.4. Moreover, condition 2. is equivalent to the function $u$ be
Lipschitz continuous on $\mathbb{R}^{n}$. In view of this, Proposition 4.4
extends Theorem 20 in [6].

###  4.2 The Algorithm

We begin this subsection by recalling the definition of a quasi-Fejér monotone
sequence. A sequence $\\{x_{k}\\}\subset\mathbb{R}^{n}$ is called _quasi-Fejér
monotone_ with respect to $M\subset\mathbb{R}^{n}$ if for every $u\in M$ there
exists a sequence $\\{\varepsilon_{k}\\}\subset\mathbb{R}_{+}$ with
$\sum\varepsilon_{k}<\infty$, such that

$\|x_{k+1}-u\|^{2}\leq\|x_{k}-u\|^{2}+\varepsilon_{k}.$

We state below an important result concerning quasi-Fejér monotone sequences
[3, Theorem 5.33].

######  Theorem 4.6.

Let $\\{x_{n}\\}$ be a sequence in a Hilbert space $H$ and let $C$ be a non-
empty subset of $H$ such that $(x_{n})_{n\in\mathbb{N}}$ is quasi-Fejér
monotone with respect to $C$. Then the following hold:

  1. 1.

$\\{x_{n}\\}$ is bounded.

  2. 2.

Suppose that every weak sequential cluster point of $\\{x_{n}\\}$ belongs to
$C$. Then $\\{x_{n}\\}$ converges weakly to a point in $C$.

We are now ready to present an algorithm for finding maximal elements of
preference relations defined in $\mathbb{R}^{n}$. In the sequel, we will
assume the following conditions:

  1. 1.

$\succeq$ is convexs,

  2. 2.

there is a function $f:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$
satisfying

    1. (a)

$f(x,y)<0$ if, and only if, $y\in U^{s}(x)$;

    2. (b)

$f(x,y)>0$ if, and only if, $x\in U^{s}(y)$;

    3. (c)

there is $L>0$ such that $|f(x,y)|\leq L\|x-y\|$, for all
$x,y\in\mathbb{R}^{n}$;

    4. (d)

for all $x,y\in\mathbb{R}^{n}$, $x\succ y$ if, and only if, $f(x,z)>f(y,z)$,
for all $z\in\mathbb{R}^{n}$, if, and only if, $f(x,z)>f(y,z)$, for some
$z\in\mathbb{R}^{n}$; and

    5. (e)

$f$ is upper semicontinuous with respect to its first variable.

  3. 3.

$\mathscr{ME}_{\succeq}(\mathbb{R}^{n})$ is non-empty.

Under these conditions we can write an analog of the classical steepest
descent method, as follows:

  * Initialization: Take $x_{1}\in\mathbb{R}^{n}$, arbitrarily.

  * Iteration: Given $x_{k}$, calculate the next iterative $x_{k+1}$ from

$x_{k+1}:=x_{k}-\theta_{k}x_{k}^{*},~{}k\in\mathbb{N},$

where $\theta_{k}$ is some positive real number such that

$\sum\theta_{k}=\infty\text{ and }\sum\theta_{k}^{2}<\infty,$

and $x_{k}^{*}\in N_{f}(x_{k})$ such that $\|x_{k}^{*}\|\leq L$ with $L>0$.

Notice that item (d) of assumption 2. in the algorithm implies assumption 1.
in Proposition 4.4. Also, items (b) and (d) of assumption 2. imply the
transitivity of $\succeq$. Moreover, assumption 2. in the algorithm does not
imply the completeness of $\succeq$ as we can see in the following example.

######  Example 4.7.

Consider $\succeq$ defined as in Remark 3.13, that is, $x\succeq y$ if, and
only if, $x=y=0$, which is clearly non-complete. However, choosing $f\equiv 0$
on $\mathbb{R}^{2}$, $f$ and $\succeq$ trivially satisfy conditions 1., 2. and
3. in the algorithm.

It is important to notice that if $x_{k}^{*}=0$, for some $k$, then by
Proposition 4.3, item 1, $x_{k}$ is a maximal element. From now on, we assume
that the sequence $\\{x_{k}\\}$ generated by the algorithm is infinite, that
is $x_{k}^{*}\neq 0$, for all $k\in\mathbb{N}$. It is clear that
$\mathscr{ME}_{\succeq}(\mathbb{R}^{n})\subset\displaystyle\bigcap_{k\in\mathbb{N}}U^{s}(x_{k})$.

######  Theorem 4.8.

The sequence $\\{x_{k}\\}$ generated by the Algorithm converges to a point
$\hat{x}\in\mathscr{ME}_{\succeq}(\mathbb{R}^{n})$.

###### Proof.

Notice that

$\|x_{k+1}-x_{k}\|^{2}=\theta_{k}^{2}\|x_{k}^{*}\|^{2}=L^{2}\theta_{k}^{2},$

which implies the series $\sum\|x_{k+1}-x_{k}\|^{2}$ is convergent. On the
other hand,

$\|x_{k+1}-u\|^{2}=\|x_{k}-u\|^{2}+\|x_{k+1}-x_{k}\|^{2}+2\theta_{k}\langle
x_{k}^{*},u-x_{k}\rangle$ (5)

for all $k\in\mathbb{N}$ and all $u\in\mathbb{R}^{n}$. Taking
$\hat{x}\in\mathscr{ME}_{\succeq}(\mathbb{R}^{n})$ in the previous equality we
have

$\|x_{k+1}-\hat{x}\|^{2}\leq\|x_{k}-\hat{x}\|^{2}+\|x_{k+1}-x_{k}\|^{2}.$

Thus, the sequence $\\{x_{k}\\}$ is quasi-Fejer monotone with respect to
$\mathscr{ME}_{\succeq}(\mathbb{R}^{n})$ with
$\varepsilon_{k}=\|x_{k+1}-x_{k}\|^{2}$. Moreover, from (5) we obtain

$\|x_{k+1}-\hat{x}\|^{2}\leq\|x_{k}-\hat{x}\|^{2}+\|x_{k+1}-x_{k}\|^{2}+2\theta_{k}f(x_{k},\hat{x}),$

which in turn implies

$-2\theta_{k}f(x_{k},\hat{x})\leq\left(\|x_{k}-\hat{x}\|^{2}-\|x_{k+1}-\hat{x}\|^{2}\right)+\|x_{k+1}-x_{k}\|^{2}.$

Since both series $\sum\|x_{k+1}-x_{k}\|^{2}$ and
$\sum\left(\|x_{k}-\hat{x}\|^{2}-\|x_{k+1}-\hat{x}\|^{2}\right)$ are
convergent, we obtain

$\sum-2\theta_{k}f(x_{k},\hat{x})<\infty.$

Hence $\lim f(x_{k},\hat{x})=0$.

Finally, part 1. of Theorem 4.6 implies that $\\{x_{k}\\}$ is bounded. Without
loss of generality, we may assume that $x_{k}$ converges to some $x_{0}$, so,
due to the upper semicontinuity of $f(\cdot,\hat{x})$, $f(x_{0},\hat{x})\geq
0$. If $x_{0}$ is not maximal, there exists $z\in X$ such that $z\succ x_{0}$.
Thus, $f(z,\hat{x})>f(x_{0},\hat{x})\geq 0$, which in turn implies
$z\succ\hat{x}$ and we get a contradiction because $\hat{x}$ is a maximal
element. The result now follows from part 2. of Theorem 4.6. ∎∎

######  Remark 4.9.

Assume that $\succeq$ is represented by a utility function
$u:\mathbb{R}^{n}\to\mathbb{R}$, and consider $f_{u}$ as in Remark 4.5, that
is, $f_{u}(x,y):=u(x)-u(y)$. If $u$ is quasiconcave and Lipschitz, and
possesses a maximum on $\mathbb{R}^{n}$, then $f_{u}$ satisfy assumptions 1.,
2. and 3. of our algorithm. In view of this, our algorithm generalizes
Algorithm A1 in [14], as we do not require differentiability of $u$.

##  5 Conclusions

In this work, we improved some existence results on maximal elements via a
variational approach. In particular, we compared our results with Proposition
2.4 in [12] and Theorem 3 in [13]. We have identified a flaw in these results,
which we described in Remarks 3.3 and 3.10. Moreover, we pointed out that our
main result is not a consequence of Theorems 28 and 33 in [8]. We also
characterized the uniqueness of maximal elements, by studying the solutions of
a certain Minty variational inequality problem.

Finally, we adapted the classical steepest descent method, where we use a new
kind of normal cone, called Plastria-like normal cone, to replace the usual
derivative. In this way, we established an algorithm to find maximal elements.
This algorithm extends Algorithm A1 in [14].

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