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A note on spherical maxima sharing the same Lagrange multiplier
Fixed Point Theory and Applications volume 2014, Article number: 25 (2014)
Abstract
In this paper, we establish a general result on spherical maxima sharing the same Lagrange multiplier of which the following is a particular consequence: Let X be a real Hilbert space. For each $r>0$, let ${S}_{r}=\{x\in X:{\parallel x\parallel}^{2}=r\}$. Let $J:X\to \mathbf{R}$ be a sequentially weakly upper semicontinuous functional which is Gâteaux differentiable in $X\setminus \{0\}$. Assume that ${lim\hspace{0.17em}sup}_{x\to 0}\frac{J(x)}{{\parallel x\parallel}^{2}}=+\mathrm{\infty}$. Then, for each $\rho >0$, there exists an open interval $I\subseteq \phantom{\rule{0.2em}{0ex}}]0,+\mathrm{\infty}[$ and an increasing function $\phi :I\to \phantom{\rule{0.2em}{0ex}}]0,\rho [$ such that, for each $\lambda \in I$, one has $\mathrm{\varnothing}\ne \{x\in {S}_{\phi (\lambda )}:J(x)={sup}_{{S}_{\phi (\lambda )}}J\}\subseteq \{x\in X:x=\lambda {J}^{\prime}(x)\}$.
Here and in what follows, X is a real Hilbert space and $J:X\to \mathbf{R}$ is a functional, with $J(0)=0$. For each $r>0$, set
A point $\stackrel{\u02c6}{x}\in {S}_{r}$ such that
is called a spherical maximum of J. Assuming that J is ${C}^{1}$, spherical maxima are important in connection with the eigenvalue problem
Actually, if $\stackrel{\u02c6}{x}$ is a spherical maximum of J, by the classical Lagrange multiplier theorem, there exists ${\mu}_{\stackrel{\u02c6}{x}}\in \mathbf{R}$ such that
More specifically, one could be interested in the multiplicity of solutions for (1), in the sense of finding some $\mu \in \mathbf{R}$ for which there are more points x satisfying (1). In this connection, however, just because of dependence of ${\mu}_{\stackrel{\u02c6}{x}}$ on $\stackrel{\u02c6}{x}$, the existence of more spherical maxima in ${S}_{r}$ does not imply automatically the existence of some $\mu \in \mathbf{R}$ for which (1) has more solutions. So, in order to the multiplicity of solutions of (1), it is important to know when, at least for some $r>0$, the spherical maxima in ${S}_{r}$ share the same Lagrange multiplier.
The aim of the present note is to give a contribution along such a direction.
Here is our basic result.
Theorem 1 For some $\rho >0$, assume that J is Gâteaux differentiable in $int({B}_{\rho})\setminus \{0\}$ and that
where
and
Assume also that, for some $a>0$, with
if ${\delta}_{\rho}<+\mathrm{\infty}$, the restriction of the functional ${\parallel \cdot \parallel}^{2}aJ(\cdot )$ to ${B}_{\rho}$ is sequentially weakly lower semicontinuous.
For each $r\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho},+\mathrm{\infty}[$, put
and
Then the following assertions hold:

(i)
the function η is convex and decreasing in $]{\beta}_{\rho},+\mathrm{\infty}[$, with ${lim}_{r\to +\mathrm{\infty}}\eta (r)=0$;

(ii)
for each $r\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho}+\frac{\rho}{a},\rho {\delta}_{\rho}[$, the set $\mathrm{\Gamma}(r)$ is nonempty and, for every $\stackrel{\u02c6}{x}\in \mathrm{\Gamma}(r)$, one has
$$0<{\parallel \stackrel{\u02c6}{x}\parallel}^{2}<\rho $$and
$$\begin{array}{rcl}\stackrel{\u02c6}{x}& \in & \{x\in {S}_{{\parallel \stackrel{\u02c6}{x}\parallel}^{2}}:J(x)=\underset{{S}_{{\parallel \stackrel{\u02c6}{x}\parallel}^{2}}}{sup}J\}\\ \subseteq & \{x\in int({B}_{\rho}):{\parallel x\parallel}^{2}\eta (r)J(x)=\underset{y\in {B}_{\rho}}{inf}({\parallel y\parallel}^{2}\eta (r)J(y))\}\\ \subseteq & \{x\in X:x=\frac{\eta (r)}{2}{J}^{\prime}(x)\};\end{array}$$ 
(iii)
for each ${r}_{1},{r}_{2}\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho}+\frac{\rho}{a},\rho {\delta}_{\rho}[$, with ${r}_{1}<{r}_{2}$, and each $\stackrel{\u02c6}{x}\in \mathrm{\Gamma}({r}_{1})$, $\stackrel{\u02c6}{y}\in \mathrm{\Gamma}({r}_{2})$, one has
$$\parallel \stackrel{\u02c6}{y}\parallel <\parallel \stackrel{\u02c6}{x}\parallel ;$$ 
(iv)
if A denotes the set of all $r\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho}+\frac{\rho}{a},\rho {\delta}_{\rho}[$ such that $\mathrm{\Gamma}(r)$ is a singleton, then the function $r\to \mathrm{\Gamma}(r)$ ($r\in A$) is continuous with respect to the weak topology; if, in addition, J is sequentially weakly upper semicontinuous in ${B}_{\rho}$, then ${\mathrm{\Gamma}}_{A}$ is continuous with respect to the strong topology.
Before proving Theorem 1, let us recall a proposition from [1] that will be used in the proof.
Proposition 1 Let Y be a nonempty set, $f,g:Y\to \mathbf{R}$ two functions, and a, b two real numbers, with $a<b$. Let ${y}_{a}$ be a global minimum of the function $f+ag$ and ${y}_{b}$ a global minimum of the function $f+bg$.
Then one has $g({y}_{b})\le g({y}_{a})$.
Proof of Theorem 1 By definition, the function η is the upper envelope of a family of functions which are decreasing and convex in $]{\beta}_{\rho},+\mathrm{\infty}[$. So, η is convex and nonincreasing. We also have
for all $r>{\beta}_{\rho}$ and so
In turn, this implies that η is decreasing as it never vanishes. Now, fix $r\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho}+\frac{\rho}{a},\rho {\delta}_{\rho}[$. So, we have
Consequently, by (3),
Observe that, for each $\lambda \in \phantom{\rule{0.2em}{0ex}}]0,a[$, the restriction to ${B}_{\rho}$ of the functional ${\parallel \cdot \parallel}^{2}\lambda J(\cdot )$ is sequentially weakly lower semicontinuous. In this connection, it is enough to notice that
Fix a sequence $\{{x}_{n}\}$ in ${B}_{\rho}$ such that
Up to a subsequence, we can suppose that $\{{x}_{n}\}$ converges weakly to some ${\stackrel{\u02c6}{x}}_{r}\in {B}_{\rho}$. Fix $\u03f5\in \phantom{\rule{0.2em}{0ex}}]0,\eta (r)[$. For each $n\in \mathbf{N}$ large enough, we have
and so
But then, by sequential weak lower semicontinuity, we have
Hence, since ϵ is arbitrary, we have
and so
that is, ${\stackrel{\u02c6}{x}}_{r}\in \mathrm{\Gamma}(r)$. Now, let $\stackrel{\u02c6}{x}$ be any point of $\mathrm{\Gamma}(r)$. Let us show that $\stackrel{\u02c6}{x}\ne 0$. Indeed, since $\frac{r}{\rho}<{\delta}_{\rho}$, there exists $\tilde{x}\in {B}_{\rho}\setminus \{0\}$ such that
Clearly, this is equivalent to
So
and hence, since $J(0)=0$, we have $\stackrel{\u02c6}{x}\ne 0$, as claimed. Clearly, ${\parallel \stackrel{\u02c6}{x}\parallel}^{2}<\rho $ as $\eta (r)>0$. Moreover, if $x\in {S}_{{\parallel \stackrel{\u02c6}{x}\parallel}^{2}}$, we have
from which we get
Now, let u be any global maximum of ${J}_{{S}_{{\parallel \stackrel{\u02c6}{x}\parallel}^{2}}}$. Then we have
and so
for all $x\in {B}_{\rho}$. Hence, as ${\parallel u\parallel}^{2}<\rho $, the point u is a local minimum of the functional ${\parallel \cdot \parallel}^{2}\eta (r)J(\cdot )$. Consequently, we have
and the proof of (ii) is complete. To prove (iii), observe that
As a consequence, for each ${r}_{1},{r}_{2}\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho}+\frac{\rho}{a},\rho {\delta}_{\rho}[$, with ${r}_{1}<{r}_{2}$, and for each $\stackrel{\u02c6}{x}\in \mathrm{\Gamma}({r}_{1})$, $\stackrel{\u02c6}{y}\in \mathrm{\Gamma}({r}_{2})$, we have
and
Therefore, in view of Proposition 1, we have
and so
We claim that
Arguing by contradiction, assume that $\parallel \stackrel{\u02c6}{y}\parallel =\parallel \stackrel{\u02c6}{x}\parallel $. In view of (ii), this would imply that $J(\stackrel{\u02c6}{y})=J(\stackrel{\u02c6}{x})$ and so, at the same time,
and
In turn, this would imply $\eta ({r}_{1})=\eta ({r}_{2})$ and hence ${r}_{1}={r}_{2}$, a contradiction. So, (iii) holds. Finally, let us prove (iv). For each $r\in A$, continue to denote by $\mathrm{\Gamma}(r)$ the unique point of $\mathrm{\Gamma}(r)$. Let $r\in A$ and let $\{{r}_{k}\}$ be any sequence in A converging to r. Up to a subsequence, $\{\mathrm{\Gamma}({r}_{k})\}$ converges weakly to some $\tilde{x}\in {B}_{\rho}$. Moreover, for each $k\in \mathbf{N}$, $x\in {B}_{\rho}$, one has
From this, after easy manipulations, we get
Since the sequence $\{J(\mathrm{\Gamma}({r}_{k}))\}$ is bounded above, we have
On the other hand, by sequential weak semicontinuity, we also have
Now, passing in (4) to the lim inf, in view of (5) and (6), we obtain
which is equivalent to
Since this holds for all $x\in {B}_{\rho}$, we have $\tilde{x}=\mathrm{\Gamma}(r)$. So, ${\mathrm{\Gamma}}_{A}$ is continuous at r with respect to the weak topology. Now, assuming also that J is sequentially weakly upper semicontinuous, in view of the continuity of η in $]{\beta}_{\rho},+\mathrm{\infty}[$, we have
and hence
from which
Since X is a Hilbert space and $\{\mathrm{\Gamma}({r}_{k})\}$ converges weakly to $\mathrm{\Gamma}(r)$, this implies that
which shows the continuity of ${\mathrm{\Gamma}}_{A}$ at r in the strong topology. □
Remark 1 Clearly, when J is sequentially weakly upper semicontinuous in ${B}_{\rho}$, the assertions of Theorem 1 hold in the whole interval $]{\beta}_{\rho},\rho {\delta}_{\rho}[$, since a can be any positive number.
Remark 2 The simplest way to satisfy condition (2) is, of course, to assume that
Another reasonable way is provided by the following proposition.
Proposition 2 For some $s>0$, assume that J is Gâteaux differentiable in ${B}_{s}\setminus \{0\}$ and that there exists a global maximum $\stackrel{\u02c6}{x}$ of ${J}_{{B}_{s}}$ such that
Then (2) holds with $\rho ={\parallel \stackrel{\u02c6}{x}\parallel}^{2}$.
Proof For each $t\in \phantom{\rule{0.2em}{0ex}}]0,1]$, set
Clearly, ω is derivable in $]0,1]$. In particular, one has
So, by assumption, ${\omega}^{\prime}(1)<0$ and hence, in a left neighborhood of 1, we have
which implies the validity of (2) with $\rho ={\parallel \stackrel{\u02c6}{x}\parallel}^{2}$. □
Also, notice the following consequence of Theorem 1.
Theorem 2 For some $\rho >0$, let the assumptions of Theorem 1 be satisfied.
Then there exists an open interval $I\subseteq \phantom{\rule{0.2em}{0ex}}]0,+\mathrm{\infty}[$ and an increasing function $\phi :I\to \phantom{\rule{0.2em}{0ex}}]0,\rho [$ such that, for each $\lambda \in I$, one has
Proof Take
Clearly, I is an open interval since η is continuous and decreasing. Now, for each $r\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho}+\frac{\rho}{a},\rho {\delta}_{\rho}[$, pick ${v}_{r}\in \mathrm{\Gamma}(r)$. Finally, set
for all $\lambda \in I$. Taking (iii) into account, we then realize that the function φ (whose range is contained in $]0,\rho [$) is the composition of two decreasing functions, and so it is increasing. Clearly, the conclusion follows directly from (ii). □
We conclude deriving from Theorem 1 the following multiplicity result.
Theorem 3 For some $\rho >0$, assume that J is sequentially weakly upper semicontinuous in ${B}_{\rho}$, Gâteaux differentiable in $int({B}_{\rho})\setminus \{0\}$ and satisfies (2). Moreover, assume that there exists $\tilde{\rho}$ satisfying
where
such that ${J}_{{S}_{\tilde{\rho}}}$ has either two global maxima or a global maximum at which ${J}^{\prime}$ vanishes.
Then there exists $\tilde{\lambda}>0$ such that the equation
has at least two nonzero solutions which are global minima of the restriction of the functional $\frac{1}{2}{\parallel \cdot \parallel}^{2}\tilde{\lambda}J(\cdot )$ to $int({B}_{\rho})$.
Proof For each $r\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho},\rho {\delta}_{\rho}[$, in view of (7), we can pick ${v}_{r}\in \mathrm{\Gamma}(r)$ (recall Remark 1), so that
where
Two cases can occur. First, assume that $\tilde{\rho}\in \psi (]{\beta}_{\rho},\rho {\delta}_{\rho}[)$. So, $\psi (\tilde{r})=\tilde{\rho}$ for some $\tilde{r}\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho},\rho {\delta}_{\rho}[$. So, by (ii), for each global maximum u of ${J}_{{S}_{\tilde{\rho}}}$, we have ${J}^{\prime}(u)\ne 0$. As a consequence, in this case, ${J}_{{S}_{\tilde{\rho}}}$ has at least two global maxima which, by (ii) again, satisfies the conclusion with $\tilde{\lambda}=\frac{1}{2}\eta (\tilde{r})$. Now, suppose that $\tilde{\rho}\notin \psi (]{\beta}_{\rho},\rho {\delta}_{\rho}[)$. In this case, in view of (8), the function ψ is discontinuous and hence, in view of (iv), there exists some ${r}^{\ast}\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho},\rho {\delta}_{\rho}[$ such that $\mathrm{\Gamma}({r}^{\ast})$ has at least two elements which, by (ii), satisfy the conclusion with $\tilde{\lambda}=\frac{1}{2}\eta ({r}^{\ast})$. □
References
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Ricceri B: Uniqueness properties of functionals with Lipschitzian derivative. Port. Math. 2006, 63: 393–400.
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Ricceri, B. A note on spherical maxima sharing the same Lagrange multiplier. Fixed Point Theory Appl 2014, 25 (2014). https://doi.org/10.1186/16871812201425
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Keywords
 Lagrange Multiplier
 Global Minimum
 Lower Semicontinuous
 Global Maximum
 Weak Topology