# A note on spherical maxima sharing the same Lagrange multiplier

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## 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 $S r ={x∈X: ∥ x ∥ 2 =r}$. Let $J:X→R$ be a sequentially weakly upper semicontinuous functional which is Gâteaux differentiable in $X∖{0}$. Assume that $lim sup x → 0 J ( x ) ∥ x ∥ 2 =+∞$. Then, for each $ρ>0$, there exists an open interval $I⊆]0,+∞[$ and an increasing function $φ:I→]0,ρ[$ such that, for each $λ∈I$, one has $∅≠{x∈ S φ ( λ ) :J(x)= sup S φ ( λ ) J}⊆{x∈X:x=λ J ′ (x)}$.

Here and in what follows, X is a real Hilbert space and $J:X→R$ is a functional, with $J(0)=0$. For each $r>0$, set

$S r = { x ∈ X : ∥ x ∥ 2 = r } , B r = { x ∈ X : ∥ x ∥ 2 ≤ r } .$

A point $x ˆ ∈ S r$ such that

$J( x ˆ )= sup S r J$

is called a spherical maximum of J. Assuming that J is $C 1$, spherical maxima are important in connection with the eigenvalue problem

$J ′ (x)=μx.$
(1)

Actually, if $x ˆ$ is a spherical maximum of J, by the classical Lagrange multiplier theorem, there exists $μ x ˆ ∈R$ such that

$J ′ ( x ˆ )= μ x ˆ x ˆ .$

More specifically, one could be interested in the multiplicity of solutions for (1), in the sense of finding some $μ∈R$ for which there are more points x satisfying (1). In this connection, however, just because of dependence of $μ x ˆ$ on $x ˆ$, the existence of more spherical maxima in $S r$ does not imply automatically the existence of some $μ∈R$ 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 $ρ>0$, assume that J is Gâteaux differentiable in $int( B ρ )∖{0}$ and that

$β ρ ρ < δ ρ ,$
(2)

where

$β ρ = sup B ρ J$

and

$δ ρ = sup x ∈ B ρ ∖ { 0 } J ( x ) ∥ x ∥ 2 .$

Assume also that, for some $a>0$, with

$a> ρ ρ δ ρ − β ρ$

if $δ ρ <+∞$, the restriction of the functional $∥ ⋅ ∥ 2 −aJ(⋅)$ to $B ρ$ is sequentially weakly lower semicontinuous.

For each $r∈] β ρ ,+∞[$, put

$η(r)= sup y ∈ B ρ ρ − ∥ y ∥ 2 r − J ( y )$

and

$Γ(r)= { x ∈ B ρ : ρ − ∥ x ∥ 2 r − J ( x ) = η ( r ) } .$

Then the following assertions hold:

1. (i)

the function η is convex and decreasing in $] β ρ ,+∞[$, with $lim r → + ∞ η(r)=0$;

2. (ii)

for each $r∈] β ρ + ρ a ,ρ δ ρ [$, the set $Γ(r)$ is non-empty and, for every $x ˆ ∈Γ(r)$, one has

$0< ∥ x ˆ ∥ 2 <ρ$

and

$x ˆ ∈ { x ∈ S ∥ x ˆ ∥ 2 : J ( x ) = sup S ∥ x ˆ ∥ 2 J } ⊆ { x ∈ int ( B ρ ) : ∥ x ∥ 2 − η ( r ) J ( x ) = inf y ∈ B ρ ( ∥ y ∥ 2 − η ( r ) J ( y ) ) } ⊆ { x ∈ X : x = η ( r ) 2 J ′ ( x ) } ;$
3. (iii)

for each $r 1 , r 2 ∈] β ρ + ρ a ,ρ δ ρ [$, with $r 1 < r 2$, and each $x ˆ ∈Γ( r 1 )$, $y ˆ ∈Γ( r 2 )$, one has

$∥ y ˆ ∥<∥ x ˆ ∥;$
4. (iv)

if A denotes the set of all $r∈] β ρ + ρ a ,ρ δ ρ [$ such that $Γ(r)$ is a singleton, then the function $r→Γ(r)$ ($r∈A$) is continuous with respect to the weak topology; if, in addition, J is sequentially weakly upper semicontinuous in $B ρ$, then $Γ | A$ is continuous with respect to the strong topology.

Before proving Theorem 1, let us recall a proposition from  that will be used in the proof.

Proposition 1 Let Y be a non-empty set, $f,g:Y→R$ two functions, and a, b two real numbers, with $a. 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 $g( y b )≤g( y a )$.

Proof of Theorem 1 By definition, the function η is the upper envelope of a family of functions which are decreasing and convex in $] β ρ ,+∞[$. So, η is convex and non-increasing. We also have

$η(r)≤ ρ r − β ρ$
(3)

for all $r> β ρ$ and so

$lim r → + ∞ η(r)=0.$

In turn, this implies that η is decreasing as it never vanishes. Now, fix $r∈] β ρ + ρ a ,ρ δ ρ [$. So, we have

$ρ r − β ρ

Consequently, by (3),

$η(r)

Observe that, for each $λ∈]0,a[$, the restriction to $B ρ$ of the functional $∥ ⋅ ∥ 2 −λJ(⋅)$ is sequentially weakly lower semicontinuous. In this connection, it is enough to notice that

$a a − λ ( ∥ x ∥ 2 − λ J ( x ) ) = ∥ x ∥ 2 + λ a − λ ( ∥ x ∥ 2 − a J ( x ) ) .$

Fix a sequence ${ x n }$ in $B ρ$ such that

$lim n → ∞ ρ − ∥ x n ∥ 2 r − J ( x n ) =η(r).$

Up to a subsequence, we can suppose that ${ x n }$ converges weakly to some $x ˆ r ∈ B ρ$. Fix $ϵ∈]0,η(r)[$. For each $n∈N$ large enough, we have

$ρ − ∥ x n ∥ 2 r − J ( x n ) >η(r)−ϵ$

and so

$∥ x n ∥ 2 + ( η ( r ) − ϵ ) ( r − J ( x n ) ) <ρ.$

But then, by sequential weak lower semicontinuity, we have

$∥ x ˆ r ∥ 2 + ( η ( r ) − ϵ ) ( r − J ( x ˆ r ) ) ≤ lim inf n → ∞ ( ∥ x n ∥ 2 + ( η ( r ) − ϵ ) ( r − J ( x n ) ) ) ≤ρ.$

Hence, since ϵ is arbitrary, we have

$∥ x ˆ r ∥ 2 +η(r) ( r − J ( x ˆ r ) ) ≤ρ$

and so

$ρ − ∥ x ˆ r ∥ 2 r − J ( x ˆ r ) =η(r),$

that is, $x ˆ r ∈Γ(r)$. Now, let $x ˆ$ be any point of $Γ(r)$. Let us show that $x ˆ ≠0$. Indeed, since $r ρ < δ ρ$, there exists $x ˜ ∈ B ρ ∖{0}$ such that

$J ( x ˜ ) ∥ x ˜ ∥ 2 > r ρ .$

Clearly, this is equivalent to

$ρ r < ρ − ∥ x ˜ ∥ 2 r − J ( x ˜ ) .$

So

$ρ r < ρ − ∥ x ˆ ∥ 2 r − J ( x ˆ )$

and hence, since $J(0)=0$, we have $x ˆ ≠0$, as claimed. Clearly, $∥ x ˆ ∥ 2 <ρ$ as $η(r)>0$. Moreover, if $x∈ S ∥ x ˆ ∥ 2$, we have

$1 r − J ( x ) ≤ 1 r − J ( x ˆ )$

from which we get

$J( x ˆ )= sup S ∥ x ˆ ∥ 2 J.$

Now, let u be any global maximum of $J | S ∥ x ˆ ∥ 2$. Then we have

$ρ − ∥ u ∥ 2 r − J ( u ) =η(r)$

and so

$∥ u ∥ 2 −η(r)J(u)=ρ−rη(r)≤ ∥ x ∥ 2 −η(r)J(x)$

for all $x∈ B ρ$. Hence, as $∥ u ∥ 2 <ρ$, the point u is a local minimum of the functional $∥ ⋅ ∥ 2 −η(r)J(⋅)$. Consequently, we have

$u= η ( r ) 2 J ′ (u),$

and the proof of (ii) is complete. To prove (iii), observe that

$1 η ( r ) = inf ∥ x ∥ 2 < ρ r − J ( x ) ρ − ∥ x ∥ 2 .$

As a consequence, for each $r 1 , r 2 ∈] β ρ + ρ a ,ρ δ ρ [$, with $r 1 < r 2$, and for each $x ˆ ∈Γ( r 1 )$, $y ˆ ∈Γ( r 2 )$, we have

$r 1 − J ( x ˆ ) ρ − ∥ x ˆ ∥ 2 = inf ∥ x ∥ 2 < ρ r 1 − J ( x ) ρ − ∥ x ∥ 2$

and

$r 2 − J ( y ˆ ) ρ − ∥ y ˆ ∥ 2 = inf ∥ x ∥ 2 < ρ r 2 − J ( x ) ρ − ∥ x ∥ 2 .$

Therefore, in view of Proposition 1, we have

$1 ρ − ∥ y ˆ ∥ 2 ≤ 1 ρ − ∥ x ˆ ∥ 2$

and so

$∥ y ˆ ∥≤∥ x ˆ ∥.$

We claim that

$∥ y ˆ ∥<∥ x ˆ ∥.$

Arguing by contradiction, assume that $∥ y ˆ ∥=∥ x ˆ ∥$. In view of (ii), this would imply that $J( y ˆ )=J( x ˆ )$ and so, at the same time,

$y ˆ = η ( r 2 ) 2 J ′ ( y ˆ )$

and

$y ˆ = η ( r 1 ) 2 J ′ ( y ˆ ).$

In turn, this would imply $η( r 1 )=η( r 2 )$ and hence $r 1 = r 2$, a contradiction. So, (iii) holds. Finally, let us prove (iv). For each $r∈A$, continue to denote by $Γ(r)$ the unique point of $Γ(r)$. Let $r∈A$ and let ${ r k }$ be any sequence in A converging to r. Up to a subsequence, ${Γ( r k )}$ converges weakly to some $x ˜ ∈ B ρ$. Moreover, for each $k∈N$, $x∈ B ρ$, one has

$ρ − ∥ x ∥ 2 r k − J ( x ) ≤ ρ − ∥ Γ ( r k ) ∥ 2 r k − J ( Γ ( r k ) ) .$

From this, after easy manipulations, we get

$∥ Γ ( r k ) ∥ 2 − ρ − ∥ x ∥ 2 r − J ( x ) J ( Γ ( r k ) ) − ( ρ − ∥ x ∥ 2 r k − J ( x ) − ρ − ∥ x ∥ 2 r − J ( x ) ) J ( Γ ( r k ) ) ≤ ρ − ρ − ∥ x ∥ 2 r k − J ( x ) r k .$
(4)

Since the sequence ${J(Γ( r k ))}$ is bounded above, we have

$lim sup k → ∞ ( ρ − ∥ x ∥ 2 r k − J ( x ) − ρ − ∥ x ∥ 2 r − J ( x ) ) J ( Γ ( r k ) ) ≤0.$
(5)

On the other hand, by sequential weak semicontinuity, we also have

$∥ x ˜ ∥ 2 − ρ − ∥ x ∥ 2 r − J ( x ) J( x ˜ )≤ lim inf k → ∞ ( ∥ Γ ( r k ) ∥ 2 − ρ − ∥ x ∥ 2 r − J ( x ) J ( Γ ( r k ) ) ) .$
(6)

Now, passing in (4) to the lim inf, in view of (5) and (6), we obtain

$∥ x ˜ ∥ 2 − ρ − ∥ x ∥ 2 r − J ( x ) J( x ˜ )≤ρ− ρ − ∥ x ∥ 2 r − J ( x ) r,$

which is equivalent to

$ρ − ∥ x ∥ 2 r − J ( x ) ≤ ρ − ∥ x ˜ ∥ 2 r − J ( x ˜ ) .$

Since this holds for all $x∈ B ρ$, we have $x ˜ =Γ(r)$. So, $Γ | A$ 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 $] β ρ ,+∞[$, we have

$lim k → ∞ ρ − ∥ Γ ( r k ) ∥ 2 r k − J ( Γ ( r k ) ) = ρ − ∥ Γ ( r ) ∥ 2 r − J ( Γ ( r ) ) ,$

and hence

$lim inf k → ∞ ( ρ − ∥ Γ ( r k ) ∥ 2 ) = ρ − ∥ Γ ( r ) ∥ 2 r − J ( Γ ( r ) ) lim inf k → ∞ ( r k − J ( Γ ( r k ) ) ) = ρ − ∥ Γ ( r ) ∥ 2 r − J ( Γ ( r ) ) ( r − lim sup k → ∞ J ( Γ ( r k ) ) ) ≥ ρ − ∥ Γ ( r ) ∥ 2 r − J ( Γ ( r ) ) ( r − J ( Γ ( r ) ) ) = ρ − ∥ Γ ( r ) ∥ 2$

from which

$lim sup k → ∞ ∥ Γ ( r k ) ∥ ≤ ∥ Γ ( r ) ∥ .$

Since X is a Hilbert space and ${Γ( r k )}$ converges weakly to $Γ(r)$, this implies that

$lim k → ∞ ∥ Γ ( r k ) − Γ ( r ) ∥ =0,$

which shows the continuity of $Γ | A$ at r in the strong topology. □

Remark 1 Clearly, when J is sequentially weakly upper semicontinuous in $B ρ$, the assertions of Theorem 1 hold in the whole interval $] β ρ ,ρ δ ρ [$, since a can be any positive number.

Remark 2 The simplest way to satisfy condition (2) is, of course, to assume that

$lim sup x → 0 J ( x ) ∥ x ∥ 2 =+∞.f$

Another reasonable way is provided by the following proposition.

Proposition 2 For some $s>0$, assume that J is Gâteaux differentiable in $B s ∖{0}$ and that there exists a global maximum $x ˆ$ of $J | B s$ such that

$〈 J ′ ( x ˆ ) , x ˆ 〉 <2J( x ˆ ).$

Then (2) holds with $ρ= ∥ x ˆ ∥ 2$.

Proof For each $t∈]0,1]$, set

$ω(t)= J ( t x ˆ ) ∥ t x ˆ ∥ 2 .$

Clearly, ω is derivable in $]0,1]$. In particular, one has

$ω ′ (1)= 〈 J ′ ( x ˆ ) , x ˆ 〉 − 2 J ( x ˆ ) ∥ x ˆ ∥ 2 .$

So, by assumption, $ω ′ (1)<0$ and hence, in a left neighborhood of 1, we have

$ω(t)>ω(1),$

which implies the validity of (2) with $ρ= ∥ x ˆ ∥ 2$. □

Also, notice the following consequence of Theorem 1.

Theorem 2 For some $ρ>0$, let the assumptions of Theorem  1 be satisfied.

Then there exists an open interval $I⊆]0,+∞[$ and an increasing function $φ:I→]0,ρ[$ such that, for each $λ∈I$, one has

$∅≠ { x ∈ S φ ( λ ) : J ( x ) = sup S φ ( λ ) J } ⊆ { x ∈ X : x = λ J ′ ( x ) } .$

Proof Take

$I= 1 2 η ( ] β ρ + ρ a , ρ δ ρ [ ) .$

Clearly, I is an open interval since η is continuous and decreasing. Now, for each $r∈] β ρ + ρ a ,ρ δ ρ [$, pick $v r ∈Γ(r)$. Finally, set

$φ(λ)= ∥ v η − 1 ( 2 λ ) ∥ 2$

for all $λ∈I$. Taking (iii) into account, we then realize that the function φ (whose range is contained in $]0,ρ[$) 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 $ρ>0$, assume that J is sequentially weakly upper semicontinuous in  $B ρ$, Gâteaux differentiable in $int( B ρ )∖{0}$ and satisfies (2). Moreover, assume that there exists $ρ ˜$ satisfying

$inf x ∈ D ∥ x ∥ 2 < ρ ˜ < sup x ∈ D ∥ x ∥ 2 ,$
(7)

where

$D= ⋃ r ∈ ] β ρ , ρ δ ρ [ Γ(r),$

such that $J | S ρ ˜$ has either two global maxima or a global maximum at which $J ′$ vanishes.

Then there exists $λ ˜ >0$ such that the equation

$x= λ ˜ J ′ (x)$

has at least two non-zero solutions which are global minima of the restriction of the functional $1 2 ∥ ⋅ ∥ 2 − λ ˜ J(⋅)$ to $int( B ρ )$.

Proof For each $r∈] β ρ ,ρ δ ρ [$, in view of (7), we can pick $v r ∈Γ(r)$ (recall Remark 1), so that

$inf ] β ρ , ρ δ ρ [ ψ< ρ ˜ < sup ] β ρ , ρ δ ρ [ ψ,$
(8)

where

$ψ(r)= ∥ v r ∥ 2 .$

Two cases can occur. First, assume that $ρ ˜ ∈ψ(] β ρ ,ρ δ ρ [)$. So, $ψ( r ˜ )= ρ ˜$ for some $r ˜ ∈] β ρ ,ρ δ ρ [$. So, by (ii), for each global maximum u of $J | S ρ ˜$, we have $J ′ (u)≠0$. As a consequence, in this case, $J | S ρ ˜$ has at least two global maxima which, by (ii) again, satisfies the conclusion with $λ ˜ = 1 2 η( r ˜ )$. Now, suppose that $ρ ˜ ∉ψ(] β ρ ,ρ δ ρ [)$. In this case, in view of (8), the function ψ is discontinuous and hence, in view of (iv), there exists some $r ∗ ∈] β ρ ,ρ δ ρ [$ such that $Γ( r ∗ )$ has at least two elements which, by (ii), satisfy the conclusion with $λ ˜ = 1 2 η( r ∗ )$. □

## References

1. 1.

Ricceri B: Uniqueness properties of functionals with Lipschitzian derivative. Port. Math. 2006, 63: 393–400.

## Author information

Correspondence to Biagio Ricceri.

### Competing interests

The author declares that he has no competing interests.

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Ricceri, B. A note on spherical maxima sharing the same Lagrange multiplier. Fixed Point Theory Appl 2014, 25 (2014) doi:10.1186/1687-1812-2014-25

• #### DOI

https://doi.org/10.1186/1687-1812-2014-25

### Keywords

• Lagrange Multiplier
• Global Minimum
• Lower Semicontinuous
• Global Maximum
• Weak Topology 