Open Access

A note on spherical maxima sharing the same Lagrange multiplier

Fixed Point Theory and Applications20142014:25

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

Received: 14 October 2013

Accepted: 10 January 2014

Published: 31 January 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 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 ) = μ x .
(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 a J ( ) 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 [1] 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 < b . Let y a be a global minimum of the function f + a g and y b a global minimum of the function f + b g .

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 < a .
Consequently, by (3),
η ( r ) < a .
Observe that, for each λ ] 0 , a [ , the restriction to B ρ of the functional 2 λ J ( ) 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 r η ( 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 < 2 J ( 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 ) . □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, University of Catania

References

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

Copyright

© Ricceri; licensee Springer. 2014

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