CRITICAL POINT THEOREMS

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CRITICAL POINT THEOREMS

H. BOUKHRISSE and M. MOUSSAOUI Received 12 December 2000

LetHbe a Hilbert space such thatH=V⊕W, whereVandW are two closed subspaces ofH. We generalize an abstract theorem due to Lazer et al. (1975) and a theorem given by Moussaoui (1990-1991) to the case whereVandWare not necessarily finite dimensional.

We give two mini-max theorems where the functionalΦ:H→Ris of classᏯ²andᏯ¹, respectively.

2000 Mathematics Subject Classification: 58E05.

1. Introduction. Our purpose in this note is to generalize a mini-max theorem due to Lazer et al. [3]. Their theorem is as follows.

Theorem1.1. LetXandY be two closed subspaces of a real Hilbert spaceHsuch thatXis finite dimensional andH=X⊕Y (XandY not necessarily orthogonal). Let Φ:H→Rbe aC²functional and let∇ΦandD²Φdenote the gradient and Hessian of Φ, respectively. Suppose that there exist two positive constantsm1andm2such that

D²Φ(u)h,h

≤ −m1h²,

D²Φ(u)k,k

≥m2k² (1.1) for allu∈H,h∈X, andk∈Y. ThenΦhas a unique critical point, that is, there exists a uniquev0∈Hsuch that∇Φ(v0)=0. Moreover, this critical point is characterized by the

Φv0

=max

x∈Xmin

y∈YΦ(x+y). (1.2)

Bates and Ekeland in [1] generalizedTheorem 1.1 to the case whereXand Y are not necessarily finite dimensional. Via a reduction method, Manasevich considered the same case in [4], but he supposed weaker conditions on Hessian of Φ. On the other hand, Tersian [7] studied the case where X and Y are not necessarily finite dimensional, ∇Φ:H→H is everywhere defined and hemicontinuous on H, which means that

limt→0∇Φ(u+tv)= ∇Φ(u) ∀u,v∈H. (1.3) Instead of the conditions on the Hessian ofΦ, they supposed

(1) (∇Φ(h1+y)−∇Φ(h2+y),h1−h2)≤ −m1h1−h2²h1,h2∈X,y∈Y, (2) (∇Φ(x+k1)−∇Φ(x+k2),k1−k2)≥m2k1−k2²k1,k2∈Y,x∈X, whereH=X⊕Y,m1andm2are strictly positive.

Their result rests heavily upon two theorems onα-convex functionals and an ex- istence theorem for a class of monotone operators due to Browder. By a completely

different method, the second author gave another version of Theorem 1.1 (see [5]) with convexity conditions that are weaker than those assumed above.

Theorem1.2. LetHbe a Hilbert space such that H=V⊕W whereV is a finite- dimensional subspace ofHandW its orthogonal. LetΦ:H→Rbe a functional such that

(i) Φis of classᏯ¹. (ii) Φis coercive onW.

(iii) For fixedw∈W,vΦ(v+w)is concave onV.

(iv) For fixedw∈W,Φ(v+w)→ −∞whenv → +∞,v∈V; and the convergence is uniform on bounded subsets ofW.

(v) For allv∈V,Φis weakly lower semicontinuous onW+v.

ThenΦadmits a critical point inH.

We consider the case where X andY are not necessarily finite dimensional. Our proofs contain many steps used in [5] and our convexity conditions are weaker than those given by other authors. First, we prove a mini-max theorem whereΦ:H→Ris of classᏯ². Next, we prove the existence theorem for a particular class ofᏯ¹functional Φ:H→R.

2. First abstract result. The next two propositions are used in this work. For a proof ofProposition 2.1, see [2], and for a proof ofProposition 2.2, see [6].

Proposition2.1. LetXbe a reflexive Banach space and letΦ:X→Rbe a functional such that

(i) Φis weakly lower semicontinuous onX,

(ii) Φis coercive, that is,Φ(u)→ +∞whenu → +∞, thenΦis lower bounded and there existsu0∈Xsuch that

Φu0

=inf

X φ. (2.1)

Proposition2.2. LetHbe a real Hilbert space and letLbe a bounded linear oper- ator onH. Suppose that

(Lx,x)≥ax², (2.2)

for allx∈Handais a strictly positive real number. ThenLis an isomorphism ontoH andL⁻¹ ≤a⁻¹.

Theorem2.3. LetHbe a Hilbert space such thatH=V⊕W whereV andW are two closed and orthogonal subspaces ofH. LetΦ:H→Rbe a functional such that

(i) Φis of classᏯ².

(ii) There exists a continuous nonincreasing functionγ:[0,+∞)→]0,∞)such that D²Φ(v+w)g,g

≤ −γ v

g² (2.3)

for allv∈V,w∈W, andg∈V.

(iii) Φis coercive onW.

(iv) For allw∈W,Φ(v+w)→ −∞whenv → +∞,v∈V. (v) Φis weakly lower semicontinuous onW+v.

ThenΦ admits at least a critical pointu∈H. Moreover, this critical point ofΦ is characterized by the equality

Φ(u)=min

w∈Wmax

v∈V Φ(v+w). (2.4)

In the proof ofTheorem 2.3, we will use the following three lemmas.

Lemma2.4. For allw∈W, there exists a uniquev∈Vsuch that Φ(v+w)=max

g∈VΦ(g+w). (2.5)

Proof. FromTheorem 2.3(ii), forwfixed inW,vΦ(v+w)is continuous and strictly concave onV. Then, it is weakly upper semicontinuous onV. Moreover, from Theorem 2.3(iv), it is anticoercive onV. So that it admits a maximum onV. We affirm that this maximum is unique, otherwise we suppose that there exists two maximums v1andv2. Letvλ=λv1+(1−λ)v2for 0< λ <1, then

Φ v_λ+w

> λΦ v1+w

+(1−λ)Φ v2+w

=Φ v1+w

=Φ v2+w

. (2.6) For the rest of the note, we will adopt the notations

V (w)¯ =

v∈V:Φ(v+w)=max

g∈V Φ(g+w) , S=

u=v+w, w∈W , v∈V (w)¯ .

(2.7)

Lemma2.5. There existsu∈Ssuch that Φ(u)=inf

S Φ. (2.8)

Proof. There exists a sequence(u_n)ofS such thatΦ(u_n)→inf_SΦ=a. For alln, un=vn+wnwithwn∈W, andvn∈V (w¯ n).

Claim

wn ≤c. (2.9)

Otherwise,

Φu_n

=Φv_n+w_n

≥Φw_n. (2.10)

FromTheorem 2.3(iii),Φ(w_n)→ +∞, henceΦ(u_n)→ +∞. This gives a contradiction.

Moreover, from (2.9), there exists a subsequence also denotedw_nsuch thatw_n w.

TakevinV, byTheorem 2.3(v), we have Φ

v+w

≤lim inf

n Φ

v+wn

≤lim inf

n Φ

vn+wn

=a. (2.11)

This is true for allv∈V, in particular, forv∈V (w). Then¯ u=v+w satisfies (2.8).

Lemma2.6. The applicationV¯:W→V such that Φ

w+V (w)¯

=max

g∈V Φ(g+w) (2.12)

is of classC¹.

Proof ofTheorem2.3. For eachw∈W, letΦw :V→Rbe defined byΦw(v)= Φ(v+w). ThenΦw∈C²(V ,R)and forv∈V, we have

∇φw(v),v

=

∇Φ(v+w),v D²Φw(v)v,v ,

=D²Φ(v+w)v,v. (2.13) ByLemma 2.4, we conclude that for allw∈W, there exists a uniquevwinVsuch that

∇Φw(v_w)=0. To prove that ¯V∈C¹(W ,V ), we will use the implicit function theorem.

To see this, letP denote the orthogonal projection ofHontoV. Then

v=V (w)¯ iffP∇Φ(w+v)=0. (2.14) Next, we defineE:W×V→V by

E(w,v)=P∇Φ(w+v). (2.15) ThenE is of classC¹and given any pairw0∈W,v0∈V such thatE(w0,v0)=0, it follows thatv0=V (w¯ 0).

IfE_v denotes the partial derivative ofEwith respect tov, and ifv∈V, we have Ev

w0,v0

v=PD²Φ w0+v0

v. (2.16)

The mappingEv(w0,v0):V→Vis linear and bounded we have fromTheorem 2.3(ii) Ev

w0,v0

v,v

= D²Φ

w0+v0

v,v

≤ −γ v0 v², (2.17) for allv∈V. ByProposition 2.2, Ev(w0,v0)is an isomorphism ontoV. Then from the implicit function theorem [2], there exists aC¹mappingffrom a neighborhoodU ofw0inWintoV such thatE(w,f (w))=0 for allw∈U. Moreover, from (2.14) and (2.15),f (w)=V (w)¯ for allw∈W. Hence, sincew0was arbitrarily chosen, it follows thatf can be defined over all ofW. Then we conclude that ¯V∈C¹(W ,V ).

Remark2.7. The proof ofLemma 2.6relies on the implicit function theorem. This theorem was used by Thews in [8] to prove the existence of a critical point for a particular class of functionals. It was also used by Manasevich in [4].

Proof. Letw∈W andu∈S_w. We will prove that ifusatisfies (2.8), thenuis a critical point ofΦ. ByLemma 2.4, it is easy to see that(∇Φ(u),g)=0 for allg∈V, so it suffices to prove that

∇Φ(u),h

=0 ∀h∈W . (2.18)

Recall thatu∈Scan be writtenu=w+vwherew∈W andv∈V (w). Take¯ h∈W and letw_t=w+thfor|t| ≤1. For eachtsuch that 0<|t| ≤1, there exists a unique vt∈V (wt). ByLemma 2.6, we conclude thatvtn converge to a certainv0 and that v0∈V (w). Then, by¯ Lemma 2.4,v0=v. Fort >0, we have

Φ w_t+v_t

−Φ v_t+w

t ≥Φ

w_t+v_t

−Φ v0+w

t ≥0. (2.19)

Then,

∇Φ

vt+w+λtth ,h

≥0 for 0< λt<1. (2.20) At the limit, we obtain

∇Φ(u),h

=0 ∀h∈W . (2.21)

Hence,uis a critical point ofΦ.

3. Second abstract result. LetHbe a Hilbert space such thatH=V⊕W whereV andW are two closed and orthogonal subspaces ofH. LetΦ:H→Rbe such that

Φ=q+ψ,

q(v+w)=q(v)+q(w) ∀(v,w)∈V×W ψis weakly continuous onH.

(3.1)

Theorem3.1. LetHbe a Hilbert space such thatH=V⊕W whereV andW are two closed and orthogonal subspaces ofH. LetΦ:H→Rbe a functional verifying (3.1) such that

(i) qandψare of classᏯ¹. (ii) ∇Φis weakly continuous onH.

(iii) Φis coercive onW.

(iv) For a fixedw∈W,vΦ(v+w)is concave onV.

(v) For a fixedw∈W,Φ(v+w)→ −∞whenv → +∞,v∈V; and the convergence is uniform on the bounded sets ofW.

(vi) For a fixedv∈V,Φis weakly lower semicontinuous onW+v.

ThenΦ admits a critical pointu∈H. Moreover, this critical point is characterized by the equality

Φ(u)=min

w∈Wmax

v∈V Φ(v+w). (3.2)

For the proof ofTheorem 3.1, we use some results of Lemmas2.4and2.5and we need also the following lemmas.

Lemma3.2. For eachw∈W,V (w)¯ is convex.

Proof. Takev₁,v₂∈V (w)¯ andv_λ=λv₁+(1−λ)v₂withλ∈[0,1]. So that from Theorem 3.1(iv), we haveΦ(v_λ+w)≥λΦ(v1+w)+(1−λ)Φ(v2+w)=Φ(v1+w)= Φ(v2+w). Then

Φ vλ+w

=Φ v1+w

=Φ v2+w

. (3.3)

Sovλ∈V (w).¯

Lemma3.3. LetL(w)= {∇Φ(v+w):v∈V (w)¯ }. For eachw∈W, (i) L(w)is convex.

(ii) L(w)is closed.

Proof. (i) Leth∈Wandv1,v2∈V (w). From¯ Theorem 3.1(iv) andLemma 3.2, we have for allt >0,

Φv_λ+w+th

−Φv_λ+w

≥λ

Φv1+th+w

−Φv1+w +(1−λ)

Φv2+th+w

−Φv2+w. (3.4) Divide bytand letttend to 0, then

∇Φ vλ+w

,h

≥λ

∇Φ v1+w

,h

+(1−λ)

∇Φ v2+w

,h

. (3.5)

Since this is true for allh∈W, we conclude that

∇Φ vλ+w

=λ∇Φ v1+w

+(1−λ)∇Φ v2+w

. (3.6)

(ii) Forw∈W, letSw= {v+w:v∈V (w)¯ }.

First, we show thatS_wis closed. Letv_n+w∈S_w such thatv_n+w→v0+w.Φ(v_n+ w)→Φ(v0+w)andΦ(v_n+w)=max_g∈VΦ(g+w). Thenv0+w∈S_w.

Next, we affirm that Sw is bounded. If not, there exists vn of ¯V (w) such that v_n → +∞, and we conclude fromTheorem 3.1(v) thatΦ(v_n+w)→ −∞. This gives a contradiction.

Consequently,Swis closed and bounded. SinceSwis convex, we conclude thatSwis weakly compact. FromTheorem 3.1(ii), it follows thatL(w)is weakly compact. Then L(w)is weakly closed. ThusL(w)is closed.

Proof ofTheorem3.1. Letw∈W andu∈Sw. Ifusatisfies (2.8), we will show thatL(w)contains 0 and there existsv∈V (w)¯ such that

∇Φ(v+w)=0. (3.7)

By contradiction, suppose thatL(w)does not contain 0. Since it is convex and closed in the Hilbert space, there existsh1∈L(w)such that

0≠ h1 =inf

h:h∈L(w)

. (3.8)

Leth∈L(w),h1+λ(h−h1)∈L(w)forλ∈[0,1], thus h1+λh−h1,h1+λh−h1

≥ h1 ². (3.9) Hence

h1 ²+2λ

h−h1,h1

+λ² h−h1 ²≥ h1 ², (3.10)

so

2h−h1,h1

+λ h−h1 ²≥0. (3.11) Whenλtends to 0. We obtain(h−h1,h1)≥0. So that(h,h1)≥ h1²>0. Equivalently, ∇Φ(v+w),h1>0 ∀v∈V (w).¯ (3.12)

Denotewt=w+th1for|t| ≤1. We note thatwt∈W. ByLemma 2.4, for each 0<

|t| ≤1, there existsvt∈V (wt). Sincewt ≤ w+h1,Theorem 3.1(v) implies that there exists a constantA >0 such that

Φ v+wt

<inf

W Φ≤Φ wt

, (3.13)

forv∈V,v ≥A, and|t| ≤1.(SinceΦis coercive and weakly lower semicontinuous in the reflexive spaceW, it reaches its minimum.) It follows that

v_t ≤A. (3.14)

Otherwise, we would have

Φ vt+wt

<Φ wt

, (3.15)

which contradicts the fact thatv_t∈V (w¯ _t). We conclude then asV is reflexive that there exists a subsequencetn→0 andtn<0 such thatvtn v0∈V.

Claim

v0∈V (w).¯ (3.16)

We havev_tn v0andw_tn→w, so

vtn+wtn v0+w. (3.17)

Sinceψis weakly upper semicontinuous onH, we have ψv0+w

≥lim sup

n→∞ ψv_tn+w_tn. (3.18) ByLemma 2.4,Φis weakly upper semicontinuous onVand we know thatψis weakly lower semicontinuous onV, soq=Φ−ψis weakly upper semicontinuous onV. Then

qv0

≥lim sup

n→∞ qv_tn. (3.19)

Moreover, the continuity ofqimplies that q(w)=lim

n→∞q(w_tn)=lim sup

n→∞ q(w_tn). (3.20)

Then

q v0+w

=q v0

+q(w)

≥lim sup

n→∞ q vtn

+lim sup

n→∞ q wtn

≥lim sup

n→∞

q vtn

+q wtn

≥lim sup

n→∞ qv_tn+w_tn.

(3.21)

On the other hand,v_tn∈V (w_tn)implies that Φ

vtn+wtn

≥Φ v+wtn

∀v∈V . (3.22)

We then obtain qv0+w

+ψv0+w

≥lim sup

n→∞ qv_tn+w_tn

+lim sup

n→∞ ψv_tn+w_tn

≥lim sup

n→∞

q

vtn+wtn

+ψ

vtn+wtn

≥lim sup

n→∞

q v+wtn

+ψ v+wtn

∀v∈V

≥q(v+w)+ψ(v+w) ∀v∈V .

(3.23)

Thus

Φ v0+w

≥Φ(v+w) ∀v∈V . (3.24)

Equivalently,v0∈V (w).¯ Therefore, we have

−Φ

w_tn+v_tn

−Φ

v_tn+w

tn ≥ −Φ

w_tn+v_tn

−Φ v0+w

tn ≥0, (3.25)

and so

∇Φ

vtn+w+εntnh1

,h1

≤0 for 0< εn<1. (3.26) Whentntend to 0, by (ii), we deduce finally that

∇Φ v0+w

,h1

≤0. (3.27)

Which contradicts (3.8). Then there existsv1∈V (w)¯ such that∇Φ(v1+w)=0 and Φv1+w

=min

w∈Wmax

v∈V Φ(v+w). (3.28)

Remark3.4. In the proof ofTheorem 3.1, (3.1) allows us to show thatv0∈V (w).¯ Or, we remark that we do not need to introduceψandqifΦ(v+w)=Φ(v)+Φ(w).

Indeed,wtn→wandvtn v0imply that lim supΦv_tn+w_tn

=lim sup Φv_tn

+Φw_tn

≤lim supΦ v_tn

+lim supΦ w_tn

. (3.29)

ByLemma 2.4,Φis weakly upper semicontinuous onV, thus lim supΦv_tn+w_tn

≤Φv0

+Φ(w)=Φv0+w. (3.30) On the other hand,vtn∈V (w¯ tn)implies that

Φ

vtn+wtn

≥Φ v+wtn

∀v∈V . (3.31)

So

lim supΦ

v_tn+w_tn

≥Φ(v+w) ∀v∈V . (3.32) Then

Φv0+w

≥Φ(v+w) ∀v∈V , (3.33)

that is,v0∈V (w).¯

Remark3.5. We can also proveTheorem 3.1for any functionalΦ:H→Rwithout introducingψandqifΦis weakly upper semicontinuous onH.

Another version ofTheorem3.1. LetAbe a convex set. The functionf:A→R is quasiconcave if for allx1,x2inA, and for allλin]0,1[, then

f

λx1+(1−λ)x2

≥min f

x1

,f x2

. (3.34)

The functionfis quasiconvex if(−f )is quasiconcave, and it is strictly quasiconcave if the inequality above is strict.

It is clear that any strictly concave function is strictly quasiconcave.

Proposition3.6. LetE be a reflexive Banach space. IfΦ:E→Ris quasiconcave and upper semicontinuous, thenΦis weakly upper semicontinuous.

Theorem3.7. LetEbe a reflexive Banach space such thatE=V⊕WwhereVandW are two closed subspaces ofEnot necessarily orthogonal. LetΦ:H→Rbe a functional satisfying (3.1) such that

(i) qandψare of classᏯ¹. (ii) ∇Φis weakly continuous.

(iii) Φis coercive onW.

(iv) For allw∈W,vΦ(v+w)is strictly quasiconcave onV.

(v) For allw∈W,Φ(v+w)→ −∞whenv → +∞,v∈V; and the convergence is uniform on bounded subsets ofW.

(vi) For allv∈V,Φis lower weakly semicontinuous onW+v.

ThenΦ admits a critical pointu∈H. Moreover, this critical point is characterized by the equality

Φ(u)=min

w∈Wmax

v∈V Φ(v+w). (3.35)

In the proof of this theorem, we need Lemmas 2.4 and 2.5. We note that by Proposition 3.6, the result ofLemma 2.4is still true in this case.

Proof. We will prove thatu∈S obtained in Lemma 2.5is a critical point of Φ. We haveΦ(u),vfor allv∈V, so it is sufficient to show thatΦ(u),h =0 for all h∈W. Recall thatu∈S can be written asu=v+w wherew∈W andv∈V (w).¯ Leth∈W andw_t=w+thfor|t| ≤1. For alltsuch that 0<|t| ≤1, there exists a uniquevt∈V (w¯ t). In the same way as in the proof ofTheorem 3.1, we can extract a subsequencevtn such thatvtn v0andv0∈V (w). By¯ Lemma 2.4, we deduce that v0=v. Hence fort >0, we have

Φ(u),h

=0 ∀h∈W . (3.36)

Then,uis a critical point ofΦ.

References

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[3] A. C. Lazer, E. M. Landesman, and D. R. Meyers,On saddle point problems in the calculus of variations, the Ritz algorithm, and monotone convergence, J. Math. Anal. Appl.52 (1975), no. 3, 594–614.

[4] R. F. Manasevich,A min max theorem, J. Math. Anal. Appl.90(1982), 64–71.

[5] A. M. Moussaoui,Questions d’existence dans les problèmes semi-lineaires elliptiques, Ph.D.

thesis, Université libre de Bruxelles, 1990–1991.

[6] J. T. Schwartz,Nonlinear Functional Analysis, Gordon and Breach Science Publishers, New York, 1969.

[7] S. A. Tersian,A minimax theorem and applications to nonresonance problems for semilinear equations, Nonlinear Anal.10(1986), no. 7, 651–668.

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H. Boukhrisse: Department of Mathematics, Faculty of Sciences, University Mohammed I, Oujda, Morocco

E-mail address:[email protected]

M. Moussaoui: Department of Mathematics, Faculty of Sciences, University Mohammed I, Oujda, Morocco