A generalization of Ekerland's variational principle and application to the study of the relation between the weak P.S. condition and coercivity

Nonlinear Analysis, Theory, Methods&Applications, Vol. 29, No. 12, pp. 1421-1431, 1997 © 1997ElsevierScience Ltd Printed in Great Britain. All rightsreserved 0362-546X/97 $17.00+ 0.00

Pergamon PII: S0362-546X(96)00180-0

A G E N E R A L I Z A T I O N OF E K E L A N D ' S V A R I A T I O N A L P R I N C I P L E A N D A P P L I C A T I O N TO T H E STUDY OF T H E R E L A T I O N B E T W E E N T H E W E A K P.S. C O N D I T I O N A N D C O E R C I V I T Y ZHONG CHENG-KUI Department of Mathematics, Lanzhou Univeristy, Lanzhou, 730000, Gansu, People's Republic of China

(Received 15 November 1995; received in revised form 20 September 1996; received for publication 24 October 1996) Key words and phrases: Metric space, Ekeland's variational principle, lower semicontinuity, the weak P.S. condition, coercivity.

1. I N T R O D U C T I O N

Let M be a complete metric space, the distance of two points x, y e M being denoted by d(x, y). Let f : M ~ R U {+o0} be a lower semicontinuous function, not identically +oo and bounded from below. Then Ekeland's variational principle states that for every e > 0, ,~ > 0 and every Xo e M such that

f(xo) < inf f ( x ) + e, xEM

there exists some point x~ e M such that

f(x~) <--f(Xo),

d(x,, Xo) <- .a., f ( x ) > f(xe) - ~ d(x, x~),

V x ~ M \ I x , J.

Ekeland's variational principle has many applications to optimization, optimal control, fixed points, differential geometry, differential equations and critical point theory, see [1-3]. If X is a Banach space, f : X ~ R is a G~teaux-differential function, lower semicontinuous and bounded from below, then Ekeland's variational principle implies that for every e > 0 and for every Xo e M such that f(xo) < infx~mf(x) + e 2, there exists some xe e M such that f ( x , ) <-f(Xo), IIx~ - Xoll <- e and IIf'(x,)ll ~ e. On the other hand, if f : X ~ R is a continuously differentiable function, bounded from below, then by use of the deformation theorem in critical point theory, one can easily obtain that for every e > 0, there exists some x, e X such that f(x~) < i n f x ~ u f ( x ) + •2 and lif'(x~)ll -< e/(1 + h(llxoll)), where h: [0, +o0) ~ [0, + ~ ) is a continuous nondecreasing function, for which I~°(1/(1 + h(r))dr = +o0. This result is obviously more precise than Ekeland's variational principle. But it needs that f has stronger smoothness. In Section 2 of this paper, we generalize Ekeland's variational principle to the case which coincides with the result derived from the deformation theorem. In Section 3, we apply the generalization to consider the relation between the weak P.S. condition and the coercivity of functional. 1421

1422

ZHONG CHENG-KUI 2. A G E N E R A L I Z A T I O N

OF EKELAND'S

VARIATIONAL

PRINCIPLE

In this section we will prove the following generalization o f E k e l a n d ' s variational principle. TrmOREM 2.1. Let h: [0, +oo) -o [0, +oo) be a continuous nondecreasing function suclh that J~(1/(1 + h(r)) dr = +co. Let M be a complete metric space, x 0 e M fixed, f : M ~ R k3 1+oo} a lower semicontinuous function, not identically + oo and b o u n d e d f r o m below. Then, f o r every e > 0, every y e M such that

f ( y ) < i n f f + e,

(2.1)

M

and every 2 > 0, there exists some point z ~ M such that

f(z) < f(y),

(2.2)

d(z, Xo) < f + ro,

(2.3)

and

f ( x ) >_f ( z ) - 2(1 + h(d(xo, z))) d(x, z),

v x ~ M,

(2.4)

where r o = d(xo, y) a n d f is such that Ir0+¢ lh(r) dr _ 4. ro 1 +

(2.5)

Remark 2.2. I f we take h(r) =-- 0 and Xo = Y, then this result is just E k e l a n d ' s variational principle.

Proof. Let us define inductively a sequence {x~} in M as follows. Take xl = Y. I f x~ is k n o w n , then xn is such that either (i) f ( x ) > f ( x . ) - e/(2(1 + h(d(xo,x.))))d(X, Xn), V x ~ M , or

(ii) E~ = {x ~ M [ f ( x ) < f ( x . ) - e/(2(1 + h(d(x o, xn))))d(x, x~)] # • . I f case (i) holds, we take X.+l = x . , and if case (ii) holds, we choose x.+l ~ E . such that 1

f(x.+l) < i n f f + - - . E.

n+

(2.6)

1

Thus we obtain a sequence Ix,] in M satisfying

f(Xn+l) < f ( x n ) -

C

4(1 + h(d(xo, x.)))

d(Xn,xn+l),

n = 1,2 .....

(2.7)

In the following we first show that

d(xo, xn) < ro + f,

n = 1, 2 . . . . .

(2.8)

The weak P.S. condition and coercivity

1423

Clearly, (2.8) holds for n = 1. If there is some k such that (2.8) holds whenever 1 _ n < k, but d(xo, Xk) >- ro + ?. This implies that x2 . . . . . Xk are defined as case (ii). Hence f(Xk)

<-- f ( X k _ l )

--

2(1 + h(d(xo, xk-l)))

k-1

d(xk- 1, Xk )

C

< f ( x l ) - n=12 A(1 + h(d(xo, xn))) d(xn' x.+x). C o m b i n i n g with (2.1), we get k-1

2~ ;t(1 + h(d(xo, x.)))

d(xn, Xn+l) < f ( x O - f ( X k )

,'l=l

<--f ( y ) -- inf f < e, M

which implies 1

k-1

~" n=ll+

h(d(xo, xn)) d(xn, Xn+O < 2.

(2.9)

W i t h o u t loss o f generality, we can assume that d ( x o , X , ) , n = 1, 2 . . . . . k, is nondecreasing. In fact, if there is some n, 1 <__ n _< k - 1, such that

d(xo,X.) > d(xo,x.+O, which implies that n < k - 2 since d(xo, Xk) >- ro + f > d(xo, x . ) for every n < k, then by the m o n o t o n i c i t y o f h, we get 1

1

1 + h(d(xo,xn))-

1 + h(d(xo,xn+l) ) "

Hence 1

1 + h(d(xo, Xn))

d(xn, Xn+2) --<

1

1 + h(d(xo, xn)) 1

<_

1 + h(d(xo, x . ) ) +

(d(x., x.+l) + d(X.+l,x.+z)) d(xo , x . +l )

1 d(x.+ l , Xn+ 2). 1 + h(d(xo,X.+l))

Therefore (2.9) still holds if we take

1 + h(U(xo, x.))

d(x~, x.+ 2)

instead o f

1 + h(d(xo, x.))

d(xn, x . +1) +

Thus we can delete X.+l f r o m [Xilik=1.

1 + h(d(xo, xn+0)

d(x.+ l , xn+ 2) .

1424

ZHONG

CHENG-KUI

Since h and d(xo, Xn), n = 1, 2 . . . . . k, are nondecreasing, we have k-1 n=l

1

k-1

1

1 + h(d(xo,Xn)) d(x"'x"+l) >- n ~ 1 + h(d(xo,x.)) (d(xn+l,Xo) - d(x.,Xo)) =l k -

I d(xO'xn+l) - -

, = 1 Jd(x0,x~) l ro+~

ro

1 + h(r)

1

1 + h(r) dr > 2'

which contradicts to (2.9), and completes the p r o o f o f (2.8). N o w we start to prove that Ix.} converges to some point z for which (2.2)-(2.4) hold, By the definition o f {x,}, we k n o w that if there exists some Xk defined as case (i), then x. --- xk for any n _ k, and Xk satisfies (2.2)-(2.4). Therefore, without loss o f generality, we can assume that all o f x, are defined as case (ii). By (2.7) and (2.1), we get k = 1 2(1 + h(d(xo, Xk))) d(Xk, Xk+O < f ( x O -- f(X.÷l) --< f ( x 1 )

<

-- inff M

e.

Letting n --, oo, we obtain ~o

E

1

k = 1 1 + h(d(xo, Xk))

d(Xk, Xk+O --< 2.

(2.10)

Take a constant c such that c _> 1 + h(r o + f). Then, by (2.8), for every n,

1 + h(d(xo, x.))

I.

(2.11)

Hence n+p-I

d(x.+p,x.) <-

~

d(xk,xk+l)

k=n n+p-

< ¢

1

1

~ d(xk, k =. 1 + h(d(xo, Xk))

Xk+

1)

oo

<_c~ 1 k =. 1 + h(d(x o, Xk)) d(xk, Xk+ 1). C o m b i n i n g with (2.10), we k n o w that {x.] is a C a u c h y sequence. So [x.] converges to some point denoted by z. It follows f r o m (2.8) that z satisfies (2.3). Using (2.7) and the lower semicontinuity o f f , we get

f ( z ) <- lim f(xn) <-f(x.) <-f(xl). n~oo

This shows that (2.2) holds. N o w if z does not satisfy (2.4), then there exists some zl ~ M such that

f ( z l ) < f ( z ) - 2(1 + h(d(xo, z))) d(z, Zl).

(2.12)

The weak P.S. condition and coercivity

1425

Since litany® xn = z, and f(z) <-f(x,), there exists some n o such that, for n _ n o,

f(zl) < f(Xn) - A(1 + h(d(xo, xn))) d(xn, ZO.

(2.13)n

This shows that z I E E~ as n _> no. C o m b i n i n g (2.6) and (2.13)n+~, we obtain 1

E

inff + - >_ f(x~+l) > f(Zl) ÷ d(xn+l, Zl) e~ n + 1 2(1 + h(d(xo, xn+l))) E

>_ infenf + J.(1 + h(d(xo, xn+ 1))) d(x#+ 1, zl). C o m b i n i n g with (2.11), we have

d(Zl, Xn+ 1) Letting n ~ oo, we get l i m ~ _ . ~ x ~ completes the p r o o f . •

=

A

c

<: - - - .

en+l

zl. Hence z = z ~ . But this contradicts to (2.12) and

3. T H E W E A K P . S . C O N D I T I O N , THE EXISTENCE MINIMAL POINT AND COERCIVITY

OF

T h r o u g h o u t this section X denotes a B a n a c h space, recall that a f u n c t i o n f : X --, R LI I+oo} is called Gfiteaux differentiable if at every point Xo with f(Xo) < + oo, there exists a c o n t i n u o u s linear functional f'(Xo) such that, for every y ~ X : lira f(xo + ty) - f(Xo) = 0, every y e X such that

f(y) < inff + e x

(3.1)

and every 2 > 0, there exists z e X such that

f(z) <- f ( y ) ,

(3.2)

Ilzl[ -< Ilyll + ~,

(3.3)

8

IIf'(z)ll ~ A(1 ÷ h(llzll))'

(3.4)

where f is such that I HyII+~

Ilyll

1

1 + h(r) dr

>_ ~.

(3.5)

1426

ZHONG CHENG-KUI

Proof. Take Xo = 0 and use Theorem 2.1 directly. Inequality (2.4) gives us, for every y e X with Ilyl[ = 1 and every t > 0 f ( z + ty) - f ( z ) t

e

&(1 + h(llzll))"

(3.6)

Letting t ~ 0, we obtain

( f ' ( z ) , Y) >-

~.(1 + h([Izll))

for all y e X with [[yl[ = 1, and hence (3.4). COROLLARY 3.2. I f f is bounded from below, then for every e > O, there exists some point z, such that

f(z~) < i n f f + e 2,

(3.7)

Ilf'(zAll < -- 1 + h(llzA)'

(3.8)

X

Proof. Just take e 2 instead of e and e instead of 2 in the preceding theorem.

•

COROLLARY 3.3. I f f is bounded from below, then there exists a minimizing sequence [z~} o f f such that (3.9)

llf'(x~)ll(1 + h(llxnll)) ~ o. Proof. Take e = 1/n, n = 1, 2 . . . . in the preceding corollary.

•

Definition 3.4. f is said to satisfy the weak P.S. condition if the existence of a sequence [x~} in X such that {f(x~)] is bounded and Ilf'(x.)ll(1 + h(llx.ll)) --' 0 implies that [x,] has a convergent subsequence. Remark 3.5. If we take h(r) =- 0 and h(r) = r respectively, then the weak P.S. condition :is just the famous P.S. condition and (C) condition respectively, see [4, 5]. In critical point theory, many results still hold, if we take the weak P.S. condition instead of P.S. condition. THEOREM 3.6. I f f is bounded from below and satisfies the weak P.S. condition, t h e n f has a minimal point.

Proof. By Corollary 3.3, there is a minimizing sequence [xn} in X such thatf(xn) ~ infx f a n d IIf'(x.)ll(1 + h(llx.ll)) --' 0. The weak P.S. condition implies that [xn} has a subsequence [xnk} convergent to some point x*. Since f is lower semicontinuous, we get i n f f < _ f ( x * ) < inf f(xnk) < i n f f . k~ao

Therefore f ( x * ) = infx f .

•

The weak P.S. condition and coercivity

1427

In the following we consider the relation between the weak P.S. condition and coercivity. THEOREM 3.7. If a = lim inf f ( x ) > - - o o , r~*~ Ilxll> r then there exists a sequence [xn} in X such that IlXnll -~ oo, f ( X . ) -" ~ and

Ilf'(x.)ll(1 + h(llx.ll))--' o. If we take h(r) - O, then this theorem is just the result proved by Brezis and Nirenberg [1] using Ekeland's variational principle.

Proof. Set, for r _> 0, m(r)= infllxll>rf(X). Clearly m is nondecreasing function and limr~.~ re(r) = ~. Then, for any positive e < 1/3, there exists some number ~ __ I/e, such that c~ - e 2 <_ m(r)

for r _> t~.

(3.10)

Since I~(1/(1 + h(r))dr = +oo, choose r* > ~ such that It* lh(r) dr >_ 1. ~ 1+

(3.11)

By the definition of Riemann integral, for the positive e given above, we take a partition A:~=

r N < rN_l < ... < rl < ro = r*,

such that for any ~k e [rk+l, rk],

lh(r)d r

Ifi* ,

-

1 +

iv-1 ~

1

(rk-

rk+l)

I

< e.

k=o 1 + h(~k)

Especially, using the monotonicity o f h, we have

f r"

lh(r) dr

N-1 ~ f

I -4-

k=O

1 +lh(r)dr rk+l

N-1

1

>- k~= 0 1 + _> Set

l

~*

h(rk ) (rk - rk+l) 1

- - d r

Me = [x e X] Ilxll --- rk},

1 + h(r)

- e.

(3.12)

k = O, 1,2 . . . . . N.

Applying Theorem 2.1 in Mo, we find some point z0 e Mo such that

f(zo) < m(ro) + t 2 <- a + e z, and for every x e M 0,

f(x) >--f(Zo)

1 + h(llzoll)

Ilx - zoll.

(3.13)

1428

If

[Izoll >

ZHONG

CHENG-KUI

r0 or (3.13) holds for every x e M~, then (3.13) implies

IIf'(zo)ll(1 + h(llzoll)) -< E, and completes the proof. So, without loss o f generality, we assume that Ilzolt : r0 and there exists some x-~ e MlX,M0 such that

f(x-~l) < f(Zo)

1 + h(llzoll)lit - zoll.

(3.14)

Replacing M, e and 2 by M1, e 2 and e respectively and taking x x = Zo, by the process o f the p r o o f o f Theorem 2.1, there exists a sequence Ix,} in M~ such that 1

f(xn+l) < f(xn)

+ h(llx.ll)IIx.

- X.+lll,

(3.15)

lim xn = Zl,

n~t~

f ( z l ) < lim f ( x n ) < f ( x n ) < f ( x l ) n~oo

<_ ol + e 2 <_ m ( r O + 2e 2

and for every x e M~,

f(x) >-f(zO

1 + h(llz~ll)IIx - z, ll.

(3.16)

If Ilzlll > rl or (3.16) holds for every x e M 2, then (3.16) implies IIf'(zl)ll(1 + h([Iz~ll)) ~ ~, f r o m which the p r o o f is complete. Therefore we can assume that Ilz, tl = r~ and there exists some x2 e M2kMI, such that

1 + h(llz~ll) I1~ - z, ll.

f(x--22) < f(Zl)

(3.17)

Before the next step, we first show that, in this case, z~ satisfies 1 + h(llzoll)I[zo

- z~ll ~ f(zo) - f ( z l ) .

In fact, by (3.15) and the lower semicontinuity of f , we have f ( Z l ) < lim f ( X n ) < f ( x n ) n-1

<-f(zo)-

e

k=l ~ 1 + h(lixk[)

IIxk - x~÷lll.

Hence k~ x c IlXk - Xk+, l[ ~ f(Zo) -- f ( z O . = 1 + h(l[Xk[I)

(3.18)

The w e a k P.S. c o n d i t i o n a n d c o e r c i v i t y

If

1429

IIx~ll ~ Llzoll = ro for every k _> 1, then, by the m o n o t o n i c i t y o f h, E

oo

C

IIzo - zlll -< E = 1 + h(]lZo[[)

1 + h ( l l x , ll) IIx~ - x~+,ll -< f ( Z o ) - f ( z t ) .

Therefore, in order to p r o v e (3.18), we have only to prove inductively the following inequality: IIx~ll <

IIz011 = ro,

k = 2, 3 . . . . .

(3.19)

First, b y (3.14) and the process of the p r o o f o f T h e o r e m 2.1, we have

X2

e E1=

Ix I

e M 1 f ( x ) < f(Zo)

1 +

xzol

h(llzoll)

C o m b i n i n g with (3.13), we k n o w that x2 e MlXag/o, so IIx~ll < to. Now, if there exists some k > 2 such that [[x, II < ro for 2 <_ n < k and Ilxk[[ -> ro, since x k # xk_ 1, it follows f r o m the process o f T h e o r e m 2.1 that x 2 . . . . , xk are all defined as in case (ii). H e n c e f(xk) < f(Xk_,)

1 + h(llx~ll)Ilxk-, k-'


O n the other hand, since

E 1+

~=I

IIxA ~ ro, that is xk f ( x k ) >- f ( x l )

- xkl[

h(llx~]i)lx~ -

x,+,It.

(3.20)

e Mo, using (3.13), we have

1 + h(llxlil)I[xl, - x,[[.

(3.21)

C o m b i n i n g (3.20) and (3.21), we get k-1

~,

.=,

e

I + h(llx.ll)

IlXn -- X.+III <

1 + h(llxlll)

Ilxk -- X, II k-1

1 +

h(llxdl)

2

IIx. - x.+,ll

n=l

k-I

~

n=l

1 + h(llx, ll)IIx. - x,+,ll.

This is impossible. T h e r e f o r e (3.19) holds. Replacing M 1, Zo by M 2 and Zl respectively and repeating the process above, there exists some point z2 e M2 such that re(r2) -< f ( z z ) < - f ( z l ) -< ot + e 2 <__ re(r2) + 2e 2 and for every x e M 2 , f ( x ) > f(z2)

1 +

h(llz21t)llx - z211.

(3.22)

1430

Z H O N G CHENG-KUI

Then either (i) I[z2[[ > r2 or (3.22) holds for every x e M 3, or

(ii) [Iz2[[ = r2 and there exists some ~ e M 3 ~ M 2 such that

f(~)

< f(z2)

1 + h(Jlz2ll)I[~ - z2l[,

and

1 + h(llz~ll)Ilz2 - Zlll ~ f(zl)

-f(z2).

If case (i) holds, then (3.22) implies that

Ilf'(z2)ll(1 + h(llz2ll)) ~ c, and completes the p r o o f at this moment. If case (ii) holds, we continue step by step. N o w we assert that there exists some kth step, 0 _< k _< N, such that the case (i) holds. Otherwise, the case (ii) holds for every step. T h e n there exists Zo . . . . . Zk . . . . . ZN such that, for every 0 _< k __
Ilzkl[ =

(3.23)

rk

and (3.24)

1 + h(llzA)Ilzk - z~+,ll ~ f(zk) -f(Zk+l). Therefore, N-1

k=O

C

1 + h(llzA)IJz~- z~+~ll
(3.25)

< 2£ 2.

By (3.11), (3.12), (3.23), (3.24) and (3.25), we get t*

1 -- e<--

I

1

N-1

=

N-I

- - d r - e < _ 1 + h(r)

k=O 1 + h(HZkl[)

<- k ~ =ol+

1

k=O

1 + h(rk)

(r k -

rk+ 0

1

E

N-I

~

(/Iz,,ll-

[IZk+l[I)

l

h(llZkll)(IlZk

-- Z,+tll)

_< 2e, Hence e >- {. It contradicts to e < {. Thus the p r o o f is complete.

•

The weak P.S. condition and coercivity

1431[

COROLLARY 3.8. I f f is b o u n d e d f r o m b e l o w a n d satisfies t h e w e a k P . S . c o n d i t i o n , t h e n f ( x ) ~ +oo as I[xn[I -~ +=o. T h i s c o r o l l a r y g e n e r a l i z e s t h e r e s u l t p r o v e d b y L i [6] u s i n g a g r a d i e n t f l o w a n d b y C o s t a a n d de Silva [7] u s i n g E k e l a n d ' s v a r i a t i o n a l p r i n c i p l e . REFERENCES 1. 2. 3. 4.

Brezis, H. and Nirenberg, L., Remarks on finding critical points. Comm. PureAppl. Math., 1991, 64, 939-963. Ekeland, I., On the variational principle. J. Math. Anal. Applic., 1974, 47, 324-357. Shi, S. Z., Ekeland's variational principle and the mountain pass lemma. Acta Math. Sinca (NS), 1985, 1, 348-358. Bartolo, P., Benci, V. and Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with "strong" resonance at infinity. Nonlinear Analysis, 1983, 7(9), 981-1012. 5. Cerami, G., Un criterio di esistenza per i punti critici su varieta illimitate. Re. Ist. lomb. Sci. Lett., 1978, 112., 332-336. 6. Caklovic, L., Li, S. J. and Willem, M., A note on Palais-Smale condition and coercivity. Diff. Int. Eqn., 1990, 3, 799-800. 7. Costa, D. G. and Elves, de B. e Silva, The Palais-Smale condition versus coercivity. Nonlinear Analysis, 1991, 16, 371-381.