On Smooth Variational Principles in Banach Spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

197, 153]172 Ž1996.

0013

On Smooth Variational Principles in Banach Spaces M. Fabian* Department of Mathematics, Miami Uni¨ ersity, Oxford, Ohio 45056

P. Hajek ´ Department of Mathematics, Uni¨ ersity of Alberta, Edmonton, Alberta T6G 2G1, Canada

and J. Vanderwerff*, † Department of Mathematics and Statistics, Simon Fraser Uni¨ ersity, Burnaby, British Columbia V5A 1S6, Canada Submitted by Richard M. Aron Received November 22, 1993

A new smooth variational principle for spaces admitting Frechet differentiable ´ bump functions is proved. Further it is shown that each proper lower semicontinuous bounded below function can be supported by a smooth function with locally Holder derivative if and only if the space is superreflexive. Some geometrical ¨ refinements of the Borwein]Preiss smooth variational principle using Deville’s techniques are obtained. Q 1996 Academic Press, Inc.

INTRODUCTION Let f : X ª Žy`, q`x be a lower semicontinuous Žshortly lsc. bounded below function on a Banach space Ž X, 5 ? 5.. By a ¨ ariational principle we mean an assertion ensuring the existence of a function f : X ª R belonging to a given class such that f supports, that is, touches f from below at a point ¨ g X. The first such principle, based on the Bishop]Phelps theorem, was established by Ekeland in 1974. It says, roughly speaking, that for the supporting function f one can take a shift of the function ye 5 ? 5, where e is an arbitrarily small positive number w7, 8x. This * Research completed while visiting University of Alberta and University of Waterloo. † NSERC ŽCanada. postdoctoral fellow. 153 0022-247Xr96 $12.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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FABIAN, HAJEK ´ , AND VANDERWERFF

principle has proved to be very useful in various areas of nonlinear analysis, in particular, in optimization and Žsub.differentiability. Let us mention one of its important consequences: a Frechet smooth Banach ´ space is Asplund w9x. If f is required to be smooth, then we speak about a smooth ¨ ariational principle. The first result of this type was shown by Stegall in 1978 w19x; see also w17, Corollary 5.22x. It asserts that if f Ž x . grows enough when 5 x 5 ª q` and, moreover, if X has the Radon]Nikodym ´ property Žin . particular, if X is reflexive , then for f one can take even a linear functional, with arbitrarily small norm. Thus, if X has the Radon]Nikodym ´ property and if moreover X admits a nonzero function b with bounded nonempty support Žsuch a b will be called a bump function. with some degree of smoothness, then, after performing some easy gymnastics f can be supported by a function f having the same smoothness as b does Žsee the proof of Theorem 1.6.. However, this principle does not cover some important Banach spaces. Indeed, c 0 does not have the Radon]Nikodym ´ property while it, in fact, admits a C ` -smooth norm w1x Žand hence has a C ` -smooth bump function.. In 1987, Borwein and Preiss w2x published a smooth variational principle imposing no additional conditions on the space, except, of course, the presence of some smooth norm. More concretely and roughly speaking: if X admits an equivalent norm with some kind of first-order smoothness, then f can be supported by a concave f with the same smoothness w17, Chap. 4x. In their paper they show several applications of this principle to Žsub.differentiability. In particular, it is proved that a space admitting a Gateaux differentiable norm is a Gateaux differentiability space. It should be noted that using different techniques it was proved in w18x that such a space is in fact a weak Asplund space. For further consequences of the Borwein]Preiss principle to subdifferentiability of lsc functions, see w11x. Recently the Borwein]Preiss principle was extended by Deville et al. to spaces admitting smooth bump functions with bounded highest derivatives w4x. ŽHaydon’s work shows that this is a real extension; see w13, 5x. Their paper has two other interesting features. Namely, the proof of its principle uses the Baire category theorem and applications to viscosity solutions of Hamilton]Jacobi equations are given. However, some questions have remained open in this area. The main purpose of the present paper is twofold: to consider the cases not covered by w4x and to show the limitations of this theory. In particular, Theorem 1.1 provides a smooth variational principle for spaces admitting a Frechet smooth bump function whose derivative may not be bounded. On ´ the other hand one cannot expect a C 2-smooth or even a Holder-smooth ¨ variational principle, unless the space in question is superreflexive; see Theorem 1.6. And it should be noted that once the space is superreflexive,

SMOOTH VARIATIONAL PRINCIPLES

155

or even has the Radon]Nikodym ´ property, then Stegall’s principle is available and it is no longer necessary to use the techniques from w2x or w4x. For related results see also w12x. In the second section we do not resist the temptation to include some subtler supporting results when both f and f are convex or even norms. We proceed here in the spirit of w2x, but instead we use the Baire category technique from w4x. A main result here, Theorem 2.4, says more than that, if X has a Frechet smooth norm, then for S X the unit sphere with respect ´ to a fixed norm on X, there are equivalent Frechet differentiable norms ´ that attain their minimum on S X . We now list some of the notations and conventions that we use. Unless otherwise specified, Ž X, 5 ? 5. always denotes a real Banach space with dual X *. For a fixed norm 5 ? 5, we let S X s  x: 5 x 5 s 14 , BX s  x: 5 x 5 F 14 , and B Ž x 0 , d . s  x: 5 x y x 0 5 - d 4 for a given x 0 g X and d ) 0. A norm 5 ? 5 is said to be locally uniformly rotund ŽLUR. if 5 x n y x 5 ª 0 whenever 2 5 x n 5 2 q 2 5 x 5 2 y 5 x n q x 5 2 ª 0; see w5x for more information on renorming. The adjective ‘‘smooth’’ means the same as ‘‘differentiable.’’ By abuse of language, we say a norm is differentiable, or smooth, if it is so at every nonzero point. We use F k Žresp. G k . to denote the classes of k-times Žresp. k-times Gateaux. differentiable functions on Banach spaces, Frechet ´ as defined in w5, 16x. The class of all f g F k with f Ž k . continuous is denoted by C k . We sometimes say that a function f is S-smooth if it belongs to S , where S is a specified class of smooth functions. For a function f : X ª Žy`, q`x, we let DŽ f . s  x: f Ž x . - q`4 . A function f is said to attain its strong minimum at ¨ if f Ž ¨ . s inf f Ž x .: x g X 4 and 5 yn y ¨ 5 ª 0 whenever f Ž yn .x f Ž ¨ .. For a function c : X ª Ž Y, 5 ? 5., we let 5 c 5 ` s sup5 c Ž x .5: x g X 4 . We use some results on smooth partitions of unity; the reader is referred to w20, 5, VIII.3x for more information on smooth partitions of unity.

1. SMOOTH BUMP FUNCTIONS AND VARIATIONAL PRINCIPLES We begin with our main positive result. Limitations on the availability of higher-order principles are given later in this section. THEOREM 1.1. Let X be a Banach space admitting a Frechet differentiable ´ bump function Ž with possibly unbounded deri¨ ati¨ e . and consider an lsc function f : X ª Žy`, `x. Then for e¨ ery e ) 0 and e¨ ery u g DŽ f . there is a ¨ g DŽ f . with 5 ¨ y u 5 - e , < f Ž ¨ . y f Ž u.< - e , a number D ) 0, and a Frechet differentiable function f : B Ž ¨ , D . ª Žy`, `. such that f < BŽ ¨ , D . y f ´

FABIAN, HAJEK ´ , AND VANDERWERFF

156

attains its strong minimum at ¨ . Moreo¨ er, if f is bounded below, we can take D s y` and f may be a bump function. We need the next three lemmas in the proof of Theorem 1.1. The first is well known and can be proved by composing the bump function with an appropriate smooth real-valued function; see e.g. w15, p. 34x. LEMMA 1.2. Let X be a Banach space and let S be one of the following classes of smooth functions on X: C k , F k , G k , functions with Ž locally . aHolder deri¨ ati¨ e. Suppose there is a bump function b g S and d ) 0 is ¨ gi¨ en. Then there exists ˜ b: X ª w0, 1x belonging to S and such that ˜ bŽ0. s 1 and ˜ bŽ x . s 0 when 5 x 5 G d . If, moreo¨ er, b Ž k . is bounded, then we may also ha¨ e ˜ b Ž k . bounded. LEMMA 1.3. Consider a function r : w0, q`. ¬ w0, q`x such that r Ž0. s s 0. Then there exist d ) 0 and a nondecreasing C 1-smooth function w s : 0, d . ª w0, q`. such that s Ž0. s sqX Ž0. s 0 and s Ž t . G r Ž t . q t 2 for all t g w0, d .. X Ž0. rq

Proof. We choose a sequence  a i 4 ; Ž0, q`. such that a iq1 - a ir2 and

rŽ t. q t2 -

t 2 iq3

if t g w 0, a i x

Ž 1.1.

for all i s 1, 2, . . . . Put d s a1 and define r: w0, d x ª w0, q`. by r Ž0. s 0, r Ž a i . s 2yi and r is linear on w a iq1 , a i x for i s 1, 2, . . . . Clearly r is continuous. Define s : w0, d . ª w0, q`. by

s Ž x. s

x

H0 r Ž t . dt

for x g 0, d . .

Fix x g Ž0, d .. Then x g w a iq1 , a i . for some i, and thus by Ž1.1. and the definition of r,

s Ž x. G s

x

Ha

r Ž t . dt q

iq1

x y a iq1 2

iq1

q

a iq1

HŽ a

a iq1 2

iq3

iq1 .r2

G

r Ž t . dt G x

2

iq3

x

Ha

2yiy1 dt q

iq1

a iq1

HŽ a

iq1 .r2

2yiy2 dt

) r Ž x. q x2.

The remaining properties of s can be verified easily. LEMMA 1.4. Assume that a Banach space X admits a Frechet differen´ tiable bump function. Then there exist K ) 1 and a function d: X ª w0, q`. that is Frechet differentiable at each x g X _  04 and which also satisfies ´ 5 x 5 F dŽ x . F K 5 x 5 if 5 x 5 F 1 and dŽ x . s 2 if 5 x 5 ) 1.

157

SMOOTH VARIATIONAL PRINCIPLES

Proof. Omit the Lipschitz part from the proof of w5, Lemma VIII.1.3x. Proof of Theorem 1.1. Because X admits a Frechet differentiable bump ´ function it is an Asplund space w9x. Fix any e ) 0 and any u g DŽ f .. By w11, Theorem 1x, we find ¨ g DŽ f ., with 5 ¨ y u 5 - e , < f Ž ¨ . y f Ž u.< - e , such that f is Frechet subdifferentiable at ¨ ; that is, there exists L g X * ´ satisfying the inequality lim inf hª0

1 5 h5

f Ž ¨ q h . y f Ž ¨ . y ² L , h: G 0.

Define r : w0, q`. ª Žy`, q`x by

r Ž t . s yinf  f Ž ¨ q h . y f Ž ¨ . y ² L , h: : h g X , 5 h 5 F t 4

for t G 0.

X Ž0. s 0. Then we can easily see that r is nonnegative and r Ž0. s rq Let s : w0, d . ª w0, q`. for some d ) 0 be the function corresponding to our r in Lemma 1.3. Take d and K from Lemma 1.4. Put D s minŽ1, Ž drK . and define c Ž x . s s Ž dŽ x y ¨ .. for x g B Ž ¨ , D .. By the chain rule, c is Frechet differentiable at each point of B Ž ¨ , D . _  ¨ 4 . Also, ´

0 F c Ž ¨ q h . y c Ž ¨ . s s Ž d Ž h . . F s Ž K 5 h 5.

if 5 h 5 - D .

Consequently, c is Frechet differentiable at ¨ with c 9Ž ¨ . s 0. Moreover, ´ for x g B Ž ¨ , D . we have

c Ž x . s s Ž d Ž x y ¨ . . G s Ž 5 x y ¨ 5. G r Ž 5 x y ¨ 5. q 5 x y ¨ 5 2 G yf Ž x . q f Ž ¨ . q ² L , x y ¨ : q 5 x y ¨ 5 2 . Putting f Ž x . s ² L, x y ¨ : y c Ž x . for x g B Ž ¨ , D ., we have that f is Frechet differentiable, f Ž ¨ . s 0, and hence the above inequality yields ´ f Ž x. y fŽ x. G f Ž ¨ . y fŽ ¨ . q 5 x y ¨ 52

for x g B Ž ¨ , D . . Ž 1.2.

Thus, if  yn4 ; B Ž ¨ y D . and if f Ž yn . y f Ž yn . ª f Ž ¨ . y f Ž ¨ ., then 5 yn y ¨ 5 2 ª 0 and so f y f attains its strong minimum at ¨ . Finally, assume that inf f ) y`. By Lemma 1.2 choose a Frechet ´ differentiable bump function ˜ b: X ª w0, 1x such that ˜ bŽ0. s 1 and ˜ bŽ x . s 0 if 5 x 5 G Dr2. Now define f 1: X ª R by

f 1Ž x . s

½

˜b Ž x y ¨ . f Ž x . s f Ž ¨ . y f Ž ¨ . y inf f q 1 0

if 5 x y ¨ 5 - D otherwise.

FABIAN, HAJEK ´ , AND VANDERWERFF

158

differentiable bump function. Further, if It is clear that f 1 is a Frechet ´ 5 x y ¨ 5 - Dr2, we have by Ž1.2. f Ž x . y f 1Ž x . s f Ž x . y ˜ b Ž x y ¨ . f Ž x . q f Ž ¨ . y f Ž ¨ . y inf f q 1 G f Ž x. y ˜ b Ž x y ¨ . f Ž x . y inf f q 1 G inf f y 1 s f Ž ¨ . y f 1 Ž ¨ . . If 5 x y ¨ 5 G Dr2, then f Ž x . s f 1Ž x . s f Ž x . ) inf f y 1 s f Ž ¨ . q f 1Ž ¨ .. It follows ¨ is a minimum of f y f 1. We show that it is a strong minimum. So let f Ž yn . y f 1Ž yn . ª f Ž ¨ . y f 1Ž ¨ .. The definition of f 1 ensures that 5 yn y ¨ 5 - Dr2 for all large n; assume for simplicity that this is so for all n. Then inf f y 1 s f Ž ¨ . y f 1 Ž ¨ . s lim

nª`

Ž f Ž yn . y f 1 Ž yn . .

s lim  f Ž yn . y ˜ b Ž yn y ¨ . nª`

= f Ž yn . q f Ž ¨ . y f Ž ¨ . y inf f q 1

4

G lim sup  f Ž yn . y ˜ b Ž yn y ¨ . nª`

= f Ž yn . y inf f q 1 s lim sup nª`

G lim sup nª`

4

by Ž 1.2.

½ 1 y ˜b Ž y

n

y ¨ . f Ž yn . q ˜ b Ž yn y ¨ . w inf f y 1 x

½ 1 y ˜b Ž y

n

y ¨ . inf f q ˜ b Ž yn y ¨ . w inf f y 1 x

5

5

s inf f y lim inf ˜ b Ž yn y ¨ . . nª`

Hence lim inf n ª` ˜ bŽ yn y ¨ . G 1 and so ˜ bŽ yn y ¨ . ª 1. Thus inf f y 1 s lim  f Ž yn . y f Ž yn . q f Ž ¨ . y f Ž ¨ . y inf f q 1 nª`

4

and therefore f Ž yn . y f Ž yn . ª f Ž ¨ . y f Ž ¨ .. But from the preceding paragraph we already know that the last convergence implies yn ª ¨ . This means f y f 1 attains its strong minimum at ¨ . For a g Ž0, 1x, a function f on X is said to be a-Holder subdifferentiable ¨ at ¨ g X if there are d ) 0, C ) 0, and L g X * such that f Ž ¨ q h . y f Ž ¨ . y ² L , h: G yC 5 h 5 1q a

whenever h g X and 5 h 5 F d .

SMOOTH VARIATIONAL PRINCIPLES

159

We say that a differentiable function f has a pointwise a-Holder ¨ deri¨ ati¨ e if for any ¨ g X there are C ) 0 and d ) 0 Žboth depending on ¨ . such that 5 f 9Ž ¨ q h. y f 9Ž ¨ . 5 F C 5 h5 a whenever h g X and 5 h 5 - d . The procedure from the proof of Theorem 1.1 is applicable also for pointwise Holder smoothness, which we state without proof in the next ¨ proposition. PROPOSITION 1.5. Let f : X ª Žy` q `x be a lsc function and ¨ g X. deri¨ ati¨ e If X admits a Lipschitz bump function b with pointwise a-Holder ¨ and f is a-Holder subdifferentiable at ¨ , then there is a f : X ª R with the ¨ same smoothness properties as b such that f y f attains a strong local minimum at ¨ . Remark. If the conclusion of Proposition 1.5 holds for C 1-smoothness, then y5 ? 5 2 can be supported locally by a C 1-smooth function f at ¨ s 0. In this case, one can construct a C 1-smooth Lipschitz bump on the space by composing f with an appropriate real function. However, it is unknown if there is a space with a C 1-smooth bump function but no C 1-smooth Lipschitz bump function or if a C 1-smooth variational principle would be valid on such a space. However, the proof of w10, Proposition 3.5x shows if X r c 0 and X admits a C 1-smooth bump function, then X admits a Lipschitz and C 1-smooth bump function. This statement can also be shown with the compact variational principle of Chapter V.2. of w5x through the proof of Theorem V.3.1 of the same book. The next theorem shows that one cannot expect higher-order smooth variational principles unless the space in question is superreflexive. Thus, in particular, in c 0 , there is no C 2-smooth nor even Holder-smooth ¨ variational principle. The statement Ža. « Žd. and its proof, in the following theorem, are very similar to w5, Theorem IV.5.4x. THEOREM 1.6.

For a Banach space X, the following are equi¨ alent.

Ža. X is superreflexi¨ e. Žb. For e¨ ery lsc bounded below function f : X ª Žy`, q`x there are ¨ g X, d ) 0 and a differentiable function f : B Ž ¨ , d . ª R with Holder ¨ deri¨ ati¨ e such that f Ž x. y fŽ x. G f Ž¨ . y fŽ¨ .

for all x g B Ž ¨ , d . .

Žc. For e¨ ery lsc bounded below function f : X ª Žy`, q`x there are ¨ g X and a differentiable function f : X ª R with locally Holder deri¨ ati¨ e ¨ such that f Ž x. y fŽ x. G f Ž¨ . y fŽ¨ .

for all x g X .

FABIAN, HAJEK ´ , AND VANDERWERFF

160

Žd. For e¨ ery lsc function f : X ª Žy`, q`x, for e¨ ery e ) 0, and for e¨ ery u g DŽ f ., there are ¨ g X with 5 ¨ y u 5 - e , < f Ž ¨ . y f Ž u.< - e and a differentiable function f : X ª R with locally Holder deri¨ ati¨ e such that ¨ f y f attains its strong minimum at ¨ . Moreo¨ er, if f is bounded below, then f may be chosen to be a bump function with Holder deri¨ ati¨ e. ¨ Proof. Ža. « Žd.: Let f, e g Ž0, 1. and u be given as in Žd.. Let us first assume that f is bounded below. Since f is lsc, we choose d g Ž0, e . such that f Ž x . ) f Ž u. y e

for 5 x y u 5 - d .

Ž 1.3.

According to Pisier’s deep result Žsee, e.g., w5, Corollary IV.4.8x., X admits an equivalent smooth norm with a-Holder derivative for some a g Ž0, 1x. ¨ hence there exists a differentiable bump function b: X ª w0, 1x with aHolder derivative such that bŽ0. s 1 and bŽ x . s 0 if 5 x 5 G d . Let c s ¨ 2 e q f Ž u. y inf f and define h: X ª Žy`, q`x by hŽ x . s

f x q c Ž 1 y b Ž x y u. . q`

½Ž.

if 5 x 5 F 5 u 5 q 2, otherwise.

Since hŽ x . s q` except on a bounded set, we can use Stegall’s variational principle w17, Corollary 5.22x to find L g X * with 5 L 5 - er2Ž5 u 5 q 2. such that h q L attains its strong minimum at some ¨ g X, clearly 5 ¨ 5 F 5 u 5 q 2. If 5 ¨ y u 5 G d , then h Ž ¨ . s f Ž ¨ . q c ) inf f q c s 2 e q f Ž u . and so hŽ u. q ²L , u y ¨ : G hŽ ¨ . ) e q f Ž u. s e q hŽ u. ; this contradicts ² L, u y ¨ : - 5 L 5Ž5 u 5 q 5 ¨ 5. - e . Hence 5 ¨ y u 5 - d - e and, by Ž1.3., f Ž ¨ . ) f Ž u. y e . Also f Ž ¨ . F hŽ ¨ . F hŽ u. q ² L, u y ¨ : f Ž u. q e . So < f Ž ¨ . y f Ž u.< - e . Since every norm on X can be approximated uniformly on bounded sets derivatives Žsee w5, Chapter IVx., we can construct by norms with a-Holder ¨ a differentiable bump function ˜ b: X ª w0, 1x with a-Holder derivative such ¨ that ˜ bŽ x . s 1 if 5 x 5 F 5 u 5 q 1 and ˜ bŽ x . s 0 if 5 x 5 G 5 u 5 q 2. Define f : X ª R by

f Ž x . y c Ž 1 y bŽ x y u. . y ˜ bŽ x . ²L , x:,

x g X.

161

SMOOTH VARIATIONAL PRINCIPLES

Then f is a Žshifted. bump function with the same smoothness properties as b and Ž f y f .Ž x . s hŽ x . q ² L, x : if 5 x 5 F 5 u 5 q 1. On the other hand, if 5 x 5 ) 5 u 5 q 1 Ž) 5 u 5 q d ., we have

Ž f y f . Ž x . G inf f q c y ˜b Ž x . <² L , x :< G inf f q c y s 2 e q f Ž u. y

e 2

e 2

s 32 e q h Ž u .

G 32 e q h Ž ¨ . q ² L , ¨ : y ² L , u: G 32 e q f Ž ¨ . y f Ž ¨ . y ² L , u: ) e q f Ž ¨ . y f Ž ¨ . . Thus if Ž f y f .Ž yn . ª Ž f y f .Ž ¨ ., we have 5 yn 5 F 5 u 5 q 1 for large n and so for such n we get Ž f y f .Ž yn . s hŽ yn . q ² L, yn :. Consequently f y f attains its strong minimum at ¨ . Now suppose f is not bounded below. Because X is superreflexive, it admits an LUR norm Žsee, e.g., w5, Theorem 4.1x. and it also admits another norm with a-Holder derivative for some a g Ž0, 1x Žsee, e.g., w5, ¨ Corollary IV.4.8x.. By an averaging technique of w14x, X admits an LUR norm with a-Holder derivative. According to w20, Theorem 1 and State¨ ment III on p. 48x, X admits partitions of unity formed by functions with locally a-Holder derivative. By standard techniques in obtaining approxi¨ mations from partitions of unity, there is a differentiable function f 1: X ª R with locally a-Holder derivative such that f Ž x . y f 1Ž x . ) 0 for all ¨ x g X; see, e.g., w5, Theorem VIII.3.2 Žiii., Živ.x where C k-smoothness can be replaced by local a-Holder smoothness. Since f 1 is continuous, we ¨ choose d g Ž0, er2. so that < f 1Ž x . y f 1Ž u.< - er2 whenever 5 x y u 5 - d . Now by the first part, there is a differentiable function f on X with a-Holder derivative such that f y f 1 y f attains its strong minimum at ¨ ¨ g X where 5 ¨ y u 5 - d - e and <Ž f y f 1 .Ž ¨ . y Ž f y f 1 .Ž u.< - d . Hence < f Ž ¨ . y f Ž u.< - er2 q < f 1Ž ¨ . y f 1Ž u.< - e ; this proves Žd.. Žd. « Žc. and Žc. « Žb. are trivial. Žb. « Ža.: Let Y be any separable subspace of X. Because the definition of superreflexivity w3, p. 180x ensures that this property is separably determined, it suffices to show that Y is superreflexive. Since Y is separable, it admits an equivalent LUR norm 5 ? 5 Žsee, e.g., w5, Theorem II.2.9x.. Define f : X ª Ž0, q`x as follows f Ž x. s

½

5 x 5y1 q`

if x g Y _  0 4 otherwise;

FABIAN, HAJEK ´ , AND VANDERWERFF

162

this is an lsc function. By Žb., there are ¨ g X, d ) 0 and a function f : B Ž ¨ , d . ª R with Holder derivative such that f Ž x . y f Ž x . G f Ž ¨ . y ¨ f Ž ¨ . for all x g B Ž ¨ , d .; defining c by c Ž x . s f Ž x . q f Ž ¨ . y f Ž ¨ . we have

c Ž ¨ . s f Ž ¨ . and c Ž x . F f Ž x .

for all x g B Ž ¨ , d . .

Ž 1.4.

Observe that ¨ g Y _  04 . Now choose L g S Y * such that ² L, ¨ : s 5 ¨ 5 and let H s  y g Y: ² L, y : s 04 . Now suppose  h n4 ; H satisfies 5 ¨ q h n 5 ª 5 ¨ 5. Then 2 5 ¨ 5 s ²L , ¨ q hn q ¨ : F 5 ¨ q hn q ¨ 5 F 5 ¨ q hn 5 q 5 ¨ 5 ª 2 5 ¨ 5. Thus by the local uniform rotundity of 5 ? 5, we have 5 h n 5 s 5 ¨ q h n y ¨ 5 ª 0. From this, we conclude that there is a g ) 0 for which 5 ¨ q h5 G 5 ¨ 5 q g

whenever h g H and 5 h 5 G

d 2

.

Ž 1.5.

Now Ž1.4. and Ž1.5. imply that

c Ž ¨ q h. F

1 5¨ 5 q g

whenever

d 2

F 5 h5 - d .

Let t : R ª w0, 1x be a bump function with Lipschitz derivative such that t Ž1r5 ¨ 5. s 1 and t Ž t . s 0 for t F 1rŽ5v 5 q g .. Define b: H ª R by bŽ h. s

½

t Ž c Ž ¨ q h. . 0

if 5 h 5 - d , if 5 h 5 G d .

It is clear that b is a bump function on H with Holder derivative. Finally ¨ w5, Theorem V.3.2x says H is superreflexive and hence Y s H = R is also superreflexive. Note that Theorem 1.6 completely answers the question concerning supporting lsc functions by C k-smooth functions for k G 2. COROLLARY 1.7. equi¨ alent.

For a Banach space X and k G 2, the following are

Ža. X has the Radon]Nikodym ´ property and admits a C k-smooth bump function.

SMOOTH VARIATIONAL PRINCIPLES

163

Žb. X has a superreflexi¨ e and admits a C k-smooth bump function. Žc. For e¨ ery lsc bounded below f : X ª Žy`, q`x there are ¨ g X, d ) 0 and a C k-smooth function f : B Ž ¨ , d . ª R such that f Ž x. y fŽ x. G f Ž¨ . y fŽ¨ .

for all x g B Ž ¨ , d . .

Proof. Ža. « Žc.: Let b be a C k-smooth bump function on X and consider f q by2 . By w17, Corollary 5.22x, there are a ¨ g X and a L g X * such that f q by2 q L attains its strong minimum at ¨ . Choose d ) 0 so that B Ž ¨ , d . ;  x: bŽ x . / 04 and set f s yby2 y L. Žc. « Žb.: For k G 2, a C k-smooth function has locally Lipschitz first derivative; consequently Theorem 1.6 ensures that X is superreflexive. In particular, X admits an LUR norm. Hence copying the proof of Žb. « Ža. in Theorem 1.6, by replacing t with a C k-smooth bump, we obtain a C k-smooth bump on a hyperplane of X. Thus X admits a C k-smooth bump function. Žb. « Ža. is trivial. Remark. Day’s norm 5 ? 5D on c 0 is LUR and has points of Lipschitz smoothness Žsee, e.g., w6x.. Let ¨ be a point of Lipschitz smoothness of 5 ? 5D . Because c 0 admits a C ` -smooth norm w1x, according to Proposition 1.5 there is a Lipschitz function f with pointwise Lipschitz derivative such that 5 ? 5y1 y f attains a strong local minimum at ¨ . This contrasts with the proof of Theorem 1.6. Note that using smooth partitions of unity Žsee w5, Chapter VIIIx. and proceeding as in Theorem 1.6, one could replace Žc. in Corollary 1.7 with a statement analogous to Theorem 1.6Žd.. The question of supporting lsc functions by functions which have pointwise Holder derivative in the ¨ spaces without the Radon]Nikodym ´ property seems to still be open. Related to this question Ž4. from w6x and the following question from w10x. Does every norm on c 0 have a point of Lipschitz smoothness? 2. SMOOTH CONVEX VARIATIONAL PRINCIPLES Suppose X admits a Frechet differentiable norm and we are given a ´ convex function f which is bounded on bounded sets. Then by finding a C 1-smooth convex function such that g y f G 0 we can apply w2x to obtain a C 1-smooth convex function f such that f y f attains its strong minimum. However, f y f may be quite large ever on a fixed bounded set. In this section it is shown that when X * admits a dual LUR norm, then f y f can be small on a fixed bounded set of X. Further, we show that if f is a norm, then f may also be a norm Žnote that the function f given by w2x is not even or homogeneous.. We begin by presenting what is essentially the

FABIAN, HAJEK ´ , AND VANDERWERFF

164

Borwein]Preiss smooth variational principle. However, we require that the perturbing function be Lipschitz as well as convex and C 1-smooth. Although one can obtain this using the methods of w2x, it is not entirely obvious; so we include a proof of this using the approach of w4x. Let Ž X, 5 ? 5. be a Banach space. For C 1-smooth functions f : X ª R we define

mŽ f . s

`

Ý

2yn 5 f 5 n q 5 f 9 5 ` ,

ns1

where 5 f 5 n s sup  < f Ž x . < : x g X , 5 x 5 F n4

and

5 f 9 5 ` s sup  5 f 9 Ž x . 5 : x g X 4 .

Let M be the set of all C 1-smooth functions f such that m Ž f . - q`. It is routine to check that Ž M , m . is a Banach space. Define

˜ s  f g M : f is convex and f G 0 4 , C C s  f g C˜: f Ž x . ª q` as 5 x 5 ª q` 4 .

˜. Let r be a metric on C˜ defined by r Ž f, g . s m Ž f y g . for f, g g C ˜ Clearly Ž C , r . is a complete metric space. We now show that Ž C , r . is a Baire space. It suffices to show C is open ˜. So fix f g C . Take M ) f Ž0. and 0 - e - 13 Ž M y f Ž0... As f g C , in C there is an n such that f Ž x . ) M whenever 5 x 5 ) n. Take any g g C˜ such that r Ž f, g . - 2yne . Then < f Ž x . y g Ž x .< - e whenever 5 x 5 F n and so g Ž0. - f Ž0. q e and g Ž x . ) M y e whenever 5 x 5 s n. Thus g Ž x . y g Ž0. ) M y e y f Ž0. y e ) e if 5 x 5 s n. By convexity, it follows that ˜, r .. g Ž x . ª ` as 5 x 5 ª `. This shows g g C and C is open in Ž C Therefore C is Baire. THEOREM 2.1 ŽBorwein]Preiss ‘‘Deville style’’.. Let Ž X, 5 ? 5. be a Banach space with a Frechet smooth norm and consider an lsc bounded ´ below function f : X ª Žy`, q`x. Then the set

 g g C : f q g attains its strong minimum on X 4 is residual in C . Proof. Let 5 ? 5 be a Frechet smooth norm on X. Following the method ´ of w4x, for n s 1, 2, . . . , we let

½

Gn s g g C : there exists x 0 g X with Ž f q g . Ž x 0 . - inf Ž f q g . Ž x . : 5 x y x 0 5 G

½

1 n

55

.

165

SMOOTH VARIATIONAL PRINCIPLES

We first show that Gn is open for each n. So fix n and g g Gn and find a corresponding x 0 g X. Let 0 - e - 1 satisfy

Ž f q g . Ž x 0 . q 2 e - inf Ž f q g . Ž x . : 5 x y x 0 5 G

½

1 n

5

.

Since g g C , there is k g N such that k ) 5 x 0 5 and g Ž x . y g Ž 0 . ) Ž f q g . Ž x 0 . q 3 e y inf f

whenever 5 x 5 G k.

Take any h g C satisfying r Ž h, g . - e 2yk ; then < hŽ x . y g Ž x .< - e for all 5 x 5 F k. Consequently, h Ž x . y h Ž 0 . ) g Ž x . y g Ž 0 . y 2 e ) Ž f q g . Ž x 0 . q e y inf f ) Ž f q h . Ž x 0 . y inf f Ž G 0 . whenever 5 x 5 s k. Then, from the convexity of h, one obtains hŽ x . y hŽ0. ) Ž f q h.Ž x 0 . y inf f whenever 5 x 5 G k. This implies

Ž f q h . Ž x . G inf f q h Ž x . y h Ž 0 . whenever 5 x 5 G k.

) Ž f q h. Ž x0 .

Ž 2.1.

On the other hand, if 5 x 5 F k and 5 x y x 0 5 G 1rn, then < hŽ x . y g Ž x .< - e and so

Ž f q h. Ž x0 . - Ž f q g . Ž x0 . q e - Ž f q g . Ž x . y e - Ž f q h. Ž x . . Combining this with Ž2.1. shows that h g Gn ; hence Gn is open. Now we show that Gn is dense in C . Fix n g N, g g C , and 0 - e - 1. We find c ) 0 such that Ž f q g .Ž x . ) infŽ f q g . q 1 whenever 5 x 5 ) c. Let d ) 0 satisfy `

2d q 2dc q 2d

Ý

2ym m - e .

ms1

Finally, we find x 0 g X such that Ž f q g .Ž x 0 . - infŽ f q g . q drn2 . Since drn2 - 1, we have 5 x 0 5 F x. Define u : w0, q`. ª w0, q`. by 2 u Ž t. s t 2t y 1

½

if if

0 F t F 1, t ) 1.

Put hŽ x . s du ŽI x y x 0 5., x g X. Then h g C , and

r Ž h, 0 . F 2 d q d

`

Ý ms1

2ym Ž 2 m q 2 c . - e .

FABIAN, HAJEK ´ , AND VANDERWERFF

166

Also, if 5 x y x 0 5 G 1rn, we have hŽ x . G du Ž1rn. s drn2 ; thus inf Ž f q g q h . Ž x . : 5 x y x 0 5 G

½

1 n

5

G inf Ž f q g . Ž x . : 5 x y x 0 5 G

½

1 n

5

q

d n2

) Ž f q g q h. Ž x0 . .

Thus g q h g Gn and this shows Gn is dense in C . Therefore the set l `ns 1 Gn is residual in C ; following the proof of w4x, we can show f q g attains its strong minimum for each g g F `ns 1 Gn . As an application of Theorem 2.1, we present the following result. THEOREM 2.2. Let X be a Banach space for which X * admits an equi¨ alent dual LUR norm. Let r ) 0, e ) 0, and u g rBX be gi¨ en and let f : X ª R be a con¨ ex function bounded on bounded sets. Then there exist a Frechet differentiable con¨ ex function f : X ª R and ¨ g X such that ´ 5 u y ¨ 5 - e , f y f attains its strong minimum at ¨ , f Ž x . - f Ž x . q e for all x g rBX , and f Ž x . y f Ž x . G 0 for all x g X. Before proving this theorem, we need the following approximation result. PROPOSITION 2.3. Suppose Ž X, 5 ? 5. is a Banach space such that its dual norm 5 ? 5 is LUR and let f be an lsc con¨ ex function on X with DŽ f . / B. Then for any c ) 0 the function f c Ž x . s inf  c 5 y 5 2 q f Ž x y y . : y g X 4 ,

xgX

if Frechet smooth and con¨ ex. If, moreo¨ er, f is bounded on bounded sets ´ then f c ª f uniformly on bounded sets as c ª `. Proof. The convexity of f c follows directly from its definition. The approximation properties can be shown easily Žsee, e.g., w16, Lemma 2.4x.. It remains to show that f c is Frechet differentiable. For a function g on ´ X, for e ) 0, and for x g DŽ g ., we define ­e g Ž x . by

­e g Ž x . s  L g X *: ² L , h: F g Ž x q h . y g Ž x . q e

for all h g X 4 .

Claim. Let g Ž x . s c 5 x 5 2 and suppose 5 x n 5 ª M and e n x0. If L n g ­e n g Ž x n ., then 5 L n 5 ª 2 cM. To prove this claim, let e ) 0 be fixed and choose d ) 0 so that c d - e . Note that for h g X, with 5 h 5 s d , one has ² L n , h: F c 5 x n q h 5 2 y c 5 x n 5 2 q e n F c Ž 5 x n 5 q d . 2 y c 5 x n 5 2 q e n s 2 c 5 x n 5 d q cd 2 q en .

Ž 2.2.

SMOOTH VARIATIONAL PRINCIPLES

167

On the other hand, let h n s yd Ž x nr5 x n 5. if x n / 0 Žif x n s 0, choose any h n with 5 h n 5 s d ., then ² L n , h n : F c 5 x n q h n 5 2 y c 5 x n 5 2 q en 2

s c Ž 5 x n 5 y d . y c 5 x n 5 2 q e n s y2 c 5 x n 5 d q c d 2 q e n . Ž 2.3. Because 5 x n 5 ª M, e n x0 and c d 2 - ed , it follows from Ž2.2. and Ž2.3. that 2 Mc q e G lim sup 5 L n 5 G lim inf 5 L n 5 G 2 Mc y e . n

n

Since e ) 0 was arbitrary, this proves the claim. To proceed with the proof of Proposition 2.3, fix x 0 g X and let L n g ­e n f c Ž x 0 ., where e n x0. By the definition of f c , we choose yn g x such that f c Ž x 0 . G c 5 yn 5 2 q f Ž x 0 y yn . y e n . Since DŽ f . / B, it follows that f c Ž x 0 . - `. Because f is lsc and convex, by the separation theorem there is a K ) 0 so that f Ž x . G yK 5 x 5 y K for all x g X. Thus 5 yn 54 `ns1 is bounded and so it has a subsequence converging to an M G 0. Assume, for simplicity, that 5 yn 5 ª M. Now observe that ² L n , h: F f c Ž x 0 q h . y f c Ž x 0 . q e n F c 5 yn q h 5 2 q f Ž x 0 y yn . y f c Ž x 0 . q e n F c 5 yn q h 5 2 y c 5 yn 5 2 q e n q e n . This shows that L n g ­ 2 e n g Ž yn ., where g s c 5 ? 5 2 and 5 yn 5 ª M. Consequently, the previous claim shows 5 L n 5 ª 2 Mc. In particular, for L g ­ f c Ž x 0 ., we have 5 L 5 s 2 Mc. Moreover, 12 Ž L q L n . g ­e n f c Ž x 0 . so we have 5Ž L q L n .r2 5 ª 2 Mc. Because the dual norm 5 ? 5 is LUR, it follows that 5 L y L n 5 ª 0. Thus Smulyan’s criterion for continuous convex functions Žsee, e.g., w21, Lemma 2.3x. shows that f c is Frechet differentiable at ´ x 0 Žnote that f c is bounded on bounded sets and thus is continuous.. Proof of Theorem 2.2. For convenience, we assume 5 ? 5 is Frechet ´ differentiable. We frequently use the fact that a convex function bounded on bounded sets is Lipschitz on a fixed bounded set w17, pp. 4, 5x. Put L s min e 3r16r 2 , er24 and let M ) max 1 q r, D4 . By Proposition 2.3, there is a Frechet smooth convex function ˜ g: X ª R such that < f Ž x . y ´ g˜Ž x .< - Dr8 whenever x g MBX . Then putting g 0 s ˜ g q 3Dr8, we have

168

FABIAN, HAJEK ´ , AND VANDERWERFF

Dr4 - g 0 Ž x . y f Ž x . - Dr2 for all x g MBX . let L1 be a Lipschitz constant of g 0 y f on Ž M q 1. BX and define g 1: X ª R by g 1Ž x . s L1ŽmaxŽ5 x 5 y M q 1., 04. 2 , x g X. Clearly g 1 is Frechet differentiable. If ´ x g MBX we have Ž g 0 q g 1 y f .Ž x . G Ž g 0 y f .Ž x . ) Dr4, while g 1Ž x . G L1 for M F 5 x 5 F M q 1, and so for such x we have

Ž g0 q g1 y f . Ž x . G Ž g0 y f . M

ž

x 5 x5

/

y L1 5 x y M

x 5 x5

5 q L1 )

D 4

.

Now put g 2 s L2 Žmax5 x 5 y M y 1 q 1, 04. 2 , where L2 is a Lipschitz constant of g 0 q g 1 y f on Ž M q 2. BX . Then for M q 1 F 5 x 5 F M q 2, one has

Ž g 0 q g 1 q g 2 y f . Ž x . G Ž g 0 q g 1 y f . Ž M q 1.

ž

yL2 5 x y Ž M q 1 .

x 5 x5

x 5 x5

/

5 q L2 )

D 4

,

and Ž g 0 q g 1 q g 2 y f .Ž x . ) Dr4 if x g Ž M q 1. BX as well. Continuing in a similar fashion, we construct g 3 , g 4 , . . . and let g s Ý`ns0 g n . Then g is convex and Frechet smooth because this sum is locally finite. Moreover ´ gŽ x. y f Ž x. )

D 4

for all x g X .

Also for x g rBX , we have g Ž x . y f Ž x . s g 0 Ž x . y f Ž x . - Dr2 - er2 because r - M y 1. Define h s Ž Dr8r .u ( 5 ? 5, where u is from the proof of Theorem 2.1; then h g C and h F Dr4r 5 ? 5. We now apply Theorem 2.1 to the function f [ g q er16r 2 5 ? y u 5 2 y f, to obtain a nonnegative convex Frechet ´ smooth function f 1: X ª R and ¨ g X such that r Ž f 1 , h. - Dr4 ? 1r2 rq1 and g q er16r 2 5 ? y u 5 2 y f q f 1 attains its strong minimum at ¨ . Put f s g q er16r 2 5 ? y u 5 2 q f 1. Then for x g rBX we have w 1Ž x . hŽ x . q Dr4 - Dr2 and so

fŽ x. y f Ž x. -

D 2

q

e 16 r

2

2 Ž2 r . q

D 2

sDq

e 4

- e.

Note also that if we had 5 ¨ y u 5 ) e , then

f Ž ¨ . y f Ž ¨ . ) er16r 2 5 ¨ y u 5 2 ) e 3r16r 2 G D while

f Ž u . y f Ž u . s g Ž u . y f Ž u . q f 1Ž u . ) Hence we must have 5 ¨ y u 5 - e .

D 2

q

D 2

s D.

169

SMOOTH VARIATIONAL PRINCIPLES

In the above theorem, even given that f is a norm, there is no way to guarantee that f would be a norm. We deal with this in our final application of variational techniques. Let Ž X, 5 ? 5. be a Banach space with unit ball BX and unit sphere S X with respect to the norm 5 ? 5. Let F˜ denote the set of seminorms m on X such that m2 is Frechet differentiable. We equip F˜ with a metric r ´ ˜ defined for m , n g F by

r Ž m , n . s sup  < m2 Ž x . y n 2 Ž x . < : x g BX 4 q sup  5 Ž m2 . 9 Ž x . q Ž n 2 . 9 Ž x . 5 : x g BX 4 . Now let F be the collection of equivalent Frechet differentiable norms ´ on X; then Ž F , r . is an open subset of F˜. Thus it is a Baire space. We say that n g F has a strong symmetric minimum at ¨ g S X if n Ž ¨ . s inf n Ž x .: x g S X 4 and min 5 x n " ¨ 5 ª 0 whenever x n g S X and n Ž x n . ª n Ž ¨ .. THEOREM 2.4. Let a Banach space Ž X, 5 ? 5. admit an equi¨ alent Frechet ´ differentiable norm and let Ž F , r . be as abo¨ e. Then Ža. the set  n g F : n attains its strong symmetric minimum on S X 4 is residual in F Žb. for e¨ ery u g S X and e¨ ery e ) 0, there are n g F and ¨ g S X such that 5 ¨ y u 5 - e and n attains its strong symmetric minimum on S X at ¨ . Proof. Ža. We follow the technique of w4x. For n s 1, 2, . . . , let

½

Gn s n g F : there exists x 0 g S X satisfying

n Ž x 0 . - inf n Ž x . : x g S X , min 5 x " x 0 5 G

½

1 n

55

.

It is clear that each Gn is open. Let us show tht each Gn is dense in F. So fix n g N, e ) 0, and n g F. Let dŽ x, R y . denote the distance from x to the one-dimensional space R y in the n norm. Because n is Frechet ´ differentiable and the distance is attained, it follows that d 2 Ž?, R y . if Frechet differentiable. Now choose c1 , c 2 g R such that c1 5 ? 5 F n F c1 5 ? 5 ´ and choose d ) 0 satisfying 3 d c 22 - e . Further find x 0 g S X such that

n 2 Ž x 0 . - inf  n 2 Ž x . : x g S X 4 q

d c12 8 n2

Ž 2.4.

FABIAN, HAJEK ´ , AND VANDERWERFF

170

and let m s Ž n 2 q d d 2 Ž?, R x 0 ..1r2 . Clearly m g F. Also, using the fact that dŽ?, R x 0 . is c 2-Lipschitz with respect to 5 ? 5, we obtain

r Ž m , n . s sup  d d 2 Ž x, R x 0 . : x g BX 4 q sup  2 d d Ž x, R x 0 . ? d Ž ?, R x 0 . 9 Ž x . : x g BX 4 F d sup  n 2 Ž x . : x g BX 4 q 2 d c 2 sup  c 2 n Ž x . : x g BX 4 F d c 22 q 2 d c 22 - e .

Ž 2.5.

Consider, now, x g S X and a g R such that 5 x y ax 0 5 - 1r2 n. If a ) 0, we have 5 x y x 0 5 F 5 x y ax 0 5 q 5 ax 0 y x 0 5 - 1r2 n q < a y 1 < - 1rn. Likewise if a - 0, we get 5 x q x 0 5 - 1rn. Hence if x g S X and min 5 x " x 0 5 G 1rn, then d Ž x, R x 0 . G c1 min  5 x y ax 0 5 : a g R 4 G c1

1 2n

.

Consequently, using Ž2.4., for 5 x " x 0 5 G 1rn, we obtain

m2 Ž x . s n 2 Ž x . q d d 2 Ž x, R x 0 . Ginf  n 2 Ž x . : x g S X 4 q

d c12 4 n2

) n 2 Ž x0 .

s m2 Ž x 0 . . This means that m g Gn and so Gn is dense since, by Ž2.5., r Ž m , n . - e . From here, an argument similar to that in w4x shows that n attains its strong symmetric minimum on S X whenever n g F `ns1 Gn . Žb. Let u g S X and e ) 0 be given. Take n 1 g F satisfying n 1 G 5 ? 5. Let M s sup n 1Ž x .: x g S X 4 and define m by

m s 2

n 12

q

5M 2

e2

d 2 Ž ?, R u . ,

where the distance function is measured with respect to the norm n 1. Since n 1 G 5 ? 5, we can check as in part Ža., that dŽ x, R u. G er2 whenever x g S X and min 5 x " u 5 G e . Therefore, for x g S X with min 5 x " u 5 G e , we have

m2 Ž x . G

5M 2 e 2

5M 2

M2

4

n 2 Ž x . ) m2 Ž x . y

4

M2 8

G

4

q n 12 Ž u . s

M2

q m 2 Ž u . . Ž 2.6. 4 By part Ža., there are z g S X and n g F such that < n 2 Ž x . y m2 Ž x .< - M 2r8 for all x g S X and n attains its strong symmetric minimum on S X at z. Now if x g S X and min 5 x " u 5 G e , then using Ž2.6.

e2

s

G m2 Ž u . q

M2 8

) n 2 Ž u. G n 2 Ž z . .

SMOOTH VARIATIONAL PRINCIPLES

171

Hence min 5 z " u 5 - e . If 5 z y u 5 - e , put ¨ s z, while if 5 z q u 5 - e , put ¨ s yz; this is allowed since n Ž z . s n Žyz .. In some cases, we can be more precise than in Theorem 2.4Žb.. THEOREM 2.5. Suppose Ž X, 5 ? 5. is a Banach space and X * admits an equi¨ alent dual LUR norm. Let Ž F , r . be as abo¨ e. Then for e¨ ery u g S X and e¨ ery e ) 0, there is a n g F such that 1 F n Ž x . F 1 q e for all x g S X and n attains its strong symmetric minimum at ¨ g S X , where 5 ¨ y u 5 - e . If X admits a dual LUE norm, then the Frechet differentiable norms ´ are dense among all norms on X Žsee, e.g., w5, Theorem II.4.1x.. Thus analogous to the proof of Theorem 2.2, we can take a smooth norm very close to 5 ? 5, add a smooth distance penalty function to it Žmuch smaller than in Žb. above., and then apply Theorem 2.4Ža. to prove Theorem 2.5. We omit the details.

ACKNOWLEDGMENT The authors are indebted to V. Zizler for several suggestions and helpful discussions concerning this paper.

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