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A Generalization of Ekeland's ϵ-Variational Principle and Its Borwein–Preiss Smooth Variant
Journal of Mathematical Analysis and Applications 246, 308᎐319 Ž2000. doi:10.1006rjmaa.2000.6813, available online at http:rrwww.idealibrary.com on
A Generalization of Ekeland’s ⑀-Variational Principle and Its Borwein᎐Preiss Smooth Variant 1 Li Yongxin and Shi Shuzhong Nankai Institute of Mathematics, Tianjin 300071, People’s Republic of China Submitted by George Leitmann Received February 10, 1998
We give a generalization of Ekeland’s ⑀-Variational Principle and of its Borwein᎐Preiss smooth variant, replacing the distance and the norm by a ‘‘gauge-type’’ lower semi-continuous function. As an application of this generalization, we show that if on a Banach space X there exists a Lipschitz -smooth ‘‘bump function,’’ then every continuous convex function on an open subset U of X is densely -differentiable in U. This generalizes the Borwein᎐Preiss theorem on the differentiability of convex functions. 䊚 2000 Academic Press
In 1972, Ivar Ekeland proved the following theorem: EKELAND’S ⑀-VARIATIONAL PRINCIPLE. Let Ž X, d . be a complete metric space, F: X ª ⺢ j q⬁4 be a lower semi-continuous function bounded from below, and ) 0. Then, for e¨ ery x 0 g X and ⑀ ) 0 such that F Ž x 0 . F inf F q ⑀
Ž 1.
X
there exists an x⑀ g X such that d Ž x 0 , x⑀ . F
Ž 2.
F Ž x⑀ . q Ž ⑀r . d Ž x⑀ , x 0 . F F Ž x 0 . F inf F q ⑀
Ž 3.
F Ž x . q Ž ⑀r . d Ž x⑀ , x . ) F Ž x⑀ . .
Ž 4.
X
᭙ x / x⑀ ,
This principle has wide applications in nonlinear analysis w1᎐5x. 1
Research supported by the National Natural Science Foundation of China. 308
0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
EKELAND’S
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309
In 1987, for the application to differentiability problems of convex functions, Borwein and Preiss w6x revised this principle as the following theorem Žbut in the Banach space setting.: THE BORWEIN ᎐PREISS SMOOTH ⑀-VARIATIONAL PRINCIPLE. Let Ž X, d . be a complete metric space, F: X ª ⺢ j q⬁4 be a lower semi-continuous function bounded from below, ) 0, and p G 1. Then, for e¨ ery x 0 g X and ⑀ ) 0 such that F Ž x 0 . - inf F q ⑀
Ž 5.
X
there exist a sequence x n4 ; X which con¨ erges to some x⑀ g X and a function p : X ª ⺢ of the form
p Ž x . s
⬁
Ý n d Ž x, x n .
p
,
ns1
where ᭙ n s 1, 2, . . . ,
⬁
n ) 0
and
Ý n s 1 ns1
such that d Ž x⑀ , x 0 . -
Ž 6.
F Ž x⑀ . - inf F q ⑀
Ž 7.
X
᭙x g X,
F Ž x . q Ž ⑀r p . p Ž x . G F Ž x⑀ . q Ž ⑀r p . p Ž x⑀ . . Ž 8 .
Noting that for all x g X, 1Ž x . y 1Ž x⑀ . F dŽ x, x⑀ ., from Ž8. in the case of p s 1, we obtain that ᭙x g X,
F Ž x . q Ž ⑀r . d Ž x, x⑀ . G F Ž x⑀ .
Ž 9.
which is almost the same as Ž4.. Moreover, we can find too that between Ž1. ᎐ Ž3. and Ž5. ᎐ Ž7., there also exist some slight differences; and so Ekeland’s ⑀-Variational Principle is not an exact consequence of the Borwein᎐Preiss Smooth ⑀-Variational Principle. This means that it is possible to improve the last one. In this paper, we give a generalization of these two ⑀-variational principles as follows: THEOREM 1. Let Ž X, d . be a complete metric space and F: X ª ⺢ j q⬁4 be a lower semi-continuous function bounded from below. Suppose that
310
LI AND SHI
: X = X ª ⺢qj q⬁4 is a function, satisfying Ž x, x . s 0; Ž yn , z n . ª 0 « d Ž yn , z n . ª 0;
Ž i. ᭙ x g X , Ž ii . ᭙ yn , z n 4 g X = X , Ž iii . ᭙ z g X ,
y ¬ Ž y, z . is lower semi-continuous;
Ž 10 . and that ␦ 0 ) 0, ␦n G 0, n s 1, 2, . . . , is a nonnegati¨ e number sequence. Then, for e¨ ery x 0 g X and ⑀ ) 0 with F Ž x 0 . F inf F q ⑀
Ž 11 .
X
there exists a sequence x n4 ; X which con¨ erges to some x⑀ g X such that
Ž x⑀ , x n . F ⑀r2 n␦ 0 ,
n s 0, 1, 2, . . . ;
Ž 12 .
when for infinitely many n, ␦n ) 0, ⬁
F Ž x⑀ . q
Fq⑀ Ý ␦n Ž x⑀ , x n . F F Ž x 0 . F inf X
Ž 13 .
ns0
᭙ x / x⑀ ,
FŽ x. q
⬁
Ý
␦n Ž x, x n . ) F Ž x⑀ . q
ns0
⬁
Ý ␦n Ž x⑀ , x n . Ž 14. ns0
and when ␦ k ) 0 and for all j ) k G 0, ␦ j s 0, Ž14. is replaced by ᭙ x / x⑀ ,
᭚m G k,
ky1
FŽ x. q
Ý ␦i Ž x, x i . q ␦ k Ž x, x m . is0 ky1
) F Ž x⑀ . q
Ý ␦ i Ž x⑀ , x i . q ␦ k Ž x⑀ , x m . .
Ž 15 .
is0
We can say that in Ž10. is a ‘‘gauge-type’’ function, which, for example, may be any function f Ž dŽ x, y .. with f : ⺢qª ⺢q, which is strictly increasing and continuous and satisfies f Ž0. s 0. It is obvious that if Ž y, z . s Ž ⑀r. dŽ y, z ., ␦ 0 s 1, and ␦n s 0, for all n ) 0. Theorem 1 Žin this case the summation in Ž15. does not exist. recaptures Ekeland’s ⑀-Variational Principle with a little improvement; and if Ž y, z . s Ž ⑀r p . dŽ y, z . p , F Ž x 0 . s inf X F q ⑀ ⬘ - inf X F q ⑀ , ␦ 0 s 1 s 1 y ␥ ) ⑀ ⬘r⑀ with ␥ ) 0 and ␦n s nq1 ) 0, Theorem 1 becomes the Borwein᎐Preiss Smooth ⑀-Variational Principle with some significant improvements, which replace Ž7. and Ž8. by F Ž x⑀ . q Ž ⑀r p . p Ž x⑀ . F F Ž x 0 . - inf F q ⑀ X
EKELAND’S
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PRINCIPLE
and ᭙ x / x⑀ ,
F Ž x . q Ž ⑀r p . p Ž x . ) F Ž x⑀ . q Ž ⑀r p . p Ž x⑀ . .
Moreover, in Theorem 1, ␦n4 may be any positive number sequence, even not bounded. Therefore, Theorem 1 unifies and generalizes these two ⑀-variational principles. Now we give the proof of Theorem 1. The idea of this proof is ‘‘classic’’ as that of the proof of Ekeland’s ⑀-Variational Principle in w3x. Certainly, it also produces a new proof of the Borwein᎐Preiss theorem, which is simpler than that in w6x. Proof of Theorem 1. There are two cases for ␦n4 : Ži. infinitely many ␦n ) 0; and Žii. only finitely many ␦n ) 0. For first case, without loss of generality, we can assume that all ␦n ) 0. Then, set T Ž x 0 . [ x g X < F Ž x . q ␦ 0 Ž x, x 0 . F F Ž x 0 . 4 .
Ž 16 .
From the lower semi-continuity of F and of Ž⭈, x 0 . and x 0 g T Ž x 0 ., T Ž x 0 . is a nonempty closed subset of X, and ᭙ y g T Ž x0 . ,
␦ 0 Ž y, x 0 . F F Ž x 0 . y F Ž y . F F Ž x 0 . y inf F F ⑀ . X
Ž 17 . Take x 1 g T Ž x 0 . such that F Ž x1 . q ␦ 0 Ž x1 , x 0 . F
inf xgT Ž x 0 .
F Ž x . q ␦ 0 Ž x, x 0 . 4 q ␦ 1 ⑀r2 ␦ 0 Ž 18.
and set again 1
½
T Ž x1 . [ x g T Ž x 0 . F Ž x . q
Ý ␦i Ž x, x i . F F Ž x 1 . q ␦ 0 Ž x 1 , x 0 . is0
5
.
Ž 19 . In general, suppose that we have defined x ny 1 g T Ž x ny2 . y and T Ž x ny1 . such that ny1
½
T Ž x ny 1 . [ x g T Ž x ny2 . F Ž x . q
Ý ␦i Ž x, x i . is0
ny2
F F Ž x ny 1 . q
Ý ␦i Ž x ny1 , x i . is0
5
.
Ž 20 .
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LI AND SHI
Take x n g T Ž x ny1 . such that ny1
Ý ␦i Ž x n , x i .
F Ž xn . q
is0
F
inf xgT Ž x ny1 .
½
ny1
FŽ x. q
Ý ␦i Ž x, x i . is0
5
q ␦n ⑀r2 n␦ 0
Ž 21 .
and n
½
T Ž x n . [ x g T Ž x ny1 . F Ž x . q
Ý ␦i Ž x, x i . is0
ny1
F F Ž xn . q
Ý ␦i Ž x n , x i . is0
5
Ž 22 .
which is also non-empty and closed. From Ž21. and Ž22., we have that ny1
␦n Ž y, x n . F F Ž x n . q
᭙ y g T Ž xn . ,
Ý ␦i Ž x n , x i . is0 ny1
y FŽ y. q
Ý ␦i Ž y, x i . is0
ny1
F F Ž xn . q
Ý ␦i Ž x n , x i . is0 ny1
y
inf xgT Ž x ny1 .
FŽ x. q
Ý ␦i Ž x, x i .
F ␦n ⑀r2 n␦ 0
is0
and so, ᭙ y g T Ž xn . ,
Ž y, x n . F ⑀r2 n␦ 0 .
Ž 23 .
Hence, from Ž10.Žii., it follows dŽ y, x n . ª 0 and the diameter of T Ž x n . ª 0. Since X is complete, there exists an unique x⑀ g F ⬁ns0 T Ž x n ., which, from Ž17. and Ž23., satisfies Ž12.. In addition, x n ª x⑀ . Finally, for any x / x⑀ , we have that x f F ⬁ns0 T Ž x n . and so there exists an m g ⺞ such that m
FŽ x. q
my1
Ý ␦i Ž x, x i . ) F Ž x m . q Ý is0
is0
␦i Ž x m , x i . .
⑀-VARIATIONAL
EKELAND’S
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PRINCIPLE
But from Ž16., Ž20., and Ž21., we can deduce that for any q G m we have always my1
F Ž x0 . G F Ž xm . q
Ý
␦i Ž x m , x i .
is0 qy1
G F Ž xq . q
q
Ý ␦ i Ž x q , x i . G F Ž x⑀ . q Ý ␦ i Ž x⑀ , x i . . is0
is0
Therefore, Ž13. and Ž14. hold. Now we deal with the second case of ␦n4 . Assume that ␦ k ) 0 and ␦ j s 0 for all j ) k G 0. Without loss of generality, we suppose that ␦ i ) 0 for all i F k. Thus, when n F k, we take the same x n and T Ž x n . as above. When n ) k, we take x n g T Ž x ny1 . such that ky1
F Ž xn . q
Ý ␦i Ž x n , x i . is0
F
inf xgT Ž x ny1 .
½
ky1
FŽ x. q
Ý ␦i Ž x, x i . is0
5
q ␦ k ⑀r2 n␦ 0
Ž 24 .
and set ky1
½
T Ž x n . [ x g T Ž x ny1 . F Ž x . q
Ý ␦i Ž x, x i . q ␦ k Ž x, x n . is0 ky1
F F Ž xn . q
Ý ␦i Ž x n , x i . is0
5
. Ž 25 .
Then, by the same deduction as above, Ž12. ᎐ Ž14. also hold. But when x / x⑀ it may follow that there exists an m ) k such that ky1
FŽ x. q
Ý ␦i Ž x, x i . q ␦ k Ž x, x m . is0 ky1
) F Ž xm . q
Ý ␦i Ž x m , x i . is0 ky1
G F Ž x⑀ . q
Ý ␦ i Ž x⑀ , x i . q ␦ k Ž x⑀ , x m . , is0
i.e., Ž15. holds.
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LI AND SHI
Remark 1. In Theorem 1, we can replace the ‘‘gauge-type’’ function by a sequence of functions n : X = X ª ⺢ j q⬁4 satisfying:
n Ž x, x . s 0;
Ž i. ᭙ x g X , Ž ii . ᭙ y, z g X ,
n Ž y, z . F ␣ n « d Ž y, z . F ␥n , ␣ n ) 0, ␥n ) 0, ␥n ª 0;
where
Ž 26 .
y ¬ n Ž y, z . is lower semi-continuous.
Ž iii . ᭙ z g X ,
In this case, if we take ␦ 0 s ⑀r␣ 0 , replace Ž12. by dŽ x⑀ , x n . F ␥n , and replace Ž x, x n . by nŽ x, x n ., then Theorem 1 also holds. When X is a Banach space, from Theorem 1 we obtain that THEOREM 2. Let X be a Banach space, F: X ª ⺢ j q⬁4 be a lower semi-continuous function bounded from below, and ) 0. Suppose that : X ª ⺢qj q⬁4 is a lower semi-continuous function such that
Ž i . Ž 0 . s 0; Ž ii . ᭙ y k 4 ; X , Ž y k . ª 0 « 5 y k 5 ª 0;
Ž 27 .
and that ␦ 0 ) 0, ␦n G 0, n s 1, 2, . . . , is a nonnegati¨ e number sequence. Then, for e¨ ery x 0 g X and ⑀ ) 0 with F Ž x 0 . F inf F q ⑀
Ž 28 .
X
there exists a sequence x n4 ; X which con¨ erges to some x⑀ g X such that
Ž x⑀ y x n . F ⑀r2 n␦ 0 ,
n s 0, 1, 2, . . . ;
Ž 29 .
when for infinitely many n, ␦n ) 0, F Ž x⑀ . q
⬁
Fq⑀ Ý ␦n Ž x⑀ y x n . F F Ž x 0 . F inf X
Ž 30 .
ns0
᭙ x / x⑀ ,
FŽ x. q
⬁
⬁
ns0
ns0
Ý ␦ n Ž x y x n . ) F Ž x⑀ . q Ý ␦ n Ž x⑀ y x n . Ž 31 .
and when ␦ k ) 0 and for all j ) k G 0, ␦ j s 0, Ž31. is replaced by ky1
᭙ x / x⑀ , ᭚ m G k ,
FŽ x. q
Ý ␦i Ž x y x i . q ␦ k Ž x y x m . is0
ky1
) F Ž x⑀ . q
Ý ␦ i Ž x⑀ y x i . q ␦ k Ž x⑀ y x m . . is0
Ž 32 .
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Now we give an application of Theorem 2 to the differentiability problem of convex functions. We recall some concepts of w6, 7x. Let X be a Banach space. A bonology on X, denoted  , is any nonempty family of bounded sets of X with D S g  S s X. A function f : X ª ⺢ j "⬁4 is -subdifferentiable at x g X with -subderi¨ ati¨ e x* g X * Žthe dual of X . if, for each ␣ ) 0 and each set S g  , there exists ␦ ) 0 such that for all t g Ž0, ␦ ., ᭙h g S,
f Ž x q th . y f Ž x . ty1 y ² x*, h: G y␣ .
Ž 33 .
Denote x* g ⭸ f Ž x .. -superdifferentiability and -superderi¨ ati¨ e are similarly defined and denoted x* g ⭸  f Ž x .. If f is -subdifferentiable and -superdifferentiable at x g X, then f is called -differentiable at x. Its necessarily unique -derivative is denoted ⵜ f Ž x ., which must coincide with its Gateaux derivative ⵜf Ž x .. If a function f is -differentiable at all ˆ points in a subset A ; X, then f is called -smooth in A. PROPOSITION 1. Assume that is a Lipschitz and -smooth real-¨ alued function on B␣ s x g X ¬ 5 x 5 - ␣ 4 ; a sequence x n g X, n s 0, 1, 2, . . . , satisfies 5 x⑀ y x n 5 - ␣r2,
n s 0, 1, 2, . . . ;
␦n ) 0, n s 0, 1, 2, . . . , is a positi¨ e number sequence with Ý⬁ns0 ␦n - q⬁; and ⌽⑀ Ž x . [
⬁
Ý ␦n Ž x y x n . .
Ž 34 .
ns0
Then ⌽⑀ is -differentiable at x⑀ with ⵜ ⌽⑀ Ž x⑀ . s
⬁
Ý ␦nⵜ Ž x⑀ y x n . .
Ž 35 .
ns0
Since is Lipschitz on B␣ , w Ž x⑀ y x n q th. y Ž x⑀ y x n .x ty1 4 are bounded uniformly for t small enough and h g S, and ⵜ Ž x⑀ y x n .4 is also bounded. Hence, this proposition is a consequence of the Weierstrass M-test. From Theorem 2 and Proposition 1, we obtain THEOREM 3. Let X be a Banach space, F: X ª ⺢ j q⬁4 be a lower semi-continuous function bounded from below, and , ␣ , k ) 0. Suppose that the following assumption holds
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There exists a lower semi-continuous function : X ª ⺢qj q⬁4 such that
Ž i . Ž 0 . s 0; Ž ii . ᭙ y k 4 ; X , Ž y k . ª 0 « 5 y k 5 ª 0; Ž iii . is Lipschitz of rank L and -smooth in B , s x g X < Ž x . - 4 ; Ž iv. 5 x 5 - ␣ « x g B , and Ž x . F rk « 5 x 5 - ␣r2.
Ž P.
Then, for e¨ ery x 0 g X and ⑀ ) 0 with F Ž x 0 . F inf F q ⑀ X
Ž 36 .
there exists an x⑀ g X such that 5 x⑀ y x 0 5 F ␣r2
Ž 37 .
F Ž x⑀ . F F Ž x 0 . F inf F q ⑀
Ž 38 .
0 g ⭸ F Ž x⑀ . q 2 ⑀ kL y1 B*,
Ž 39 .
X
and
where B* s x* g X * ¬ 5 x* 5* F 14 . Proof. From Theorem 2, for any positive number sequence ␦n4 , there exists a sequence x n4 ; X which converges to some x⑀ g X such that
Ž x⑀ y x n . F ⑀r2 n␦ 0 , F Ž x⑀ . q
n s 0, 1, 2, . . . ;
Ž 40 .
⬁
Fq⑀ Ý ␦n Ž x⑀ y x n . F F Ž x 0 . F inf X
Ž 41 .
F Ž x . q ⌽⑀ Ž x . ) F Ž x⑀ . q ⌽⑀ Ž x⑀ . ,
Ž 42 .
ns0
and ᭙ x / x⑀ ,
where ⌽⑀ is defined by Ž34.. From Ž41., we obtain Ž38.. Take ␦ 0 s k ⑀r and Ý⬁ns 1 ␦n s k ⑀r. Then, Ž40. and ŽP.Živ. imply 5 x⑀ y x n 5 - ␣r2, including Ž37., and from the inequality Ž42. and Proposition 1, it is easy to deduce that 0 g ⭸ F Ž x⑀ . q ⵜ ⌽⑀ Ž x⑀ . .
Ž 43 .
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317
But from Ž35., we have that ⵜ ⌽⑀ Ž x⑀ . * F
⬁
Ý ␦n
ⵜ Ž x⑀ y x n . * F 2 ⑀ kL y1 .
Ž 44 .
ns0
Hence, Ž39. is proved. It is obvious that we can replace ŽP.Žiii., Živ. by
Ž iii . ⬘ is Lipschitz and -smooth near x s 0. Now we show that the condition ŽP. is equivalent to the existence of a Lipschitz -smooth bump function; i.e., we have the following proposition: PROPOSITION 2. In a Banach space X, the condition ŽP. is equi¨ alent to the following hypothesis: There exists a Lipschitz -smooth function : X ª w0, 1x such that
Ž i . Ž 0 . s 1; Ž ii . 5 x 5 ) 1 « Ž x . s 0.
Ž H.
Proof. ŽH. « ŽP.. Suppose that satisfies ŽH.. We take
Ž x. s
⬁
Ý ns1
1 22n
1 y Ž2n x. .
Obviously, such a is a Lipschitz -smooth function with the same Lipschitz rank as that of . In addition, for any x g X, Ž x . G Ž0. s 0 and if 5 x 5 ) 1r2 n, then Ž x . G w1 y Ž2 n x .xr2 2 n s 1r2 2 n ; it means that Ž y . ª 0 « 5 y 5 ª 0. Therefore, satisfies ŽP.. ŽP. « ŽH.. Suppose that satisfies ŽP.. From ŽP.Žii., there exists ␦ ) 0 such that 5 x5 ) ␣ « Ž x. ) ␦. We take a function ¨ such that
Ž i . ¨ : ⺢qª w 0, 1 x is continuously differentiable ; Ž ii . t G ␦ « ¨ Ž t . s 0; Ž iii . ¨ Ž 0 . s 1; and for each x g X, set Ž x . s ¨ Ž Ž ␣ x ... Then satisfies ŽH.. THEOREM 4. Let X be a Banach space and f : X ª R j "⬁4 be a lower semi-continuous function. Suppose that ŽH. holds. Then f is -subdifferentiable at a dense subset of points in its graph.
318
LI AND SHI
Proof. Without loss of generality, we suppose that there exists a satisfying ŽP.. Let ⑀ ) 0 and x 0 g X with f Ž x 0 . finite given. Since f is lower semi-continuous at x 0 and satisfies ŽP.Žii., we can choose 1 g Ž0, . small enough such that inf f Ž x . < Ž x y x 0 . F 1 4 G f Ž x 0 . y ⑀ .
Ž 45 .
We apply Theorem 3, replacing by 1 , to f1 s f q ␦1, where
␦1Ž x . s
½
0
if Ž x y x 0 . F 1
q⬁
if Ž x y x 0 . ) 1 .
Then, there exists an x⑀ with Ž x⑀ y x 0 . F 1rk such that ⭸ f1Ž x⑀ . is nonempty. Since f1 coincides with f in a neighborhood of x⑀ , ⭸ f Ž x⑀ . is also nonempty. Moreover, since x⑀ satisfies Ž41., we have that f Ž x 0 . y ⑀ F f Ž x⑀ . s f1Ž x⑀ . F f Ž x 0 . . Thus we obtain a point Ž x⑀ , f Ž x⑀ .. with nonempty ⭸ f Ž x⑀ . in the graph of f, which can be arbitrarily close to Ž x 0 , f Ž x 0 ... The conclusion is proved. THEOREM 5. Let X be a Banach space and f be a continuous con¨ ex function on an open subset U ; X. Suppose that ŽH. holds. Then f is densely -differentiable in U. Proof. Applying Theorem 4 to yf, we obtain that yf is densely -subdifferentiable in U; i.e., f is densely -superdifferentiable in U. Since f is always -subdifferentiable in U, the conclusion is proved. Theorems 4 and 5 generalize the corresponding results of Borwein and Preiss w6x, in which Ž x . s Ž ⑀r p .5 x 5 p, p ) 1. In particular, thanks to an example of a Banach space X, constructed by Haydon w8x, such that there exists a Lipschitz continuously Frechet-differentiable bump function on X ´ but there is no equivalent norm on X Gateaux-differentiable on X _ 04 , ˆ our Theorem 5 is not a consequence of Preiss’ theorem w9x Žor the Preiss᎐Phelps᎐Namioka theorem w10x.; this Preiss theorem Žrespectively, the Preiss᎐Phelps᎐Namioka theorem. affirms that if on a Banach space X there exists an equi¨ alent norm -smooth away from origin, then e¨ ery locally Lipschitz function Ž respecti¨ ely, continuous con¨ ex function. on an open subset U of X is densely -differentiable in U Ž respecti¨ ely, -differentiable in a G␦ dense subset of U . and implies the Borwein᎐Preiss theorem on the differentiability of convex functions.
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Remark 2. It is easy to show that ŽH. is also equivalent to There exists a Lipschitz -smooth function : X ª ⺢ such that
Ž i . Ž 0 . ) 0; Ž ii . 5 x 5 ) 1 « Ž x . F 0.
Ž H⬘.
Remark 3. After finishing the first version of this paper, we learned that Deville et al. w11, 12x also proved Theorems 4 and 5 by a completely different method.
ACKNOWLEDGMENTS We thank Professors S. Simons and G. Godefroy for showing us w12x. We also thank an anonymous referee for posing that it is possible to show ŽP. m ŽH. thereby greatly simplifying the expression of this paper.
REFERENCES 1. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 Ž1974., 324᎐353. 2. I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. Ž N.S.. 1 Ž1979., 193᎐216. 3. J.-P. Aubin and I. Ekeland, ‘‘Applied Nonlinear Analysis,’’ Wiley, New York, 1984. 4. I. Ekeland and G. Lebourg, Generic Frechet differentiability and perturbed optimization ´ problems in Banach spaces, Trans. Amer. Math. Soc. 224 Ž1976., 193᎐216. 5. F. H. Clarke, ‘‘Optimization and Nonsmooth Analysis,’’ Wiley, New York, 1983. 6. J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 Ž1987., 517᎐527. 7. R. R. Phelps, ‘‘Convex Functions, Monotone Operators and Differentiability,’’ Lecture Notes in Math., Vol. 1364, Springer-Verlag, Berlin, 1989. 8. R. Haydon, A counterexample to several questions about scattered compact spaces, Bull. London Math. Soc. 22 Ž1990., 261᎐268. 9. D. Preiss, Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal. 91 Ž1990., 312᎐345. 10. D. Preiss, R. R. Phelps, and I. Namioka, Smooth Banach spaces and monotone or usco mappings, Israel J. Math. 72 Ž1990., 257᎐279. 11. R. Deville, G. Godefroy, and V. Zizler, Un principe variationnel utlisant des fonctions bosses, C. R. Acad. Sci. Paris Ser. ´ I 312 Ž1991., 281᎐286. 12. R. Deville, G. Godefroy, and V. Zizler, A smooth variational principle with applications to Hamilton᎐Jacobi equations in infinite dimensions, preprint.
A Generalization of Ekeland’s ⑀-Variational Principle and Its Borwein᎐Preiss Smooth Variant 1 Li Yongxin and Shi Shuzhong Nankai Institute of Mathematics, Tianjin 300071, People’s Republic of China Submitted by George Leitmann Received February 10, 1998
We give a generalization of Ekeland’s ⑀-Variational Principle and of its Borwein᎐Preiss smooth variant, replacing the distance and the norm by a ‘‘gauge-type’’ lower semi-continuous function. As an application of this generalization, we show that if on a Banach space X there exists a Lipschitz -smooth ‘‘bump function,’’ then every continuous convex function on an open subset U of X is densely -differentiable in U. This generalizes the Borwein᎐Preiss theorem on the differentiability of convex functions. 䊚 2000 Academic Press
In 1972, Ivar Ekeland proved the following theorem: EKELAND’S ⑀-VARIATIONAL PRINCIPLE. Let Ž X, d . be a complete metric space, F: X ª ⺢ j q⬁4 be a lower semi-continuous function bounded from below, and ) 0. Then, for e¨ ery x 0 g X and ⑀ ) 0 such that F Ž x 0 . F inf F q ⑀
Ž 1.
X
there exists an x⑀ g X such that d Ž x 0 , x⑀ . F
Ž 2.
F Ž x⑀ . q Ž ⑀r . d Ž x⑀ , x 0 . F F Ž x 0 . F inf F q ⑀
Ž 3.
F Ž x . q Ž ⑀r . d Ž x⑀ , x . ) F Ž x⑀ . .
Ž 4.
X
᭙ x / x⑀ ,
This principle has wide applications in nonlinear analysis w1᎐5x. 1
Research supported by the National Natural Science Foundation of China. 308
0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
EKELAND’S
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PRINCIPLE
309
In 1987, for the application to differentiability problems of convex functions, Borwein and Preiss w6x revised this principle as the following theorem Žbut in the Banach space setting.: THE BORWEIN ᎐PREISS SMOOTH ⑀-VARIATIONAL PRINCIPLE. Let Ž X, d . be a complete metric space, F: X ª ⺢ j q⬁4 be a lower semi-continuous function bounded from below, ) 0, and p G 1. Then, for e¨ ery x 0 g X and ⑀ ) 0 such that F Ž x 0 . - inf F q ⑀
Ž 5.
X
there exist a sequence x n4 ; X which con¨ erges to some x⑀ g X and a function p : X ª ⺢ of the form
p Ž x . s
⬁
Ý n d Ž x, x n .
p
,
ns1
where ᭙ n s 1, 2, . . . ,
⬁
n ) 0
and
Ý n s 1 ns1
such that d Ž x⑀ , x 0 . -
Ž 6.
F Ž x⑀ . - inf F q ⑀
Ž 7.
X
᭙x g X,
F Ž x . q Ž ⑀r p . p Ž x . G F Ž x⑀ . q Ž ⑀r p . p Ž x⑀ . . Ž 8 .
Noting that for all x g X, 1Ž x . y 1Ž x⑀ . F dŽ x, x⑀ ., from Ž8. in the case of p s 1, we obtain that ᭙x g X,
F Ž x . q Ž ⑀r . d Ž x, x⑀ . G F Ž x⑀ .
Ž 9.
which is almost the same as Ž4.. Moreover, we can find too that between Ž1. ᎐ Ž3. and Ž5. ᎐ Ž7., there also exist some slight differences; and so Ekeland’s ⑀-Variational Principle is not an exact consequence of the Borwein᎐Preiss Smooth ⑀-Variational Principle. This means that it is possible to improve the last one. In this paper, we give a generalization of these two ⑀-variational principles as follows: THEOREM 1. Let Ž X, d . be a complete metric space and F: X ª ⺢ j q⬁4 be a lower semi-continuous function bounded from below. Suppose that
310
LI AND SHI
: X = X ª ⺢qj q⬁4 is a function, satisfying Ž x, x . s 0; Ž yn , z n . ª 0 « d Ž yn , z n . ª 0;
Ž i. ᭙ x g X , Ž ii . ᭙ yn , z n 4 g X = X , Ž iii . ᭙ z g X ,
y ¬ Ž y, z . is lower semi-continuous;
Ž 10 . and that ␦ 0 ) 0, ␦n G 0, n s 1, 2, . . . , is a nonnegati¨ e number sequence. Then, for e¨ ery x 0 g X and ⑀ ) 0 with F Ž x 0 . F inf F q ⑀
Ž 11 .
X
there exists a sequence x n4 ; X which con¨ erges to some x⑀ g X such that
Ž x⑀ , x n . F ⑀r2 n␦ 0 ,
n s 0, 1, 2, . . . ;
Ž 12 .
when for infinitely many n, ␦n ) 0, ⬁
F Ž x⑀ . q
Fq⑀ Ý ␦n Ž x⑀ , x n . F F Ž x 0 . F inf X
Ž 13 .
ns0
᭙ x / x⑀ ,
FŽ x. q
⬁
Ý
␦n Ž x, x n . ) F Ž x⑀ . q
ns0
⬁
Ý ␦n Ž x⑀ , x n . Ž 14. ns0
and when ␦ k ) 0 and for all j ) k G 0, ␦ j s 0, Ž14. is replaced by ᭙ x / x⑀ ,
᭚m G k,
ky1
FŽ x. q
Ý ␦i Ž x, x i . q ␦ k Ž x, x m . is0 ky1
) F Ž x⑀ . q
Ý ␦ i Ž x⑀ , x i . q ␦ k Ž x⑀ , x m . .
Ž 15 .
is0
We can say that in Ž10. is a ‘‘gauge-type’’ function, which, for example, may be any function f Ž dŽ x, y .. with f : ⺢qª ⺢q, which is strictly increasing and continuous and satisfies f Ž0. s 0. It is obvious that if Ž y, z . s Ž ⑀r. dŽ y, z ., ␦ 0 s 1, and ␦n s 0, for all n ) 0. Theorem 1 Žin this case the summation in Ž15. does not exist. recaptures Ekeland’s ⑀-Variational Principle with a little improvement; and if Ž y, z . s Ž ⑀r p . dŽ y, z . p , F Ž x 0 . s inf X F q ⑀ ⬘ - inf X F q ⑀ , ␦ 0 s 1 s 1 y ␥ ) ⑀ ⬘r⑀ with ␥ ) 0 and ␦n s nq1 ) 0, Theorem 1 becomes the Borwein᎐Preiss Smooth ⑀-Variational Principle with some significant improvements, which replace Ž7. and Ž8. by F Ž x⑀ . q Ž ⑀r p . p Ž x⑀ . F F Ž x 0 . - inf F q ⑀ X
EKELAND’S
⑀-VARIATIONAL
311
PRINCIPLE
and ᭙ x / x⑀ ,
F Ž x . q Ž ⑀r p . p Ž x . ) F Ž x⑀ . q Ž ⑀r p . p Ž x⑀ . .
Moreover, in Theorem 1, ␦n4 may be any positive number sequence, even not bounded. Therefore, Theorem 1 unifies and generalizes these two ⑀-variational principles. Now we give the proof of Theorem 1. The idea of this proof is ‘‘classic’’ as that of the proof of Ekeland’s ⑀-Variational Principle in w3x. Certainly, it also produces a new proof of the Borwein᎐Preiss theorem, which is simpler than that in w6x. Proof of Theorem 1. There are two cases for ␦n4 : Ži. infinitely many ␦n ) 0; and Žii. only finitely many ␦n ) 0. For first case, without loss of generality, we can assume that all ␦n ) 0. Then, set T Ž x 0 . [ x g X < F Ž x . q ␦ 0 Ž x, x 0 . F F Ž x 0 . 4 .
Ž 16 .
From the lower semi-continuity of F and of Ž⭈, x 0 . and x 0 g T Ž x 0 ., T Ž x 0 . is a nonempty closed subset of X, and ᭙ y g T Ž x0 . ,
␦ 0 Ž y, x 0 . F F Ž x 0 . y F Ž y . F F Ž x 0 . y inf F F ⑀ . X
Ž 17 . Take x 1 g T Ž x 0 . such that F Ž x1 . q ␦ 0 Ž x1 , x 0 . F
inf xgT Ž x 0 .
F Ž x . q ␦ 0 Ž x, x 0 . 4 q ␦ 1 ⑀r2 ␦ 0 Ž 18.
and set again 1
½
T Ž x1 . [ x g T Ž x 0 . F Ž x . q
Ý ␦i Ž x, x i . F F Ž x 1 . q ␦ 0 Ž x 1 , x 0 . is0
5
.
Ž 19 . In general, suppose that we have defined x ny 1 g T Ž x ny2 . y and T Ž x ny1 . such that ny1
½
T Ž x ny 1 . [ x g T Ž x ny2 . F Ž x . q
Ý ␦i Ž x, x i . is0
ny2
F F Ž x ny 1 . q
Ý ␦i Ž x ny1 , x i . is0
5
.
Ž 20 .
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LI AND SHI
Take x n g T Ž x ny1 . such that ny1
Ý ␦i Ž x n , x i .
F Ž xn . q
is0
F
inf xgT Ž x ny1 .
½
ny1
FŽ x. q
Ý ␦i Ž x, x i . is0
5
q ␦n ⑀r2 n␦ 0
Ž 21 .
and n
½
T Ž x n . [ x g T Ž x ny1 . F Ž x . q
Ý ␦i Ž x, x i . is0
ny1
F F Ž xn . q
Ý ␦i Ž x n , x i . is0
5
Ž 22 .
which is also non-empty and closed. From Ž21. and Ž22., we have that ny1
␦n Ž y, x n . F F Ž x n . q
᭙ y g T Ž xn . ,
Ý ␦i Ž x n , x i . is0 ny1
y FŽ y. q
Ý ␦i Ž y, x i . is0
ny1
F F Ž xn . q
Ý ␦i Ž x n , x i . is0 ny1
y
inf xgT Ž x ny1 .
FŽ x. q
Ý ␦i Ž x, x i .
F ␦n ⑀r2 n␦ 0
is0
and so, ᭙ y g T Ž xn . ,
Ž y, x n . F ⑀r2 n␦ 0 .
Ž 23 .
Hence, from Ž10.Žii., it follows dŽ y, x n . ª 0 and the diameter of T Ž x n . ª 0. Since X is complete, there exists an unique x⑀ g F ⬁ns0 T Ž x n ., which, from Ž17. and Ž23., satisfies Ž12.. In addition, x n ª x⑀ . Finally, for any x / x⑀ , we have that x f F ⬁ns0 T Ž x n . and so there exists an m g ⺞ such that m
FŽ x. q
my1
Ý ␦i Ž x, x i . ) F Ž x m . q Ý is0
is0
␦i Ž x m , x i . .
⑀-VARIATIONAL
EKELAND’S
313
PRINCIPLE
But from Ž16., Ž20., and Ž21., we can deduce that for any q G m we have always my1
F Ž x0 . G F Ž xm . q
Ý
␦i Ž x m , x i .
is0 qy1
G F Ž xq . q
q
Ý ␦ i Ž x q , x i . G F Ž x⑀ . q Ý ␦ i Ž x⑀ , x i . . is0
is0
Therefore, Ž13. and Ž14. hold. Now we deal with the second case of ␦n4 . Assume that ␦ k ) 0 and ␦ j s 0 for all j ) k G 0. Without loss of generality, we suppose that ␦ i ) 0 for all i F k. Thus, when n F k, we take the same x n and T Ž x n . as above. When n ) k, we take x n g T Ž x ny1 . such that ky1
F Ž xn . q
Ý ␦i Ž x n , x i . is0
F
inf xgT Ž x ny1 .
½
ky1
FŽ x. q
Ý ␦i Ž x, x i . is0
5
q ␦ k ⑀r2 n␦ 0
Ž 24 .
and set ky1
½
T Ž x n . [ x g T Ž x ny1 . F Ž x . q
Ý ␦i Ž x, x i . q ␦ k Ž x, x n . is0 ky1
F F Ž xn . q
Ý ␦i Ž x n , x i . is0
5
. Ž 25 .
Then, by the same deduction as above, Ž12. ᎐ Ž14. also hold. But when x / x⑀ it may follow that there exists an m ) k such that ky1
FŽ x. q
Ý ␦i Ž x, x i . q ␦ k Ž x, x m . is0 ky1
) F Ž xm . q
Ý ␦i Ž x m , x i . is0 ky1
G F Ž x⑀ . q
Ý ␦ i Ž x⑀ , x i . q ␦ k Ž x⑀ , x m . , is0
i.e., Ž15. holds.
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Remark 1. In Theorem 1, we can replace the ‘‘gauge-type’’ function by a sequence of functions n : X = X ª ⺢ j q⬁4 satisfying:
n Ž x, x . s 0;
Ž i. ᭙ x g X , Ž ii . ᭙ y, z g X ,
n Ž y, z . F ␣ n « d Ž y, z . F ␥n , ␣ n ) 0, ␥n ) 0, ␥n ª 0;
where
Ž 26 .
y ¬ n Ž y, z . is lower semi-continuous.
Ž iii . ᭙ z g X ,
In this case, if we take ␦ 0 s ⑀r␣ 0 , replace Ž12. by dŽ x⑀ , x n . F ␥n , and replace Ž x, x n . by nŽ x, x n ., then Theorem 1 also holds. When X is a Banach space, from Theorem 1 we obtain that THEOREM 2. Let X be a Banach space, F: X ª ⺢ j q⬁4 be a lower semi-continuous function bounded from below, and ) 0. Suppose that : X ª ⺢qj q⬁4 is a lower semi-continuous function such that
Ž i . Ž 0 . s 0; Ž ii . ᭙ y k 4 ; X , Ž y k . ª 0 « 5 y k 5 ª 0;
Ž 27 .
and that ␦ 0 ) 0, ␦n G 0, n s 1, 2, . . . , is a nonnegati¨ e number sequence. Then, for e¨ ery x 0 g X and ⑀ ) 0 with F Ž x 0 . F inf F q ⑀
Ž 28 .
X
there exists a sequence x n4 ; X which con¨ erges to some x⑀ g X such that
Ž x⑀ y x n . F ⑀r2 n␦ 0 ,
n s 0, 1, 2, . . . ;
Ž 29 .
when for infinitely many n, ␦n ) 0, F Ž x⑀ . q
⬁
Fq⑀ Ý ␦n Ž x⑀ y x n . F F Ž x 0 . F inf X
Ž 30 .
ns0
᭙ x / x⑀ ,
FŽ x. q
⬁
⬁
ns0
ns0
Ý ␦ n Ž x y x n . ) F Ž x⑀ . q Ý ␦ n Ž x⑀ y x n . Ž 31 .
and when ␦ k ) 0 and for all j ) k G 0, ␦ j s 0, Ž31. is replaced by ky1
᭙ x / x⑀ , ᭚ m G k ,
FŽ x. q
Ý ␦i Ž x y x i . q ␦ k Ž x y x m . is0
ky1
) F Ž x⑀ . q
Ý ␦ i Ž x⑀ y x i . q ␦ k Ž x⑀ y x m . . is0
Ž 32 .
EKELAND’S
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315
Now we give an application of Theorem 2 to the differentiability problem of convex functions. We recall some concepts of w6, 7x. Let X be a Banach space. A bonology on X, denoted  , is any nonempty family of bounded sets of X with D S g  S s X. A function f : X ª ⺢ j "⬁4 is -subdifferentiable at x g X with -subderi¨ ati¨ e x* g X * Žthe dual of X . if, for each ␣ ) 0 and each set S g  , there exists ␦ ) 0 such that for all t g Ž0, ␦ ., ᭙h g S,
f Ž x q th . y f Ž x . ty1 y ² x*, h: G y␣ .
Ž 33 .
Denote x* g ⭸ f Ž x .. -superdifferentiability and -superderi¨ ati¨ e are similarly defined and denoted x* g ⭸  f Ž x .. If f is -subdifferentiable and -superdifferentiable at x g X, then f is called -differentiable at x. Its necessarily unique -derivative is denoted ⵜ f Ž x ., which must coincide with its Gateaux derivative ⵜf Ž x .. If a function f is -differentiable at all ˆ points in a subset A ; X, then f is called -smooth in A. PROPOSITION 1. Assume that is a Lipschitz and -smooth real-¨ alued function on B␣ s x g X ¬ 5 x 5 - ␣ 4 ; a sequence x n g X, n s 0, 1, 2, . . . , satisfies 5 x⑀ y x n 5 - ␣r2,
n s 0, 1, 2, . . . ;
␦n ) 0, n s 0, 1, 2, . . . , is a positi¨ e number sequence with Ý⬁ns0 ␦n - q⬁; and ⌽⑀ Ž x . [
⬁
Ý ␦n Ž x y x n . .
Ž 34 .
ns0
Then ⌽⑀ is -differentiable at x⑀ with ⵜ ⌽⑀ Ž x⑀ . s
⬁
Ý ␦nⵜ Ž x⑀ y x n . .
Ž 35 .
ns0
Since is Lipschitz on B␣ , w Ž x⑀ y x n q th. y Ž x⑀ y x n .x ty1 4 are bounded uniformly for t small enough and h g S, and ⵜ Ž x⑀ y x n .4 is also bounded. Hence, this proposition is a consequence of the Weierstrass M-test. From Theorem 2 and Proposition 1, we obtain THEOREM 3. Let X be a Banach space, F: X ª ⺢ j q⬁4 be a lower semi-continuous function bounded from below, and , ␣ , k ) 0. Suppose that the following assumption holds
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There exists a lower semi-continuous function : X ª ⺢qj q⬁4 such that
Ž i . Ž 0 . s 0; Ž ii . ᭙ y k 4 ; X , Ž y k . ª 0 « 5 y k 5 ª 0; Ž iii . is Lipschitz of rank L and -smooth in B , s x g X < Ž x . - 4 ; Ž iv. 5 x 5 - ␣ « x g B , and Ž x . F rk « 5 x 5 - ␣r2.
Ž P.
Then, for e¨ ery x 0 g X and ⑀ ) 0 with F Ž x 0 . F inf F q ⑀ X
Ž 36 .
there exists an x⑀ g X such that 5 x⑀ y x 0 5 F ␣r2
Ž 37 .
F Ž x⑀ . F F Ž x 0 . F inf F q ⑀
Ž 38 .
0 g ⭸ F Ž x⑀ . q 2 ⑀ kL y1 B*,
Ž 39 .
X
and
where B* s x* g X * ¬ 5 x* 5* F 14 . Proof. From Theorem 2, for any positive number sequence ␦n4 , there exists a sequence x n4 ; X which converges to some x⑀ g X such that
Ž x⑀ y x n . F ⑀r2 n␦ 0 , F Ž x⑀ . q
n s 0, 1, 2, . . . ;
Ž 40 .
⬁
Fq⑀ Ý ␦n Ž x⑀ y x n . F F Ž x 0 . F inf X
Ž 41 .
F Ž x . q ⌽⑀ Ž x . ) F Ž x⑀ . q ⌽⑀ Ž x⑀ . ,
Ž 42 .
ns0
and ᭙ x / x⑀ ,
where ⌽⑀ is defined by Ž34.. From Ž41., we obtain Ž38.. Take ␦ 0 s k ⑀r and Ý⬁ns 1 ␦n s k ⑀r. Then, Ž40. and ŽP.Živ. imply 5 x⑀ y x n 5 - ␣r2, including Ž37., and from the inequality Ž42. and Proposition 1, it is easy to deduce that 0 g ⭸ F Ž x⑀ . q ⵜ ⌽⑀ Ž x⑀ . .
Ž 43 .
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PRINCIPLE
317
But from Ž35., we have that ⵜ ⌽⑀ Ž x⑀ . * F
⬁
Ý ␦n
ⵜ Ž x⑀ y x n . * F 2 ⑀ kL y1 .
Ž 44 .
ns0
Hence, Ž39. is proved. It is obvious that we can replace ŽP.Žiii., Živ. by
Ž iii . ⬘ is Lipschitz and -smooth near x s 0. Now we show that the condition ŽP. is equivalent to the existence of a Lipschitz -smooth bump function; i.e., we have the following proposition: PROPOSITION 2. In a Banach space X, the condition ŽP. is equi¨ alent to the following hypothesis: There exists a Lipschitz -smooth function : X ª w0, 1x such that
Ž i . Ž 0 . s 1; Ž ii . 5 x 5 ) 1 « Ž x . s 0.
Ž H.
Proof. ŽH. « ŽP.. Suppose that satisfies ŽH.. We take
Ž x. s
⬁
Ý ns1
1 22n
1 y Ž2n x. .
Obviously, such a is a Lipschitz -smooth function with the same Lipschitz rank as that of . In addition, for any x g X, Ž x . G Ž0. s 0 and if 5 x 5 ) 1r2 n, then Ž x . G w1 y Ž2 n x .xr2 2 n s 1r2 2 n ; it means that Ž y . ª 0 « 5 y 5 ª 0. Therefore, satisfies ŽP.. ŽP. « ŽH.. Suppose that satisfies ŽP.. From ŽP.Žii., there exists ␦ ) 0 such that 5 x5 ) ␣ « Ž x. ) ␦. We take a function ¨ such that
Ž i . ¨ : ⺢qª w 0, 1 x is continuously differentiable ; Ž ii . t G ␦ « ¨ Ž t . s 0; Ž iii . ¨ Ž 0 . s 1; and for each x g X, set Ž x . s ¨ Ž Ž ␣ x ... Then satisfies ŽH.. THEOREM 4. Let X be a Banach space and f : X ª R j "⬁4 be a lower semi-continuous function. Suppose that ŽH. holds. Then f is -subdifferentiable at a dense subset of points in its graph.
318
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Proof. Without loss of generality, we suppose that there exists a satisfying ŽP.. Let ⑀ ) 0 and x 0 g X with f Ž x 0 . finite given. Since f is lower semi-continuous at x 0 and satisfies ŽP.Žii., we can choose 1 g Ž0, . small enough such that inf f Ž x . < Ž x y x 0 . F 1 4 G f Ž x 0 . y ⑀ .
Ž 45 .
We apply Theorem 3, replacing by 1 , to f1 s f q ␦1, where
␦1Ž x . s
½
0
if Ž x y x 0 . F 1
q⬁
if Ž x y x 0 . ) 1 .
Then, there exists an x⑀ with Ž x⑀ y x 0 . F 1rk such that ⭸ f1Ž x⑀ . is nonempty. Since f1 coincides with f in a neighborhood of x⑀ , ⭸ f Ž x⑀ . is also nonempty. Moreover, since x⑀ satisfies Ž41., we have that f Ž x 0 . y ⑀ F f Ž x⑀ . s f1Ž x⑀ . F f Ž x 0 . . Thus we obtain a point Ž x⑀ , f Ž x⑀ .. with nonempty ⭸ f Ž x⑀ . in the graph of f, which can be arbitrarily close to Ž x 0 , f Ž x 0 ... The conclusion is proved. THEOREM 5. Let X be a Banach space and f be a continuous con¨ ex function on an open subset U ; X. Suppose that ŽH. holds. Then f is densely -differentiable in U. Proof. Applying Theorem 4 to yf, we obtain that yf is densely -subdifferentiable in U; i.e., f is densely -superdifferentiable in U. Since f is always -subdifferentiable in U, the conclusion is proved. Theorems 4 and 5 generalize the corresponding results of Borwein and Preiss w6x, in which Ž x . s Ž ⑀r p .5 x 5 p, p ) 1. In particular, thanks to an example of a Banach space X, constructed by Haydon w8x, such that there exists a Lipschitz continuously Frechet-differentiable bump function on X ´ but there is no equivalent norm on X Gateaux-differentiable on X _ 04 , ˆ our Theorem 5 is not a consequence of Preiss’ theorem w9x Žor the Preiss᎐Phelps᎐Namioka theorem w10x.; this Preiss theorem Žrespectively, the Preiss᎐Phelps᎐Namioka theorem. affirms that if on a Banach space X there exists an equi¨ alent norm -smooth away from origin, then e¨ ery locally Lipschitz function Ž respecti¨ ely, continuous con¨ ex function. on an open subset U of X is densely -differentiable in U Ž respecti¨ ely, -differentiable in a G␦ dense subset of U . and implies the Borwein᎐Preiss theorem on the differentiability of convex functions.
EKELAND’S
⑀-VARIATIONAL
PRINCIPLE
319
Remark 2. It is easy to show that ŽH. is also equivalent to There exists a Lipschitz -smooth function : X ª ⺢ such that
Ž i . Ž 0 . ) 0; Ž ii . 5 x 5 ) 1 « Ž x . F 0.
Ž H⬘.
Remark 3. After finishing the first version of this paper, we learned that Deville et al. w11, 12x also proved Theorems 4 and 5 by a completely different method.
ACKNOWLEDGMENTS We thank Professors S. Simons and G. Godefroy for showing us w12x. We also thank an anonymous referee for posing that it is possible to show ŽP. m ŽH. thereby greatly simplifying the expression of this paper.
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