Generic Gateaux differentiability via smooth perturbations - CiteSeerX

Generic Gateaux differentiability via smooth perturbations∗ Pando Gr. Georgiev, Nadia P. Zlateva University of Sofia Department of Mathematics and Informatics 5 James Bourchier Blvd. 1126 Sofia, BULGARIA

Abstract We prove that in a Banach space with an uniformly Gateaux smooth bump function, every continuous function which is directionally differentiable on a dense Gδ subset of the space, is Gateaux differentiable on a dense Gδ subset of the space. Applications of this result are given.

The usual applications of the variational principles in Banach spaces refer to differentiability of real valued functions. For example the papers of [B-P] and [D-G-Z] contain results about Gateaux differentiability on dense sets. An application of Ekeland’s variational principle to generic Frechet differentiability is given in the proff of famous Ekeland-Lebourg’s theorem (see [E-L]). In [Ge] an application of the smooth variational principle to generic Gateaux differentiability is presented. In this paper we prove some results about generic Gateaux differentiability of directionally differentiable functions. The tool for proving the main ∗

The research was partially supported by the Bulgarian Ministry of Education and Science under contract number MM 506/1995.

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result (Theorem 2) is Proposition 1, which localizes precisely the δ-minimum point of the perturbed function. The estimation of this localization is the same as in the Ekeland variational principle. Denote by Lf the Lipschitz constant of a Lipschitz function f : E → R and by S, B[x; r] (resp. B(x; r)) - the unit sphere of E and the closed (resp. open) ball with center x and radius r. A function b : E → R is said to be a bump function, if there exists a bounded subset supp b, such that b(x) = 0 for every x 6∈ supp b. Proposition 1 Let b be a bump function, such that supp b ⊂ B(0, 1), b(0) = 1 and 0 ≤ b(x) ≤ 1 ∀x ∈ E. Let f : E → R ∪ {+∞} be a function, bounded below and such that D(f ) = {x ∈ E : f (x) < +∞} 6= ∅. Let ε > 0, λ > 0 be given. Suppose that y0 ∈ X satisfies the condition f (y0 ) < inf f + ε. E

Then for every δ > 0 there exists a point x0 ∈ E, such that: x − y0 x0 − y0 ) < inf {f (x) − εb( )} + δ; (a) f (x0 ) − εb( E λ λ x 0 − y0 (b) ∈ supp b; λ (c) kx0 − y0 k < λ. Proof. Denote h(x) = f (x) − εb(

x − y0 ). λ

Then h(y0 ) = f (y0 ) − ε < inf f. E

Let δ1 = inf f − h(y0 ). E

There exists a point x0 ∈ E such that h(x0 ) < inf h + min{δ, δ1 }. E

x0 − y0 0 ∈ / supp b. Then using that b( ) = 0 we have Assume that x0 −y λ λ that δ1 > h(x0 ) − h(y0 ) = f (x0 ) − inf f + δ1 ≥ δ1 , E

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which is a contradiction. So (b) is fulfilled. It is clear that (c) follows immediately from (b). Now we suppose that the bump function b is uniformly Gateaux differentiable. This means b(x + th) − b(x) = h∇b(x), hi, t→0 t

lim

where ∇b(x) ∈ E ∗ is the Gateaux derivative of b at x , and for every h ∈ E this limit is uniform with respect to the points x ∈ E. Recall that the function f : E → R is said to be directionally differentiable at point x0 if for every h ∈ E the one-sided directional derivative f 0 (x0 ; h) = lim t↓0

f (x0 + th) − f (x0 ) t

exists. The following theorem extends the main result in [Ge]. Problems in a similar setting are considered in [Zh], [F-P]. Theorem 2 Let the Banach space E admits a bounded Lipschitz uniformly Gateaux differentiable bump function ˜b. Then every continuous function f , defined on an open subset D ⊂ E, which is directionally differentiable on a dense Gδ subset G of D, is Gateaux differentiable on a dense Gδ subset of D. Proof. From Proposition 2.1 of [Zh] it follows that f is locally Lipschitz on a dense and open subset D1 of D. Let U ⊂ D1 be an open subset, such that f is Lipschitz on U . If we prove that f is Gateaux differentiable on a dense Gδ subset of U , then the theorem will be proved, having in mind the localization principle (see [K, Chapter I, Section 10, V]), stating that a subset P of a topological space is of first Baire category, if for every point p ∈ P there exists an open set H 3 p such that P ∩ H is of first Baire category in H. Without loss of generality we can assume that ˜b(0) 6= 0. Define the bump function b : E → R by b(x) := τ (˜b(dx)), where d =

sup ksk, τ : R → R+ is a differentiable bump function with s∈supp ˜b

Lipschitz derivative, such that τ (˜b(0)) = max τ (t) = 1, t∈R

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0∈ / supp τ . Then b

is a bounded Lipschitz uniformly Gateaux differentiable bump function, such that supp b ⊂ B(0; 1), b(0) = 1, 0 ≤ b(x) ≤ 1 ∀x ∈ E. Define the sets n

Xn := x ∈ U : ∃xn ∈ E, ∃tn ∈ (0, √ x − xn f (x) − 2 tn b( √ ) < tn

√ 1 x − xn √ ∈ supp b, B[x , 2 tn ] ⊂ U, ) : n n2 tn

inf √

z∈B[xn ,2

o √ z − xn {f (z) − 2 tn b( √ )} + t2n . tn tn ]

Since f and b are continuous functions, the sets Xn are open. We shall prove that Xn is dense in U . Let x∗ ∈ U and ε0 > 0 be fixed. For every n ≥ 1, choose ε ∈ (0, min{ε0 , 1 }) in such a way, that B[x∗ ; ε] ⊂ U. We may assume without loss of genern ality that the Lipschitz constant Lf of f on U is less than 1. Then f (x∗ ) ≤

inf

z∈B[x∗ ,ε]

{f (z) + Lkx∗ − zk}
t 2 for every x ∈ supp b and for every t ∈ (0, δ). For every ε > 0 and such δ, for n > 1δ and h ∈ S, since √ √ √ kx0 + tn h − xn k ≤ kx0 − xn k + tn < tn + tn = 2 tn , √ we have x0 + tn h ∈ B[xn , 2 tn ] ⊂ U. Then √ f (x0 + tn h) − f (x0 ) 2 tn  x0 + tn h − xn x0 − xn  √ ≥ b( ) − b( √ ) − tn tn tn tn tn  x0 − xn ε ≥ 2 h∇b( √ ), hi − − tn ≥ tn 2 x0 − xn ≥ h2∇b( √ ), hi − ε − tn . tn n Since b is Lipschitz, k∇b( x0√−x )k ≤ Lb and we can choose a w∗ -converging tn n )}n≥1 , which w∗ -limit generalized subsequence from the sequence {∇b( x0√−x tn b∗ is denoted by 1 . 2 After passing to limits, we obtain f 0 (x0 ; h) ≥ hb∗1 , hi − ε and since this is valid for every ε > 0 and h ∈ S we have f 0 (x0 ; h) ≥ hb∗1 , hi, ∀h ∈ E. Repeating this reasoning for the function (−f ) we obtain a dense Gδ subset in U , at every point x of which it is fulfilled: there exists b∗1 , b∗2 ∈ E ∗ such that hb∗2 , hi ≥ f 0 (x; h) ≥ hb∗1 , hi, ∀h ∈ E. Hence b∗1 = b∗2 = ∇f (x) and the proof is completed. Since the convex functions are directionally differentiable in the interior of its domain, the above theorem implies that the Banach spaces with bounded uniformly Gateaux differentiable Lipschitz bump are weak Asplund spaces. In the following Lemma we shall show that a Banach space with uniformly Gateaux differentiable norm admits a Lipschitz uniformly Gateaux differentiable bump function. 5

Lemma 3 Let the Banach space E has an uniformly Gateaux differentiable norm. Then E admits a Lipschitz uniformly Gateaux differentiable bump function. Proof. Let k.k be an uniformly Gateaux differentiable norm on E and let r : R → R be a function with Lipschitz derivative, such that r = 0 on (−∞, 1] ∪ [3, +∞) and r(2) 6= 0. Denote by Lr0 the Lipschitz constant of r0 and | r0 |= sup | r0 (t) | . Then the function t∈[1,3]

b(x) := r(kxk) is Lipschitz and uniformly Gateaux differentiable. Indeed, let h ∈ S, ε > 0 ε ε . There exists t0 < such that for every are fixed and 0 < γ < 0 2|r | Lr 0 0 < t < t0 and y ∈ S ky + thk − 1 − hkyk0 , hi < γ. t From the definition of the function r it is clear that it is enough to restrict t0 our considerations only on x ∈ E such that 1 ≤ kxk ≤ 3. Let t ∈ (0, ). By 2 the mean value theorem we have b(x + th) − b(x) − hb0 (x), hi =

t

r(kx + thk) − r(kxk) − r0 (kxk)hkxk0 , hi =

t (α = α(x, t) is between kxk and kx + thk)

kx + thk − kxk 0 r (α) − r0 (kxk)hkxk0 , hi =

t

x k kxk + 0 r (α)

t hk kxk t kxk

−1



± hkxk0 , hi − r0 (kxk)hkxk0 , hi ≤

t0 ε ε Lr0 < + = ε. 2 2 2 Since every separable Banach space admits an uniformly Gateaux differentiable norm (see [D-G-Z], Corollary 6.9 (i)), for such spaces Theorem 2 is valid. | r0 | γ + Lr0 | α − kxk |≤| r0 | γ +

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Now we shall give some applications. Let A be an arbitrary non-void set and {gα : E → R, α ∈ A} be a family of functions. Define the function f : E → R by f (x) = inf gα (x). α∈A We need the following lemma. Lemma 4 Let the family {gα : E → R, α ∈ A} be such that (i) for every α ∈ A, gα is K-Lipschitz; (ii) for every x ∈ E, h ∈ S, there exists ε > 0 such that the limit gα0 (x; h) = lim t↓0

gα (x + th) − gα (x) t

exists and it is uniform with respect to α ∈ Mε (x), where Mε (x) := {α ∈ A : gα (x) ≤ f (x) + ε}. Then the function f (x) = inf gα (x) is K-Lipschitz, f 0 (x; h) exists and it is α∈A

equal to sup

inf

ε>0 α∈Mε (x)

gα0 (x; h).

Proof. At first let us note that ∀x0 ∈ E, ∀ε > 0, ∃δ > 0 such that Mδ (x) ⊂ Mε (x0 ), ∀x ∈ B(x0 ; δ).

(1)

Indeed, from (i) we have | gα (x) − gα (x0 ) |≤ Kkx − x0 k, for every α ∈ A and |f (x) − f (x0 )| ≤ Kkx − x0 k. Then, for δ = min{ε/4K, ε/2}, x ∈ B(x0 ; δ) and α ∈ Mδ (x) we have gα (x0 ) ≤ gα (x) + Kkx − x0 k ≤ f (x) + δ + Kkx − x0 k ≤ ≤ f (x0 ) + δ + 2Kkx − x0 k ≤ f (x0 ) + ε, which means α ∈ Mε (x0 ), and (1) is proved. Let x ∈ E be fixed. It is enough to consider only the case when k h k= 1. Denote a := sup inf gα0 (x; h) . From (ii), for every ε > 0, there exist ε>0 α∈Mε (x)

ε0 > 0 and t0 = t0 (ε) such that g (x + th) − g (x) α α − gα0 (x; h) < ε/3

t

for every α ∈ Mε0 (x) and t ∈ (0, t0 ). 7

Let t1 ∈ (0, t0 ) and 0 < ε1 < min{t1 ε/3, ε0 } . There exists α1 ∈ Mε1 (x) such that inf gα0 (x; h) > gα0 1 (x; h) − ε/3. α∈Mε1 (x)

Since ε1 < ε0 , we have Mε1 (x) ⊂ Mε0 (x) and : f (x + t1 h) − f (x) gα1 (x + t1 h) − gα1 (x) − ε1 ≤ < t1 t1 ε ε1 < gα0 1 (x; h) + + < 3 t1 < inf gα0 (x; h) + ε ≤ a + ε. α∈Mε1 (x)

Hence lim sup t↓0

f (x + th) − f (x) ≤ a+ε t

and since ε > 0 is arbitrary small , lim sup t↓0

f (x + th) − f (x) ≤ a. t

Now we shall prove the oposite inequality. Let ε3 > 0 be such that inf gα0 (x; h) > a − ε/3. From (1), there exists δ > 0 such that

α∈Mε3 (x) Mδ (x0 ) ⊂

Mε3 (x) whenever k x0 − x k< δ . Let 0 < t2 < min{t0 , δ}, 0 < ε2 < min{t2 ε/3, ε0 , δ} and α2 ∈ Mε2 (x + t2 h). Then Mε2 (x + t2 h) ⊂ Mδ (x + t2 h) ⊂ Mε3 (x) and we can write: f (x + t2 h) − f (x) gα (x + t2 h) − gα2 (x) − ε2 ≥ 2 ≥ t2 t2 ε ε2 ≥ gα0 2 (x; h) − − > 3 t2 ε > inf{gα0 (x; h) : α ∈ Mε2 (x + t2 h)} − 2 ≥ 3 ε ≥ inf gα0 (x; h) − 2 > a − ε. α∈Mε3 (x) 3 f (x + th) − f (x) ≥ a − ε and since ε > 0 is arbitrary t f (x + th) − f (x) small, lim inf ≥ a and the proof is completed. t↓0 t Hence lim inf t↓0

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If A is a non-empty subset of a E, the distance function to A is defined by dist(x, A) := inf kx − ak. a∈A

Corollary 5 Let the norm of the Banach space E be such that ∀h ∈ S, ∀ε > 0, ∃δ > 0 : kx + thk − kxk − k.k0 (x; h) < ε, ∀t ∈ (0, δ), ∀x ∈ S, t where S is the unit sphere of E. Then for every non-empty closed subset A ⊂ E the distance function dist(., A) is directionally differentiable on E \A. The following theorem is a direct consequence of Theorem 2 and Lemma 4, and can be considered as a generalization of a ”Gateaux” version of the Ekeland-Lebourg theorem [E-L] (see also [Za], where another variant is announced). Theorem 6 If the Banach space E admits a bounded Lipschitz uniformly Gateaux differentiable bump function, then the function f (x) = inf gα (x), α∈A

where for the family {gα , α ∈ A} the assumptions (i) and (ii) from Lemma 4 are fulfilled, is Gateaux differentiable on a dense Gδ subset of E. Recall that from an uniformly Gateaux differentiable norm we can construct an uniformly Gateaux differentiable Lipschitz bump function (see Lemma 3) and therefore, we have the following fact, announced in [Za]. Corollary 7 . In a Banach space E with an uniformly Gateaux differentiable norm, any distance function is Gateaux differentiable on a dense Gδ subset of E.

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REFERENCES [B-P] Borwein J.M. and Preiss D., A smooth variational principle with appli cations to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), 517-527. [D-G-Z] Deville R., Godefroy G. and Zizler V., A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, Journal of Funct. Anal. 111 (1993), 197-212. [E] Ekeland I., Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 443-474. [E-L] Ekeland I. and Lebourg G., Generic Frechet differentiability and perturbed optimization problems in Banach spaces, Trans. Amer. Math. Soc. 224 (1976), 193-216. [F-P] Fabian M. and Preiss D., On intermediate differentiability of Lipschitz functions on certain spaces, Proc. Amer. Soc. 113 (1991), 733-740. [Ge] Georgiev P.G., The smooth variational principle and generic differentiability, Bull. Austral. Math. Soc. 43 (1991), 169-175. [K] Kuratowski K., Topology I, Academic Press, New York and London, 1966. [Za] Zajicek L. A generalization of an Ekeland-Lebourg theorem and the differentiability of distance functions, Proc. 11-th Winter School on Abstract Analysis, Suppl. Rend. Circolo Mat. di Palermo, Ser. II, 3, 1984. [Zh] Zhivkov N. V., Generic Gateaux differentiability of directionally differentiable mappings, Rev. Roumaine Math. Pures Appl. 32 (1987), 179188.

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