Nonlinear Analysis 75 (2012) 380–383

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Nonlinear Analysis

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A note on an approximate mean value theorem for Fréchet subgradients

Nguyen Thi Quynh TrangInstitute of Mathematics, Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi 10307, Viet Nam

a r t i c l e i n f o

Article history:Received 13 June 2011Accepted 14 August 2011Communicated by S. Ahmad

Keywords:Asplund spacesApproximation mean value theoremCharacterization

a b s t r a c t

In this note, we derive a new characterization of Asplund spaces and give a clarificationof the proof of the approximate mean value theorem in Mordukhovich (2006) [5] andMordukhovich and Shao (1996) [1].

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The first result of full statements of the approximate mean value theorem for Fréchet subgradients in Asplund spacesgiven by Mordukhovich and Shao in [1] reads as follows.

Theorem 1.1 (See [1, Theorem 8.2]). Let X be an Asplund space, and let ϕ : X → R ∪ +∞ be a proper lower semicontinuousfunction finite at two given points a = b. Consider any point c ∈ [a; b) at which the function ψ(x) := ϕ(x)+

ϕ(b)−ϕ(a)‖b−a‖ ‖x − b‖

attains its minimum on [a; b]. Then there are sequences xkϕ−→ c and x∗

k ∈ ∂ϕ(xk) satisfyinglim infk→∞

⟨x∗

k , b − xk⟩ ≥ϕ(b)− ϕ(a)

‖b − a‖‖b − c‖, (1.1)

lim infk→∞

⟨x∗

k , b − a⟩ ≥ ϕ(b)− ϕ(a). (1.2)

Moreover, when c = a one has

limk→∞

⟨x∗

k , b − a⟩ = ϕ(b)− ϕ(a). (1.3)

In an earlier version of the approximate mean value theorem, Loewen [2] established the theorem for the Fréchetsubgradients in Banach space whose norm is Fréchet smooth. Observe that a Banach space whose norm is Fréchet smoothis an Asplund space but the inverse is not true, and there exist Banach spaces without Asplund property. Inspired by thefact that the approximatemean value theorem for Clarke subgradients (more general, for presubdifferentials in sense of [3])holds in any Banach space [3,4], we want to know whether the approximate mean value theorem for Fréchet subgradientsis valid in an arbitrary Banach space.

The aim of this note is to establish a new characterization of Asplund spaces in terms of the approximate mean valuetheorem, which allows us to receive a satisfactory answer for the above question, and give a clarification of the proof of theapproximate mean value theorem presented in [5,1].

E-mail addresses: nqtrang@math.ac.vn, nqtrang609@gmail.com.

0362-546X/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2011.08.041

N.T.Q. Trang / Nonlinear Analysis 75 (2012) 380–383 381

2. Main results

Let X be a Banach space and ϕ : X → R ∪ ±∞ be finite at x ∈ X . The Fréchet subdifferential of ϕ at x is the set∂ϕ(x)defined in [5] by

∂ϕ(x) =

x∗

∈ X∗: lim inf

x→x

ϕ(x)− ϕ(x)− ⟨x∗, x − x⟩‖x − x‖

⩾ 0.

As usual, we put∂ϕ(x) = ∅ if |ϕ(x)| = ∞. For any nonempty subsetΩ of X and any point x ∈ X , the setN(x;Ω) := ∂δ(x;Ω)is called the Fréchet normal cone toΩ at x. Here δ(·;Ω) signifies the indicator function ofΩ , that is, δ(x;Ω) = 0 if x ∈ Ω

and δ(x;Ω) = ∞ if x ∈ X \Ω .In the sequel, we need the following difference rule for the Fréchet subdifferential which was established by

Mordukhovich et al. [6].

Lemma 2.1 ([6, Theorem 3.1]). Let ϕi : X → R ∪ ±∞ be extended-real-valued functions on a Banach space X, finite at x fori = 1, 2. Assume that∂ϕ2(x) = ∅. Then∂(ϕ1 − ϕ2)(x) ⊂

x∗∈∂ϕ2(x)

[∂ϕ1(x)− x∗].

Recall [5, p. 196] that a Banach space X is said to be Asplund if every convex continuous function ϕ : U → R definedon an open convex subset U of X is Fréchet differentiable on a dense subset of U . The class of Asplund spaces, which iswell investigated in the geometric theory of Banach spaces, plays a remarkable role in variational analysis and generalizeddifferentiation [5]. This class includes, among others, every Banach space with separable dual and every space having anequivalent Fréchet differentiable renorm (in particular, all reflexive spaces); spaces of continuous functions C(K) on ascattered compact Hausdorff space K (i.e., K is a compact Hausdorff space such that every subset of K has an isolated point);the classical space of sequences c0 with supremum norm. We refer the reader to [7,5,8,9] and the references therein forvarious characterizations, properties, and examples of Asplund spaces.

Lemma 2.2 ([5, Theorem 2.33]). Let X be an Asplund space with x ∈ X. Assume that ϕi : X → R ∪ +∞(i = 1, 2) are properfunctions such that ϕ1 is lower semicontinuous and ϕ2 is Lipschitz continuous around x. Then for any γ > 0 one has∂(ϕ1 + ϕ2)(x) ⊂

∂ϕ1(x1)+∂ϕ2(x2) : xi ∈ x + γB, |ϕi(xi)− ϕi(x)| ≤ γ , i = 1, 2

+ γB∗,

where B and B∗ are the unit balls in X and X∗, respectively.

Lemma 2.3 ([5, Example 2.19]). Let X be a Banach space with no Asplund property. Then there exists a proper closed subset Ωof X such that N(x;Ω) = 0 for every x ∈ Ω .

We are now ready to state and prove our main results.

Theorem 2.4. Let X be a Banach space. The following are equivalent.

(i) X is Asplund.(ii) For any a proper lower semicontinuous function ϕ : X → R ∪ +∞ finite at two given points a = b and for any point

c ∈ [a; b) at which the function

ψ(x) := ϕ(x)+ϕ(b)− ϕ(a)

‖b − a‖‖x − b‖

attains its minimum on [a; b], there are sequences xkϕ−→ c and x∗

k ∈ ∂ϕ(xk) satisfying (1.1) and (1.2). Moreover, if c = a,then (1.3) is valid.

Proof. (i) ⇒ (ii). Observe that ψ is a lower semicontinuous function, attains its minimum at c and ψ(a) = ψ(b). By theproof of Mordukhovich [5, pp. 308–310], one can find sequences xk

ϕ−→ c , xk = b, and x∗

k ∈ ∂ψ(xk) such that

lim infk→∞

⟨x∗

k , b − xk⟩ ≥ 0, (2.1)

lim infk→∞

⟨x∗

k , b − a⟩ ≥ 0, (2.2)

and

limk→∞

⟨x∗

k , b − a⟩ = 0 (2.3)

when c = a. Let us consider the following two cases.

382 N.T.Q. Trang / Nonlinear Analysis 75 (2012) 380–383

Case 1:ϕ(a) ≤ ϕ(b). Since ϕ(b)−ϕ(a)‖b−a‖ ‖·−b‖ is a convex Lipschitz function andϕ(x) is lower semicontinuous, by Lemma2.2,

there are points xk, vk ∈ xk + ηkB∗ with |ϕ(xk) − ϕ(xk)| < ηk satisfying x∗

k ∈ ∂ϕ(xk) +ϕ(b)−ϕ(a)

‖b−a‖ ∂(‖ · −b‖)(vk) + ηkB∗,

where ηk :=1

k(‖x∗k‖+1) and ∂ denotes the sudifferential in the sense of convex analysis. Thus we can find x∗

k ∈ ∂ϕ(xk), v∗

k ∈

∂(‖ · −b‖)(vk) and u∗

k ∈ ηkB∗ such that x∗

k = x∗

k +ϕ(b)−ϕ(a)

‖b−a‖ v∗

k + u∗

k . Since

⟨x∗

k , b − xk⟩ = ⟨x∗

k , b − xk⟩ + ⟨x∗

k , xk − xk⟩≥ ⟨x∗

k , b − xk⟩ − ‖x∗

k‖ ‖xk − xk‖≥ ⟨x∗

k , b − xk⟩ − ‖x∗

k‖ηk

≥ ⟨x∗

k , b − xk⟩ −1k ,

it follows from (2.1) that lim infk→∞

⟨x∗

k , b − xk⟩ ≥ 0. Noting that limk→∞ vk = c = b and v∗

k ∈ ∂(‖ · −b‖)(vk), we have

‖v∗

k‖ = 1 and ⟨v∗

k , vk − b⟩ = ‖vk − b‖ for any k sufficiently large. Thus

limk→∞

⟨v∗

k , b − xk⟩ = limk→∞

⟨v∗

k , b − vk⟩ + ⟨v∗

k , vk − xk⟩

= limk→∞

−‖vk − b‖

+ lim

k→∞

⟨v∗

k , vk − xk⟩

= −‖c − b‖.

Then we have

lim infk→∞

⟨x∗

k , b − xk⟩ = lim infk→∞

⟨x∗

k , b − xk⟩ −ϕ(b)− ϕ(a)

‖b − a‖⟨v∗

k , b − xk⟩ − ⟨u∗

k , b − xk⟩

= lim infk→∞

⟨x∗

k , b − xk⟩ −ϕ(b)− ϕ(a)

‖b − a‖limk→∞

⟨v∗

k , b − xk⟩ − limk→∞

⟨u∗

k , b − xk⟩

≥ϕ(b)− ϕ(a)

‖b − a‖‖b − c‖.

Since c ∈ [a, b) and

limk→∞

⟨v∗

k , b − c⟩ = limk→∞

⟨v∗

k , vk − c⟩ − ⟨v∗

k , vk − b⟩

= − limk→∞

‖vk − b‖

= −‖b − c‖,

we have limk→∞⟨v∗

k , b − a⟩ = −‖b − a‖. By virtue of (2.2),

lim infk→∞

⟨x∗

k , b − a⟩ = lim infk→∞

⟨x∗

k , b − a⟩ −ϕ(b)− ϕ(a)

‖b − a‖limk→∞

⟨v∗

k , b − a⟩ − limk→∞

⟨u∗

k , b − a⟩

≥ ϕ(b)− ϕ(a).

If c = a then by using (2.3) in place of (2.2), we get (1.3).Case 2: ϕ(a) > ϕ(b). Since x∗

k ∈ ∂ψ(xk) and ϕ(a)−ϕ(b)‖b−a‖ ‖x − b‖ = ϕ(x)− ψ(x), by Lemma 2.1, we have x∗

k +ϕ(a)−ϕ(b)

‖b−a‖ ∂(‖ ·

−b‖)(xk) ⊂ ∂ϕ(xk). Take any v∗

k ∈ ∂(‖ · −b‖)(xk). For xk := xk and x∗

k := x∗

k +ϕ(a)−ϕ(b)

‖b−a‖ v∗

k , we have x∗

k ∈ ∂ϕ(xk), ‖v∗

k‖ = 1and ⟨v∗

k , xk − b⟩ = ‖xk − b‖. Hence

lim infk→∞

⟨x∗

k , b − xk⟩ = lim infk→∞

⟨x∗

k , b − xk⟩ +ϕ(a)− ϕ(b)

‖b − a‖⟨v∗

k , b − xk⟩

= lim infk→∞

⟨x∗

k , b − xk⟩ +ϕ(a)− ϕ(b)

‖b − a‖limk→∞

(−‖xk − b‖)

≥ϕ(b)− ϕ(a)

‖b − a‖‖b − c‖.

Observe that limk→∞⟨v∗

k , b − a⟩ = −‖b − a‖. By virtual of (2.2),

lim infk→∞

⟨x∗

k , b − a⟩ = lim infk→∞

⟨x∗

k , b − a⟩ +ϕ(a)− ϕ(b)

‖b − a‖⟨v∗

k , b − a⟩

= lim infk→∞

⟨x∗

k , b − a⟩ +ϕ(a)− ϕ(b)

‖b − a‖limk→∞

⟨v∗

k , b − a⟩

≥ ϕ(b)− ϕ(a).

N.T.Q. Trang / Nonlinear Analysis 75 (2012) 380–383 383

If c = a, then by (2.3) we have

limk→∞

⟨x∗

k , b − a⟩ = limk→∞

⟨x∗

k , b − a⟩ +ϕ(a)− ϕ(b)

‖b − a‖limk→∞

⟨v∗

k , b − a⟩

= ϕ(b)− ϕ(a).

(ii)⇒ (i). Assume that X is not Asplund. By Lemma2.3,we can find a proper closed subsetΩ of X satisfyingN(x;Ω) = 0for every x ∈ Ω . Let us consider the function ϕ : X → R defined by

ϕ(x) =

0 if x ∈ Ω

1 otherwise.

It is not difficult to see that ϕ is a proper lower semicontinuous function. We will prove that∂ϕ(x) = 0 for all x ∈ X .Indeed, since ϕ is constant on the open set X \Ω,∂ϕ(x) = 0 for all x ∈ X \Ω . If x ∈ Ω , then by the Fermat rule 0 ∈ ∂ϕ(x).For any x∗

∈ ∂ϕ(x), we have

lim infu→x

ϕ(u)− ϕ(x)− ⟨x∗, u − x⟩‖u − x‖

≥ 0.

This implies

lim supuΩ−→x

⟨x∗, u − x⟩‖u − x‖

≤ 0,

which means that x∗∈ N(x;Ω) = 0. Thus ∂ϕ(x) = 0 for all x ∈ X . Take any a ∈ Ω and b ∈ X \ Ω . We have

ϕ(b)− ϕ(a) = 1 while ⟨x∗, b − x⟩ = 0 for any x ∈ X and x∗∈ ∂ϕ(x). This proves that (ii) is invalid if X is not Asplund. The

proof is complete.

Remark 2.5. From Theorem 2.4, it follows that the largest class of Banach spaces on which the approximate mean valuetheorem for Fréchet subgradients holds is the class of Asplund spaces. It is worth observing that the techniques used inthe proof of [2] are based on the fuzzy sum rule for Fréchet subgradients and some properties of smooth norm and are notdirectly extensible to the use of the theorem in the sense of [1]. Instead of the lack of the smoothness, among other things,Mordukhovich and Shao [1] employed the Moreau–Rockafellar subdifferential theorem in convex analysis for the functionϕ2(x) :=

ϕ(b)−ϕ(a)‖b−a‖ ‖x−b‖+

1k‖x− c‖+ δ(x, [a, b]) (see [1, p. 1268]). Another proof of the approximate mean value theorem

was presented in [5, pp. 309–310] for the case ϕ(b) = ϕ(a).

Acknowledgments

The author expresses her thanks to Professor Nguyen Dong Yen, Doctor Nguyen Quang Huy, and the referee for theirhelpful comments and suggestions.

References

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(2006) 685–708.[7] M. Fabian, Gâteaux Differentiability of Convex Functions and Topology. Weak Asplund Spaces, Wiley, New York, 1997.[8] B.S. Mordukhovich, Y. Shao, Extremal characterizations of Asplund spaces, Proc. Amer. Math. Soc. 124 (1996) 197–205.[9] R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd ed., Springer, Berlin, 1993.