ON TWO-SIDED UNIDIRECTIONAL MEAN VALUE INEQUALITY IN A FR´ECHET SMOOTH SPACE 1

: The paper is devoted to a new unidirectional mean value inequality for the Fr´echet subdiﬀerential of a continuous function. This mean value inequality ﬁnds an intermediate point and localizes its value both from above and from below; for this reason, the inequality is called two-sided. The inequality is considered for a continuous function deﬁned on a Fr´echet smooth space. This class of Banach spaces includes the case of a reﬂexive space and the case of a separable Asplund space. As some application of these inequalities, we give an upper estimate for the Fr´echet subdiﬀerential of the upper limit of continuous functions deﬁned on a reﬂexive space.


Introduction
Consider the following mean value inequalities.
Proposition 1 [12, Theorems 2.1 and 2.2].Let a scalar function f be defined and lower semicontinuous on a Fréchet smooth Banach space X.Let points ǔ and v in X be given.Then, for arbitrary numbers š < f (v) − f (ǔ) and κ > 0, there exist a point z − ∈ [ǔ; v] + κB and a Fréchet subgradient ζ − ∈ ∂f (z − ) such that š < ζ − (v − ǔ) and f (z − ) < f (ǔ) + max(0, š) + κ. (1.1) Furthermore, if f is continuous, there are a point z + ∈ [ǔ; v] + κB and a Fréchet subgradient Note that inequalities (1.1) and (1.2) are similar.This suggests that, in the case of continuity of f , it is possible to get a common point z + = z − such that the value f (z) is localized from both above and below.Proving the corresponding two-sided unidirectional mean inequality is the primary goal of this paper.
As part of the historical background, note that the existence of a pair (z − , ζ − ) satisfying inequalities like (1.1) has been widely studied (see, for example, [13,Subsect. 3.4.8]and [14,Sect. 4.4]).Unlike different variants of Lagrange's mean value theorem for certain classes of Lipschitz continuous functions, they ensure an upper bound of f (v) − f (u) through some subgradient ζ.These inequalities apply to any lower semicontinuous function.Furthermore, the corresponding to the Fréchet subdifferentials unidirectional mean value inequality is equivalent to the Asplund property of a Banach space [14,17], and therefore is equivalent to several basic principles of variational analysis [1,18], for example, to the inspired by [16] and [5] multidirectional mean value inequality [4,Subsection 3.6.1];for more recent references, see [10] and [8].However, the multidirectional mean value theorem as well as the previous unidirectional mean value inequality also localizes f (ẑ) on one side only.
The rest of the paper is organized as follows.In Section 3, we will prove the desired twosided unidirectional mean value inequality for continuous functions.Then, applying this result, in Section 4, we will show an upper estimate for the Fréchet subdifferential of the upper limit of continuous functions.But first, we will recall several elementary definitions and notions.

Definitions and notation
We will use elementary notions from the set-valued and variational analyses [4,13,15].For a nonempty set X of some real Banach space X, denote by cl X and co X the closure and the convex hull of X .For a point x ∈ X , the contingent (Bouligand tangent) cone to X at x is the set T (x; X ) of all v ∈ X such that one finds a decreasing to 0 sequence of positive t n and a converging to v sequence of v n ∈ X such that x + t n v n ∈ X for all positive integers n.For a point x ∈ X, we say that ζ ∈ X * is a Fréchet normal to X at x if one has x ∈ X and for all converging to x sequences of z n ∈ X .Denote by N (x; X ) the set of all Fréchet normals to X at x.We call a Banach space X Fréchet smooth if this space has an equivalent norm that is C 1smooth off the origin.Note that any reflexive Banach space and any separable Asplund space are Fréchet smooth [4,Theorem 6.1.6].It is worth mentioning that each Fréchet smooth space has a C 1 -smooth Lipschitz function with bounded nonempty support [3,Ex. 4.3.9].
Denote by B and B * the unit closed balls in X and X * , respectively.Given an extended-real-valued function g : X → R ∪ {−∞, ∞}, define its lower semicontinuous envelope lsc g by the rule: Note that this function is lower semicontinuous.In addition, a function g is lower semicontinuous iff its epigraph epi g is closed.In the case of lower semicontinuous function g, define the Fréchet subdifferential of g at x as ∂g(x)

Two-sided mean value inequality
Theorem 1.Let X be a Fréchet smooth space.Let a continuous function f : X → R and some closed interval [u; v] in X be given.Then, for a real number s < f (v) − f (u) and positive ε, there exist some point ẑ ∈ P r o o f.Without loss of generality, we can assume that u = 0 and f (u) = 0. Now, the initial inequality can be written as s < f (v).
Case s < 0. Let s be negative.Choose a positive number ε < min(|s|, f (v)− s).Define ε = ε/4 and s △ = s + 3ε.Since s < 0 = f (0) < |s| and s < f (v), one finds a positive number t < 1 such that For the same reason, there is a positive κ < ε such that We claim that there exist some ẑ ∈ [0; tv] + κB and ζ ∈ ∂f (ẑ) such that To this end, consider the continuous map Since h( t) = h(0) = 0 holds, due to the intermediate value theorem, there exists positive τ ≤ t that satisfies the equality h(τ ) = 0 and at least one of the following conditions: Now, the relations 0 < τ ≤ t < 1, h(τ ) = 0, and (3.2) yield the inequality for all positive integers n.This gives some r − , r Next, taking into account the inequalities τ > 0 and f (0 Now, in the case of τ < κ (condition (i)) and in the case of nonpositive h| Note that all rn v lie in the compact set [0; τ v].Passing to a subsequence, we can assume that this sequence converges.By ẑn − rn v → 0, the both sequences of ẑn and rn v has the common limit.The sequences of f (ẑ n ) and f (r n v) are the same by the continuity of f ; in particular, one finds a positive integer N such that So, it is required to check only the inequality Now, in the case of nonnegative h| [0;τ ] (condition (iii)), by the choice of rN = r − and ẑN = z − , we obtain 0 In the case h| [0;τ ] ≤ 0 (condition (ii)), one has Finally, in the case τ < κ (condition (i)), (3.3) and (3.7) yield Inequalities (3.4) have been proved.
Case s ≥ 0. Assume that s is nonnegative.Recall that s < f (v).Choose a positive ε such that s + ε < f (v).Define and one can choose positive ε such that Consider the map f : Then, we have f (0, 0) = 0, Since −εs < 0, we can apply the first case of our theorem to the inequality Now, the first inequality leads to s < s < ζv by s < s; on the other hand, the second inequality entails The theorem is proved.
Remark 1.As [12, Example 2.1] has shown, (1.2) can be violated if f : R → R is only lower semicontinuous.Therefore, the assumption of the continuity of f is essential in this theorem as well.
Remark 2. In the case of Lipschitz continuous function f , for its G-subdifferential, there exists a variant of unidirectional mean value inequality that guaranties the inclusion z ∈ [u; v] instead of z ∈ [u; v] + εB (see [9,Theorem 4.70]).However, this is not true for a Fréchet subdifferential.Indeed, for the Lipschitz continuous function its Fréchet subdifferential is empty on the interval [(0, 0); (0, 1)]; in particular, no Fréchet subgradient ζ satisfies (3.1).Remark 3. It may mistakenly seem that Theorem 1 does not essentially use the asymmetry of a Fréchet subdifferential and can be directly extended to the symmetric case.Indeed, Lebourg's mean value theorem [6,Theorem 2.4] for Clarke subdifferentials, the mean value theorem [2] for MP-subdifferentials, and the symmetric subdifferential mean value theorem [13,Theorem 3.47], [14,Theorem 4.11] give the corresponding gradient ζ of f at some ẑ ∈ [u; v] that satisfies the symmetric bound This bound is exactly the limit of bounds , and ε ↓ 0. Similarly, passing to the limit in |f (ẑ) − f (u)| < |s| + ε, we could hope for the eatimate

Subdifferentials of the upper limit of continuous functions
Let a family of continuous functions f θ : X → R ∪ {−∞, ∞}, θ ∈ [0; ∞) be given.Define a function f sup : X → R ∪ {−∞, ∞} by the following rule: For every positive δ, denote by Z δ (x) the set of all ζ ∈ X * for which there exists a pair The following estimate of the subdifferential of the upper limit function is the enlargement of [11,Lemma 6] on reflexive spaces as well as the refinement of [12, Theorem 6.1(a)] in the case of continuous functions; its proof is similar to that of [12,Theorem 6.1(a)].
The general case.Let a point x ∈ X and a subgradient ξ ∈ ∂ lsc f sup (x) be given.Define y = lsc f sup (x).Choose a positive number δ < 1/3.

Remark 4 .
If X △ = R d , by the famous Carathéodory theorem [15, Theorem 2.29], any finite convex sum of a (co)vectors can be represented by some finite convex sum of no more than d + 1 of them.So, we can assume that N ≤ d + 1.