The Extention of Mean Value Theorem in Asplund Spaces

In this paper a nonsmooth mean value theorem in asplund spaces, under convexity, using the properties of limiting subdifferentials is established. We research on a kind of mean value theorem and prove that this theorem for set-valued mappings under convexity of domein in banach spaces. This theorem is use full to establish new results in convex analysis.


Introduction
The mean value theorem is an important theorem in smooth and nonsmooth analysis from some theoretical and practical points of view.In many papers different versions of this theorem in terms of the different types of derivative have been studied (see, e.g., [1,5,10] and [11]).In all of them the related function is defined on a line segment (interval) and some the results are obtained for a convex domain.Antczak [1] established a mean value theorem for convex domains and locally Lipschitz mappings by using the concept of Clarke's generalized gradient [5].In fact, Antczak has considered an η path instead of an interval and has proved the result after establishing the existence of critical points (see Theorem (29) in [1]).Another proof for this theorem in convex analysis has been provided by Soleimani-damaneh [8], recently.R. Burachik proved an extended version of mean value theorem for set-valued functions.In this paper a non-smooth mean value theorem in Asplund spaces, under convexity, using the properties of limiting sub-differentials is established.This theorem is use full to establish new results in convex analysis.we extend Burachiks proof under convexity of domains.

Preliminaries and notations
We recall that for a nonempty subset S of R n , η : S × S → R n and arbitrary point u of S, S is said to be convex at with respect to η if for each S is said to be an convex set with respect to η if S is convex at each u ∈ S with respect to the same η.
A set in R n is convex with respect to η(x, u) = 0 forall x, u ∈ R n The definition of an convex set has a clear geometric interpretation.Thus, the definition essentially says that there is a path starting from u which is contained in S. We do not require that x should be one of the end points of the path.However, if we demand that x be an end point of the path for every pair of points x, u ∈ S that η(x, u) = x − u and convexity reducing to convexity.Thus, it is true that every convex set is also convex with respect to η(x, u) = x − u.
Definition 2.1.Let S ⊂ R n be a nonempty convex set with respect to η, x and u two arbitrary points of S. The set P u,v is said to be a closed η-path joining u and v = u + η(x, u) (contained in S) if Analogously, an open η-path joining the points u and v = u + η(x, u) (contained in S) is defined by holds.
Definition 2.3.Let S ⊂ R n be a nonempty convex set with respect to η. we call a function f : S :→ R be Q-pre-convex with respect to η and β if, there exists a vector-valued function η : S × S → R n and β : holds.
Remark 2.2.Every convex function is pre-convex with respect to η(x, u) = x − u, but the converse may not always be true and also every pre-convex function is Q-pre-convex with respect to β (t, s) = t − s, but the converse may not always be true.
Definition 2.4.Let S ⊂ R n be a nonempty convex set with respect to η.A differentiable function f : S :→ R is said to be convex with respect to η and β if, there exists a vector-valued function η : S × S → R n and β : C ×C → R (where C is an open subset of R)such that the relation holds.

Remark 2.3. Every convex function (convex in the case of differentiability) is Q-convex (Q-convex in the case of differentiability
)with respect to β (t, s) = t − s, but always its converse may not be true.Let X and Y be two nonempty sets and let F : X → 2 Y be a mapping defined on X which takes values in the family of subsets of Y ; that is F(x) is a subset of Y with the possibility that F(x) = ϕ for some x ∈ X is admitted.In this case F is characterized by its graph, which is defined by Gph(F)

The projection of Gph(F) onto its first argument is the domain of F, denoted by D(F); i.e., D(F)
Assume S is a nonempty open set in X and f : S → R be a real-valued mapping.The Clarke's generalized directional derivative of f at x ∈ S in the direction v ∈ X is defined by Also, the Clarke's generalized gradient of f at x, denoted by ∂ f (x), is the subset of X * given by

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It is well known that , where t is any scalar.
(iii) If f attains a local extremum at x, then 0 ∈ ∂ f (x) For a nonempty subset K of a Banach space X, the normal cone N k : X → 2 X * is defined by and S K : X → 2 X is defined as For a convex subset K of X, it is well known that S K (x) ⊆ T K (x) for all x ∈ K [6].
Definition 2.5.The derivative DF(x 0 , y 0 ) : X → 2 Y of F at a point (x 0 , y 0 ) ∈ Gph(F) is the set-valued map whose graph is the tangent cone T Gph(F) (x 0 , y 0 ), as introduced in definition ∀α ∈ [0, 1] and x 1 , x 2 ∈ D(F) [7].Also, a mapping F is η-convex if Gph(F) is η-convex.If maps X into X * , then the expression lim x→ x sup F(x) means the sequential Kuratowski-Painlev upper limit of F with respect to the norm topology in X and the weak-star topology in X * That is, Throughout this paper X is an Asplund space, i.e., a Banach space, in which every convex continuous function is generically Fechet differentiable, X * denotes the topological dual of X and ⟨0, 0⟩ exhibits the duality pairing.For an extended-real-valued function φ : we say it is proper if φ(x) > −∞ for all x ∈ X.The domain, epigraph and hypergraph of φ are as followe: Let Ω be a nonempty subset of X.Given x ∈ Ω and ε ≥ 0, the set of ε−normals to Ω at x is defined by → stands for convergence in weak * topology, and N denotes the set of all natural numbers.See the textbook [5] for more details.Considering a point x ∈ X with |φ( x)| < ∞, the see is the limiting subdifferential of φ at x and its elements are limiting subdifferentials of φ at this introduced, in the equivalent form, by Mordukhovich [9].See [9,11,14,15] for more details and applications.One of the classes of functions whose set of limiting subdifferentials is nonempty is the class of locally Lipschitz (Lipschitz continuous) functions which are also considered in this study.The limiting upper subdifferential of φ at x is defined as follows: Furthermore, the Frechet subdifferential and the Frechet upper subdifferential of φ at x, respectively, are defined as follows: These two sets may be empty simultaneously even for continuous functions, e.g., consider φ(x) = 3 √ x at x = 0.The following results are useful in the next section.The proofs of them.
Note that the above proposition is valid even in non-Asplund spaces.It is not difficult to show that ∂ φ( x) ⊆ ∂ φ( x).The case of equality signifies the concept of lower regularity, i.e., φ is said to be lower regular at we say that is sequentially normally epi-compact at x if epi(φ) is sequentially normally compact at ( x, φ( x)).

main section
In the following theorem φ : X → R is defined on an open set containing P u,v .For instance, can be defined on an open convex set with respect to η.
According to Theorem (1.26) in [10], is sequentially normally epi-compact at x = u + λ η(x, u).If epi φ is compactly epi-Lipschitzian around ( x, φ( x)) and this happen when φ is directionally Lipschitzian around x (see [12] for details concerning directionally Lipschitz functions).Hence, φ is sequentially normally epi-compact at x = u + λ η(x, u) due to Lipschitz continuity around x. Therefore g and φ satisfy the conditions of Theorem 3.1 at λ and x, respectively.

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The collection of such normals introduces the following N( x; Ω) = lim x→ x,ε↓0 sup N(x; Ω) which is the limiting normal cone to Ω at x. Put N( x; Ω) = / 0 for x ̸ ∈ Ω.Note that the symbol u Ω → x means u → x with u ∈ Ω.The symbol w *
is the segment line with the end points u and x.
Theorem 3.1.Let φ be Lipschits continuous on an open set X containing P uv and be lower regular on P 0 uv .Let β : C ×C → R, where C is a subset of R and β be Lipschitz continiuous and lower regpular on C. Let