On nonconvex subdifferential calculus in binormed spaces - m-hikari

International Mathematical Forum, 2, 2007, no. 55, 2723 - 2731

On Nonconvex Subdifferential Calculus in Binormed Spaces S. Lahrech, J. Hlal, A. Ouahab Dept. of Mathematics, Faculty of Science, Mohamed first University Oujda, Morocco, (GAFO Laboratory) A. Jaddar National School of Managment, Mohamed first University Oujda, Morocco, (GAFO Laboratory) A. Mbarki National School of Applied Sciences, Mohamed first University Oujda, Morocco, (GAFO Laboratory) Abstract We give in this paper some useful calculus results related to the limiting subdifferential in binormed spaces (generalized limiting subdifferential) which is a generalization of the limiting subdifferential in Banach spaces [5, 6].

Mathematics Subject Classification: 49J52, 49J50 Keywords: Nonconvex calculus, limiting subdifferential in binormed spaces, generalized normal cones

1

Introduction

The limiting subdifferential was first studied by Mordukhovich in [10], followed by joint work with Kruger in [5], and by work of Ioffe [1, 2]. In the beginning Ioffe, Jourani, kruger, Loewen, Mordukhovich, Shao and Thibault used the limiting subdifferential to establish full principal calculus rules and applications in Banach spaces which possess some geometric assumptions (we can see [2, 3, 4, 9, 12, 13]). Later, Mordukhovich and Shao

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S. Lahrech, J. Hlal, A. Ouahab, A. Jaddar and A. Mbarki

in [11] extend some of those results to arbitrary Banach spaces including sum rules, chain rules, product and quotient rules, however most of those results must involve the strict differentiability notion. In this paper, we generalize some interesting calculus rules of [11] for non necessarily Fr´echet strictly differentiable mappings using the notion of limiting subdifferential in binormed spaces (generalized limiting subdifferential). Thus, our results are closely related to (extending in some sense) those of [11]. The rest of this paper is organized as follows. Section 2 contains some definitions and preliminary material, while in section 3, we establish the result related to sum rule for the generalized limiting subdifferential and in section 4 we prove some other calculus results: chain rule, products and quotients rules. Finally, we give an example of a non Fr´echet strictly differentiable mapping for which we can apply our sum rule theorem, while the classical sum rule theorem of [11] cannot be used.

2

Basic definitions and properties

Throughout this paper we consider two binormed spaces (X,  . 1 ,  . 2 ) and (Y,  . 3 ,  . 4 ) such that (X,  . 2 ) and (Y,  . 4 ) are two separable Banach spaces, and for some c > 0, c > 0  . 1 ≤ c  . 2 , . 3 ≤ c  . 4 . For (x, y) ∈ X × Y we set  (x, y) 5 = x 1 +  y 3 and  (x, y) 6 = x 2 +  y 4 . Hence, X ×Y becomes a binormed space under the pair of norms ( . 5 ,  . 6 ). Moreover,  . 5 ≤ max(c, c )  . 6 . ϕ,.1

By Cl1 Ω we denote the closure of Ω in (X,  . 1 ), while x → x¯ (respectively Ω,.1

.1

x)( respectively, x → x¯ with x −→ x¯) means that x → x¯ with ϕ(x) → ϕ(¯ respect to the norm  . 1 and x, x¯ ∈ Ω). x) denotes the open ball centered at x¯ with radius δ in (X,  . 1 ), Let Bδ1 (¯ where x¯ ∈ X and δ > 0. For any multifunction Φ acting from X into its topological dual (X,  . 2 )∗ , we define the ( . 1 ,  . 2 )-sequential Kuratowski-Painlev´e upper limit (1,2)

ω∗

by: lim supΦ(x) ≡ ( . 1 ,  . 2 ) − lim supΦ(x) = {x∗ ∈ (X,  . 2 )∗ /∃ x∗k → ∗

x→¯ x .1 xk →

x→¯ x

x∗k

ω∗

x ,∃ x¯ with ∈ Φ(xk ) ∀k = 1, 2 . . .}, where → denotes the convergence for the weak-star topology ∗σ((X,  . 2 )∗ , X) of (X,  . 2 )∗ . ¯ is an extended-real-valued function, then the domain Recall that if ϕ: X −→ R domϕ of ϕ and the epigraph epi ϕ of ϕ are respectively defined by: domϕ = {x ∈ X/ | ϕ(x) |< +∞}, epiϕ = {(x, μ) ∈ X × R/μ ≥ ϕ(x)}. Definition 2.1 Let U be an open subset of (X,  . 1 ) and let ϕ : U −→ R be a real-valued function. Assume that x¯ ∈ U.

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Subdifferential calculus in binormed spaces

i) We say that ϕ is ( . 1 ,  . 2 )-strictly Fr´echet differentiable at x¯ if there exists a continuous linear operator L from (X,  . 2 ) into R such that ϕ(u) − ϕ(x) − L(u − x) = 0. .1  u − x  2 u,x → x ¯ lim

(1)

In this case, we set ∇1,2 ϕ(x) ≡ ( . 1 ,  . 2 ) − ∇ϕ(x) = L. ii) We say that ϕ is ( . 1 ,  . 2 )-locally Lipschitzian around x¯ if for some positive scalar k, we have: | ϕ(x) − ϕ(x ) |≤ k  x − x 2

(2)

for all x, x close to x¯ with respect to the norm  . 1 . Let us remark that if ϕ is ( . 1 ,  . 2 )-strictly Fr´echet differentiable at x¯, then it is necessarily  . 2 -strictly Fr´echet differentiable at x¯ and therefore, continuous with respect to the norm  . 2 . But in general, even if ϕ is ( . 1 ,  . 2 )-strictly Fr´echet differentiable at x¯, it is not necessarily  . 1 strictly Fr´echet differentiable at x¯. But the converse is true, that is, if ϕ is  . 1 -strictly Fr´echet differentiable at x¯, then it is ( . 1 ,  . 2 )-strictly Fr´echet differentiable at x¯. Definition 2.2 i) Let Ω be a nonempty subset of X and let ε ≥ 0. Assume that x ∈ Cl1 Ω. The set of Fr´echet ε-normals vectors to Ω at x with respect to the pair of norms 1.2 ( . 1 ,  . 2 ) denoted Nˆε (x, Ω) is given by : ∗

1.2

x , u − x Nˆε (x, Ω) = {x∗ ∈ (X,  . 2 )∗ /lim sup ≤ ε}.  u − x 2 Ω,.1

(3)

u −→ x

When ε=0 (3) is a cone which is called prenormal cone or Fr´echet normal cone to Ω at x with respect to the pair of norms ( . 1 ,  . 2 ) and is denoted 1,2 ˆ 1,2 (x, Ω). If x ∈ / Cl1 Ω, we set Nˆε (x, Ω)=∅ for all ε ≥0. by N ii) Let x¯ ∈ Cl1 Ω, the nonempty cone : 1,2 N 1,2 (¯ x, Ω) = lim sup Nˆε (x, Ω).

(4)

.1

x→x ¯,ε0

is called the normal cone to Ω at x¯ with respect to the pair of norms ( . 1 ,  . 2 ). We set N 1,2 (¯ x, Ω) = ∅ for x¯ ∈ / Cl1 Ω. x, Ω) the normal cone to Ω at x¯ with respect to the norm  . 2 . Denote by N 2 (¯ x, Ω) ⊂ N 2 (¯ x, Ω) and N 2 (¯ x, Ω) ⊂ N 1,2 (¯ x, Ω). In general, if x ∈ Cl2 Ω, then N 1,2 (¯ 1,2 1 2 But, if  . 1 is equivalent to  . 2 , then N (¯ x, Ω) = N (¯ x, Ω) = N (¯ x, Ω).

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S. Lahrech, J. Hlal, A. Ouahab, A. Jaddar and A. Mbarki

So, our normal cone N 1,2 (¯ x, Ω) defined in binormed space (X,  . 1 ,  . 2 ) generalizes in some sens the notion of classical normal cone defined in normed space. Let ε ≥ 0. We define the generalized Fr´echet ε-subdifferential of ϕ at x ∈ domϕ with respect to the pair of norms ( . 1 , . 2 ) by: ∗

1,2 ϕ(u) − ϕ(x) − x , u − x ≥ −ε}. (5) ∂ˆε ϕ(x) = {x∗ ∈ (X,  . 2 )∗ /lim inf .1  u − x  2 u −→x

If ε=0, then (5) is called the generalized presubdifferential or generalized Fr´echet subdifferential of ϕ at x with respect to the pair of norms ( . 1 , . 2 ) and we denoted it by ∂ˆ1,2 ϕ(x). ¯ be an extended-real-valued function and Definition 2.3 Let ϕ : X −→ R x¯ ∈ domϕ. Assume that ϕ is l.s.c. around x¯ with respect to the norm  . 1 . The set 1,2 x) = lim sup ∂ˆε ϕ(x) (6) ∂ 1,2 ϕ(¯ ϕ,.1

x −→ x ¯,ε0

is called the generalized limiting subdifferential of ϕ at x¯ with respect to the x)=∅ if x¯ ∈ / domϕ. pair of norms ( . 1 ,  . 2 ). We set ∂ 1,2 ϕ(¯ x) ⊂ ∂ 2 ϕ(¯ x) and ∂ 2 ϕ(¯ x) ⊂ ∂ 1,2 ϕ(¯ x). Let us remark that in general, ∂ 1,2 ϕ(¯ 1 2 x) = ∂ ϕ(¯ x) = ∂ 1,2 ϕ(¯ x). So, But if  . 1 is equivalent to  . 2 , then ∂ ϕ(¯ the generalized limiting subdifferential is a generalized version of the limiting subdifferential.

3

Sum rule

In this section we extend the result of sum rule for the limiting subdifferential given by Mordukhovich and Shao in [11] to generalized limiting subdifferential. Theorem 3.1 Let ϕ : X → R be a ( . 1 ,  . 2 )-Fr´echet strictly differentiable mapping at x¯ ∈ X and let ψ : X → R be a l.s.c. function around x¯ with respect to the norm  . 1 . Assume that ϕ is l.s.c. around x¯ with respect to the norm  . 1 . Then ∂ 1,2 (ϕ + ψ)(¯ x) = ∇1,2 ϕ(¯ x) + ∂ 1,2 ψ(¯ x). Proof. First we verify that: x) ⊂ ∇1,2 ϕ(¯ x) + ∂ 1,2 ψ(¯ x). ∂ 1,2 (ϕ + ψ)(¯

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Subdifferential calculus in binormed spaces

Since ϕ is ( . 1 ,  . 2 )- Fr´echet strictly differentiable at x¯, then there are sequences γv  0, δv  0, such that 



x), z − x |≤ γv  x − z 2 ∀x, z ∈ Bδ1v (¯ x), v = 1, 2 . . . | ϕ(z) − ϕ(x) − ∇1,2 ϕ(¯ Consequently, ϕ(z) − ϕ(x) − ∇1,2 ϕ(¯ x), z − x ≥ −γv  x − z 2 ∀x, z ∈ Bδ1v (¯ x), v = 1, 2 . . . (7) .1

ω∗

Let x∗ ∈ ∂ 1,2 (ϕ + ψ)(¯ x). Then, there are sequences xk → x¯, x∗k → x∗ , εk  0 as k −→ ∞ such that: (ϕ + ψ)(xk ) → (ϕ + ψ)(¯ x), and ∀k = 1, 2 . . . x∗k ∈ ∂ˆε1,2 (ϕ + ψ)(xk ). k

(8)

.1

Since xk → x¯, then we can choose a subsequence xkv (k1 < k2 < ... < kv < ... positive integers) converging to x¯ with respect to the norm  . 1 and satisfying  xkv − x¯ 1 ≤ δ2v ∀v = 1, 2 . . .. Fix ε > 0 and pick (ηv ) such that 0 ≤ ηv ≤ δ2v (∀v) and ψ(x) − ψ(xkv ) + ϕ(x) − ϕ(xkv ) − x∗kv , x − xkv  ≥ −εkv − ε.  x − xkv 2 (9) 1 1 Observe that if x ∈ Bηv (xkv ), then x ∈ Bδv (¯ x). Therefore, by virtue of (7) and (9) we deduce that ∀v positive integer ∀x ∈ Bη1v (xkv )

ψ(x)−ψ(xkv )− x∗kv −∇1,2 ϕ(¯ x), x−xkv  ≥ −(εkv +γv +ε)  x−xkv 2 ∀x ∈ Bη1v (xkv ); i.e.

∀v positive integer : x∗kv − ∇1,2 ϕ(¯ x) ∈ ∂ˆε1,2 ¯v ψ(xkv ),

where ε¯v = γv + εkv , (∀v).

.1

x), xk → x¯, ϕ and ψ are On the other hand, we have (ϕ + ψ)(xk ) → (ϕ + ψ)(¯ x). Consequently, l.s.c. with respect to the norm  . 1 . Hence, ψ(xkv ) → ψ(¯ x∗ − ∇1,2 ϕ(¯ x) ∈ ∂ 1,2 ψ(¯ x). x) ⊂ ∇1,2 ϕ(¯ x) + ∂ 1,2 ψ(¯ x). Thus, ∂ 1,2 (ϕ + ψ)(¯ Applying the last result to functions −ϕ and ϕ + ψ, we deduce the opposite x) + ∂ 1,2 ψ(¯ x) ⊂ ∂ 1,2 (ϕ + ψ)(¯ x). Thus, we achieve the proof. inclusion ∇1,2 ϕ(¯ Let us remark that if  . 1 is equivalent to  . 2 , then we recapture the result of sum rule for the limiting subdifferential given by Mordukhovich and Shao in [11]. The following result is an immediate consequence of theorem 3.1. Corollary 3.2 Let ϕ : X → R be a ( . 1 ,  . 2 )-Fr´echet strictly differentiable mapping at x¯ ∈ X and let ψ : X → R be a l.s.c. function around x¯ with x) = ∂ 2 (ϕ+ψ)(¯ x). Assume that respect to the norm  . 1 satisfying ∂ 1,2 (ϕ+ψ)(¯ 1,2 ϕ is l.s.c. around x¯ with respect to the norm  . 1 . Then ∂ ψ(¯ x) = ∂ 2 ψ(¯ x).

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4

S. Lahrech, J. Hlal, A. Ouahab, A. Jaddar and A. Mbarki

Chain rule and other calculus rules

Analogously to section 3, we give a generalization of other calculus rules established by Mordukhovich and Shao in [11]. Let us start with the theorem of chain rule. Theorem 4.1 Let Φ : (X,  . 1 ) −→ (Y,  . 3 ) be a continuous singlevalued mapping such that Φ : X −→ (Y,  . 4 ) is ( . 1 ,  . 2 )− locally Lipschitzian around x¯ ∈ X. Let ϕ : X × Y −→ R be a ( . 5 ,  . 6 )-Fr´echet strictly differentiable mapping at (¯ x, Φ(¯ x)). Assume that ϕ is l.s.c. with respect to the norm  . 5 . Then x) = ∇1,2 x, y¯) + ∂ 1,2 ∇3,4 x, y¯), Φ(¯ x) with y¯ = Φ(¯ x), ∂ 1,2 (ϕ ◦ Φ)(¯ x ϕ(¯ y ϕ(¯ where ϕ ◦ Φ is the function acting from X into R defined by: (ϕ ◦ Φ)(x) = ϕ(x, Φ(x)). Proof. Under our hypothesis, we can easily see that ϕ ◦ Φ is l.s.c. around x¯ with respect to the norm  . 1 and ∇3,4 x, y¯), Φ is continuous around x¯ y ϕ(¯ with respect to the norm  . 1 . Therefore, we can talk about ∂ 1,2 (ϕ ◦ Φ)(¯ x) 1,2 3,4 x, y¯), Φ(¯ x). and ∂ ∇y ϕ(¯ Denote by l the Lipschitz constant of Ψ at x¯. Then there exists δ > 0 such that ∀x, u ∈ Bδ1 (¯ x)  Φ(u) − Φ(x) 4 ≤ l  u − x 2 . Since ϕ is ( . 5 ,  . 6 )-Fr´echet strictly differentiable at (¯ x, y¯) with y¯ = Φ(¯ x), then there are sequences γv  0, ρv  0 such that: ρv < δ ∀v and x, y¯), u − x − ∇3,4 x, y¯), Φ(u) − Φ(x) ϕ(u, Φ(u)) − ϕ(x, Φ(x)) − ∇1,2 x ϕ(¯ y ϕ(¯ x), v = 1, 2 . . . ≥ −γv ( u − x 2 +  Φ(u) − Φ(x) 4 ) ∀x, u ∈ Bρ1v (¯ .1

ω∗

Let x∗ ∈ ∂ 1,2 (ϕoΦ)(¯ x). Then, there are sequences xk → x¯, x∗k → x∗ , and εk  x) and x∗k ∈ ∂ˆε1,2 (ϕoΦ)(xk ) ∀k = 1, 2 . . .. 0 such that: (ϕoΦ)(xk ) −→ (ϕoΦ)(¯ k .1

Due to assumption xk → x¯ as k −→ ∞, we can choose a subsequence xkv (k1 < k2 < ..... < kv < .. positive integers) such that  xkv − x¯ 1 ≤ ρ2v for all v = 1, 2 . . .. Fix ε > 0 and choose a sequence (ηv ) such that 0 ≤ ηv ≤ ρ2v and ϕ(x, Φ(x)) − ϕ(xkv , Φ(xkv )) − x∗kv , x − xkv  ≥ −(εkv + ε)  x − xkv 2 (10) for all x ∈ Bη1v (xkv ), v = 1, 2 . . . x) if x ∈ Bη1v (xkv ), we deduce that for all v Noting that x ∈ Bρ1v (¯ ϕ(xkv , Φ(xkv ))−ϕ(x, Φ(x))− ∇1,2 x, y¯), xkv −x− ∇3,4 x, y¯), Φ(xkv )−Φ(x) x ϕ(¯ y ϕ(¯

Subdifferential calculus in binormed spaces

≥ −γv ( xkv − x 2 +l  xkv − x 2 ) ∀x ∈ Bη1v (xkv ).

2729 (11)

Therefore, by virtue of (10) and (11), we obtain: for all v

∇3,4 x, y¯), Φ(x) − ∇3,4 x, y¯), Φ(xkv ) − x∗kv − ∇1,2 x, y¯), x − xkv  y ϕ(¯ y ϕ(¯ x ϕ(¯ ≥ −[εkv + ε + γv (l + 1)]  x − xkv 2 ∀x ∈ Bη1v (xkv ). Consequently, for all v 3,4 x∗kv − ∇1,2 x, y¯) ∈ ∂ˆε1,2 x, y¯), Φ(xkv ) with ε¯v = [εkv + γv (l + 1)]. x ϕ(¯ ¯v ∇y ϕ(¯

Hence,

x∗ − ∇1,2 x, y¯) ∈ ∂ 1,2 ∇3,4 x, y¯), Φ(¯ x). x ϕ(¯ y ϕ(¯

This proves the inclusion ∂ 1,2 (ϕ ◦ Φ)(¯ x) ⊂ ∇1,2 x, y¯) + ∂ 1,2 ∇3,4 x, y¯), Φ(¯ x). x ϕ(¯ y ϕ(¯ To establish the opposite inclusion we employ the similar arguments starting with a point x∗ ∈ ∂ 1,2 ∇3,4 x, y¯), Φ(¯ x). Thus, we achieve the proof. y ϕ(¯ Remark 4.2 Let us remark that if  . 1 is equivalent to  . 2 and  . 3 is equivalent to  . 4 , then we recapture the result of chain rule for the limiting subdifferential given by Mordukhovich and Shao in [11]. At the end of this section we give some additional calculus formulas for the generalized limiting subdifferential. The first one is the product rules involving ( . 1 ,  . 2 )-locally Lipschitzian functions. Theorem 4.3 Let ϕi : X −→ R, i = 1, 2 be two ( . 1 ,  . 2 )-locally Lipschitzian functions around x¯ ∈ X. Assume that all ϕi are continuous with respect to the norm  . 1 . Then we have x) = ∂ 1,2 (ϕ2 (¯ x)ϕ1 + ϕ1 (¯ x)ϕ2 )(¯ x). ∂ 1,2 (ϕ1 .ϕ2 )(¯

(12)

if in addition, one of the functions (say ϕ1 ) is ( . 1 ,  . 2 )- Fr´echet strictly differentiable at x¯, then we have: ∂ 1,2 (ϕ1 .ϕ2 )(¯ x) = ϕ2 (¯ x)∇1,2 ϕ1 (¯ x) + ∂ 1,2 (ϕ1 (¯ x)ϕ2 )(¯ x).

(13)

Proof. Consider the smooth function ϕ : X ×R2 −→ R and the ( . 1 ,  . 2 )locally Lipschitzian mapping (around x¯) Φ : X −→ R2 defined by: Φ(x) = (ϕ1 (x), ϕ2 (x)) , ϕ(x, y1 , y2 ) = y1 .y2 . Applying the above chain rule theorem to the composition (ϕoΦ)(x) ≡ (ϕ1 .ϕ2 )(x), we deduce the result. If now ϕ1 is ( . 1 ,  . 2 )-Fr´echet strictly differentiable at x¯, then by virtue of Theorem 3.1, we obtain (13). Thus, we achieve the proof. Similarly to Theorem 4.2 we obtain the following quotient rules.

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S. Lahrech, J. Hlal, A. Ouahab, A. Jaddar and A. Mbarki

Theorem 4.4 Let ϕi : X −→ R, i = 1, 2 be two ( . 1 ,  . 2 )-locally Lipschitzian functions around x¯ ∈ X. Assume that all ϕi are continuous with respect to the norm  . 1 . Assume that ϕ2 (¯ x) = 0. Then x) = ∂ 1,2 (ϕ1 /ϕ2 )(¯

∂ 1,2 (ϕ2 (¯ x)ϕ1 − ϕ1 (¯ x)ϕ2 ) . [ϕ2 (¯ x)]2

if in addition, one of the functions (say ϕ1 ) is ( . 1 ,  . 2 )-Fr´echet strictly differentiable at x¯, then: x) = ∂ 1,2 (ϕ1 /ϕ2 )(¯

∇1,2 ϕ1 (¯ x)ϕ2 (¯ x) + ∂ 1,2 (−ϕ1 (¯ x)ϕ2 )(¯ x) . 2 [ϕ2 (¯ x)]

Proof. Consider the smooth function ϕ : X ×R2 −→ R and the ( . 1 ,  . 2 )locally Lipschitzian mapping (around x¯) Φ : X −→ R2 defined by: y1 Φ(x) = (ϕ1 (x), ϕ2 (x)) , ϕ(x, y1 , y2 ) = . y2 Applying the chain rule theorem (theorem 4.1) to the composition (ϕoΦ)(x) ≡   ϕ1 ϕ2

(x) and applying the sum rule theorem in the case where one of the func-

tions (say ϕ1 ) is ( . 1 ,  . 2 )-Fr´echet strictly differentiable at x¯, we deduce the result. Finally we give an example of a non Fr´echet strictly differentiable mapping with respect to the norm  . 1 for which we can apply our sum rule theorem (theorem 3.1), while the classical sum rule theorem of [11] given by Mordukhovich and Shao cannot be used. Example. Let Ω be a bounded domain in R2 and let X = W01,2 (Ω) be the Sobolev space under its usual norm  . 2 =  . W 1,2 (Ω) . Let also p and ε such 0 that 0 < ε < 1, ε + 2 < p < ∞. Set  . 1 = . Lp (Ω) . Remark that (X,  . 2 ) is a Banach separable space. Since W01,2 (Ω) → Lp (Ω), then (X,  . 1 ,  . 2 ) is a binormed space such that  . 2 is finer than  . 1 . Set g(u)=| u |ε+2 and consider the functional G defined on X by: 

G(x) =

Ω

g(x(s))ds

Then G is  . 2 -twice Fr´echet differentiable at every x ∈ X. Moreover, G1 (x)h = and G2 (x)(h1 , h2 ) =



Ω

g (x(s))h(s)ds,

Ω

g  (x(s))h1 (s)h2 (s)ds.



It is proved in [7] that G is ( . 1 ,  . 2 )-Fr´echet strictly differentiable at every x ∈X and is not  . 1 -Fr´echet strictly differentiable at any point x of X. Therefore, if for any reason we want to apply the sum rule theorem for the limiting subdifferential involving the norm  . 1 , then our sum rule result given in theorem 3.1 is more adapted for this example than the one given by Mordukhovich and Shao in [11].

Subdifferential calculus in binormed spaces

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References [1] A. D. Ioffe: Sous-diff´erentielles approch´ees de fonctions num´eriques, Comptes Rendus de l’Acad´emie des Sciences de Paris, 292, 675-678, (1981). [2] A. D. Ioffe: Approximate subdifferentials and applications. I: The finite dimensional theory, Trans. Amer. Math. Soc. 281, 389-416, (1984). [3] A. Jourani, L. Thibault: A note on Fr´echet and approximate subdifferentials of composite functions, Bull. Austral. Math. Soc. 49, 111-116, (1994). [4] A. Y. Kruger: Properties of generalized differentials, Siberian Math. J. 26, 822-832, (1985). [5] A. Y. Kruger, B. S. Mordukhovich: Extremal points and Euler equations in nonsmooth optimization, Dokl. Akad. Nauk BSSR 24, 684-687, (1980). [6] A. Y. Kruger, B. S. Mordukhovich: Generalized normals and derivatives, and necessary optimality conditions in nondifferentiable programming, Depon. VINITI 408-80, 494-80, (1980). [7] S. Lahrech, A. Benbrik:On the Mordukhovich Subdifferential In Binormed Spaces and Some Applications. International Journal of Pure and Applied Mathematics.Volume 20, No 1, 31-39, (2005). [8] S. Lahrech:On the Topological Properties of the Generalized Clarke Subdifferential. Lobachevskii J. Math.(Submitted). [9] P. D. Loewen: Limits of Fr´echet normals in nonsmooth analysis, Optimization and Nonlinear Analysis, Pitman Research Notes Math. Ser 244, 178-188, (1992). [10] B. S. Mordukhovich: Maximum principle in the problem of time optimal response with nonsmooth contraints, Journal of Applied Mathematics and M´echanics, 40, 960-969, (1976). [11] B. S. Mordukhovich, Y. Shao: On Nonconvex Subdifferential Calculus in Banach Spaces. Journal of Convex Analysis. Volume 2, 211-227, (1995). [12] B. S. Mordukhovich, Y. Shao:Nonsmooth Sequential Analysis in Asplund Spaces. Transactions of American Mathematical Society 384, 1235-1280, (1996). [13] L. Thibault:On subdifferentials of optimal value functions, SIAM J. Control Optim. 29, 1019-1036, (1991). Received: April 7, 2007