Fuzzy Calculus for Strong Limiting Subdifferential in Banach Spaces

Int. Journal of Math. Analysis, Vol. 1, 2007, no. 9, 443 - 450

Fuzzy Calculus for Strong Limiting Subdifferential in Banach Spaces S. Lahrech Dept of Mathematics, Faculty of science, Mohamed first University Oujda, Morocco, (GAFO Laboratory) A. Benbrik 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. Ouahab Dept of Mathematics, Faculty of science, 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 fuzzy calculus results related to strong limiting subdifferential in Banach spaces.

Mathematics Subject Classification: 49J52, 49J50 Keywords: Fuzzy calculus, strong limiting subdifferential in Banach spaces, strong sequential Kuratowski-Painlev´e upper limit.

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S. Lahrech, A. Benbrik, A. Jaddar, A. Ouahab, A. Mbarki

Introduction

The strong limiting subdifferential was first introduced by S.Lahrech, J. Hlal, A. Jaddar, A.Ouahab and A. Mbarki in [1] in order to characterize the extrema of pseudoconvex functions via this a new kind of limiting subdifferential. We establish in this paper some interesting fuzzy calculus rules related to strong limiting subdifferential for extended-real-valued functions defined on Banach spaces. 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 and in section 4 we prove some other calculus results: chain rule, products and quotients rules.

2

Basic definitions and properties

In this section, we recall several definitions and results necessary in the sequel. Throughout this paper, X and Y denote two Banach spaces and X ∗ the topological dual of X equipped with the weak-star topology. By Bρ (x) we denote the open ball centered at x with radius ρ (ρ > 0). ¯ the domain of f is defined For any extended-real-valued function f : X −→ R, by: domf = {x ∈ X/ | f (x) |< +∞}. ω∗

f

The symbol x → x¯ (respectively, → ) means that x → x¯ with f (x) → f (¯ x) ∗ (respectively, the convergence for the weak-star topology of X ). Definition 2.1 [8] Let f : X −→ R ∪ {+∞} be an extended-real-valued function, and let x¯ ∈ domf . The set ∗

f (u) − f (x) − x , u − x ≥ −ε}, ε ≥ 0 ∂ˆε f (x) = {x∗ ∈ X ∗ /lim inf u→x  u−x

(1)

is called the ε-Fr´echet subdifferential of f at x. When ε = 0, then the set (1) is called the presubdifferential or Fr´echet subdifˆ (x) ferential of f at x and is denoted by ∂f Definition 2.2 Let f : X −→ R ∪ {+∞} be an extended-real-valued function, and let x¯ ∈ domf . Assume that f is l.s.c around x¯. ˘ (¯ We define the strong limiting subdifferential ∂f x) by: ˘ (¯ ∂f x) =

lim sup ∂ˆε f (x), f,ρ

ρ0,ε0,x → x ¯

(2)

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where

lim sup ∂ˆε f (x) denotes the strong sequential Kuratowski-Painlev´e upf,ρ

ρ0,ε0,x → x ¯

per limit, that is: ∗

f ω lim sup ∂ˆε f (x) = {x∗ ∈ X ∗ /∃εk ↓ 0, ∃ρk ↓ 0, ∃ xk → x¯, ∃x∗k → x∗ such that f,ρ

ρ0,ε0,x → x ¯

(ρk )k is a nonincreasing sequence satisfying: ρk > 0 and ∀ (k1 , k2 ) large enough, there is a positive integer k3 ≥ k2 such that ∀x ∈ Bρk1 (xk3 )

f (x ) − f (xk3 ) − x∗k3 , x − xk3  ≥ −2εk3  x − xk3  and  x¯ − xk3 
0) ∃δ > 0 ∃n0 ∈ N ∀n ≥ n0 ∀z ∈ Bδ (x) f (xn ) − f (z)− < x∗ , xn − z >≥ −δn  z − xn } is called the strict Fr´echet sequential subdifferential of f at x. If ∂ˆs f (x) = ∅, then we say that f is strictly sequentially Fr´echet subdifferentiable at x. Theorem 3.2 Let ϕ : X → R, ψ : X → R be two l.s.c real functions around x¯ ∈ X. Assume that ϕ is sequentially strictly Fr´echet subdifferentiable at x¯. Then ˘ + ψ)(¯ ˘ x). ∂(ϕ x) ⊂ ∂ˆs ϕ(¯ x) + ∂ψ(¯ ϕ+ψ

˘ + ψ)(¯ Proof. Let x∗ ∈ ∂(ϕ x). Then, there are sequences ρk ↓ 0, εk ↓ 0, xk → ∗ ω x¯, x∗k → x∗ such that(ρk )k is a nonincreasing sequence satisfying: ρk > 0(∀k), ∀δ > 0 ∀(k1 , k2 ) large enough, there is a positive integer k3 ≥ max(k1 , k2 ) such that ∀x ∈ Bρk1 (xk3 ) (ϕ + ψ)(x) − (ϕ + ψ)(xk3 ) − x∗k3 , x − xk3  ≥ −2εk3  x − xk3 , (3)

446

S. Lahrech, A. Benbrik, A. Jaddar, A. Ouahab, A. Mbarki ρ

and  x¯ − xk3 < k23 . Since ϕ is sequentially strictly Fr´echet subdifferentiable x). Consequently, at x¯, then there is x∗1 ∈ ∂ˆs ϕ(¯ x) ∃δn → 0 (δn > 0) ∃δ > 0 ∃n0 ∈ N ∀n ≥ n0 ∀z ∈ Bδ (¯ ϕ(xn ) − ϕ(z)− < x∗1 , xn − z >≥ −δn  z − xn  . Let k1 , k2 be two large positive integers. Then, there is a positive integer ρ k3 ≥ max(k1 , k2 , n0 ) such that ρk1 + k23 ≤ 2δ and ∀x ∈ Bρk1 (xk3 )

(ϕ + ψ)(x) − (ϕ + ψ)(xk3 ) − x∗k3 , x − xk3  ≥ −2εk3  x − xk3 

and  x¯ −xk3 < we deduce that

ρk3 . 2

Observe that if x ∈ Bρk1 (xk3 ), then x ∈ Bδ (¯ x). Therefore,

∀x ∈ Bρk1 (xk3 ) ϕ(xk3 ) − ϕ(x) − x∗1 , xk3 − x ≥ −δk3  x − xk3  .

(4)

Consequently, by virtue of (3) and (4), we obtain ∀x ∈ Bρk1 (xk3 ) ψ(x) − ψ(xk3 ) − x∗k3 − x∗1 , x − xk3  ≥ −(2εk3 + δk3 )  x − xk3  . x), xk → x¯, ϕ and On the other hand, we have (ϕ + ψ)(xk ) → (ϕ + ψ)(¯ ˘ x). Thus, x). Consequently, x∗ − x∗1 ∈ ∂ψ(¯ ψ are l.s.c. Hence, ψ(xkv ) → ψ(¯ ˘ x). ˘ + ψ)(¯ x) + ∂ψ(¯ ∂(ϕ x) ⊂ ∂ˆs ϕ(¯

4

Chain rule and other fuzzy calculus rules

Analogously to section 3, we give other fuzzy calculus rules for our strong limiting subdifferential. Definition 4.1 Let Φ : X −→ Y be a continuous single-valued mapping and let ϕ : X × Y −→ R be a real-valued function. Set y¯ = Φ(¯ x). Assume that ϕ(¯ x, .) is Fr´echet differentiable at y¯. We say that ϕ is Φ-sequentially strictly Fr´echet differentiable at x¯ if ϕ◦Φ

∀ρk ↓ 0, ∀εk ↓ 0, ∀ xk → x¯, ∀k1 , k3 such that k3 ≥ k1 the following implication holds: x) such that ∀x ∈ Bρk1 (xk3 ) ϕ(xk3 , y¯) − ϕ(x, y¯) − x∗1 , xk3 − x [∃x∗1 ∈ ∂ˆs ϕ(., y¯)(¯ ≥ −εk3  x − xk3 ] ⇒ x, y¯), Φ(xk3 )−Φ(x) ∃γk ↓ 0 such that ϕ(xk3 , Φ(xk3 ))−ϕ(x, Φ(x))−x∗1 , xk3 −x−∇y ϕ(¯ ≥ −γk3 ( xk3 − x  +  Φ(xk3 ) − Φ(x) ) ∀x ∈ Bρk1 (xk3 ).

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Fuzzy calculus

Let us start with the theorem of chain rule. Theorem 4.2 Let Φ : X −→ Y be a continuous single-valued mapping such that Φ : X −→ Y is locally Lipschitzian around x¯ ∈ X. Set y¯ = Φ(¯ x). Let ϕ : X × Y −→ R be a Φ-sequentially strictly Fr´echet differentiable mapping at x¯. Assume that ϕ is l.s.c around (¯ x, Φ(¯ x)). Suppose also that ϕ(., Φ(¯ x)) is sequentially strictly Fr´echet subdifferentiable at x¯. Then ˘ ◦ Φ)(¯ ˘ y ϕ(¯ ∂(ϕ x) ⊂ ∂ˆs ϕ(., y¯)(¯ x) + ∂∇ x, y¯), Φ(¯ x), 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, y¯), Φ is continuous around x¯. Therefore, we can talk about x¯ and ∇y ϕ(¯ ˘ ˘ y ϕ(¯ ∂(ϕ ◦ Φ)(¯ x) and ∂∇ x, y¯), Φ(¯ x). Denote by l the Lipschitz constant of Ψ at x¯. Then there exists δ > 0 such that ∀x, u ∈ Bδ (¯ x)  Φ(u) − Φ(x) ≤ l  u − x  . ∗

ϕ◦Φ ω ˘ ◦ Φ)(¯ Let x∗ ∈ ∂(ϕ x). Then, there are sequences ρk ↓ 0, εk ↓ 0, xk → x¯, x∗k → x∗ such that(ρk )k is a nonincreasing sequence satisfying: ρk > 0(∀k), ∀δ > 0 ∀(k1 , k2 ) large enough, there is a positive integer k3 ≥ max(k1 , k2 ) such that

∀x ∈ Bρk1 (xk3 ) (ϕ◦Φ)(x)−(ϕ◦Φ)(xk3 )−x∗k3 , x−xk3  ≥ −2εk3  x−xk3 , (5) ρ

and  x¯ − xk3 < k23 . Since ϕ(., y¯) is sequentially strictly Fr´echet subdifferenx). Consequently, tiable at x¯, then there is x∗1 ∈ ∂ˆs ϕ(., y¯)(¯ x) ∃δn → 0 (δn > 0) ∃δ > 0 ∃n0 ∈ N ∀n ≥ n0 ∀z ∈ Bδ (¯ ϕ(xn , y¯) − ϕ(z, y¯)− < x∗1 , xn − z >≥ −δn  z − xn  . Let k1 , k2 be two large positive integers. Then, there is a positive integer ρ k3 ≥ max(k1 , k2 , n0 ) such that ρk1 + k23 ≤ 2δ , ∀x ∈ Bρk1 (xk3 ) (ϕ◦Φ)(x)−(ϕ◦Φ)(xk3 )−x∗k3 , x−xk3  ≥ −2εk3  x−xk3  (6) and  x¯ −xk3 < we deduce that

ρk3 . 2

Observe that if x ∈ Bρk1 (xk3 ), then x ∈ Bδ (¯ x). Therefore,

∀x ∈ Bρk1 (xk3 ) ϕ(xk3 , y¯) − ϕ(x, y¯) − x∗1 , xk3 − x ≥ −δk3  x − xk3  . Since ϕ is Φ-sequentially strictly Fr´echet differentiable at x¯, then x, y¯), Φ(xk3 )−Φ(x) ∃γk ↓ 0 such that ϕ(xk3 , Φ(xk3 ))−ϕ(x, Φ(x))−x∗1 , xk3 −x−∇y ϕ(¯

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S. Lahrech, A. Benbrik, A. Jaddar, A. Ouahab, A. Mbarki

≥ −γk3 ( xk3 − x  +  Φ(xk3 ) − Φ(x) ) ∀x ∈ Bρk1 (xk3 ). Hence, ∀x ∈ Bρk1 (xk3 ) ϕ(xk3 , Φ(xk3 )) − ϕ(x, Φ(x)) − x∗1 , xk3 − x − ∇y ϕ(¯ x, y¯), Φ(xk3 ) − Φ(x) ≥ −γk3 ( xk3 − x  +l  xk3 − x ).

(7)

By virtue of (6) and (7), we obtain: ∀x ∈ Bρk1 (xk3 ) ∇y ϕ(¯ x, y¯), Φ(x) − ∇y ϕ(¯ x, y¯), Φ(xk3 ) − x∗k3 − x∗1 , x − xk3  ≥ −[2εk3 + γk3 (l + 1)]  x − xk3  . Consequently, ˘ y ϕ(¯ x, y¯), Φ(¯ x). x∗ − x∗1 ∈ ∂∇ Thus, we achieve the proof. At the end of this section we give some additional calculus formulas for the strong limiting subdifferential. The first one is the product rules involving locally Lipschitzian functions. Theorem 4.3 Let ϕi : X −→ R, i = 1, 2 be two locally Lipschitzian functions around x¯ ∈ X. Assume that all ϕi are continuous. Assume that ∀ρk ↓ 0, ∀εk ↓ 0, ∀ xk → x¯, ∀k1 , k3 such that k3 ≥ k1 ∃γk ↓ 0 such that ∀x ∈ Bρk1 (xk3 ) ϕ1 (xk3 )ϕ2 (xk3 ) − ϕ1 (x)ϕ2 (x) − ϕ2 (¯ x)(ϕ1 (xk3 ) − ϕ1 (x)) − ϕ1 (¯ x)(ϕ2 (xk3 ) − ϕ2 (x)) ≥ −γk3 ( xk3 − x  + | ϕ1 (xk3 ) − ϕ1 (x) | + | ϕ2 (xk3 ) − ϕ2 (x) |). Then ˘ 2 (¯ ˘ 1 .ϕ2 )(¯ x) ⊂ ∂(ϕ x)ϕ1 + ϕ1 (¯ x)ϕ2 )(¯ x). ∂(ϕ

(8)

Proof. Consider the smooth function ϕ : X × R2 −→ R and the locally Lipschitzian mapping (around x¯) Φ : X −→ R2 defined by: Φ(x) = (ϕ1 (x), ϕ2 (x)) , ϕ(x, y1 , y2 ) = y1 .y2 . Using the same argument as in the proof of theorem 6 and taking into account x), we deduce the result. that 0 ∈ ∂ˆs ϕ(., y¯)(¯

449

Fuzzy calculus

Theorem 4.4 Let ϕi : X −→ R, i = 1, 2 be two locally Lipschitzian funcx) = 0. tions around x¯ ∈ X. Assume that all ϕi are continuous and ϕ2 (¯ Suppose also that ∀ρk ↓ 0, ∀εk ↓ 0, ∀ xk → x¯, ∀k1 , k3 such that k3 ≥ k1 ∃γk ↓ 0 such that ∀x ∈ Bρk1 (xk3 ) ϕ1 (xk3 ) ϕ1 (x) 1 1 1 − − (ϕ1 (xk3 ) − ϕ1 (x)) − ϕ1 (¯ − ) x)( ϕ2 (xk3 ) ϕ2 (x) ϕ2 (¯ x) ϕ2 (xk3 ) ϕ2 (x) ≥ −γk3 ( xk3 − x  + | ϕ1 (xk3 ) − ϕ1 (x) | + | Then

1 1 − |). ϕ2 (xk3 ) ϕ2 (x)

˘ 2 (¯ x)ϕ1 − ϕ1 (¯ x)ϕ2 )(¯ x) ∂(ϕ ˘ ϕ1 )(¯ x) ⊂ . ∂( 2 ϕ2 [ϕ2 (¯ x)]

(9)

Proof. Consider the smooth function ϕ : X × R2 −→ R and the locally Lipschitzian mapping (around x¯) Φ : R2 −→ X defined by: Φ(x) = (ϕ1 (x), ϕ2 (x)) , ϕ(x, y1 , y2 ) =

y1 . y2

Using the same argument as in the proof of theorem 6 and taking into account that 0 ∈ ∂ˆs ϕ(., y¯)(¯ x), we deduce the result.

References [1] S.Lahrech, A.Jaddar, J.Hlal, A.Ouahab, A.Mbarki Characterization of the extrema of a pseudoconvex function in terms of limiting and strong limiting subdifferential. submitted for publication in Optimization letters. [2] A. D. Ioffe: Approximate subdifferentials and applications. I: The finite dimensional theory, Trans. Amer. Math. Soc. 281, (1984), 389-416. [3] A. Jourani, L. Thibault: A note on Fr´echet and approximate subdifferentials of composite functions, Bull. Austral. Math. Soc. 49, (1994), 111-116. [4] A. Y. Kruger: Properties of generalized differentials, Siberian Math. J. 26, (1985), 822-832. [5] A. Y. Kruger, B. S. Mordukhovich: Extremal points and Euler equations in nonsmooth optimization, Dokl. Akad. Nauk BSSR 24,(1980), 684-687.

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[6] P. D. Loewen: Limits of Fr´echet normals in nonsmooth analysis, Optimization and Nonlinear Analysis, Pitman Research Notes Math. Ser 244, (1992), 178-188. [7] B. S. Mordukhovich: Maximum principle in the problem of time optimal response with nonsmooth contraints, Journal of Applied Mathematics and M´echanics, 40, (1976), 960-969. [8] B. S. Mordukhovich, Y. Shao: On Nonconvex Subdifferential Calculus in Banach Spaces. Journal of Convex Analysis. Volume 2, (1995), 211-227. [9] B. S. Mordukhovich, Y. Shao:Nonsmooth Sequential Analysis in Asplund Spaces. Transactions of American Mathematical Society 384, (1996), 1235-1280. Received: November 30, 2006