A Note on the Weakly Quasi-Convex Functions 1 ... - Semantic Scholar

International Mathematical Forum, 2, 2007, no. 35, 1755 - 1761

A Note on the Weakly Quasi-Convex Functions S. Lahrech 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) J. Hlal Dept. of Mathematics, Faculty of Science, 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 The main purpose of this paper is to characterize weakly quasiconvex functions via the limiting subdifferential.

Mathematics Subject Classification: 49J52, 49J50 Keywords: Mean value Theorem, Weakly quasi-convex functions, Limiting subdifferential

1

Introduction

The generalized convexity have found extensive applications in areas such as mathematical programming and economics. Such notion is a natural gen-

1756

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

eralization of convexity. In particular many results have been done for the weak convexity, we can cite for instance the result given in [4] by Jourani and Th´era which used this notion (with modulus ε) in order to characterize the ε-monotonicity of the limiting Fr´echet ε-subdifferential by adopting the following definition : A real function f defined on a normed space (X, .) is said to be weakly convex (with modulus ε > 0) or ε-convex if for all x, y ∈ X and λ ∈ [0, 1] we have : f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) + ελ(1 − λ) x − y . Our aim in this paper is to extend this definition to a big class called the weak quasi-convexity (with modulus ε > 0) in order to characterize it via the limiting subdifferential. Recall that a real function f defined on a normed space (X, .) is said to be weakly quasi-convex (with modulus ε > 0) or ε-quasi-convex if for all x, y ∈ X and λ ∈ [0, 1] we have : f (λx + (1 − λ)y) ≤ max{f (x), f (y)} + ελ(1 − λ) x − y . We can easily see that any weakly convex function is weakly quasi-convex. The rest of this paper is organized as follows. After recalling some definitions and properties in section 2 we give in section 3 a necessary and sufficient conditions for a function f to be weakly quasi-convex.

2

Basic definitions and properties

In this section we recall several definitions and results necessary for further developments. Throughout this paper, (X, .) denotes a reflexive separable Banach space, X ∗ its topological dual, B ∗ the closed unit ball in X ∗ . Under the above conditions, (X, .) is an Asplund space. Moreover, B ∗ is metrizable for the weak star topology ∗σ(X ∗ , X) of X ∗ . Let f : X −→ R ∪ {+∞} be an extended-real valued function. We recall that: f ω∗ domf = {x ∈ X/f (x) < +∞}, 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 Let f : X −→ R ∪ {+∞} be an extended-real valued function and let ε ≥ 0. The Fr´echet ε-subdifferential of f at x ∈ domf is defined by: ∂εF f (x) = {x∗ ∈ X ∗ /lim inf u→x

f (u) − f (x) − x∗ , u − x ≥ −ε}. u−x

(1)

1757

A note on the weakly quasi-convex functions

The limiting Fr´echet ε-subdifferential at x ∈ dom f is defined by: ∂ˆε f (x) = lim sup ∂εF f (y),

(2)

f

y →x

where”limsup” stands the sequential Kuratowski-Painlev´e upper limit with respect to the strong topology of X and the weak-star topology of X∗, i.e.: ω∗

f

lim sup ∂εF f (y) = {x∗ ∈ X ∗ /∃ xn → x, ∃x∗n → x∗ such that x∗n ∈ ∂εF f (xn ) ∀n ∈ N}. f

y →x

ˆ (x). When ε=0, then the set (2) is denoted by ∂f Since X is an Asplund space, then we have (see [3, 6]) : ˆ (x) + εB ∗ . ∂ˆε f (x) = ∂f

(3)

Recall also (see [5]) that if the function f is l.s.c. around x ∈ domf , then the limiting subdifferential (in the Mordukhovich sense) is given by: ∂f (x) = lim sup ∂εF f (y).

(4)

f

y →x,ε0

Proposition 2.2 [1] Let f : X −→ R ∪ {+∞} be an extended-real valued function and let x ∈ domf . Then we have: +

∂ F f (x) ⊂ ∂ D f (x) ⊂ ∂ C f (x), +

where ∂ D f (x) is the Dini upper subdifferential and ∂ C f (x) is the Clarke subdifferential of f at x.

3

Characterization of weakly quasi-convex functions via the limiting subdifferential

We give in this section a necessary and sufficient conditions for a function f to be weakly quasi-convex using the limiting subdifferential. Let us start first with the necessary condition. Proposition 3.1 Let f : X −→ R be a l.s.c real function. If f is weakly quasi-convex (with modulus ε > 0), then for all x, y in X the following implication holds : [∃ x∗ ∈ ∂f (x) : x∗ , y − x > ε y − x] ⇒ f (z) ≤ f (y) + ελ(1 − λ) x − y , for any z = λx + (1 − λ)y with λ ∈ [0, 1].

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

Proof. Let x, y ∈ X and x∗ ∈ ∂f (x) such that x∗ , y − x > ε y − x. Then, there exists η > 0 such that x∗ , y − x > ε y − x+η. Since x∗ ∈ ∂f (x) f ω∗ then, by (4) there exists sequences xn → x, x∗n → x∗ , εn  0 satisfying x∗n ∈ ∂εFn f (xn ) for any n ∈ N. By the continuity of u −→ y − u and ., . there is n0 ∈ N such that ∀n ≥ n0 ∀ε close to ε(ε > ε) we have:

x∗n , y − xn  > ε y − xn  + η.

(5)

On the other hand, x∗n ∈ ∂εFn f (xn ) ⊂ ∂ˆεn f (xn ). Consequently, by virtue of (3), ˆ (xn ) such that x∗ = x˜∗ + k ∗ with k ∗  ∗ ≤ εn . From (5) there exists x˜∗n ∈ ∂f n n n n X we get : ∀n ≥ n0 ∀ε close to ε(ε > ε) ˜ x∗n , y − xn  > ε y − xn  − kn∗ , y − xn  + η. Therefore, ∀n ≥ n0 ∀ε close to ε(ε > ε) ˜ x∗n , y − xn  > (ε − εn ) y − xn  + η.

(6)

Let ε > ε such that ∀n ≥ n0 ˜ x∗n , y − xn  > (ε − εn ) y − xn  + η.

(7)

ˆ (xn ), we deduce that By virtue of (7) and taking into account that x˜∗n ∈ ∂f ∗ + f ω there exists x˜∗n,m → x˜∗n , xn,m → xn with x˜∗n,m ∈ ∂ F f (xn,m ) ⊂ ∂ D f (xn,m ) such that ∀n ≥ n0 ∃m0 ∀m ≥ m0 we have 

+



f D (xn,m , y − xn,m ) ≥ x˜∗n,m , y − xn,m > (ε − εn ) y − xn,m  + η, f (xn,m + t(y − xn,m )) − f (xn,m ) . t t0 Consequently, there are sequences (sn ) and (mr ) of positive integers satisfying the following conditions: +

where f D (xn,m , y − xn,m ) = lim sup

f

xsn ,mr → x as n, r → +∞; mr → +∞ as r → +∞; sn → +∞ as n → +∞ and for every positive integers n, r +





f D (xsn ,mr , y − xsn ,mr ) ≥ x˜∗sn ,mr , y − xsn ,mr > (ε − εsn ) y − xsn ,mr  + η. Consequently, there exists a sequence τn,r  0 and there is an integer n2 such that ∀n, r ≥ n2 f (xsn ,mr + τn,r (y − xsn ,mr )) > f (xsn ,mr ) + τn,r (1 − τn,r )(ε − εsn ) y − xsn ,mr  .

1759

A note on the weakly quasi-convex functions

Since ε > ε, then for n, r large enough we obtain f (xsn ,mr + τn,r (y − xsn ,mr )) > f (xsn ,mr ) + τn,r (1 − τn,r )ε y − xsn ,mr  . Hence, by virtue of the weak quasi-convexity of the function f it follows f max(f (xsn ,mr ), f (y)) = f (y) for n, r large enough. On the other hand, xsn ,mr → x. Therefore, max(f (x), f (y)) = f (y) and the result follows. The sufficient condition is given by the following proposition. Proposition 3.2 Let f : X −→ R be a locally Lipschitzian real-function and let ε > 0. If for all x, y ∈ X the following implication holds : ∃ x∗ ∈ ∂f (x) : x∗ , y − x > ε y − x ⇒ f (z) ≤ f (y) + ελ(1 − λ) x − y , for any z = λx + (1 − λ)y with λ ∈ [0, 1]. Then f is weakly quasi-convex (with modulus 2ε). Proof. Assume the contrary. Then there are x, y ∈ X and z = λx+(1−λ)y ∈ ]x, y[, (λ ∈]0, 1[) such that f (z) > max{f (x), f (y)} + 2ελ(1 − λ) x − y .

(8)

Assume that λ ∈ [ 12 , 1[. Applying [Theorem 18, [4]] at points x, z, we deduce that there are sequences cn → c ∈]x, z[ and c∗n ∈ ∂ˆ 1 f (cn ) satisfying n

lim inf c∗n , z − cn  ≥ n→∞

f (z) − f (x) z − c . z − x

ˆ (cn ) such that: c∗ = c˜∗ + k ∗ Since c∗n ∈ ∂ˆ 1 f (cn ), then there exists c˜∗n ∈ ∂f n n n n with kn∗  ≤ n1 . Fix an η > 0 such that f (z) > max{f (x), f (y)} + 2ελ(1 − λ) x − y + η

z − x . z − c

Then, since kn∗ → 0 for the strong topology of X ∗ and using the fact that lim inf c∗n , z − cn  ≥ n→∞

f (z) − f (x) z − c , z − x

we deduce that there exists an integer n0 such that for all n ≥ n0 :

˜ c∗n , z − cn  ≥

f (z) − f (x) z − cn  − η. z − x

(9)

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

Let us prove that there exists c∗ ∈ ∂f (c) such that c∗ , y − c > ε y − c . It is clear that there exist ε > ε and an integer n1 such that for all n ≥ n1 : f (z) − f (x) 2ελ(1 − λ) x − y η > + . z − x z − x z − cn  By virtue of (9), (10) and from z − x = (1 − λ) y − x with λ ≥ easily see that:

˜ c∗n , z − cn  > ε z − cn  , for n large enough.

(10) 1 2

we can (11) ∗

ω ˆ (cn ), then by (4) there are sequences c∗ → c˜∗n On the other hand, c˜∗n ∈ ∂f n,m f

(∀n), cn,m → cn (∀n), as m → +∞ satisfying c∗n,m ∈ ∂ F f (cn,m ) for any integers m and n. By Proposition 2, c∗n,m ∈ ∂ C f (cn,m ) (∀m, n). On the other hand, f is locally Lipschitzian around c. Hence, there is K > 0 (K the Lipschitz constant of f at c), there are sequences (sn ) and (mr ) of positive integers and there is a positive integer n2 such that f

csn ,mr → c as n, r → +∞, ∀n, r ≥ n2 ∂ C f (csn ,mr ) ⊂ KB ∗ . Moreover, mr → +∞ as r → +∞, sn → +∞ as n → +∞. Since for every integer n ω∗

c∗sn ,mr → c˜∗sn as r → +∞, then for n large enough

   ∗  c˜sn 





≤ lim inf c∗sn ,mr  ≤ k. r→∞

Consequently, (˜ c∗sn )n is bounded in X ∗ . Since X is separable, then without ω∗ loss of generality, we can assume that there is c∗ ∈ X ∗ such that c˜∗sn → c∗ . On the other hand, KB ∗ is metrizable for the weak star topology ∗σ(X ∗ , X). Therefore, we can assume that ω∗

c∗sn ,mr → c∗ as n, r → +∞. By (11), it is easy to see that there is a sequence (rn )n satisfying rn → +∞ and for n large enough 



 

 

c∗sn ,mrn , z − csn ,mrn > ε z − csn ,mrn  .

Consequently, tending n → ∞ we obtain :

c∗ , z − c ≥ ε z − c .

1761

A note on the weakly quasi-convex functions

z−c On the other hand, (z − c) = y−c (y − c) and ε > ε. Hence, c∗ , y − c > ε y − c with c∗ ∈ ∂f (c). Therefore, using the fact that z = μc + (1 − μ)y, where μ = λ y−x , it follows y−c

f (z) ≤ f (y) + εμ(1 − μ) c − y .

(12)

In this case, y − x f (z) ≤ f (y) + ελ y − c





y − c − λ y − x y − c . y − c

Since y − c ≤ y − x, then f (z) ≤ f (y) + ελ(1 − λ) y − x On the other hand,

y−x y−c

=

μ λ

y − x . y − c

≤ 2 (λ ≥ 12 ). Consequently,

f (z) ≤ f (y) + 2ελ(1 − λ) y − x , which contradicts (7). In the case where λ ∈]0, 12 [, we use the same arguments by applying [Theorem 18, [4]] at points z, y. Thus, we arrive at f (z) ≤ f (x)+2ελ(1−λ) y − x , which contradicts also (7). Thus, we achieve the proof.

References [1] Aussel. D. Mean Value Theorem and Generalized Convexity in Nonsmooth Analysis. These, Universit´e Blaise Pascal, (1994). [2] Brezis. H. Analyse fonctionnelle, th´eorie et Applications. Masson. Paris. (1983). [3] Jofr´e. A, Luc. D. T, Th´era. M. ε-subdifferential and ε- monotonicity, Nonlinear Analysis, Th. Meth. and Appl., 33, (1998), 71-90. [4] Jourani. A, Th´era. M. On Limiting Fr´echet ε-Subdifferentials. Generalized convexity generalized monotonicity: recent results (Luminy 1996), Nonconvex Optim. Appl, (1997),185-198. [5] Kruger. A. Y. Properties of Generalized Differentials, Siberian Math. J. 26, (1985), 822-832. [6] Mordukhovich. B. S, Shao. Y. Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc., 348, (1996), 1235-1280. Received: November 18, 2006