Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization

Appendix B

Elements of Nonlinear Analysis

Definition B.1. A family F of functions from R

n

to R

ℓ

is said to be equicon-

tinuous at a point x ∈ R

n

if for every ε > 0 there exists a δ > 0 such that

kf(x) − f(y)k < ε for all f ∈ F and all y such that kx − yk < δ. The family

is equicontinuous if it is equicontinuous at each point of R

n

.

Definition B.2. A collection C = {C

1

, C

2

, . . .} of subsets of K ⊆ R

n

is said

to have the finite intersection property if every finite subcollectio n of C has

nonempty intersection, that is, for every finite collection {C

1

, C

2

, . . . , C

m

} of

C we have

T

m

i=1

C

i

6= ∅.

Theorem B .1. A subset K of R

n

is compact if and only if every collection

of closed sets in K having the finite intersection proper ty has a nonempty

intersection.

Theorem B .2 (Brouwer’s Fixed Point Theorem). Let B be a closed ball in

R

n

and T : B → B be continuous. Then T admits at least one fixed po int.

Definition B.3. Let K be a nonempty convex subset of R

n

. A set-valued map

P : K → 2

K

is said to be a KKM-map if for every finite subset {x

1

, x

2

, . . . , x

m

}

of K,

co{x

1

, x

2

, . . . , x

m

} ⊆

m

[

i=1

P (x

i

),

where co{x

1

, x

2

, . . . , x

m

} denotes the convex hull of {x

1

, x

2

, . . . , x

m

}.

The following Fan-KKM theorem and the Browder-type fixed point theo-

rem for se t- valued maps w ill be the key too ls to establish existence results for

solutions o f nonsmooth vector var iational-like inequalities.

Theorem B.3 (Fan-KKM Theorem). [71] Let K be a nonempty convex sub-

set of R

n

and P : K → 2

K

be a KKM-map such that P (x) is clo sed for all

x ∈ K, and P (x) is compact fo r at least one x ∈ K. Then,

T

x∈K

P (x) 6= ∅.

Theorem B.4. [15] Let K be a nonempty convex subset of R

n

and P, Q :

K → 2

K

be two set-valued maps. Assume that the following conditions hold:

(i) For each x ∈ K, coP (x) ⊆ Q(x) and P (x) is nonempty;

(ii) For each y ∈ K, P

−1

(y) = {x ∈ K : y ∈ P (x)} is open in K;

259

© 2014 by Taylor & Francis Group, LLC

260 Elements of Nonlinear Analysis

(iii) If K is not compact, assume that there exist a nonempty compact convex

subset B of K and a nonempty compact subset D of K such that for

each x ∈ K \ D there exists ˜y ∈ B such that ˜y ∈ P (x).

Then, there exists ¯x ∈ K such that ¯x ∈ Q(¯x).

Theorem B.5. Let K be a nonempty convex subset of R

n

and P : K → 2

K

a set-valued map. Assume that the following conditions hold:

(i) for all x ∈ K, P (x) is convex;

(ii) for each finite subset A of K and for a ll y ∈ co(A), P

−1

(y) ∩ co(A) is

open in co(A);

(iii) for each finite subset A of K and all x, y ∈ co(A) and every sequence

{x

m

} in K converging to x such that λy + (1 − λ)x /∈ P (x

m

) for all

m ∈ N and all λ ∈ [0, 1], we have y /∈ P (x);

(iv) there exist a nonempty compact subset D of K a nd an element ˜y ∈ D

such that ˜y ∈ P (x) for all x ∈ K \ D;

(v) for all x ∈ D, P (x) is nonempty.

Then, there exists ˆx ∈ K such that ˆx ∈ P (ˆx).

Theorem B .6 (Kakutani). [119] Let K be a nonempty compact convex sub-

set of R

n

and P : K → 2

K

be a set-valued map such that for each x ∈ K,

P (x) is nonempty, compact, and conve x. Then P has a fixed point, that is,

there exists ¯x ∈ K such that ¯x ∈ P (¯x).

Theorem B.7 (Kneser). [131] Let K be a nonempty convex subset of R

n

and

D be a nonempty compact convex subset of R

m

. Suppose that f : K × D →

R is lower semicontinuous in the second argument and concave in the first

argument. Then

min

y∈D

sup

x∈K

f(x, y) = sup

x∈K

min

y∈Y

f(x, y).

© 2014 by Taylor & Francis Group, LLC