 Appendix B
Elements of Nonlinear Analysis
Deﬁnition B.1. A family F of functions from R
n
to R
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).  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
xK
P (x) 6= .
Theorem B.4.  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).  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).  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
yD
sup
xK
f(x, y) = sup
xK
min
yY
f(x, y).
© 2014 by Taylor & Francis Group, LLC

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