A TURNPIKE THEOREM FOR A FAMILY OF FUNCTIONS* - Ivie

and
u(k u ;k u ) ¸u(x;y) +q u (y ¡x) for all(x;y) 2X; and
u(k u ;k u )>u(x;y) +q u (y ¡x) for all(x;y) 2X;
such that(x;y) 6=(k u ;k u ):
(21)
Proof: The existence of a subsequence fu ns g of fu n g and a concave function
u 2 U ¹ such that limu ns =uis the content of i) in Lemma 3. Note that, without
s!1
loss of generality, we can supposeu n !u:
Now, fk u n
g ½ X 1 by A4, andq u n
= D 1 u n (k u n
;k u n
) for anyn¸1by (16).
Therefore, fk un g and fq un g are bounded becauseX 1 is compact and because of
the uniform bound over functions and derivatives. Hence, there exists a point
(k u ;q u ) 2X 1 £ < n + such that, without loss of generality,k un !k u andq un !q u
(taking subsequences, if necessary): Note that(k u ;k u ) 2X:
To prove that u(x;y)+ 1 2 B(x2 +y 2 ) is concave at (k u ;k u ) we will proceed
by contradiction. Hence, we suppose this is not the case, that is, there exists
(x;y) 2 X with (x;y) 6= (k u ;k u ) and ¸ 2 (0;1) such that if v¸ = (x¸;y¸) =
¸(x;y)+(1 ¡¸)(k u ;k u ),
1
©¸u(x;y)+¸B
2 (x2 +y 2 )+(1 ¡¸)u(k u ;k u )+B(1 ¡¸)(k u ) 2ª >
u(v¸)+ 1B(x2¸+y2¸)
(22)
2
Take v ņ := ¸(x;y)+(1 ¡¸)(k u n
;k u n
). Then (22) implies that for all n large
enough we have
©¸un (x;y)+¸B 1 2 (x2 +y 2 )+(1
n
¡¸)u n (k un ;k un )+B(1
o
¡¸)(k un ) 2ª >
u n (v ņ )+ 1 2 B ((x (ku n;k u n)
¸
) 2 +(y (ku n;k u n)
¸
) 2 (23)
)
where(x (kun ;k un )
¸
;y (kun ;k un )
¸
)=v ņ ; because from Lemma 3 we have that
u n (k un ;k un ) ! u(k u ;k u ); u n (x;y) ! u(x;y) and lim u n (v ņ ) = u(v¸) as well.
n!1
But (23) is in contradiction with assumption A6, thenu(x;y)+ 1 2 B(x2 +y 2 ) is
concave at(k u ;k u ):
This proves the …rst part of the Lemma.
To show thatu(k u ;k u ) ¸u(x;y) for all(x;y) 2D such thaty¸x; we will
suppose this is not the case to get a contradiction; …rst note thatuis indeed de-
…ned at(x;y) 2Dfory ¸xbecause we have that(x;y) 2X by A4; now suppose
17