A TURNPIKE THEOREM FOR A FAMILY OF FUNCTIONS* - Ivie

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³ ´³ ´ Lemma 7. Take L u;± (t) = q u;± ¡q u;± t k u;± ¡k u;± t de…ned earlier. Then for any " > 0 and 1 > ± 3 > 0 there exists L ";± 3 < 0; L ";± 3 2 ± ¸± 3 andu2U: Furthermore,limL ";± 3=0: "!0 Proof: Simply takeL ";± 3= k)± ¡1 : ¤ min (x;k;q;q 0 )2< 4n + ;jx¡kj";jxj;jkj M;jqj;jq ~ 0 jN 1 ;± 3 ±1 (q¡q 0 )(x¡ Theorem 2 (A Generalized U-neighborhood Turnpike Theorem) For any ">0 there exists N(") and 00, and± 1 (") 2 (0;1) such that, for allu2U and1>± ¸± 1 ¯ ("), whenever we have ¯k u;± t¡1 ¡k u;±¯¯¯> ", thenf u;± (k u;± t¡1 ;ku;± t ) ¸ ½("). Hence, by (33), we have that for all u 2 U and 1>± ¸± 1 (") ¯ ¯k u;± t¡1 ¡k u;±¯¯¯>"; impliesL u;± (t) ¡± ¡1 L u;± (t ¡1) ¸½(") (34) Now, letN(")=2+ sup ¡2L u;± (0)=½(") and L= ~ inf L u;± (0). 1¸±¸± 1 (");u2U 1¸±¸± 1 (");u2U ³ ´ ¡k ¢ Note that since L u;± (0) = q u;± ¡q u;± 0 u;± ¡k 0 , N(") and L ~ are well de…ned 22
ecause of the uniform bound over functions and derivatives, andk 0 ;k u;± 2X 2 . Now we will show that the reasoning given by McKenzie ([7], page. 1313) can be adapted to our problem: Let ¹ ± 2 (") be such that (± ¡1 ¡1) ~ L>¡½(")=2 for all1>± ¸¹± 2 ("); (35) then for allu 2U and1>±¸¹± 2 ("), we have (± ¡1 ¡1)L u;± (0)>¡½(")=2 (36) Suppose ¯¯k0 n ¡k u;±¯¯ > ", then, summing (34) fort=1and (36), for all1> ± ¸max ± 1 ("); ¹ o ± 2 (") andu 2U we have Then L u;± (1) ¡L u;± (0)>½(")=2 (37) (± ¡1 ¡1)L u;± (1)>¡½(")=2 (38) also £ holds, because if(± ¡1 ¡1)L u;± (1) ¡ ½(")=2, using (36), we would obtain L u;± (1) ¡L u;± (0) ¤£ ± ¡1 ¡1 ¤ < 0, so L u;± (1) ¡L u;± (0) < 0 since £ 1 ¡± ¡1¤ < 0, contradicting (37). Then we have proved the following statement: ¯ ¯k0 ¡k u;±¯¯>" and(± ¡1 ¡1)L u;± (0)>¡½(")=2; implies L u;± (1) ¡L u;± (0)>½(")=2 and(± ¡1 ¡1)L u;± (39) (1)>¡½(")=2: Thus we may apply induction to obtainL u;± (t)¡L u;± (t¡1)>½(")=2, if ¯¯kl ¡k u;±¯¯> " for all0l±¸max ± 1 ("); ¹ o ± 2 (") , we have ¯ ¯k l ¡k u;±¯¯>" for all0l½(")=2 for all1 lt: (40) From (40), and sinceL u;± (t) 0for allt 2N; if we …x(u;±); ¯ ¯kt ¡k u;±¯¯>" (41) cannot occur for ever, becauseL u;± (t) would become positive. Indeed, if we denote byN(u;±;") the maximal number of consecutive periods that (41) is satis…ed, then L u;± (N(u;±;")) ¡L u;± (0)>N(u;±;")½(")=2 23
Page 1 and 2: A TURNPIKE THEOREM FOR A FAMILY OF
Page 3 and 4: 1. Introduction Many dynamic models
Page 5 and 6: able to rule out the kind of phenom
Page 7 and 8: To avoid details related with deriv
Page 9 and 10: i A7 i)for all±2 ³^±;1 ;(k u;±
Page 11 and 12: will have as a particular case of o
Page 13 and 14: this paper, which is a uniform turn
Page 15 and 16: With these preliminaries in place,
Page 17 and 18: and u(k u ;k u ) ¸u(x;y) +q u (y
Page 19 and 20: ¹k=k u , contradiction. Then the l
Page 21: On the othernhand, as noted o above
Page 25 and 26: we have proved that, given " 0 arbi
Page 27 and 28: force optimal paths to be close to
Page 29 and 30: (B ¡¹") j¹xj 2 ¡N 1 j¹xj 3 . L
Page 31 and 32: Remark 6 Theorem 2 admits weaker as
Page 33 and 34: e less and less important as the di
Page 35: [12] K. Nishimura and M. Yano, Nonl

A TURNPIKE THEOREM FOR A FAMILY OF FUNCTIONS* - Ivie

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