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For these problems we establish the existence of optimal (good) solutions on infinite horizon. 1 1 1 Let RC D fx 2 R1 W x 0g, v 2 Œ0; 1/; f W RC ! x/ > 0 for all x > 0 1 1 ! xt /: T2 1 2 Let integers T1 0, T2 > T1 . yt / ! fxt gTtD0 ; fyt gTtD01 / is a program such that x0 D z; tD0 1 . yt / evaluates consumption at moment t . In Chap. 4 we establish the existence of good programs. 1 v/x : In Chap. yt / T D 1I let 0 < m0 < x < M0 . T2 T1 / /j Ä M : tDT1 In Chap. 4 analogous results are also obtained for a class of two-dimensional optimal control problems.
Xi ; x/ N W i D 1; : : : ; T g Ä holds. Proof. Assume the contrary. 2 Auxiliary Results 29 Let k be a natural number. 11) It is easy to see that the sequence fxt g1 tD0 is a program. v/-good program. It follows N D 0 holds. 12). The contradiction we have reached proves the lemma. 13 and (A1) imply the following result. 14. Let M be a positive number. Then there exists an integer T such that the inclusion XM YNT holds. v; x; y; T / is finite. 15. Let be a positive number. xt ; x/ N Ä holds for all integers t D 0; : : : ; T .
90) holds. Assume that T is a natural number and that 1; 2 2 X; . 87) holds. 87) is true if T D 1. Therefore we may consider only the case with T > 1. 86) that fyt gTtD0 is a program. yt ; ytC1 / D v. 95) C2 is a program. x; N 1 / C v. 87). 20 is proved. 6. 96) and that for each natural number i , each natural number T , and each pair of points N Ä ıi , j D 1; 2 we have 1 ; 2 2 X satisfying . 97) Assume that the assertion of the theorem does not hold. 101) 46 2 Turnpike Properties of Discrete-Time Problems and for all natural numbers k.