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**Extra resources for Aubry-Mather theory**

**Example text**

16. 16) are uniformly ultimately bounded. Proof. 16). Let > 0 be sufficiently large so that M c. M /º be given. ˛//º. Let t0 2 R and '0 2 PCŒŒ r; 0; Rn . 16) with k'0 kr < ˛. t I t0 ; '0 /k ˇ. 36) 48 2 Lyapunov stability and boundedness (A) ¤ tk , k D 1; 2; : : : . t// is continuous at 0; x. C 0// D V . ; x. ˇ/ and C V . ; x. 16) , we have V . V . ; x. /// > V . ; x. kx. /k/ V . ; x. ˛/; and hence kx. 16) V . ; x. kx. M /º/ D M max¹M; c. 37). (B) D tj for some j 2 ¹1; 2; : : : ; k; : : :º. 14.

45). D/ . 44) are equi-bounded. 54 2 Lyapunov stability and boundedness Proof. 47). 48). tI t0 ; '0 / 2 v ˛ for t0 r Ä t Ä t0 C r. tI t0 ; '0 / 2 v ˛ for t > t0 C r, too. Suppose that this is not true. e. t/ cannot leave v ˛ by jump. T / 2 @v ˛ . 44). T // < 0. e. t/ 2 v ˛ for all t t0 r. 1 C a/ 2kDk is valid. t/k < for all t t0 . D/ are equi-bounded. 16). Here, the results from the previous section will be used. 16). 26. t t0 / W t0 /: In the next theorems, we shall use Lyapunov functions of the class V0 , whose derivatives are estimated by the elements of set P for Á Rn .

23) for t t0 . t t0 / as t 2 Œt0 r; t0 , and this solution does not leave the domain . Let t1 be the first moment of impulsive perturbation. t1 / D ˆC 1 . t1 / D ˆC 1 . t/ for t1 < t Ä t2 . tk 1 ; tk , k D 3; 4; : : :, respectively. tk / D ˆC . t / for tk < t Ä tkC1 . t0 ; '0 /. t0 r; t0 / are the points of discontinuity of first kind of the function '0 at which it is continuous from the left, the proof of Assertion 1 is similar. We shall note that, in this case, it is possible that tk D Âl C r for some k D 1; 2; : : : and l D 1; 2; : : : ; s.