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Sequences and Their Limits
Published in John Srdjan Petrovic, Advanced Calculus, 2020
Definition 2.8.9. If {an} is a bounded sequence, we call its largest accumulation point the limit superior of the sequence {an}, and we write lim sup an (or lim an). We denote the smallest accumulation point of {an} by lim inf an (or liman), and we call it the limit inferior of the sequence {an}. When {an} is not bounded above, we will say that lim sup an = +∞. When {an} is not bounded below, we will say that lim inf an = —∞.
A note on the marginal instability rates of two-dimensional linear cocycles
Published in Dynamical Systems, 2023
We remark that the situation described in Theorem 1.1 is quite delicate: if the dimension of the linear maps is raised from 2 to 3, or if a shift over a compact infinite alphabet is allowed in place of the finite alphabet , then the conclusion no longer holds and the above sequences may grow at a rate strictly intermediate between linear growth and boundedness (see Refs. [13,21,23]). In this article, we give an alternative proof of the above result which is due to the second named author and which was previously presented in the thesis [29]. We remark that the actual existence of the limit (2) is a new contribution originating in this article: in Refs. [13,29], it was shown that the limit inferior and limit superior of this sequence are finite and nonzero, but it was not shown that they are equal to one another.
Input-to-state contraction for impulsive systems
Published in International Journal of Control, 2022
Bin Liu, Zhijie Sun, Bo Xu, Dong-Nan Liu
(i) For , from the S-I.F. condition in and by the definition of limit superior, for any satisfying , there exists a positive real number such that Let . Then we have and By , , and , we get that for some and , and and .
Nonsmooth optimization by successive abs-linearization in function spaces
Published in Applicable Analysis, 2022
Andrea Walther, Olga Weiß, Andreas Griewank, Stephan Schmidt
Suppose, , i.e. ϕ is also locally Lipschitz continuous. Let be given. Then the limit superior exists and is called Clarke derivative of ϕ at in direction h. Since this limit superior exists for all , the function ϕ is called Clarke differentiable at . The set denotes the Clarke generalized gradient or subdifferential of ϕ at , where refers to the dual space of V.