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Preliminaries
Published in Hugo D. Junghenn, Principles of Analysis, 2018
Cantor Intersection Theorem 1 A metric space X is complete iff the intersection of any decreasing sequence of nonempty closed sets Cn $ C_n $ in X with d(Cn)→0 $ d(C_n) \rightarrow 0 $ consists of a single point.
Cantor Set Experiments
Published in James K. Peterson, Basic Analysis IV: Measure Theory and Integration, 2020
Finally, each of these closed intervals has length an which we know goes to 0 in the limit on n. So z and fa(x) are both in a decreasing sequence of sets whose lengths go to 0. Hence z and fa(x) must be the same (this uses what is called the Cantor Intersection Theorem).
Foundations of mathematics under neuroscience conditions of lateral inhibition and lateral activation
Published in International Journal of Parallel, Emergent and Distributed Systems, 2018
Andrew Schumann, Alexander V. Kuznetsov
The idea was to define the property , natural for , on the elements of , by transforming this property from the derived theorem to the preexisted axiom, also by transforming from the theorem to the axiom and by expecting that the final object will behave like numbers with almost the same structural theorems. Surprisingly, it became true! For example, for there exists the compactness of the bounded and closed set and for the complete metric space of functions, continuous on [0, 1], with the norm , also there exists the compactness of the bounded set (with an additional condition of equicontinuity) – it is the claim of the Arzelà-Ascoli theorem, denoted by , the analogue of . The Cauchy-Cantor intersection theorem, i.e. , gets its counterpart in the form of the topological Cauchy-Cantor intersection theorem, denoted by , for the closed non-empty nested subsets of X.
Solutions for a class of fractional Hamiltonian systems with exponential growth
Published in Applicable Analysis, 2022
Associated with the eigenvalues of , there exists an orthonormal basis of corresponding eigenfunctions in . We set, Let be a fixed nonnegative function and where and are given in Lemma 3.2. We recall that these constants depend of y only. We will use the following notation: Furthermore, define the class of mappings and set Using an intersection theorem (see Proposition 5.9 in [33]), we have which in combination with Lemma 3.1 implies that . On the other hand, an upper bound for the mini-max level can be obtained as follows. Since the identity mapping belongs to , we have that for , Consequently, We remark that the upper bound does not depend of n, but it depends on y.
A unified study of existence theorems in topologically based settings and applications in optimization
Published in Optimization, 2022
Phan Quoc Khanh, Nguyen Hong Quan
A turning point leading to developments of KKM theorems was [13], where Fan studied intersection points, sectional points and generalized fixed points in a vector topological setting, still under convexity assumptions, with the proofs based on his infinite dimensional extension of the famous Knaster–Kuratowski–Mazurkiewicz theorem [14] (known also as the KKM theorem or the three Polish lemma), which is an intersection theorem for subsets of a simplex. Inspired by [13], beginning with [2], many contributions have been made to the study of the existence of such points and other important ones, by using continuous maps from simplices to the topological space under consideration to replace convex hulls, see [15–22].