Arzela-Ascoli Research Articles

Nash Equilibria for Noncooperative n-Person Games in Normal Form

This paper concentrates on the problem of the existence of $\varepsilon $-equilibrium points in the Nash sense for noncooperative n-person games in normal form. Firstly, a survey of known $\varepsilon $-equilibrium.point theorems and proof techniques is given. Then various new $\varepsilon $-equilibrium point theorems are derived, in which one of the players has a big strategy space and where nonbounded, payoff functions are allowed. To prove these theorems, results of Arzela–Ascoli are extended.

Applications of convergence spaces

Convergence notions are used extensively in the areas of probability and statistics. Many times proofs can be simplified by considering an appropriate convergence structure on the space and using well-known results from the theory of convergence spaces; for example, compactness arguments are sometimes simplified by using a generalized Ascoli theorem in the convergence space setting. The theory of convergence spaces is also used to generalize some results in probability and statistics.

Elements of Intuitionistic Analysis II the Stone‐Weierstrass Theorem and Ascoli's Theorem

Mathematical Logic QuarterlyVolume 22, Issue 1 p. 501-508 Article Elements of Intuitionistic Analysis II the Stone-Weierstrass Theorem and Ascoli's Theorem H. de Swart, H. de Swart Nijmegen (Netherlands)Search for more papers by this author H. de Swart, H. de Swart Nijmegen (Netherlands)Search for more papers by this author First published: 1976 https://doi.org/10.1002/malq.19760220158Citations: 4AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume22, Issue11976Pages 501-508 RelatedInformation

A multifunction generalization of Gale's Ascoli theorem

Our purpose is to improve the Gale-type multifunction Ascoli theorem of Mancuso (1971; page 470). This latter supposes the range space to be normal and Hausdorff, and therefore does not contain Gale's theorem (1950; page 304). To obtain a multifunction theorem containing Gale's theorem (also Mancuso's theorem), we return to Gale's essential hypotheses. Thus, we assume the regularity of the range space in the sufficiency direction, and, in the necessity direction, weassume the domain to be a k-space and the range to be a regular Hausdorff space. We dispense with the “point-like” condition imposed by Mancuso. Unexplained terminology and notation is that of Mancusopos;s paper.

A Non-Hausdorff Multifunction Ascoli Theorem for 𝓴3-Spaces

A non-Hausdorff Ascoli theorem for continuous functions was established in [6]. The present purpose is to extend this result to point-compact continuous multifunction, using Levine's generalization for closed subsets [12]. The paper is organized as follows: the object of section 2 is to establish the necessary multifunction lemmas and to introduce the notion of a Tychonoff set; section 3 generalizes to multifunction context the partial exponential law of R. H. Fox [9, p. 430], and establishes a special exponential law for multifunctions; section 4 concerns the crucial properties of even continuity for multifunctions, introduced in [8]; the main theorem of the paper is established in section 5.

Generalizations of Ascoli's theorems

Generalizations of Ascoli's theorems

Ascoli theorems and the pseudocharacter of mapping spaces

The pseudocharacter of the space YX of continuous functions from X to Y is studied. For certain topologies on YX sufficient conditions, which are shown in some cases also to be necessary, for YX to have a specified pseudocharacter are given. An Ascoli theorem is a theorem which characterizes the compact subsets of YX. Several Ascoli theorems are obtained, including ones which utilize results on pseudocharacter.

A non-Hausdorff Ascoli theorem for 𝑘₃-spaces

The paper establishes an Ascoli theorem in the space of continuous functions on a k 3 {k_3} -space to a regular space. The theorem, in terms of even continuity and the compact open topology τ c {\tau _c} , properly contains the Ascoli theorems of Myers, Gale, Kelley-Morse, Bagley-Yang, and the k 3 {k_3} -space theorem of Noble.

An ascoli theorem for multi-valued functions

The concept of simultaneous or collective continuity of a family of single valued functions was introduced by Gale [3] for regular spaces to replace equicontinuiry in metric spaces. Smithson [6] extended the standard point-open and compact-open function space topologies to include multi-valued functions. The aim of this paper is to use these topologies and extend the notion of collective continuity in order to obtain an Ascoli type theorem for multi-valued functions analogous to Theorem 1 in [3, p. 304]. We have the following theorem in mind:

A generalization of the Ascoli theorem and an application to functional differential equations

A generalization of the Ascoli theorem and an application to functional differential equations

Free boundary problems for the heat equation in which the moving interface coincides initially with the fixed face

A semi-infinite solid at the melting temperature 0 occupies the region 0 ⩽ x < ∞. At the fixed face x = 0 there is prescribed either ( a) the temperature f( t) ⩾ 0 or ( b) the heat flux f( t) directed into the solid, where f( t) is not identically zero in the vicinity of t = 0. There is a free boundary s( t) separating the liquid phase from the solid phase, and a temperature distribution u( x, t) in the liquid phase. We prove existence, uniqueness, and continuous dependence theorems for problems ( a) and ( b). Let F( t) be the integral, from 0 to t, of f(τ). Then we prove that for problem ( a) s( t) behaves, in the vicinity of t = 0, essentially like the square root of F( t) while for problem ( b) s( t) behaves essentially like F( t). We obtain also upper bounds for s′( t) for both problems. The existence theorems are proved as follows: the region 0 ⩽ x ⩽ a is taken to be liquid at the melting temperature and the region a < x < ∞ solid at the melting temperature. Problems ( a) and ( b) are known to have solutions u a and s a in this case which are bounded, equicontinuous families. By Arzela's theorem uniformly convergent subsequences can be extracted. These subsequences converge to the solutions of ( a) and ( b) for the case a = 0.

A noncontractive fixed point theorem

The sequence Txn so constructed contains a subsequence TXnk which converges to some x X and which, by Ascoli's theorem [3], is equicontinuous. It follows in turn from (1) that the sequence Xnk is equicontinuous. To complete the proof it suffices to show xn,(t)-x(t) in B. For then the sequence Xnk is compact, again by Ascoli's theorem, so that Xn,kX; hence Txnk->Tx, which together with Txnk-)x yields Tx=x. That indeed Xnk(t)->x(t) is immediate from the fact that xnk(0) =u(0) =x(0) and that, for t50 and sufficiently large k,

Foundations of the Fredholm Alternative Theorem, Equicontinuous Operators and Completely Continuous Operators

The Fredholm Alternative Theorem gives the notion of bounded and continuous operator which makes a family F of elements of C[a, b] bounded and equicontinuous. A continuous operator of finite rank is completely continuous but every continuous operator is not completely continuous operator. Keywords: Arzela-Ascoli's theorem; completely continuous; equicontinuous and Fredholm alternative theorem DOI: http://dx.doi.org/10.3126/bibechana.v8i0.5700 BIBECHANA 2012; 8: 127-130

On uniform convergence of measures

A new and simpler proof is given in Section 3 for the sufficiency part of Theorem 3.1 in Ranga Rao [6] and its generalization by Billingsley and TopsOe [1]. Essential for the proof, which does not require the topological space X to be metric, is Lemma 2.1. As examples of possible wider application of this lemma, simple proofs are given for a well known result on uniformity in convergence of distribution functions (Example 2.3) and of Theorem 4.2 in Ranga Rao [6]. The derivation of the latter from the lemma is substantially simpler than the derivation from Theorem 3.1 in Ranga Rao [6]. Another result on uniformity, given in Rubin [7], is closely related to the Ascoli theorem, but outside the scope of applicability of our lemma.

Ascoli Theorems and the Exponential Map

Ascoli Theorems and the Exponential Map

A note on 𝑧-closed projections

All hypothesized spaces are assumed to be completely regular Hausdorff. Recall that the projection rx: X X Y->X is called z-closed if it maps zero sets to closed sets. This notion has recently been used to give conditions under which v(X X Y) = vX X v Y (vX denotes the Hewitt-Nachbin realcompactification of X)[CN]; to characterize products XX Y for which the product uniformity is fine[N1] and [H]; and to prove exponential laws and Ascoli Theorems [N2]. Some further applications will be given in [N3]. In this note, using the more general notion of relatively z-closed projections, we give some new conditions under which a projection on a product will be z-closed.

Ascoli theorems and the exponential map

Ascoli theorems and the exponential map

Another Proof of Ascoli's Theorem

Another Proof of Ascoli's Theorem

On 𝑘-spaces and function spaces

does compactness of (F, e) or, more general, local compactness imply that e is jointly continuous? It is shown that the product of a locally compact Hausdorff space with a Hausdorff k-space is a k-space. This is used to prove that e is jointly continuous when (F, e) is locally compact and X is a Hausdorff k-space. These results are combined with a theorem of R. Arens [1] to construct an example of two kspaces whose product is not a k-space. We also prove a generalization of the Ascoli Theorem 7.21 [2, Theorem 21, Chapter 7]. In a remark following this theorem Kelley points out that it can be extended to k-spaces by weakening the condition on even continuity. We show that the theorem holds for Hausdorff k-spaces without alteration, Theorem 4. The same remark holds for [2, Theorem 7.17]. THEOREM 1. If X is a locally compact Hausdorff space and Y is a Hausdorff k-space, then X X Y is a k-space.

An Extension of Ascoli's Theorem and Its Applications to The Theory of Optimal Control

An Extension of Ascoli's Theorem and Its Applications to The Theory of Optimal Control