DOCSLIB.ORG

Appendix. the Theorem of Lustemik Andschnirelmann

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix. the Theorem of Lustemik Andschnirelmann Appendix. The Theorem of Lustemik andSchnirelmann One owes to Lusternik and Schnirelmann a general axiomatic theory of the calculus of variations; see [LS 2], [Ly]. The greatest achievements of this theory, as applied to closed geodesics, are the so-called Lusternik-Schnirelmann Theorem that there always exist three non-self-intersecting closed geodesics on a surface of genus O (see [LS 1]) and the Lyusternik-Fet Theorem on the existence of at least one closed geodesic on every compact Riemannian manifold (see [LF]). In this appendix we prove these theorems using the Lusternik-Schnirelmann theory. We have made this appendix completely independent of the main part of these Lectures, where we developed the extended Morse theory of the Rie­ mannian Hilbert manifold AM of closed H1-curves with its functional E. It will become evident at once-and this is true in particular for the proof of the Lyusternik-Fet Theorem-that the Lusternik-Schnirelmann approach to the theory of closed geodesics is much more elementary and less complicated than Morse's approach in its extended version dealing with (AM,E). But then, the Morse theory of (AM, E) is much more sensitive to the finer aspects of the closed geodesic problem - witness the existence theorem of infinitely many closed geodesics in Chapter 4. The source of the power of this approach is the Morse complex (see (2.5)). So, upon comparison, the extension of Morse's ideas must be considered better adapted to the theory of closed geodesics than the Lusternik-Schnirelmann theory, with one noteworthy exception, namely the Lusternik-Schnirelmann Theorem on the existence of non-self-intersecting closed geodesics. Morse's theory does not seem to be suita bie for finding such special prime closed geo­ desics. Section 1 of this appendix gives a brief, elementary introduction to the Lusternik-Schnirelmann theory as relevant to the closed geodesics problem. We obtain a short proof of the Lyusternik-Fet Theorem. In the second section, we consider the length-decreasing (or rather, energy­ decreasing) deformations in the class of non-self-intersecting closed curves on the 2-sphere. Our deformations are considerably more elementary than the one used in [LS 2] and [Ly]. In Section 3, finaIly, we give a complete elementary proof of the Lusternik­ Schnîrelmann Theorem. The topological prerequisites are kept to a minimum. In particular, we have avoided any reference to Pontrjagin's Theorem of the "removal of cycles", which plays an essential role in the earlier proofs. We condude this introduction by drawing attention to the fact that the Lusternik-Schnirelmann theory can be applied immediately to Finsler manifolds, 204 Appendix. The Theorem of Lustemik and Schnirelmann even with a non-symmetric Finsler structure. It is doubtful that the Morse theory of (AM,E) can also be extended fully from Riemannian manifolds to Finsler manifolds. For a partial extension, we refer to Mercuri [Mer]. In the case of a non-symmetric Finsler structure on S2 the methods of Luster­ nik and Schnirelmann yield the existence of only two non-self-intersecting closed geodesics; this reflects the fact that the space of oriented unparameterized great circles on S2 is isomorphic to S2 and therefore its Z2-homology is 2-dimensional, whereas the space of non-oriented unparameterized great circles on S2 is isomor­ phic to the projective plane p 2 and thus, its Z2-homology is 3-dimensional. An example due to Katok [Kat] shows that there actually exist non-symmetric Finsler structures on S2 with only two closed geodesics (not counting the multiple coverings). See Matthias [Mat] for an elementary presentation of this example. Using the methods of Lusternik-Schnirelmann, as presented in this appendix, Thorbergsson [Thr] has investigated the existence of closed geodesics on non­ compact Riemannian manifolds. In preparing this appendix the assistance of W. Ballmann was very valuable. He has applied the Lusternik-Schnirelmann method to construct non-self-inter­ secting closed geodesics on surfaces of arbitrary genus; see [Ba]. A.l The Space PM and the Theorem of Lyusternik and Fet Let M be a compact Riemannian manifold. The set CO(S,M) of continuous maps c: S= [O, 1]/{0, 1 } ..... M is in a natural way a metric space (Banach manifold) by taking as distance d",(c, c') = sUP d(c(t), c'(t)}. IeS Denote by P M the subspace of CO (S, M) formed by the piecewise differentiable closed curves. For cEPM we have the functions L(c) = J Icldt, and E(c)=tj Icl2dt. s s They are related by L(c).:;;V2E(c) with equality if and only if c is parameterized proportionally to arc length, i.e. if L(cl[O,tJ)=tL(c). Since M is compact, there exists an '1 > O such that any two points p and q on M with d(p,q)':;;2'1 can be joined by an uniquely determined geodesic cpq =cpq(t), 0.:;;t.:;;1, L(cpq )=V2E(cpq )=d(p,q). Here, as always, we assume that geodesics are parameterized proportionally to arc length. A.1 The Space PM and the Theorem of Lyusternik and Fet 205 A.I.I Proposition. Let {Cn} be a sequence of piecewise differentiable paths cn:[O,l]--+M. Put cn(O)=Pn, cn(1)=qn and assume that d(Pn,qn)<o;'t/, for ali n. If the sequences {E(cn)} and {d2(Pn,qn)j2} are both convergent with the same /imit, then {cn} possesses a convergent subsequence whose /imit is a geodesic seg­ ment c: [O, l]--+M of length <o;'t/ equal to the distance from p = c(O) to q= c(1). Proof. Consider the sequence {cpnqJ of unique minimizing geodesic segments from Pn to qn' Since M is compact there exists a subsequence, which we denote again by {cMJ, such that limPn=P and Iim qn=q exist and hence {cMJ con­ verges to the minimizing segment Cpq • Consider toE]O,l[. We have to show that for every sequence {tn} on [0,1] with Iim tn=tO:lim Cn(tn) = cpq(to)' To see this put cn(tn)=rn and let {rn(k)} be a convergent subsequence of {rn}. Let r be its limit. The sequences {cpn(k)rn(k): [O,tn(k)]-+M} {crn(k)qn(k): [tn(k),l]-+M} of unique minimizing geodesics converge to cpr:[O,to]--+M and crq :[to,l]--+M, respectively. From our hypothesis we have E(cpr ) + E(crq ) = Iim {E(cpn(k)rn(k») + E(crn(k)qn(k)} <O;.lim E(Cn(k») =E(cpq) Since cpq : [O, 1]--+ M is the unique geodesic segment of minimal E-value from P to q it follows that the curve cprucrq from P to q coincides with CPq , In particular, the value r for the parameter t=to must be equal to cpq(to)' This completes the proof of (A.1.1). D Let tI> Obe as above. FixK> Oand choose an even integer k> Owith 4Klk<o;'t/2. For every cEPKM={CEPM;E(c) <O;.K} and every toES, we have Here, as always, t is to be taken modulo 1. We now proceed to define a continuous E-decreasing deformation from PKM into itself. Letj be an even integer, O<O;.j<O;.k-2. For aEU/k, U+2)jk], we define f1fl aC, CEPK M, by f1fl aC(t)=C(t), for tE [O,jjk] or tE[a,1], f1fl .. clU/k, a] = cCWk)c( .. )IU/k, a]. Here, as always, cpq denotes the minimizing geodesic from P to q with the indi­ cated parameter interval. Note that d(c(jjk), c(a))<o;'t/. We continue by defining f1fl ..c for O"E[1,2]. Let j again be even, O<O;.j<O;.k-2, and take tE[l,l+ljk] to mean tE[O,ljk]. Then, for aE[l+jjk, 1 + (j+2)jk], cEPKM, put 206 Appendix. The Theorem of Lusternik and Schnirehnann PiJ.,.c(t)=c(t), for tE[1/k,0'+1)/k] or tE[0'-1 +l/k, 1 +l/k], PiJ .,.cl [0' + 1)/k, O' -1 + 1/k] = cC((j+l )/k)c(.,.-l +l/k) I[0' + 1)/k, O' -1 + 1/k]. We now define PiJ(O',c) for O'E[O,2], cEP"M, to be the subsequent application of the mappings PiJ2/k , ••• , PiJ21 /k , PiJ., where 21 is the even integer determined by 21~ kO'< 2'+ 2. A.t.2 Proposition. The mapping PiJ: [0,2] x P"M ...... P"M is continuous. Moreover, E(PiJ(2,c))~E(c), with equality if and only if c is either a constant map or a closed geodesic, i.e. c:8...... M is an immersion satisfying Vc=O, c#O. Proo! The continuity of PiJ, withP"Mbeing considered as subspace of CO (8, M), is obvious from the definition. Let p, q be points with d(p, q) ~ 1'/. If c is a piecewise differentiable path from p to q then with equality if and only if cpq = c. This implies the last statement of (A. 1.2). O A.1.3 Lemma. Choose K > O and denote by PiJ the deformation PiJ (2, ) introduced above. Let {cn} be a sequence in P"M such that {E(cn)} and {E(PiJcn )} are both convergent with the same /imit Ko > O. Then{ cn} possesses a convergent subsequence whose /imit is a closed geodesic co, E (co) = Ko . Proo! To define PiJ we chose an even k>O such that 4K/k~1'/2. Therefore every PiJcn consists ofthe k/2 geodesic segments ~cnl[0'+1)/k, 0'+3)/k],j=0,2, ..
Load more

Recommended publications
  • On Summability Domains
    Proc. Camb. Phil. Soc. (1973), 73, 327 327 PCPS 73-37 Printed in Great Britain On summability domains BY N. J. KALTON Department of Mathematics, University College of Swansea, Singleton Park, Swansea SA2 8PP (Received 12 June 1971) 1. Introduction. We denote by w the space of all complex sequences with the topology given by the semi-norms n where 8 (x) = xn. An FK-space, E, is a subspace of o) on which there exists a complete metrizable locally convex topology T, such that the inclusion (E, T) G O) is con- tinuous; if T is given by a single norm then E is a BK-space. Let A = (a^ii = 1,2, ...;j = 1,2,...) be an infinite matrix; then if a;, yew are such that 00 we write • y = Ax. If E is a sequence space (i.e. a subspace of w) then the summability domain EA is defined to be the set of all a; e o> such that Ax (exists and) belongs to E. If E is an FK- space, with a fundamental sequence of semi-norms (pn), then EA is an FK-space with the topology given^by the semi-norms x ->• q^x) = sup k x-+pn(Ax) (»=1,2,...). We shall denote by 0 the space of all sequences eventually equal to zero. For we write if ^ is an FK-space containing <f> we say that (i) E is a BS-space if (Pnx; n = 1,2,...) is bounded for each xeE, (ii) E is an AK-space if Pnx -> x for each a; G^/.
    [Show full text]
  • A Riesz Decomposition Theorem on Harmonic Spaces Without Positive Potentials
    Hiroshima Math. J. 38 (2008), 37–50 A Riesz decomposition theorem on harmonic spaces without positive potentials Ibtesam Bajunaid, Joel M. Cohen, Flavia Colonna and David Singman (Received October 17, 2006) (Revised May 24, 2007) Abstract. In this paper, we give a new definition of the flux of a superharmonic function defined outside a compact set in a Brelot space without positive potentials. We also give a new notion of potential in a BS space (that is, a harmonic space without positive potentials containing the constants) which leads to a Riesz decomposition theorem for the class of superharmonic functions that have a harmonic minorant outside a compact set. Furthermore, we give a characterization of the local axiom of proportionality in terms of a global condition on the space. 1. Introduction The Riesz decomposition theorem for positive superharmonic functions states that any positive superharmonic function on a region of hyperbolic type in C can be expressed uniquely as the sum of a nonnegative potential and a harmonic function. This result is of no interest when the region is parabolic because there are no nonconstant positive superharmonic functions. By a Riesz decomposition theorem for a class of superharmonic functions we mean a unique representation of each member of this class as a global harmonic function plus a member of the class of a special type. For a harmonic space with a Green function, this special class is the class of nonnegative potentials. In this case the most natural class to consider is the admissible superharmonic functions, which are the superharmonic functions that have a harmonic minorant outside a compact set.
    [Show full text]
  • Graduate Prospectus2014 Institute of Space Technology
    Graduate Prospectus2014 Institute of Space Technology we HELP YOU ACHIEVE YOUR AMBITIONS P R O S P E C T U S 2 141 INSTITUTE OF SPACE TECHNOLOGY w w w . i s t . e d u . p k To foster intellectual and economic vitality through teaching, research and outreach in the field of Space Science & Technology with a view to improve quality of life. www.ist.edu.pk 2 141 P R O S P E C T U S INSTITUTE OF SPACE TECHNOLOGY CONTENTS Welcome 03 Location 04 Introduction 08 The Institute 09 Facilities 11 Extra Curricular Activities 11 Academic Programs 15 Department of Aeronautics and Astronautics 20 Local MS Programs 22 Linked Programs with Beihang University 31 Linked Programs with Northwestern Polytechnic University 49 Department of Electrical Engineering 51 Local MS Programs 53 Linked Programs with University of Surrey 55 Department of Materials Science & Engineering 72 Department of Mechanical Engineering 81 Department of Remote Sensing & Geo-information Science 100 w w w . i s t . e d u . p k Department of Space Science 106 ORIC 123 Admissions 125 Fee Structure 127 Academic Regulations 130 Faculty 133 Administration 143 Location Map 145 1 20 P R O S P E C T U S 2 141 INSTITUTE OF SPACE TECHNOLOGY w w w . i s t . e d u . p k 2 141 P R O S P E C T U S INSTITUTE OF SPACE TECHNOLOGY Welcome Message Vice Chancellor The Institute of Space Technology welcomes all the students who aspire to enhance their knowledge and specialize in cutting edge technologies.
    [Show full text]
  • Fixed Points in Convex Cones
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 4, Pages 68–93 (July 19, 2017) http://dx.doi.org/10.1090/btran/15 FIXED POINTS IN CONVEX CONES NICOLAS MONOD Abstract. We propose a fixed-point property for group actions on cones in topological vector spaces. In the special case of equicontinuous representations, we prove that this property always holds; this statement extends the classical Ryll-Nardzewski theorem for Banach spaces. When restricting to cones that are locally compact in the weak topology, we prove that the property holds for all distal actions, thus extending the general Ryll-Nardzewski theorem for all locally convex spaces. Returning to arbitrary actions, the proposed fixed-point property becomes a group property, considerably stronger than amenability. Equivalent formu- lations are established and a number of closure properties are proved for the class of groups with the fixed-point property for cones. Contents 1. Introduction and results 68 2. Proof of the conical fixed-point theorems 74 3. Integrals and cones 77 4. Translate properties 78 5. Smaller cones 80 6. First stability properties 81 7. Central extensions 82 8. Subexponential groups 84 9. Reiter and ratio properties 85 10. Further comments and questions 86 Acknowledgments 91 References 91 1. Introduction and results As an art, mathematics has certain fundamental techniques. In particular, the area of fixed-point theory has its specific artistry. Sister JoAnn Louise Mark [Mar75, p. iii] Traiter la nature par le cylindre, la sph`ere, le cˆone... Paul C´ezanne, letter to Emile´ Bernard, 15 April 1904 [Ber07, p. 617] Received by the editors January 23, 2017 and, in revised form, February 10, 2017.
    [Show full text]
  • Some Classes of Operators Related to the Space of Convergent Series Cs
    Filomat 31:20 (2017), 6329–6335 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1720329P University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat Some Classes of Operators Related to the Space of Convergent Series cs Katarina Petkovi´ca aFaculty of Civil Engineering and Architecture, University of Niˇs,Serbia Abstract. Sequence space of convergent series can also be seen as a matrix domain of triangle. By using the theory of matrix domains of triangle, as well as the fact that cs is an AK space we can give the representation of some general bounded linear operators related to the cs sequence space. We will also give the conditions for compactness by using the Hausdorff measure of noncompactness. 1. Basic Notations The set ! will denote all complex sequences x = (xk)1 and ` , c, c0 and φ will denote the sets of all k=0 1 bounded, convergent, null and finite sequences. As usual, let e and e(n),(n = 0; 1; :::) represent the sequences (n) (n) with ek = 1 for all k, and en = 1 and ek = 0 for k , n. A sequence (bn)1 in a linear metric space X is called a Schauder basis if for every x X exists a unique n=0 P 2 sequence (λn)n1=0 of scalars such that x = n1=0 λnbn. A subspace X of ! is called an FK space if it is a complete linear metric space with continuous coordinates Pn : X C ,(n = 0; 1; :::) where Pn(x) = xn. An FK space (n) ! X φ is said to have AK if his Schauder basis is (e )1 , and a normed FK space is BK space.
    [Show full text]
  • Subalgebras to a Wiener Type Algebra of Pseudo-Differential Operators Tome 51, No 5 (2001), P
    R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Joachim TOFT Subalgebras to a Wiener type algebra of pseudo-differential operators Tome 51, no 5 (2001), p. 1347-1383. <http://aif.cedram.org/item?id=AIF_2001__51_5_1347_0> © Association des Annales de l’institut Fourier, 2001, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 51, 5 (2001), 13471347-1383 SUBALGEBRAS TO A WIENER TYPE ALGEBRA OF PSEUDO-DIFFERENTIAL OPERATORS by Joachim TOFT 0. Introduction. In 1993 J. Sjostrand introduced in [Sl] without explicit reference to any derivatives, a normed space of symbols, denoted by Sw in [S2], which contains the Hormander class 88,0 (smooth functions bounded together with all their derivatives), and such that the corresponding space of pseudo- differential operators of the type when a E is stable under composition and is contained in the space of bounded operators on L 2(R’) - (Throughout the paper we are going to use the same notations as in [H] for the usual spaces of functions and distributions.) Here 0 t 1.
    [Show full text]
  • Approachability and Fixed Points for Non-Convex Set-Valued Maps*
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 170, 477-500 (1992) Approachability and Fixed Points for Non-convex Set-Valued Maps* H. BEN-EL-MECHAIEKH Department of Mathematics, Brock University, St Catharines, Ontario L2S 3A1, Canada AND P. DEGUIRE Departemeni de MathPmatiques et de Physique, UniversitP de Moncton, Moncfon, New-Brunswick EIA 3E9, Canada Submitted by C. Foias Received September 4, 1990 TO THE MEMORY OF PROFESSORJAMES DUGUNDJI INTRODUCTION This paper studies the approachability of upper semicontinuous non- convex set-valued maps by single-valued continuous functions and its applications to fixed point theory. The use of “approximate selections” was initiated by J. von Neumann [41] in the proof of his well-known minimax principle. This remarkable property was extended to the normed space setting by Cellina [S] and Cellina and Lasota [9] in the context of a degree theory for upper semicontinuous convex compact-valued maps. It also holds for maps with compact contractible values defined on finite polyhedra of Iw” (Ma+Cole11 [40]) and more generally defined on compact ANRs (McLennan [39]). It was recently taken up by Gbrniewicz, Granas, and Kryszewski [22-241 in the context of an index theory for non-convex maps defined on compact ANRs. The content of this paper is divided into three parts. The first part is devoted to background material concerning the types of spaces and set-valued maps studied herein. In the second part we define the abstract class d of approachable set-valued maps; that is, the class of maps A: X+ Y * Supported by the Natural Sciencesand Engineering Research Council of Canada.
    [Show full text]
  • Genericity and Universality for Operator Ideals
    GENERICITY AND UNIVERSALITY FOR OPERATOR IDEALS KEVIN BEANLAND AND RYAN M. CAUSEY Abstract. A bounded linear operator U between Banach spaces is universal for the complement of some operator ideal J if it is a member of the complement and it factors through every element of the complement of J. In the first part of this paper, we produce new universal operators for the complements of several ideals and give examples of ideals whose complements do not admit such operators. In the second part of the paper, we use Descriptive Set Theory to study operator ideals. After restricting attention to operators between separable Banach spaces, we call an operator ideal J generic if anytime an operator A has the property that every operator in J factors through a restriction of A, then every operator between separable Banach spaces factors through a restriction of A. We prove that many of classical operator ideals are generic (i.e. strictly singular, weakly compact, Banach-Saks) and give a sufficient condition, based on the complexity of the ideal, for when the complement does not admit a universal operator. Another result is a new proof of a theorem of M. Girardi and W.B. Johnson which states that there is no universal operator for the complement of the ideal of completely continuous operators. In memory of Joe Diestel (1943 - 2017) 1. Introduction The study of operator ideals, as initiated by A. Pietsch [47], has played a prominent role in the development of the theory of Banach spaces and operators on them. To quote J. Diestel, A.
    [Show full text]
  • Fixed Point Theorems for Middle Point Linear Operators in L1
    Fixed point theorems for middle point linear operators in L1 Milena Chermisi ∗ , Anna Martellotti † Abstract In this paper we introduce the notion of middle point linear operators. We prove a fixed point result for middle point linear operators in L1. We then present some examples and, as an application, we derive a Markov-Kakutani type fixed point result for commuting family of α-nonexpansive and middle point linear operators in L1. 1 Introduction Furi and Vignoli [11] proved that any α-nonexpansive map T : K → K on a nonempty, bounded, closed, convex subset K of a Banach space X satisfies inf kT (x) − xk = 0, x∈K where α is the Kuratowski measure of non compactness on X. It is of great importance to obtain the existence of fixed points for such mappings in many applications such as eigenvalue problems as well as boundary value problems, including approximation theory, variational inequalities, and complementarity problems. Such results are used in applied mathematics, engineering, and economics. In this paper we give optimal sufficient conditions for T to have a fixed point on K in case that X = L1(µ), where µ is a σ-finite measure, and as a minor application, in case that X is a reflexive Banach space. The study of fixed point theory has been pursued by many authors and many results are known in literature. In order to have an overview of the problem, we present a brief survey of most relevant fixed point theorems. Darbo [7] showed that any α-contraction T : K → K has at least one fixed point on every nonempty, bounded, closed, convex subset K of a Banach space.
    [Show full text]
  • Completeness, Equicontinuity, and Hypocontinuity in Operator Algebras
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS I, 419-442 (1967) Completeness, Equicontinuity, and Hypocontinuity in Operator Algebras ROBERT T. MOORE* Department of Mathematics, University of California, Berkeley, California 94720 Communicated by Edward Nelson Received March 28. 1967 1. INTRODUCTION The results presented here provide three different types of operator- algebraic characterizations of barreled spaces. The relationship between completeness conditions on the underlying locally convex space and completeness conditions on its operator algebra is discussed for various operator topologies. In particular, if “large” algebras A of operators on a locally convex space SEare required to have properties necessary for a reasonable theory of A-valued functions, then these results show that A must be barreled. For more detailed discussion, we require the following familiar notions. (i) A set B C X in a locally convex space is bounded if? it is absorbed by every neighborhood N of 0: ANIB for some h > 0. (ii) Let b : r)r x (D2+ gs be a bilinear function, where the !& are locally convex. For each u E’I)~ , write b, : gz --+ 9s for the function b,(e) = b(u, v). Similarly, b,(u) = b(u, V) maps r)i into & . Then b is Zeft (respectively, right) hypocontinuous 8 every b, and b, is continu- ous, while for every bounded B, C !& , the set {b,, 1u E BL} is equi- continuous in L(g2 , g3) (respectively, for every bounded B, CT&, (b, 1e, E B,) is equicontinuous in L(z)i , 9,)).
    [Show full text]
  • Full Characterizations of Minimax Inequality, Fixed-Point Theorem
    Full Characterizations of Minimax Inequality, Fixed-Point Theorem, Saddle-Point Theorem, and KKM Principle in Arbitrary Topological Spaces Guoqiang Tian∗ Department of Economics Texas A&M University College Station, Texas 77843 Abstract This paper provides necessary and sufficient conditions for the existence of solutions of some important problems from optimization and nonlinear analysis by replacing two typical conditions — continuity and quasiconcavity with a unique condition, weakening topological vector spaces to arbitrary topological spaces that may be discrete, continuum, non-compact or non-convex. We establish a single condition, γ-recursive transfer lower semicontinuity, which fully characterizes the existence of γ-equilibrium of minimax inequality without imposing any restrictions on topological space. The result is then used to provide full characterizations on fixed-point theorem, saddle-point theorem, and KKM principle. 2010 Mathematical subject classification: 49K35, 90C26, 55M20 and 91A10. Keywords: Fixed-point theorem, minimax inequality, saddle-point theorem, FKKM theorem, recursive transfer continuity. ∗Financial support from the National Natural Science Foundation of China (NSFC-71371117) and the Key Laboratory of Mathematical Economics (SUFE) at Ministry of Education of China is gratefully acknowledged. 1 1 Introduction Ky Fan minimax inequality [1, 2] is probably one of the most important results in mathematical sciences in general and nonlinear analysis in particular, which is mutually equivalent to many im- portant
    [Show full text]
  • Fixed Point Theorems for Infinite Dimensional Holomorphic Functions
    University of Kentucky UKnowledge Mathematics Faculty Publications Mathematics 2004 Fixed Point Theorems for Infinite Dimensional Holomorphic Functions Lawrence A. Harris University of Kentucky, [email protected] Follow this and additional works at: https://uknowledge.uky.edu/math_facpub Part of the Mathematics Commons Right click to open a feedback form in a new tab to let us know how this document benefits ou.y Repository Citation Harris, Lawrence A., "Fixed Point Theorems for Infinite Dimensional Holomorphic unctions"F (2004). Mathematics Faculty Publications. 40. https://uknowledge.uky.edu/math_facpub/40 This Article is brought to you for free and open access by the Mathematics at UKnowledge. It has been accepted for inclusion in Mathematics Faculty Publications by an authorized administrator of UKnowledge. For more information, please contact [email protected]. Fixed Point Theorems for Infinite Dimensional Holomorphic unctionsF Notes/Citation Information Published in Journal of the Korean Mathematical Society, v. 41, no. 1, p. 175-192. This article, Harris, L. A. (2004). Fixed point theorems for infinite dimensional holomorphic functions. Journal of the Korean Mathematical Society, 41(1), 175-192., was published by the Korean Mathematical Society and is available online at http://jkms.kms.or.kr. © Korean Mathematical Society. Proprietary Rights Notice for Journal of the Korean Mathematical Society Online: The person using Journal of the Korean Mathematical Society Online may use, reproduce, disseminate, or display the open access version of content from this journal for non-commercial purpose. This article is available at UKnowledge: https://uknowledge.uky.edu/math_facpub/40 J. Korean Math. Soc. 41 (2004), No. 1, pp.
    [Show full text]