Showing papers on "Fixed-point theorem published in 1971"

On the basic theorem of complementarity

Abstract: Using a fixed point theorem of Browder, the basic existence theorem of Lemke in linear complementarity theory is generalized to the nonlinear case.

Fixed points of order preserving multifunctions

Abstract: Let F: X--X be a multifunction on a partially ordered set (X, f). Suppose for each pair x1 ?x2 and for each y1eF(x1) there is a y2eF(Y2) such that Y16Y2. Then sufficient conditions are given such that multifunctions F satisfying the above condition will have a fixed point. These results generalize the Tarski Theorem on complete lattices, and they also generalize some results of S. Abian and A. B. Brown, Canad. J. Math 13 (1961), 78-82. By similar techniques two selection theorems are obtained. Further, some related results on quasi-ordered and partially ordered topological spaces are proved. In particular, a fixed point theorem for order preserving multifunctions on a class of partially ordered topological spaces is obtained. 1. Multifunctions on partially ordered sets. Tarski's result on the existence of a fixed point for an isotone function on a complete lattice is well known (see Birkhoff [3, p. 115, Theorem 11]), and a number of related results have also been published. For example, S. Abian and A. B. Brown [2] published results on nondecreasing maps on a partially ordered set, and A. Abian [1] obtained a result for nonincreasing functions on a totally ordered set. Then in 1954, L. E. Ward, Jr. [6] published several results for continuous order preserving functions on quasi-ordered and partially ordered topological spaces. The purpose of the present paper is to present results analogous to these for multivalued functions. In this paper a multifunction F:X-+Y is a correspondence such that 0# F(x) C Y for each xEX where 0 is the empty set. Multifunctions will be denoted by F, G, etc. Let P be a property of sets. Then a multifunction F is said to be point P in case F(x) has property P for each x in the domain. Finally a fixed point of F is a point x such that x F(x). Let F:X-* Y and let < denote a relation on X and a relation on Y. Then we shall use the following two conditions throughout this paper. Received by the editors March 13, 1970. AMS 1969 subject classifications. Primary 9620, 5485, 5465; Secondary 0630, 5456.

Fixed Points and Stability for a Sum of Two Operators in Locally Convex Spaces

Abstract: : Some fixed point theorems for a sum of two operators are proved, generalizing to locally convex spaces a fixed point theorem of M. A. Krasnoselskii, for a sum of a completely continuous and a contraction mapping, as well as some of its recent variants. A notion of stability of solutions of nonlinear operator equations in linear topological spaces is formulated in terms of specific topologies on the set of nonlinear operators, and a theorem on the stability of fixed points of a sum of two operators is given. As a byproduct, sufficient conditions for a mapping to be open or to be onto are obtained.

Some fixed point theorems

Abstract: Introduction. We wish to summarize here some new asymptotic fixed point theorems. By an asymptotic fixed point theorem we mean roughly a theorem in functional analysis in which the existence of fixed points of a map ƒ is proved with the aid of assumptions on the iterates f of ƒ. Such theorems have proved of use in the theory of ordinary and functional differential equations (see [7], [8], [9] and [l5]). I t also seems likely that many of the fixed point theorems which have been used in nonlinear functional analysis can be unified by obtaining them as corollaries of general asymptotic fixed point theorems. These theorems also give new results, of course. In our first section we restrict attention to continuous maps ƒ defined on closed, convex subsets of Banach spaces. We obtain a general fixed point theorem (Theorem 1 below), and we prove that certain fixed point theorems of R. L. Frum-Ketkov [5], F. E. Browder [ l ] , [2], W. A. Horn [ó], G. Darbo [3], the author [ l l ] , [12], [13] and others follow as corollaries. In the second section we consider maps defined on more general spaces than closed, convex subsets of Banach spaces, and we generalize some of the results of §1.

Generalization of the stolper-samuelson theorem

Abstract: This paper is concerned with the generalization of the Stolper-Samuelson theorem from the 2 x 2 case to the n x n case. We start by proving theorems establishing the validity of the factor price equalization theorem and the Stolper-Samuelson theorem for the n x n case. The conditions established in these theorems are then interpreted economically in terms of the generalized versions of factor intensity. It may be noted that the above results, apart from being more readily interpretable in economic terms, are of basic mathematical interest.

Formal Cohomology: III. Fixed Point Theorems

Abstract: Let (R, (wi)) be a complete discrete valuation ring of characteristic 0 with quotient field K and finite residue class field k = GF(q). In our previous papers, [1] and [2], (which we shall refer to as FC I and FC II), we constructed the formal cohomology groups, Hi(A; K), of certain smooth k algebras A and developed several of their basic properties. The Frobenius endomorphism F: X Xq of A operates on the groups H'(A; K). In this paper we shall prove a fixed point theorem for the action of F, which generalizes results of Dwork and Reich and leads via techniques of Dwork to a new proof of the rationality of the zeta function of a k-variety. Applying the same techniques to the study of Fod, where j is an automorphism of A of finite order, we shall obtain a new and rather simple proof of the rationality of the Artin L-functions. Let A be a w. c. f. g. algebra over (R, (wr)). (See FC I for terminology.) A Frobenius endomorphism of A is a lifting of F: A d A, where A = A/wA. Such liftings exist provided A is very smooth. Suppose then that A is very smooth, that A is pure n dimensional and that F is a fixed Frobenius endomorphism of A. By FC I, Th. 8.6 Cor. 1, p 216, F induces a bijection F* on H(A, K). In fact Corollary 2 of the same theorem constructs an operator A: D(A) D(A) such that A* = qn(Fh)on H(A; K). The main object of this paper is to show that qlfF~l behaves like a compact operator on H(A, K) and that the alternating sum of the traces of (qnfF;l)s on the Hi(A; K) is equal to the number of points of Spec A rational over GF(qs). (It is likely that the H'(A; K) are always finite dimensional but we have only been able to prove this in general for i < 1.) In outline the paper goes as follows. Section 1 introduces a class of linear operators on vector spaces over K, the "nuclear operators". The axiomatics here are essentially the Riesz-Serre decomposition theorem for compact endomorphisms of a Banach space. In particular, we are able to define the trace and characteristic power series of a nuclear operator. Section 2 studies "Dwork operators". If M is a finite A-module, a Dwork

The Converse to the Fixed Point Theorem of P. A. Smith: I

Abstract: Microwave energy is used for sealing polymeric film material particularly the sealing of film material in the form of flexible containers.

Some fixed point theorems in metric spaces

Abstract: 3.

Fixed points for contractive multifunctions

Abstract: Let F:X->X be a point closed multifunction on the bounded metric space (X, d). Let d denote the Hausdorff metric for the nonempty closed subsets of X. Then F is contractive iff J(F(x), F(y))

A fixed point theorem for plane continua

Abstract: In this paper it is proved that every bounded arcwise connected plane continuum which does not separate the plane has the fixed point property. A set X is said to have the fixed point property if each continuous function ƒ on X into itself leaves some point fixed (that is, there is a point x belonging to X such that f(x)=x). The problem "Must a bounded plane continuum which does not separate the plane have the fixed point property?" has motivated a great deal of research in plane topology. K. Borsuk in 1932 proved that every Peano continuum which lies in the plane and does not separate the plane has the fixed point property [2]. Since that time, other general conditions have been found which insure that a plane continuum has this property. In 1967, H. Bell proved that every bounded plane continuum which does not separate the plane and has a hereditarily decomposable boundary has the fixed point property [ l ] . The following question is still outstanding. If a bounded plane continuum is arcwise connected and does not separate the plane, then must it have the fixed point property? Here an affirmative answer is given to this question by proving the following theorem. If M is a bounded arcwise connected plane continuum which does not have infinitely many complementary domains, then the boundary of M does not contain an indecomposable continuum. Throughout this paper 5 is the set of points of a simple closed surface (that is, a 2-sphere). The proof of the following theorem is based on techniques which are closely related to the folded complementary domain concept defined by F. Burton Jones [3, p. 173]. THEOREM 1. Suppose M is a continuum in Sy S—M does not have AMS 1970 subject classifications. Primary 54H25, 57A05, 47H10, 55C20; Secondary 54F22.

On the Brauer-Speiser theorem

Abstract: Let 9C be an absolutely irreducible rational valued character of a finite group G. The component of the group algebra QG corresponding to 9C is central simple over Q and the QG-irreducible module of this component affords the character w@(9C)9C which is also rational valued; hence this module is isomorphic to its dual, whence its e n d o morphism ring (i.e. the division algebra appearing in the simple component) is isomorphic to its opposite and so is a quaternion algebra (Albert-Hasse-Brauer-Nöether). This result is known as the Brauer-Speiser Theorem [ l ] , [2]. We ask: Does every quaternion division algebra central simple over Q appear in some QG? The answer is yes: Let G be generated by x, y, c subject to the relations x = l (p odd), y~xy=x (a is primitive mod p), y~ = c, c = l and c central. Then QG contains as a simple component the cyclic algebra (Q(£p), (r), — 1), which is c.s. over Q and has Hasse-invariant 1/2 at (R and p. The quaternion group of order 8 yields the ordinary quaternion algebra (Hasseinvariant 1/2 at (R and 2) and so, by taking tensor products, we see that every quaternion algebra is available.

On the Controllability of Nonlinear Systems.

Abstract: : Controllability of several classes of nonlinear systems described by nonlinear differential equations is studied. In a similar approach to the one used in the theory of differential equations to establish the existence of a solution, the fixed point theorem is used to establish the existence of a control function for a certain system which transfers the initial state of the system to the origin in a finite time. (Author)

A fixed point theorem for mappings with a nonexpansive iterate

Abstract: Let X be a reflexive Banach space which has strictly convex norm and suppose K is a nonempty, bounded, closed and convex subset of X. Suppose T:K->K has the property that, for some positive integer N, TN is nonexpansive (11 TNx-TNyll 1, such that if I1Tix-Tiyj|i!kjIx-yjI for all x, yEK, 1 _j X is called nonexpansive if I ITx-Ty| I ||x-y|| for all x, y &K. F. Browder [3 ], D. Gohde [5 ], and the author [6 ] proved independently that if X is a uniformly convex Banach space, K a nonempty, bounded, closed and convex subset of X, and T is a nonexpansive mapping of K into K, then T always has at least one fixed point. (The slightly weaker assumptions in [6] are that X be reflexive and that K possess the property Brodskil and Mil'man [2] call "normal structure.") If one merely assumes that for some positive integer N the Nth iterate, TN, of T is nonexpansive then T need not have a fixed point since, in particular, a periodic homeomorphism of the unit ball of Hilbert space may be fixed point free (Klee [7]). In this paper we respond to a question raised recently by K. Goebel in [4] and obtain conditions sufficient to guarantee existence of fixed points for mappings T such that TN is nonexpansive. Our principal result is the following: THEOREM 1. Let X be a reflexive Banach space which has strictly convex norm and suppose K is a nonempty, bounded, closed and convex subset of X which possesses normal structure. Suppose the mapping T: K->K has the property that for some integer N> 1, TN is nonexpansive, and suppose further that there is a constant k satisfying (i) AT-2[(N 1)(N 2)k2 + 2(N 1)k] < 1 such that ||Tjx-Tjy|| _k||x-y|| for all x, yEK, 1

Construction of a fixed point for contractions in Banach space

Abstract: A method for constructing fixed points of contractions in uniformly convex Banach spaces is developed. The fixed point obtained is the limit of one sequence that always converges (provided that a fixed point exists).