Showing papers on "Fixed-point theorem published in 1973"

A Generalization of a Fixed Point Theorem of Reich

Abstract: The following theorem is the principal result of this paper. Let (M, d) be a metric space and T a self-mapping of M satisfying the condition for x,y ∊ M 1 where a, b, c, e,f are nonnegative and we set α=a+b+c+e+f.

Properties of fixed-point sets of nonexpansive mappings in Banach spaces

Abstract: Let C be a closed convex subset of the Banach space X. A subset F of C is called a nonexpansive retract of C if either F = 0 or there exists a retraction of C onto F which is a nonexpansive mapping. The main theorem of this paper is that if T : C -» C is nonexpansive and satisfies a conditional fixed point property, then the fixed-point set of T is a nonexpansive retract of C. This result is used to generalize a theorem of Belluce and Kirk on the existence of a common fixed point of a finite family of commuting nonexpansive mappings. Introduction. In this paper, A' always denotes a Banach space (either real or complex) and C a nonempty, closed and convex subset of X. Most published results on nonexpansive mappings have centered on existence theorems for fixed points of nonexpansive F : C —> X. In this paper we initiate the study of the structure of the fixed-point set F(T) = {x : Tx = x}. In this connection it is useful to assume that a conditional fixed point property is satisfied: Either F has no fixed points, or F has a fixed (CFP) point in every nonempty bounded closed convex set that F leaves invariant. It is obvious that existence theorems serve to define classes of nonexpansive mappings which satisfy (CFP). However, (CFP) holds even in contexts where no existence theorem can be hoped for. For example (Theorem 4), if C is locally weakly compact and X is strictly convex, then every nonexpansive F : C -» X satisfies (CFP). Our principal structure result is Theorem 2: if C is locally weakly compact, T : C -» C is nonexpansive, and F satisfies (CFP), then F(T) is a nonexpansive retract of C. Therefore our approach is to study the structure of F(T) by studying nonexpansive retracts of C. Although we are not primarily interested in existence results, our main structure theorem does permit us to prove an existence result (Theorem 7) under more general hypotheses than before, by a more transparent argument. Received by the editors November 13, 1970 and, in revised form, May 19, 1972. A MS (MOS) subject classifications (1970). Primary 47H10.

Multivalued nonexpansive mappings and opial's condition

Abstract: We give relations between a condition introduced by Z. Opial which characterizes weak limits by means of the norm in some Banach spaces and approximations of the identity, in particular for systems of projections. Finally a fixed point theorem for multivalued nonexpansive mappings in a Banach space satisfying this condition is proved; this result generalizes those of J. Markin and F. Browder. © 1973 American Mathematical Society.

Fixed point theorems for various classes of 1-set-contractive and 1-ball-contractive mappings in Banach spaces

Abstract: Let X be a real Banach space, D a bounded open subset of X, and D the closure of D. In ?1 of this paper we establish a general fixed point theorem (see Theorem 1 below) for I-set-contractions and 1-ball-contractions T: D? X under very mild conditions on T. In addition to classical fixed point theorems of Schauder, Leray and Schauder, Rothe, Kransnoselsky, Altman, and others for T compact, Theorem 1 includes as special cases the earlier theorem of Darbo as well as the more recent theorems of Sadovsky, Nussbaum, Petryshyn, and others (see ?1 for further contributions and details) for T k-set-contractive with k < 1, condensing, and l-set-contractive. In ??2, 3, 4, and 5 of this paper Theorem 1 is used to deduce a number of known, as well as some new, fixed point theorems for various special classes of mappings (e.g. mappings of contractive type with compact or completely continuous perturbations, mappings of semicontractive type introduced by Browder, mappings of pseudo-contractive type, etc.) which have been recently extensively studied by a number of authors and, in particular, by Browder, Krasnoselsky, Kirk, and others (see ?1 for details), Introduction. Let X be a real Banach space, D a bounded open subset of X9 D and OD the closure and the boundary of D9 respectively. The object of this paper is two-fold. First, in ?1 we extend our main fixed point result (see Theorem 7' in Petryshyn [34]) by proving (see Theorem 1 below) that if T: D X is either a I-set or 1-ball contraction which satisfies the Leray-Schauder condition on OD, then T has a fixed point in D if and only if T satisfies condition (c). As will be seen9 Theorem 1 unifies and extends in some cases to nonconvex domains and/or to more general boundary conditions a number of known9 as well as some new, fixed Received by the editors October 4, 1971 and, in revised form, November 2, 1972. AMS (MOS) subject classifications (1970). Primary 47H10; Secondary 47H99.

Redundancies in the Hilbert-Bernays Derivability Conditions for Godel's Second Incompleteness Theorem

Abstract: This paper presents three generalizations of the Second Incompleteness Theorem of Godel (see [2]) which apply to a broader class of formal systems than previous generalizations (as, e.g., the generalization in [1]).The content of all three of our results is that it is versions of the third derivability condition of Hilbert and Bernays (see p. 286 of [3]) which are crucial to Godel's theorem. The second derivability condition plays some role in certain logics, but even there, only in the far weaker form of a definability condition (see Theorem 2).The elimination of the first derivability condition allows the application of the Consistency Theorem to cut-free logics which cannot prove that they are closed under cut.It is Theorem 1 which will probably have primary interest for readers who are not concerned with technical proof theory or with foundations, for it treats logics with quantifiers, and in that case one can dispose entirely of the first and second derivability conditions. The derivation of this result requires a “new twist” on old arguments. Specifically, we use a new variation on the standard self-referential lemma (this is Lemma 5.1 of [l]) to obtain a somewhat different self-referential construction than has previously been employed (this is our lemma in §2 below). With this new construction, we consider the sentence φ which “expresses” the fact: “My negation is provable.” Then, using hypotheses associated with Godel's First Incompleteness Theorem, one shows that ⊢¬φ is impossible in a consistent logic. Finally, using only a version of the third Hilbert-Bernays derivability condition, one shows that the provability of the consistency statement implies ⊢¬φ, and hence that consistency is unprovable.

Commuting analytic functions without fixed points

Abstract: Let A be the set of nonidentity analytic functions which map the open unit disk into itself. Wolff has shown that the iterates of fe A converge uniformly on compact sets to a constant T(f), unless/is an elliptic conformai automorphism of the disk. This paper presents a proof that if/and g are in A and commute under composition, and if /is not a hyperbolic conformai automorphism of the disk, then T(f)=T(g). This extends, in a sense, a result of Shields. The proof involves the so-called angular derivative of a function in A at a boundary point of the disk. Let D be the open unit disk in the complex plane. Let D be its closure. Shields [5] has proved the following result. Theorem 1. If f and g are continuous in D, analytic in D, and map D into itself, andiff°g=g °fi then fand g have a common fixed point. Let A be the set of all analytic functions which map D into £>, except for the identity function, which we exclude. This paper presents an extension of the result of Shields to the set A. For f eA we define the iterates of/recursively by/1=/, and/n+1 = f°fn when n eZ+. A member of A which maps D univalently onto D will be called a conformai automorphism (c.a.) of D. We shall assume that the reader is acquainted with the standard classification of linear fractional transformations as elliptic, hyperbolic, parabolic, or loxodromic, as given in [3, p. 70]. Each c.a. of D is of one of the first three types mentioned. The elliptic transformations yield noneuclidean rotations of D with the hyperbolic metric, while the hyperbolic and parabolic transformations have their fixed points on the boundary of D. Theorem 2 (Wolff [7]). Iffe A is not an elliptic c.a. of D, then there is a constant T(f) e D for which hmn_cafn = T(f) uniformly on compact sets. Presented to the Society, January 24, 1971 ; received by the editors March 1, 1971. AMS (MOS) subject classifications (1970). Primary 30A20, 30A76; Secondary 39A15, 30A72.

A fixed point theorem-free approach to weak almost periodicity

Abstract: In this paper we present a generalization of the Eberlein, de Leeuw and Glicksberg decomposition theorem for weakly almost periodic functions which does not rely on any fixed point theorem for its proof. A generalization of the Ryll-Nardzewski fixed point theorem is given.

Nonlinear Stochastic Integral-Equation of Hammerstein Type

Abstract: A nonlinear stochastic integral equation of the Hammerstein type in the form x(t; c) = h(t; co) + f k(t, s; co)f (s, x(s; co)) dy (s) is studied where t E S, a v-finite measure space with certain properties, co E Q, the supporting set of a probability measure space (Q, A, P), and the integral is a Bochner integral. A random solution of the equation is defined to be a second order vector-valued stochastic process x(t; co) on S which satisfies the equation almost certainly. Using certain spaces of functions, which are spaces of second order vector-valued stochastic processes on S, and fixed point theory, several theorems are proved which give conditions such that a unique random solution exists.

Borel’s fixed point theorem for Kaehler manifolds and an application

Abstract: A short proof of a generalization of the Borel fixed point theorem to the case of Kaehler manifolds is given and, as an application, a short proof of Wang's theorem that compact simply connected homogeneous manifolds are projective and of the form GIP, where G is a complex semisimple Lie group and P is a parabolic subgroup. I will give a short proof of a generalization of the Borel fixed point theorem to the case of Kaehler manifolds and, as an application, give a short proof of Wang's theorem that compact simply connected homogeneous Kaehler manifolds are projective and of the form GIP, where G is a complex semisimple Lie group and P is a parabolic subgroup. I would like to thank Phillip Griffiths who suggested trying to find a short proof of Wang's theorem. I would also like to thank Professor Yozo Matsushima for his comments. PROPOSITION I. Let X be a compact Kaehler manifold with H'(X, C)=O, and let S be a solvable connected complex Lie group acting holomorphically on X. Let Y be a subvariety of X invariant under S. Then S has a fixed point on Y and the fixed points form a subvariety. PROOF. First assume Y is a manifold, S is one dimensional and has no fixed points on Y. Associated to S we have a holomorphic tangent field on X and by invariance of Y under S, also on Y; call it A. By assumption A has no zeroes on Y. We have short exact sequences where C'y, O% are the holomorphic structure sheaves and Qx Q2 are the holomorphic one forms and M is a subsheaf of Ox. A and AIy as sections of the dual sheaves of Q2 and Received by the editors December 7, 1972. AMS (MOS) subject classifications (1970). Primary 14C30, 14M15, 32J25, 32M05, 32M10.

A generalization of Kodaira's embedding theorem

Abstract: It was shown in [5] that a normal Moi~ezon space carries an almost positive torsion free coherent analytic sheaf of rank 1. The "if' part was proved for the case of isolated singularities, Since each compact complex analytic space can be desingularized and coherent analytic sheaves can be made free (modulo torsion) by proper modifications, it would be enough to prove Conjecture I for the case X a manifold and 6 e an almost positive invertible sheaf (associated to an almost positive line bundle L). The purpose of this note is to prove it under the additional assumption that X is a K~ihler manifold. In that case, the assumptions on L may be weakend. We prove the following

Über die Verallgemeinerten Fixpunktindizes von Iterierten Verdichtender Abbildungen

Abstract: The main result of this paper is the following theorem on the generalized fixed point index for locally condensing mappings: Let X be a subset of a Banach space such that there exists a locally finite covering {Ci|i∈I} of X by closed, convex subsets of X. If s=pt with p prime and t∈N, M an open subset of X and g: D[g]→X with D[g]⊂X and M⊂D[gS] such that g|M and gS|M are locally condensing and the fixed point set F of gS|M is compact and g(F)⊂M, then iX(gS,M)≡iX(g,M) mod p. This theorem can be applied in the theory of asymptotic fixed point theorems: An example may be found at the end of this paper.

Simple proof of the Jahn-Teller theorem

Abstract: The Jahn-Teller theorem is established directly by using a particular case of the Frobenius reciprocity theorem.

A fixed point theorem for weakly uniformly strict contractions

Abstract: In Meir and Keeler [3], the authors proved a fixed point theorem in a complete metric space (X, d) for a mapping f that satisfies the following condition of weakly uniformly strict contraction: Given ∊ > 0, there exists δ > 0 such that (A)

Degree theory for noncompact multivalued vector fields

Abstract: Introduction. In this note we indicate the development and state the properties of a degree theory for a rather general class of multivalued mappings, the so-called ultimately compact vector fields, and then use this degree to obtain fixed point theorems. As will be seen, these results unite and extend the degree theory for single-valued ultimately compact vector fields in [13] and the degree theory for multivalued compact vector fields in ([5], [8]) and also serve to extend to multivalued mappings the fixed point theorems for single-valued mappings obtained in [1], [2], [3], [9], [10], [13], and others (see [13]) and to more general multivalued mappings the fixed point theorems in [4], [6], [8]. The detailed proofs of the results presented in this note will be published elsewhere.

The existence and uniqueness of nonstationary ideal incompressible flow in bounded domains in

Abstract: It is shown here that the mixed initial-boundary value problem for the Euler equations for ideal flow in bounded domains of R3 has a unique solution for a small time interval. The existence of a solution is shown by converting the equations to an equivalent system involving the vorticity and applying Schauder's fixed point theorem to an appropriate mapping. Introduction. The purpose of this paper is to prove the existence and uniqueness of classical solutions to the mixed initial-boundary value problem for the Euler equations for an ideal incompressible fluid in bounded domains of R3 for a small time interval. This has been shown by Ebin and Marsden [3] for compact Riemannian manifolds possibly with boundary. In this paper we obtain a new proof by converting the equations to an equivalent system involving the vorticity and solving this by using the Schauder fixed point theorem. The technique is similar to that of Kato [4] where he proved existence of ideal flow in bounded domains of the plane for an arbitrary time interval. The author also used a similar method to obtain an existence result for ideal flow in all R3 by showing that a solution is the limit of Navier-Stokes flow as the viscosity goes to zero [7]. Existence of a solution for a short time in all R3 was originally shown by Lichtenstein [5]. The Euler equations for ideal incompressible flow in a domain D in R3 with boundary bD are (Et) ar/at + (r grad)r =-grad P + B, V * U = O, with constraints U n = 0 on bD and r(0) = A, where x is a point in D U bD; t E [O, T], Iv(X, t) is the velocity vector, P(x, t) is the (scalar) pressure, B(x, t) is the external force field vector, n(x) is the outward normal vector to bD and A(x) is the initial velocity vector. By formally computing the curl of (E') we get the system Presented to the Society, January 18, 1972 under the title The existence and uniqeness of nonstationary idealflow in bounded domains in R3; received by the editors October 12, 1971 and, in revised form, March 20, 1972. A J1S (MOS) subject classifications (1970). Primary 35F25, 35F30, 35Q05, 76C05.