A substitute for the Sard-Smale theorem in theC 1 case

IfF is a Fredholm mapping of indexΝ ∃ ℤ and classC ^max(Ν,0)+1 between separable Banach spaces, the Sard—Smale theorem yields the existence of arbitrarily small perturbations ofF having 0 as a regular value. The smoothness requirement cannot be weakened in the Sard—Smale theorem itself, at least whenΝ ≥ 0, but we prove that the approximation result remains valid irrespective of the indexΝ whenF is only of classC ¹ and satisfies appropriate properness-like conditions. The separability of the spaces is not needed either. Everything carries over to the setting of Banach manifolds modeled on spaces with a norm of classC ¹ away from the origin. We also show that in Banach spaces, theC ¹ norm assumption can be dropped without major prejudice. The application to degree theory forC ¹ Fredholm mappings of index 0 is developed in a separate paper.     

On global bifurcation for quasilinear elliptic systems on bounded domains

General second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated into nonlinear mappings between Sobolev spaces. It is shown that the linearized mapping is a Fredholm operator of index zero. This and the abstract global bifurcation theorem of [Jacobo Pejsachowicz, Patrick J. Rabier, Degree theory for C¹ Fredholm mappings of index 0, J. Anal. Math. 76 (1998) 289-319] allow us to carry out bifurcation analysis directly on these elliptic systems. At the abstract level, we establish a unilateral global bifurcation result that is needed when studying positive solutions. Finally, we supply two examples of cross-diffusion population model and chemotaxis model to demonstrate how the theory can be applied.     

Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds

Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x ₀)⊂E→F be C¹ nonlinear map, whereU (x ₀) is an open set containing pointx ₀∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x ₀) near x₀,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C¹ Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.     

TheK-theory of AF embeddings of the rational rotation algebras

Our main result is the construction of an embedding ofC(T²) into an approximately finite-dimensionalC ^*-algebra which induces an injection onK ₀(C(T²)). The existence of this embedding implies that Cech cohomology cannot be extended to a stable, continuous homology theory forC ^*-algebras which admits a well-behaved Chern character. Homotopy properties ofC ^*-algebras are also considered. For example, we show that the second homotopy functor forC ^*-algebras is discontinuous. Similar embeddings are constructed for all the rational rotation algebras, with the consequence that none of the rational rotation algebras satisfies the homotopy property called semiprojectivity.     

Zeros of Fredholm Operator Valued Hp–Functions

This paper is concerned with Fredholm operator valued H^p – functions on the unit disc, where the Fredholm operators action a Banach space. Sufficient conditions are presented which guarantee that Fatou's theorem is valid. Using the theory of traces and determinants on quasi – Banach operator ideals, we develop conditions that guarantee that the zeros of Fredholm operator valued H^p – functions satisfy the Blaschke condition.     

On the size of balls covered by analytic transformations

Two quantitative forms of the inverse function theorem giving estimates on the size of balls covered biholomorphically are proved for holomorphic mappings of a ball in a Banach space into the space. Also, a Bloch theorem forK-quasiconformal mappings on the open unit ball of a Banach space is given and some mapping properties ofK-quasiconformal mappings are deduced.Research supported by N. S. F. Grant GP-33117A-2.     

On smooth, nonlinear surjections of banach spaces

It is shown that (1) every infinite-dimensional Banach space admits aC ¹ Lipschitz map onto any separable Banach space, and (2) if the dual of a separable Banach spaceX contains a normalized, weakly null Banach-Saks sequence, thenX admits aC ^∞ map onto any separable Banach space. Subsequently, we generalize these results to mappings onto larger target spaces. Supported by an NSF Postdoctoral Fellowship in Mathematics.     

Moduli of non-dentability and the radon-nikodým property in banach spaces

We show that for a separable Banach spaceX failing the Radon-Nikodym property (RNP), andε > 0, there is a symmetric closed convex subsetC of the unit ball ofX such that every extreme point of the weak-star closure ofC in the bidualX** has distance fromX bigger than 1 −ε. An example is given showing that the full strength of this theorem does not carry over to the non-separable case. However, admitting a renorming, we get an analogous result for this theorem in the non-separable case too. We also show that in a Banach space failing RNP there is, forε > 0, a convex setC of diameter equal to 1 such that each slice ofC has diameter bigger than 1 −ε. Some more related results about the geometry of Banach spaces failing RNP are given.     

An Extension of Set-Valued Contraction Principle for Mappings on a Metric Space with a Graph and Application

In this paper, we obtain an existence theorem for fixed points of contractive set-valued mappings on a metric space endowed with a graph. This theorem unifies and extends several fixed point theorems for mappings on metric spaces and for mappings on metric spaces endowed with a graph. As an application, we obtain a theorem on the convergence of successive approximations for some linear operators on an arbitrary Banach space. This result yields the well-known Kelisky–Rivlin theorem on iterates of the Bernstein operators on C[0,1].     

Extreme points for convex mappings ofB n

We exhibit a collection of extreme points of the family of normalized convex mappings of the open unit ball of ℂ ⁿ forn≥2. These extreme points are defined in terms of the extreme points of a closed ball in the Banach space of homogeneous polynomials of degree 2 in ℂ ⁿ⁻¹, which are fully classified. Two examples are given to show that there are more convex mappings than those contained in the closed convex hull of the set of extreme points here exhibited.     

Uniformly Gâteaux smooth approximations on

Every Lipschitz mapping from c₀(Γ) into a Banach space Y can be uniformly approximated by Lipschitz mappings that are simultaneously uniformly Gâteaux smooth and C^∞-Fréchet smooth.     

Functor D and strong homotopy

This paper studies functor D and strong homotopy, introduced earlier by the author [1]. A theorem is proven on mappings, and the connection is established between the concepts of strong homotopy of DGA-mapping of coalgebras and functor D. As topological applications, in particular, it is shown that continuous mappings of the sphere f, g: S^2n–1Sⁿ have one and the same Hopf invariant if and only if the induced chain of mappings is strongly homotopic.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 557–564, April, 1977.     

Extremal K -Theory and Index for C*-Algebras

We develop the general theory for a new functor K _e on the category of C ^*-algebras. The extremal K-set, K _e (A), of a C ^*-algebra A is defined by means of homotopy classes of extreme partial isometries. It contains K ₁ (A) and admits a partially defined addition extending the addition in K ₁ (A), so that we have an action of K ₁ (A) on K _e (A). We show how this functor relates to K ₀ and K ₁, and how it can be used as a carrier of information relating the various K-groups of ideals and quotients of A. The extremal K-set is then used to extend the classical theory of index for Fredholm and semi-Fredholm operators.     

On Ω-blow-ups in the simplest <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth skew products of interval mappings

The authors prove a criterion (necessary and sufficient condition) for the emergence of the C ⁰-Ω-blow-up for C ¹-smooth skew products of interval mappings with closed set of periodic points. An example of the mapping with given properties that admits the C ⁰-Ω-blow-up is presented. It is proved that the C ¹-Ω-blow-up is impossible for mappings of such a type (in the space of C ¹-smooth skew products of interval mappings). It is proved that there is no one-parameter family of C ¹-smooth skew products of interval mappings with closed set of periodic points C ¹-smoothly depending on the parameter in which from one fixed point, periodic orbits with periods 2 and 4 simultaneously arise. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

Global continuation for first order systems over the half‐line involving parameters

Let X be one of the functional spaces W^1,p ((0, ∞), ?^N ) or C₀¹ ([0, ∞), ?^N ), we study the global continuation in λ for solutions (λ, u, ξ) ∈ ? × X × ?^k of the following system of ordinary differential equations: where ?^N = X₁ ⊕ X₂ is a given decomposition, with associated projection P: ?^N → X₁. Under appropriate conditions upon the given functions F and φ, this problem gives rise to a nonlinear Fredholm operator which is proper on the closed bounded subsets of ? × X × ?^k and whose zeros correspond to the solutions of the original problem. Using a new abstract continuation result, based on a recent degree theory for proper Fredholm mappings of index zero, we reduce the continuation problem to that of finding a priori estimates for the possible solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The cohomology of homotopy categories and the general linear group

A homotopy categoryC (of co-H-groups resp.H-groups) represents an element C in the third cohomology ofC. This element determines all Toda brackets and secondary homotopy operations inC. Moreover, in caseC =VS ⁿ consists of all one-point unions ofn-spheres, the bracket is actually a /2-generator which restricts to Igusa's class(1) in casen3; an explicit new cocycle for(1) is obtained by automorphisms of free nil(2)-groups.     

Leggett-Williams norm-type theorems for coincidences

We study the existence of positive solutions to the operator equation Lx = Nx, where L is a linear Fredholm mapping of index zero and N is a nonlinear operator. Using the properties of cones in Banach spaces and Leray-Schauder degree for completely continuous operators, k-set contractions and condensing mappings, we obtain some refinements of the results established in [3] and [14]. Received: 9 July 2005; revised: 12 January 2006     

Univalence Criteria Starting from the Method of Loewner Chains

The paper continues the work of Royster (Duke Math J 19:447–457, 1952), Mocanu [Mathematica (Cluj) 22(1):77–83, 1980; Mathematica (Cluj) 29:49–55, 1987], Cristea [Mathematica (Cluj) 36(2):137–144, 1994; Complex Var 42:333–345, 2000; Mathematica (Cluj) 43(1):23–34, 2001; Mathematica (Cluj), 2010, to appear; Teoria Topologica a Functiilor Analitice, Editura Universitatii Bucuresti, Romania, 1999] of extending univalence criteria for complex mappings to C ¹ mappings. We improve now the method of Loewner chains which is usually used in complex univalence theory for proving univalence criteria or for proving quasiconformal extensions of holomorphic mappings f : B → C ⁿ to C ⁿ . The results are surprisingly strong. We show that the usual results from the theory, like Becker’s univalence criteria remain true for C ¹ mappings and since we use a stronger form of Loewner’s theory, we obtain results which are stronger even for holomorphic mappings f : B → C ⁿ . In our main result (Theorem 4.1) we end the researches dedicated to quasiconformal extensions of K-quasiregular and holomorphic mappings f : B → C ⁿ to C ⁿ . We show that a C ¹ quasiconformal map f : B → C ⁿ can be extended to a quasiconformal map F : C ⁿ → C ⁿ , without any metric condition imposed to the map f.     

Geometrical implications of the existence of very smooth bump functions in Banach spaces

We show that ifX is a Banach space and if there is a non-zero real-valuedC ^∞-smooth function onX with bounded support, then eitherX contains an isomorphic copy ofc ₀(N), or there is an integerk greater than or equal to 1 such thatX is of exact cotype 2k and, in this case,X contains an isomorphic copy ofl ^2k(N). We also show that ifX is a Banach space such that there is onX a non-zero real-valuedC ⁴-smooth function with bounded support and ifX is of cotypeq forq<4, thenX is isomorphic to a Hilbert space.     

Uniqueness of <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroups on a general locally convex vector space and an application

The main purpose is to generalize a theorem of Arendt about uniqueness of C ₀-semigroups from Banach space setting to general locally convex vector spaces. More precisely, we show that cores are the only domains of uniqueness for C ₀-semigroups on locally convex spaces. As an application, we find a necessary and sufficient condition for that the mass transport equation has one unique L ¹(ℝ ^d ,dx) weak solution.