Approximable Maps Research Articles

Compactness results for triholomorphic maps

We consider triholomorphic maps from an almost hyper-Hermitian manifold $\mathcal{M}^{4m}$ into a hyperKahler manifold $\mathcal{N}^{4n}$. This means that $u \in W^{1,2}$ satisfies a quaternionic del-bar equation. We work under the assumption that $u$ is locally strongly approximable in $W^{1,2}$ by smooth maps: then such maps are almost stationary harmonic (when $\mathcal{M}$ is hyperKahler as well, then stationary harmonic). We show that in this more general situation the classical $\epsilon$-regularity result still holds. We then address compactness issues for a weakly converging sequence $u_\ell \rightharpoonup u_\infty$ of strongly approximable triholomorphic maps $u_\ell:\mathcal{M} \to \mathcal{N}$ with uniformly bounded Dirichlet energies. The blow up analysis leads, as in the usual stationary setting, to the existence of a rectifiable blow-up set $\Sigma$ of codimension $2$, away from which the sequence converges strongly. The defect measure $\Theta(x) {\mathcal{H}}^{4m-2} \llcorner \Sigma$ encodes the loss of energy in the limit; we prove that for a.e. point on $\Sigma$ the value of $\Theta$ is given by the sum of energies of a (finite) number of smooth non-constant holomorphic bubbles (here the holomorphicity is understood w.r.t. a complex structure on $\mathcal{N}$ that depends on the chosen point on $\Sigma$). In the case that $\mathcal{M}$ is hyperKahler this result was established by C. Y. Wang (2003) with a different proof; we rely on Lorentz space estimates. By means of a calibration and a homological argument we further prove that for each portion of $\Sigma \setminus \text{Sing}_{u_\infty}$ contained in a Lipschitz graph we find a unique alm. compl. st. on $\mathcal{M}$ that makes the portion pseudoholomorphic and smooth, with $\Theta$ constant; moreover the bubbles originating at points of such a smooth piece are holomorphic for a common complex structure.

Note on essential fixed points of approximable multivalued mappings

AbstractA new definition of essential fixed points is introduced for a large class of multivalued maps. Two abstract existence theorems are presented for approximable maps on compact ANR-spaces in terms of a nontrivial fixed point index, or a nontrivial Lefschetz number and a zero topological dimension of the fixed point set. The second one is applied to the periodic dissipative Marchaud differential inclusions for obtaining the existence of a discretely essential subharmonic solution. Three simple illustrative examples are supplied.

On Bernstein classes of well approximable maps

We study the structure of families of “well approximable” elements of tensor products of Banach spaces including analogs of the classical quasianalytic classes in the sense of Bernstein and Beurling. As in the case of quasianalytic functions, we prove for members of these families variants of the Mazurkiewicz and Markushevich theorems and in some particular cases, if such elements are Banach-valued continuous maps on a compact metric space, estimate massivity of their graphs and level sets.

An oriented coincidence index for nonlinear Fredholm inclusions with nonconvex‐valued perturbations

We suggest the construction of an oriented coincidence index for nonlinear Fredholm operators of zero index and approximable multivalued maps of compact and condensing type. We describe the main properties of this characteristic, including applications to coincidence points. An example arising in the study of a mixed system, consisting of a first-order implicit differential equation and a differential inclusion, is given.

Leray–Schauder results for inward acyclic and approximable maps defined on Fréchet spaces

New Leray–Schauder type results are presented for inward type mappings defined on a Fréchet space E . The proofs rely on Leray–Schauder results in Banach spaces and viewing E as the projective limit of a sequence of Banach spaces.

Recent results in analytical fixed point theory

We survey recent results in analytical fixed point theory. Firstly, we state resolutions of long-standing problems in infinite dimensional topology—the Schauder conjecture, the compact AR problem, and the Banach problem on the Hilbert cube. Secondly, we list some fixed point theorems on Kakutani maps, generalized upper hemicontinuous maps, Fan–Browder maps, approximable maps, acyclic maps, and admissible or better admissible class B of maps. Some related results are also given.

Fixed point theory and generalized Leray-Schauder alternatives for approximable maps in topological vector spaces

Some new fixed point theorems for approximable maps are obtained in this paper. Homotopy results, via essential maps, are also presented for approximable maps.

Leray-Schauder alternatives for approximable maps in topological vector spaces

Using a fixed-point theorem for compact approximable maps defined on an admissible convex subset of a topological vector space, we prove Leray-Schauder alternatives for compact or pseudo-condensing approximable maps.

A Leray–Schauder type theorem for approximable maps in topological vector spaces

A Leray–Schauder type theorem for approximable maps in topological vector spaces

Fixed points and random fixed points for weakly inward approximable maps

In this paper we obtain new fixed point and random fixed point theory for approximable maps.

A Leray-Schauder type theorem for approximable maps: A simple proof

We present a simple and direct proof for a Leray-Schauder type alternative for a large class of condensing or compact set-valued maps containing convex as well as nonconvex maps.

Fixed points forM ? and regularly approximable maps

Several new fixed point results are presented for set valued maps. The maps we discuss are either ofM* type or of regularly approximable type.

Fixed points of approximable maps

We present a simple proof of the Leray-Schauder type theorem for approximable multimaps given recently by Ben-El-Mechaiekh and Idzik. We apply this theorem to obtain a Schaefer type theorem, the Birkhoff-Kellogg type theorems, a Penot type theorem for non-self-maps, and quasi-variational inequalities, all related to compact closed approximable maps.

A Leray-Schauder type theorem for approximable maps

We present an elementary proof of a Leray-Schauder type theorem for approximable set-valued maps. Our theorem generalizes many results for convex as well as nonconvex maps. Our argument is not based on a homotopy invariance property but, quite surprisingly, on a matching theorem of Ky Fan on closed covers of convex sets.