A selection theorem for quasi-lower semicontinuous mappings in hyperconvex spaces

We prove a continuous selection theorem for quasi-lower semicontinuous mappings with values that are closed sub-admissible subsets of a hyperconvex metric space and apply this result to obtain fixed point theorems in these spaces.     

Unbounded critical points for a class of lower semicontinuous functionals

For a general class of lower semicontinuous functionals, we prove existence and multiplicity of critical points, which turn out to be unbounded solutions to the associated Euler equation. We apply a nonsmooth critical point theory developed in [10], [12] and [13] and applied in [8], [9] and [20] to treat the case of continuous functionals.     

A selection theorem for a new class of set-valued maps

A new class of set-valued maps that includes all upper and lower semicontinuous set-valued maps is introduced. For this class, a selection theorem having applications in the theory of differential inclusions is presented. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 503–507, October, 1999.     

Berry–Esseen theorem and local limit theorem for non uniformly expanding maps

In Young towers with sufficiently small tails, the Birkhoff sums of Hölder continuous functions satisfy a central limit theorem with speed , and a local limit theorem. This implies the same results for many non uniformly expanding dynamical systems, namely those for which a tower with sufficiently fast returns can be constructed.     

A minimax theorem for functions with possibly nonconnected intersections of sublevel sets

We apply the selection theorem for multivalued mappings with paraconvex values (rather than various versions of KKM-principle) to prove several minimax theorems. In contrast with well-known minimax theorems for coordinatewise semicontinuous functions, in our theorems finite intersections of sublevel or uplevel sets can be nonempty and nonconnected.     

Jensen's inequality for the lower semicontinuous quasiconvex envelope and relaxation of multidimensional control problems

Assume that K⊂Rnm is a convex body with o∈int(K) and is a function with f|K∈C⁰(K,R) and f|(Rnm?K)≡+∞. We show that its lower semicontinuous quasiconvex envelope

     

On fixed point and almost fixed point problems of lower semicontinuous type multivalued mappings

In the present paper, some new almost fixed point theorems and fixed point theorems for lower semicontinuous type multivalued mappings are obtained in metrizable H-spaces.     

Existence of solutions for integral inclusions of Urysohn type with nonconvex-valued orientor field

In this paper, we prove the existence of solutions for an integral inclusion of Urysohn type with nonconvex orientor field and with delay. We make standard boundedness and continuity assumptions on the data, and we assume that the orientor field is l.s.c. in the state variable. Using a selection theorem of Fryszkowski, we are able to prove the existence of solutions, extending an earlier result of Angell.This research was supported by NSF Grant No. DMS-86-02313.     

A topological position of the set of continuous maps in the set of upper semicontinuous maps

Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q \ Σ)), i.e., there is a homeomorphism h :↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q \ Σ...     

Hyperspaces and open monotone maps of hereditarily indecomposable continua

We prove the following theorems:

Theorem 1. Let be an -dimensional hereditarily indecomposable continuum. Then there exist -dimensional hereditarily indecomposable continua and monotone maps such that is an embedding and the space of all subcontinua of is embeddable in by .

Theorem 2. For every open monotone map with non-trivial sufficiently small fibers on a finite dimensional hereditarily indecomposable continuum with there exists a -dimensional subcontinuum such that and the restriction of to is also monotone and open.

The connection between these theorems and other results in Hyperspace theory is studied.

     

A finiteness theorem for harmonic maps into Hilbert Grassmannians

In this article we demonstrate that every harmonic map from a closed Riemannian manifold into a Hilbert Grassmannian has image contained within a finite-dimensional Grassmannian.

     

A Bernstein type theorem for minimal volume preserving maps

We show that any minimal volume preserving map from the Euclidean plane into itself is a linear diffeomorphism. We derive this from a similar result on minimal diffeomorphisms. We also show that the classical Bernstein theorem on minimal graphs is a corollary of our result.

     

Uniform factorization for compact sets of operators

We prove a factorization result for relatively compact subsets of compact operators using the Bartle and Graves Selection Theorem, a characterization of relatively compact subsets of tensor products due to Grothendieck, and results of Figiel and Johnson on factorization of compact operators. A further proof, essentially based on the Banach-Dieudonné Theorem, is included. Our methods enable us to give an easier proof of a result of W.H. Graves and W.M. Ruess.

     

Individual ergodic theorem for unitary maps of random matrices

Using simple techniques of finite von Neumann algebras, we prove a limit theorem for random matrices.

     

Best approximation, coincidence and fixed point theorems for quasi-lower semicontinuous set-valued maps in hyperconvex metric spaces

Suppose X is a compact admissible subset of a hyperconvex metric spaces M, and suppose F:XM is a quasi-lower semicontinuous set-valued map whose values are nonempty admissible. Suppose also G:XX is a continuous, onto quasi-convex set-valued map with compact, admissible values. Then there exists an x₀X such that

As applications, we give some coincidence and fixed point results for weakly inward set-valued maps. Our results, generalize some well-known results in literature.     

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 point theorems of upper semicontinuous multivalued mappings with applications in hyperconvex metric spaces

In the present paper, we establish two fixed point theorems for upper semicontinuous multivalued mappings in hyperconvex metric spaces and apply these to study coincidence point problems and minimax problems.     

The triangular maps with closed sets of periodic points

In a recent paper we provided a characterization of triangular maps of the square, i.e., maps given by F(x,y)=(f(x),gx(y)), satisfying condition (P1) that any chain recurrent point is periodic. For continuous maps of the interval, there is a list of 18 other conditions equivalent to (P1), including (P2) that there is no infinite ω-limit set, (P3) that the set of periodic points is closed and (P4) that any regularly recurrent point is periodic, for instance. We provide an almost complete classification among these conditions for triangular maps, improve a result given by C. Arteaga [C. Arteaga, Smooth triangular maps of the square with closed set of periodic points, J. Math. Anal. Appl. 196 (1995) 987-997] and state an open problem concerning minimal sets of the triangular maps. The paper solves partially a problem formulated by A.N. Sharkovsky in the eighties. The mentioned open problem, the validity of (P4) ⇒ (P3), is related to the question whether some regularly recurrent point lies in the fibres over an f-minimal set possessing a regularly recurrent point. We answered this question in the positive for triangular maps with nondecreasing fiber maps. Consequently, the classification is completed for monotone triangular maps.     

Approximate selections of almost lower semicontinuous multimaps in C-spaces

We study the relations of almost lower semicontinuity, lower semicontinuity and other generalized lower semicontinuity; then we establish a new approximate selection theorem for almost lower semicontinuous multimaps with the generalized Zima type condition in C$C$-spaces. Our result unify and extend the approximate selection theorems in many published works.     

A topological invariant for pairs of maps

In this paper we develop the notion of contact orders for pairs of continuous self-maps (f, g) from ℝⁿ, showing that the set Con(f, g) of all possible contact orders between f and g is a topological invariant (we remark that Con(f, id) = Per(f)). As an interesting application of this concept, we give sufficient conditions for the graphs of two continuous self-maps from ℝ intersect each other. We also determine the ordering of the sets Con(f, 0) and Con(f, h), for h ∈ Hom(ℝ) such that f ∘ h = h ∘ f. For this latter set we obtain a generalization of Sharkovsky’s theorem.