Functionals defined by the maximum of a Lagrangian and their applications to ordinary differential equations

This paper considers the extremal problem for a maximized Lagrangian functionalF(u)=maxL(x,u(x),...,u ^(k) (x)) instead of the more usual extremal problem for an integrated Lagrangian density

Piecewise linear approximations in nonconvex nonsmooth optimization

We present a bundle type method for minimizing nonconvex nondifferentiable functions of several variables. The algorithm is based on the construction of both a lower and an upper polyhedral approximation of the objective function. In particular, at each iteration, a search direction is computed by solving a quadratic program aiming at maximizing the difference between the lower and the upper model. A proximal approach is used to guarantee convergence to a stationary point under the hypothesis of weak semismoothness. This research has been partially supported by the Italian “Ministero dell’Istruzione, dell’Università e della Ricerca”, under PRIN project Ottimizzazione Non Lineare e Applicazioni (20079PLLN7_003).     

A tauberian theorem for distributional point values

We give a tauberian theorem for boundary values of analytic functions. We prove that if is the distributional limit of the analytic function F defined in a region of the form (a, b) × (0, R), if F (x ₀ + iy)→ γ as y → 0⁺, and if f is distributionally bounded at x = x ₀, then f (x ₀) = γ distributionally. As a consequence of our tauberian theorem, we obtain a new proof of a tauberian theorem of Hardy and Littlewood. Received: 10 December 2007     

Lipschitz regularity of the minimizers of autonomous integral functionals with discontinuous non-convex integrands of slow growth

Let be a Borelian function and let (P) be the problem of minimizing

Brownian motion characterization of some Besov-Lipschitz spaces on domains

We characterize the Besov-Lipschitz spaces with zero boundary conditions on bounded smooth domains. We prove that the appropriate first and second difference norms are equivalent to the norm given in terms of the transition kernel of the Brownian motion killed upon exit from the domain.     

On a theorem about orderability

The purpose of this note is proving a result of Nogura and Shakhmatov [4, Theorem 4]. Furthermore, a characterization of connected spaces with a selection is given. Work supported by the research project “Metodi e problemi in Analisi Reale” of the Italian Ministero dell'Università e della Ricerca Scientifica e Tecnologica.     

On a stability theorem of liapunov

A complex function theory argument enables us to give a simple and elegant proof of a stability theorem of Liapunov.     

Bounded point evaluations and cyclic polynomials on the spaces $$\hat F$$

LetF be aBK space withAK and denote the set of all formal power series with such that ε F for the sequence of coefficients of . We give a necessary and sufficient condition for a point to be a bounded point evaluation on , and for a polynomial to be cyclic in . As special cases, we obtain the results for the space ℓ _p (β) in [7]. Research of the authors supported under the research project #1232 of the Serbian Ministry of Sciences and Tecnology and, in the case of the second author, also by the DAAD foundation (German Academic Exchange Service), grant 911 103 102 8.     

The Balian-Low theorem and regularity of Gabor systems

For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L²(ℝ), such that the generating function g∈L²(ℝ) satisfies the condition:∫_ℝ|g(x)|²(1+|x| ^A )/log ^d (2+|x|)dx < ∞ and .     

A convergence theorem for asymptotic contractions

We show that to each asymptotic contraction T with a bounded orbit in a complete metric space X, there corresponds a unique point x _* such that all the iterates of T converge to x _*, uniformly on any bounded subset of X. If, in addition, some power of T is continuous at x _*, then x _* is a fixed point of T. Dedicated to Professor Felix E. Browder with admiration and respect     

A CRITERION OF THE NON-EXISTENCE OF PERIODIC SOLUTIONS FOR A GENERALIZED LI ENARD SYSTEM AND ITS APPLICATIONS

In this paper, a new criterion of the non-existence of periodic solutions for a generalized liénard system

A fixed point theorem

We establish a fixed point theorem for a Lie group of isometries acting on a Riemannian manifold with nonnegative curvature.     

A useful fixpoint theorem

A sufficient condition for each extensive mapping of an ordered set to have a fixpoint is presented. Financial support of the Grant Agency of the Czech Republic under the grant no. 201/93/0950 is gratefully acknowledged.     

On Schwarz-type smoothers for saddle point problems with applications to PDE-constrained optimization problems

In this paper we consider a (one-shot) multigrid strategy for solving the discretized optimality system (KKT system) of a PDE-constrained optimization problem. In particular, we discuss the construction of an additive Schwarz-type smoother for a certain class of optimal control problems. A rigorous multigrid convergence analysis is presented. Numerical experiments are shown which confirm the theoretical results. The work was supported by the Austrian Science Fund (FWF) under grant SFB 013/F1309.     

Lipschitz selections of set-valued mappings and Helly’s theorem

We prove a Helly-type theorem for the family of all m-dimensional convex compact subsets of a Banach space X. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M, ρ) into this family. Let M be finite and let F be such a mapping satisfying the following condition: for every subset M′ ⊂ M consisting of at most 2^m+1 points, the restriction F|_M′ of F to M′ has a selection f_M′ (i. e., f_M′(x) ∈ F(x) for all x ∈ M′) satisfying the Lipschitz condition ‖ƒ_M′(x) − ƒ_M′(y)‖_X ≤ ρ(x, y), x, y ∈ M′. Then F has a Lipschitz selection ƒ: M → X such that ‖ƒ(x) − ƒ(y)‖_X ≤ γρ(x,y), x, y ∈ M where γ is a constant depending only on m and the cardinality of M. We prove that in general, the upper bound of the number of points in M′, 2^m+1, is sharp. If dim X = 2, then the result is true for arbitrary (not necessarily finite) metric space. We apply this result to Whitney’s extension problem for spaces of smooth functions. In particular, we obtain a constructive necessary and sufficient condition for a function defined on a closed subset of R ² to be the restriction of a function from the Sobolev space W _∞ ² (R ²).A similar result is proved for the space of functions on R ² satisfying the Zygmund condition.     

Existence theorems for nonlinear functional differential equations of neutral type

Conditions are found upon satisfaction of which the differential equation

On the almost sure central limit theorem and domains of attraction

We give necessary and sufficient criteria for a sequence (X _n) of i.i.d. r.v.'s to satisfy the a.s. central limit theorem, i.e.,

Discontinuous extensions and almost resolvability

It is proven that in large classes of topological spaces each real-valued continuous function on aG _δ-setG has an extension to the whole space which is continuous exactly at the points ofG. AmongG _δ-spaces this property characterizes the almost resolvable normal spaces.     

Quasiconformal groups and a theorem of Biship and Jones

We provide new bounds on the exponent of convergence of a planar discrete quasiconformal group in terms of the associated dilatation and the Hausdorff dimension of its conical limit set. In doing so, we use these bounds to realize a theorem of C. Bishop and P. Jones as an asymptotic limit in the dilatation.