On the solvable operators generated by uniformly correct problems

A first order differential-operator equation with an operator-coefficient generating a C₀ class semi-group is studied. Boundary conditions for which this equation possesses a unique solution dependent continuously on its right-hand side are derived. Two theorems are formulated and proved.     

On some set-valued iteration semigroups generated by interval-valued functions

Let X be an arbitrary set. We characterize all interval-valued functions \({A:X\to 2^\mathbb{R}}\) for which a multifunction \({F:(0,\infty)\times X\to 2^X}\) of the form \({F(t,x)=A^{-}\big(A(x)+\min \{t,q-\inf A(x)\}\big)}\), where \({q=\sup A(X)}\), is an iteration semigroup. The multifunction F is the set-valued counterpart of the fundamental form of continuous iteration semigroups of single-valued functions on an interval.     

Lattices of uniformly continuous functions

Abstract

We obtain an internal characterization of the lattice U (X) of real-valued uniformly continuous functions on a uniform space X.     

Upper semicontinuity of Nemytskij operators

Summary The upper semicontinuity of the Nemytskij map from L^p to L^q, associated to a multivalued map G(, x) upper semicontinuous in x, si proved.Work performed while the authors were visiting S.I.S.S.A., Trieste.     

Positive linear operators generated by analytic functions

Let φ be a power series with positive Taylor coefficients {a _k } _k=0^∞ and non-zero radius of convergence r ≤ ∞. Let ξ _x , 0 ≤ x < r be a random variable whose values α _k , k = 0, 1, …, are independent of x and taken with probabilities a _k x ^k /φ(x), k = 0, 1, …. The positive linear operator (A _φ f)(x):= E[f(ξ _x )] is studied. It is proved that if E(ξ _x ) = x, E(ξ _x ²) = qx ² + bx + c, q, b, c ∈ R, q > 0, then A _φ reduces to the Szász-Mirakyan operator in the case q = 1, to the limit q-Bernstein operator in the case 0 < q < 1, and to a modification of the Lupaş operator in the case q > 1.     

On the density of the space of continuous and uniformly continuous functions

For X a metrizable space and (Y,ρ) a metric space, with Y pathwise connected, we compute the density of (C(X,(Y,ρ)),σ)—the space of all continuous functions from X to (Y,ρ), endowed with the supremum metric σ. Also, for (X,d) a metric space and (Y,‖⋅‖) a normed space, we compute the density of (UC((X,d),(Y,ρ)),σ) (the space of all uniformly continuous functions from (X,d) to (Y,ρ), where ρ is the metric induced on Y by ‖⋅‖). We also prove that the latter result extends only partially to the case where (Y,ρ) is an arbitrary pathwise connected metric space.To carry such an investigation out, the notions of generalized compact and generalized totally bounded metric space, introduced by the author and A. Barbati in a former paper, turn out to play a crucial rôle. Moreover, we show that the first-mentioned concept provides a precise characterization of those metrizable spaces which attain their extent.     

On algebras generated by positive operators

We study algebras generated by positive matrices, i.e., matrices with nonnegative entries. Some of our results hold in more general setting of vector lattices. We reprove and extend some theorems that have been recently shown by Kandi? and ?ivic. In particular, we give a more transparent proof of their result that the unital algebra generated by positive idempotent matrices E and F such that \(E F \ge F E\) is equal to the linear span of the set \(\{I, E, F, E F, F E, E F E, F E F, (E F)^2, (F E)^2\}\), and so its dimension is at most 9. We give examples of two positive idempotent matrices that generate unital algebra of dimension 2n if n is even, and of dimension \((2n - 1)\) if n is odd. We also prove that the algebra generated by positive matrices \(B_1\), \(B_2, \ldots , B_k\) is triangularizable if \(A B_i \ge B_i A\) (\(i=1,2, \ldots , k\)) for some positive matrix A with distinct eigenvalues.     

On topologies generated by some operators

The paper concerns topologies introduced in a topological space (X, τ) by operators which are much weaker than the lower density operators. Some properties of the family of sets having the Baire property and the family of meager sets with respect to such topologies are investigated.     

On uniformly convex functions

In this paper we study uniformly convex functions and uniformly convex functions at a point, giving some properties and characterizations of them. Further, we give some examples and applications of these types of functions.     

Concave iteration semigroups of linear continuous set-valued functions

Let {F ^t : t ≥ 0} be a concave iteration semigroup of linear continuous set-valued functions defined on a convex cone K with nonempty interior in a Banach space X with values in cc(K). If we assume that the Hukuhara differences F ⁰(x) ? F ^t (x) exist for x ∈ K and t > 0, then D _t F ^t (x) = (?1)F ^t ((?1)G(x)) for x ∈ K and t ≥ 0, where D _t F ^t (x) denotes the derivative of F ^t (x) with respect to t and $G(x) = \mathop {\lim }\limits_{s \to 0} {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} \mathord{\left/ {\vphantom {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} {\left( { - s} \right)}}} \right. \kern-0em} {\left( { - s} \right)}}$ for x ∈ K.     

On uniformly convex functions

Non-convex functions that yet satisfy a condition of uniform convexity for non-close points can arise in discrete constructions. We prove that this sort of discrete uniform convexity is inherited by the convex envelope, which is the key to obtain other remarkable properties such as the coercivity. Our techniques allow to retrieve Enflo's uniformly convex renorming of super-reflexive Banach spaces as the regularization of a raw function built from trees. Among other applications, we provide a sharp estimation of the distance of a given function to the set of differences of Lipschitz convex functions. Finally, we prove the equivalence of several possible ways to quantify the super weakly noncompactness of a convex subset of a Banach space.     

Uniformly bounded Nemytskij operators between the Banach spaces of functions of bounded n-th variation

The main result says that the generator of any uniformly bounded composition operator acting between Banach algebras of functions of bounded n-th variation is an affine function with respect to the function variable.