BVφ-solutions of nonlinear integral equations

In this paper we investigate solutions of nonlinear Hammerstein and Volterra-Hammerstein integral equations in the space of functions of bounded φ-variation in the sense of Young. We prove the existence and in some cases the existence and uniqueness of local and global solutions in this class. Real-valued as well as vector-valued functions are under our consideration. The method of our proofs is based on an application of the Banach contraction principle as well as the Leray-Schauder alternative for contractions.     

Complemented copies of c0 in spaces of operators

We study the existence of complemented copies of c ₀ in certain Banach spaces of bounded linear operators and apply our results to spaces of bounded vector valued measures. This revised version was published online in June 2006 with corrections to the Cover Date.     

The mass-transfer vector problem

The mass-transference problem is studied for Banach valued cost functions and operator-valued measures. The solvability of the primal problem is stated under certain natural conditions, for general measurable functions and measures of bounded variation. The continuous case is also studied, and the solvability and the absence of a duality gap are established for continuous vector functions and regular operator valued measures.     

Stochastic integration for Lévy processes with values in Banach spaces

A stochastic integral of Banach space valued deterministic functions with respect to Banach space valued Lévy processes is defined. There are no conditions on the Banach spaces or on the Lévy processes. The integral is defined analogously to the Pettis integral. The integrability of a function is characterized by means of a radonifying property of an integral operator associated with the integrand. The integral is used to prove a Lévy–Itô decomposition for Banach space valued Lévy processes and to study existence and uniqueness of solutions of stochastic Cauchy problems driven by Lévy processes.     

Bochner integrals in ordered vector spaces

We present a natural way to cover an Archimedean directed ordered vector space E by Banach spaces and extend the notion of Bochner integrability to functions with values in E. The resulting set of integrable functions is an Archimedean directed ordered vector space and the integral is an order preserving map.     

An extension of a theorem of Banach

Summary Via porosity a well-known theorem of Banach on bounded real valued continuous functions defined on [0, 1] is extended to higher dimensions. Summary. Via porosity a well-known theorem of Banach on bounded real valued continuous functions defined on [0, 1] is extended to higher dimensions.     

Vector-valued interval functions and the Dedekind completion of <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>, <Emphasis Type="Italic">E</Emphasis>)

We consider interval valued functions with values in a Banach lattice E. Certain notions of continuity introduced earlier for real interval valued functions are generalised to the more general case considered here. As an application, we characterise the Dedekind completion of the space of continuous, E-valued functions on a paracompact \(T_{1}\)-space, extending a result of Anguelov.     

Coordinatewise multiple summing operators in Banach spaces

We invent the new notion of coordinatewise multiple summing operators in Banach spaces, and use it to study various vector valued extensions of the well-know Bohnenblust-Hille inequality (which originally extended Littlewood's 4/3-inequality). Our results have application on the summability of monomial coefficients of m-homogeneous polynomials P:?_∞→?p, as well as for the convergence theory of products of vector valued Dirichlet series.     

On the Strong McShane Integral of Functions with Values in a Banach Space

The classical Bochner integral is compared with the McShane concept of integration based on Riemann type integral sums. It turns out that the Bochner integrable functions form a proper subclass of the set of functions which are McShane integrable provided the Banach space to which the values of functions belong is infinite-dimensional. The Bochner integrable functions are characterized by using gauge techniques. The situation is different in the case of finite-dimensional valued vector functions.     

The New Extensions of the Henstock–Kurzweil and the McShane Integrals of Vector-Valued Functions

In this paper, we define the Hake–Henstock–Kurzweil and the Hake–McShane integrals of Banach space valued functions defined on an open and bounded subset G of m-dimensional Euclidean space \(\mathbb {R}^{m}\). These are ”natural” extensions of the McShane and the Henstock–Kurzweil integrals from m-dimensional closed non-degenerate intervals to bounded and open subsets of \(\mathbb {R}^{m}\). Our goal is not a generalization for the sake of generalization. Indeed, we will show theorems which reduce the study of our integrals to the study of McShane and Henstock–Kurzweil integrals. As applications, we will present Hake-type theorems for the Henstock–Kurzweil and the McShane integrals in terms of our integrals.     

A fixed point result in strictly convex Banach spaces

We define an alternate convexically nonexpansive map T on a bounded, closed, convex subset C of a Banach space X and prove that if X is a strictly convex Banach space and C is a nonempty weakly compact convex subset of X, then every alternate convexically nonexpansive map T : C → C has a fixed point. As its application, we give an existence result for the solution of an integral equation.     

Surjective isometries on spaces of vector valued continuous and Lipschitz functions

For X a compact Hausdorff topological space and F a strictly convex and reflexive Banach space we characterize the surjective isometries of certain subspaces of C(X, F). It follows from this characterization that surjective isometries on spaces of vector valued Lipschitz functions equipped with a p-norm are also weighted composition operators.     

Periodic decomposition of measurable integer valued functions

We study those functions that can be written as a sum of (almost everywhere) integer valued periodic measurable functions with given periods. We show that being (almost everywhere) integer valued measurable function and having a real valued periodic decomposition with the given periods is not enough. We characterize those periods for which this condition is enough. We also get that the class of bounded measurable (almost everywhere) integer valued functions does not have the so-called decomposition property. We characterize those periods a₁,…,ak for which an almost everywhere integer valued bounded measurable function f has an almost everywhere integer valued bounded measurable (a₁,…,ak)-periodic decomposition if and only if Δa₁?Δakf=0, where Δaf(x)=f(x+a)−f(x).     

Quasi-stationary approximation for reaction-diffusion systems

Let S denote an idempotent semigroup, let W denote a Banach space. The space BV (S, W), which is the space of functions of bounded variation from S into W, is considered. It is shown that if f is in BV (S, W) and if W⁷ contains no copy of l_∞ then the value of f at every point is ∫_Γγ(s) dμ_f(γ), where Γ is the structure space of S and μ_f is an appropriate W valued measure. The hypothesis that $\text{W}{}^7$ has no copy of l_∞ is then dropped and necessary and sufficient conditions are given for μ_f to still have values in W. An application is made to Lipschitz functions and conditions are derived for μ_ff to be a Gelfand or a Pettis indefinite integral. Another application is made to product measures.     

Application of bases to the study of measures and operators on measure spaces

Summary Under investigation are measures m defined on a σ-algebraA with range in a Banach space X having a Schauder basis {x_n}. Utilizing the corresponding coefficient functionals and coefficient measures, we study the interaction between properties for these measures and properties for the existing measure m. For the spaceCA(A, X) of all countably additive measures of finite variation fromA into X, we show that certain separable subspaces of it are isomorphic-isometric with certain separable subspaces of functions of bounded variation on the interval [0, 1]. For {v_n} inCA(A, X) such that {v_n(A)} converges to v(A) for all A εA a certain ? control measure ? is constructed. An integral for set functions relative to a measure is defined and necessary and sufficient conditions are given for weak convergence. Bounded operators on the space FAC(A,R) of finitelly additive scalar valued set functions onA which are absolutely continuous with respect to λ are considered. Necessary and sufficient conditions for continuity and for compactness are given for Banach space valued operators T defined on such spaces. Weak compactness of T is also studied. Finally, a different kind of representation is given for Banach space valued operators defined on the space FA_v(A, X) of finitely additive set functions fromA into X. Entrata in Redazione il 26 novembre 1976. The first author expresses his gratitude to the Italian Consiglio Nazionale delle Ricerche and the Università degli Studi di Param for the support of his work during his tenure as Visiting Professor of Mathematics. This work was supported by NATO Research Grant No. 835.     

A characterization of bounded convex approximation properties

The characterization of bounded approximation properties defined by arbitrary operator ideals due to Oja is extended to bounded convex approximation properties. As an application, it is shown that the unique extension property of a Banach space X enables to lift the metric convex approximation property from a Banach space X to its dual space X*.     

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.