Existence theorem for the Hammerstein integral equation

Mieczysław Cichoń; Ireneusz Kubiaczyk

Displaying similar documents to “Existence theorem for the Hammerstein integral equation”

Weak Distances between Random Subproportional Quotients of $ℓ ₁^{m}$

Piotr Mankiewicz (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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Lower estimates for weak distances between finite-dimensional Banach spaces of the same dimension are investigated. It is proved that the weak distance between a random pair of n-dimensional quotients of $ℓ ₁^{n ²}$ is greater than or equal to c√(n/log³n).

Metric spaces admitting only trivial weak contractions

Richárd Balka (2013)

Fundamenta Mathematicae

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If (X,d) is a metric space then a map f: X → X is defined to be a weak contraction if d(f(x),f(y)) < d(x,y) for all x,y ∈ X, x ≠ y. We determine the simplest non-closed sets X ⊆ ℝⁿ in the sense of descriptive set-theoretic complexity such that every weak contraction f: X → X is constant. In order to do so, we prove that there exists a non-closed $F_{σ}$ set F ⊆ ℝ such that every weak contraction f: F → F is constant. Similarly, there exists a non-closed $G_{δ}$ set G ⊆ ℝ such that every weak...

Dual spaces of compact operator spaces and the weak Radon-Nikodým property

Keun Young Lee (2012)

Studia Mathematica

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We deal with the weak Radon-Nikodým property in connection with the dual space of (X,Y), the space of compact operators from a Banach space X to a Banach space Y. First, under the weak Radon-Nikodým property, we give a representation of that dual. Next, using this representation, we provide some applications to the dual spaces of (X,Y) and $_{w * w} (X *, Y)$ , the space of weak*-weakly continuous operators.

Approximate and weak amenability of certain Banach algebras

P. Bharucha, R. J. Loy (2010)

Studia Mathematica

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The notions of approximate amenability and weak amenability in Banach algebras are formally stronger than that of approximate weak amenability. We demonstrate an example confirming that approximate weak amenability is indeed actually weaker than either approximate or weak amenability themselves. As a consequence, we examine the (failure of) approximate amenability for $ℓ^{p}$ -sums of finite-dimensional normed algebras.

Weak Baire measurability of the balls in a Banach space

José Rodríguez (2008)

Studia Mathematica

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Let X be a Banach space. The property (∗) “the unit ball of X belongs to Baire(X, weak)” holds whenever the unit ball of X* is weak*-separable; on the other hand, it is also known that the validity of (∗) ensures that X* is weak*-separable. In this paper we use suitable renormings of $ℓ^{\infty} (ℕ)$ and the Johnson-Lindenstrauss spaces to show that (∗) lies strictly between the weak*-separability of X* and that of its unit ball. As an application, we provide a negative answer to a question raised...

Weak Type Inequality for the Square Function of a Nonnegative Submartingale

Adam Osękowski (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let f be a nonnegative submartingale and S(f) denote its square function. We show that for any λ > 0, $λ ℙ (S (f) \geq λ) \leq π / 2 ∥ f ∥ ₁$ , and the constant π/2 is the best possible. The inequality is strict provided ∥f∥₁ ≠ 0.

On weak minima of certain integral functionals

Gioconda Moscariello (1998)

Annales Polonici Mathematici

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We prove a regularity result for weak minima of integral functionals of the form $\int_{Ω} F (x, D u) d x$ where F(x,ξ) is a Carathéodory function which grows as ${| ξ |}^{p}$ with some p > 1.

On bases, compactness and weak convergence in the Banach space $A_{p}$

J. T. Marti (1971)

Colloquium Mathematicae

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On diffeomorphisms deleting weak compacta in Banach spaces

Daniel Azagra, Alejandro Montesinos (2004)

Studia Mathematica

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We prove that if X is an infinite-dimensional Banach space with $C^{p}$ smooth partitions of unity then X and X∖ K are $C^{p}$ diffeomorphic for every weakly compact set K ⊂ X.

A Weak-Type Inequality for Orthogonal Submartingales and Subharmonic Functions

Adam Osękowski (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let X be a submartingale starting from 0, and Y be a semimartingale which is orthogonal and strongly differentially subordinate to X. The paper contains the proof of the sharp estimate $ℙ (s u p_{t \geq 0} | Y_{t} | \geq 1) \leq 3.375 . . . ∥ X ∥ ₁$ . As an application, a related weak-type inequality for smooth functions on Euclidean domains is established.

Weak amenability of weighted group algebras on some discrete groups

Varvara Shepelska (2015)

Studia Mathematica

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Weak amenability of ℓ¹(G,ω) for commutative groups G was completely characterized by N. Gronbaek in 1989. In this paper, we study weak amenability of ℓ¹(G,ω) for two important non-commutative locally compact groups G: the free group ₂, which is non-amenable, and the amenable (ax + b)-group. We show that the condition that characterizes weak amenability of ℓ¹(G,ω) for commutative groups G remains necessary for the non-commutative case, but it is sufficient neither for ℓ¹(₂,ω) nor for...

Proper cocycles and weak forms of amenability

Paul Jolissaint (2015)

Colloquium Mathematicae

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Let G and H be locally compact, second countable groups. Assume that G acts in a measure class preserving way on a standard space (X,μ) such that $L^{\infty} (X, μ)$ has an invariant mean and that there is a Borel cocycle α: G × X → H which is proper in the sense of Jolissaint (2000) and Knudby (2014). We show that if H has one of the three properties: Haagerup property (a-T-menability), weak amenability or weak Haagerup property, then so does G. In particular, we show that if Γ and Δ are measure equivalent...

Weak compactness in the space of operator valued measures $M_{b} a (Σ, (X, Y))$ and its applications

N.U. Ahmed (2011)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures $M_{b a} (Σ, (X, Y))$ . This generalizes a recent result of the author characterizing conditionally weakly compact subsets of the space of nuclear operator valued measures $M_{b a} (Σ, ₁ (X, Y))$ . This result has interesting applications in optimization and control theory as illustrated by several examples.

A note on copies of $c_{0}$ in spaces of weak* measurable functions

Juan Carlos Ferrando (2000)

Commentationes Mathematicae Universitatis Carolinae

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If $(Ω, Σ, μ)$ is a finite measure space and $X$ a Banach space, in this note we show that $L_{w^{*}}^{1} (μ, X^{*})$ , the Banach space of all classes of weak* equivalent $X^{*}$ -valued weak* measurable functions $f$ defined on $Ω$ such that $∥ f (ω) ∥ \leq g (ω)$ a.e. for some $g \in L_{1} (μ)$ equipped with its usual norm, contains a copy of $c_{0}$ if and only if $X^{*}$ contains a copy of $c_{0}$ .

Factorization of vector measures and their integration operators

José Rodríguez (2016)

Colloquium Mathematicae

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Let X be a Banach space and ν a countably additive X-valued measure defined on a σ-algebra. We discuss some generation properties of the Banach space L¹(ν) and its connection with uniform Eberlein compacta. In this way, we provide a new proof that L¹(ν) is weakly compactly generated and embeds isomorphically into a Hilbert generated Banach space. The Davis-Figiel-Johnson-Pełczyński factorization of the integration operator $I_{ν} : L ¹ (ν) \to X$ is also analyzed. As a result, we prove that if $I_{ν}$ is both completely...

On the class of positive almost weak $^{☆}$ Dunford-Pettis operators

Abderrahman Retbi (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we introduce and study the class of almost weak $^{☆}$ Dunford-Pettis operators. As consequences, we derive the following interesting results: the domination property of this class of operators and characterizations of the wDP $^{☆}$ property. Next, we characterize pairs of Banach lattices for which each positive almost weak $^{☆}$ Dunford-Pettis operator is almost Dunford-Pettis.

Relaxation of the incompressible porous media equation

László Székelyhidi Jr (2012)

Annales scientifiques de l'École Normale Supérieure

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It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular $T 4$ configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding $T 4$ configurations. We then use this to construct weak solutions to the unstable interface...

Exact $^{\infty}$ covering maps of the circle without (weak) limit measure

Roland Zweimüller (2002)

Colloquium Mathematicae

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We construct $^{\infty}$ maps T on the interval and on the circle which are Lebesgue exact preserving an absolutely continuous infinite measure μ ≪ λ, such that for any probability measure ν ≪ λ the sequence ${(n^{- 1} \sum_{k = 0}^{n - 1} ν \circ T^{- k})}_{n \geq 1}$ of arithmetical averages of image measures does not converge weakly.

A uniqueness result for the continuity equation in two dimensions

Giovanni Alberti, Stefano Bianchini, Gianluca Crippa (2014)

Journal of the European Mathematical Society

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We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation $\partial_{t} u + \frac{.}{\dot{}} (b u) = 0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$ . As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with...

Asymptotic behaviour of solutions of difference equations in Banach spaces

Anna Kisiołek (2008)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper we consider the first order difference equation in a Banach space $Δ x_{n} = \sum_{i = 0}^{\infty} a_{n}^{i} f (x_{n + i})$ . We show that this equation has a solution asymptotically equal to a. As an application of our result we study the difference equation $Δ x_{n} = \sum_{i = 0}^{\infty} a_{n}^{i} g (x_{n + i}) + \sum_{i = 0}^{\infty} b_{n}^{i} h (x_{n + i}) + y_{n}$ and give conditions when this equation has solutions. In this note we extend the results from [8,9]. For example, in [9] the function f is a real Lipschitz function. We suppose that f has values in a Banach space and satisfies some conditions with respect to the measure...

A Weak-Type Inequality for Submartingales and Itô Processes

Adam Osękowski (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate $∥ Y ∥_{W} \leq (2 (α + 1) ²) / (2 α + 1) ∥ X ∥_{L^{\infty}}$ . Here W is the weak- $L^{\infty}$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

Order-bounded operators from vector-valued function spaces to Banach spaces

Marian Nowak (2005)

Banach Center Publications

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Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space $(X, | | \cdot | |_{X})$ let E(X) be a subspace of the space L⁰(X) of μ-equivalence classes of strongly Σ-measurable functions f: Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function ${| | f (\cdot) | |}_{X}$ belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let $D_{u} (= {f \in E (X) : | | f (\cdot) | |}_{X} \leq u)$ stand for the order interval in E(X). For a real Banach space $(Y, | | \cdot | |_{Y})$ a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈...

Weak difference properties of higher orders for the class $L_{p} (G)$

Zbigniew Gajda (1988)

Colloquium Mathematicae

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