Existence of solutions of the dynamic Cauchy problem on infinite time scale intervals

Ireneusz Kubiaczyk; Aneta Sikorska-Nowak

Displaying similar documents to “Existence of solutions of the dynamic Cauchy problem on infinite time scale intervals”

Regularized cosine existence and uniqueness families for second order abstract Cauchy problems

Jizhou Zhang (2002)

Studia Mathematica

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Let $C_{i}$ (i = 1,2) be two arbitrary bounded operators on a Banach space. We study (C₁,C₂)-regularized cosine existence and uniqueness families and their relationship to second order abstract Cauchy problems. We also prove some of their basic properties. In addition, Hille-Yosida type sufficient conditions are given for the exponentially bounded case.

The distance between fixed points of some pairs of maps in Banach spaces and applications to differential systems

Cristinel Mortici (2006)

Czechoslovak Mathematical Journal

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Let $T$ be a $γ$ -contraction on a Banach space $Y$ and let $S$ be an almost $γ$ -contraction, i.e. sum of an $(ε, γ)$ -contraction with a continuous, bounded function which is less than $ε$ in norm. According to the contraction principle, there is a unique element $u$ in $Y$ for which $u = T u .$ If moreover there exists $v$ in $Y$ with $v = S v$ , then we will give estimates for $∥ u - v ∥ .$ Finally, we establish some inequalities related to the Cauchy problem.

Comparison of solutions and successive approximations in the theory of the equation $\partial^{2} z / \partial x \partial y = f (x, y, z, \partial z / \partial x, \partial z / \partial y)$

J. Kisyński, A. Pelczar

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CONTENTSIntroduction........................................................................................................................................................................................................... 5I. THE CAUCHY-DARBOUX PROBLEM IN FUNCTION CLASSES $C^{1}' * (Δ_{a, b}; E)$ AND $L_{1}^{1, *} (Δ_{a, b}; E)$ ......................... 71. Basic function classes ......................................................................................................................................................................................

On the Banach-Mazur distance between continuous function spaces with scattered boundaries

Jakub Rondoš (2023)

Czechoslovak Mathematical Journal

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We study the dependence of the Banach-Mazur distance between two subspaces of vector-valued continuous functions on the scattered structure of their boundaries. In the spirit of a result of Y. Gordon (1970), we show that the constant $2$ appearing in the Amir-Cambern theorem may be replaced by $3$ for some class of subspaces. We achieve this by showing that the Banach-Mazur distance of two function spaces is at least 3, if the height of the set of weak peak points of one of the spaces differs...

Multiplying balls in the space of continuous functions on [0,1]

Marek Balcerzak, Artur Wachowicz, Władysław Wilczyński (2005)

Studia Mathematica

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Let C denote the Banach space of real-valued continuous functions on [0,1]. Let Φ: C × C → C. If Φ ∈ +, min, max then Φ is an open mapping but the multiplication Φ = · is not open. For an open ball B(f,r) in C let B²(f,r) = B(f,r)·B(f,r). Then f² ∈ Int B²(f,r) for all r > 0 if and only if either f ≥ 0 on [0,1] or f ≤ 0 on [0,1]. Another result states that Int(B₁·B₂) ≠ ∅ for any two balls B₁ and B₂ in C. We also prove that if Φ ∈ +,·,min,max, then the set $Φ^{- 1} (E)$ is residual whenever E is...

An inequality for spherical Cauchy dual tuples

Sameer Chavan (2013)

Colloquium Mathematicae

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Let T be a spherical 2-expansive m-tuple and let $T^{}$ denote its spherical Cauchy dual. If $T^{}$ is commuting then the inequality $\binom{\sum_{| β | = k} {(β!)}^{- 1} {(T^{})}^{β} (T^{}) *^{β} \leq (k + m - 1}{k) \sum_{| β | = k} {(β!)}^{- 1} (T^{}) *^{β} {(T^{})}^{β}}$ holds for every positive integer k. In case m = 1, this reveals the rather curious fact that all positive integral powers of the Cauchy dual of a 2-expansive (or concave) operator are hyponormal.

Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki (2004)

Studia Mathematica

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Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets $_{ν} (X)$ and $_{ν} (X)$ of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient...

On the semilinear integro-differential nonlocal Cauchy problem

Piotr Majcher, Magdalena Roszak (2005)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, we prove an existence theorem for the pseudo-non-local Cauchy problem $x^{'} (t) + A x (t) = f (t, x (t), \int_{t ₀}^{t} k (t, s, x (s)) d s)$ , x₀(t₀) = x₀ - g(x), where A is the infinitesimal generator of a C₀ semigroup of operator ${T (t)}_{t > 0}$ on a Banach space. The functions f,g are weakly-weakly sequentially continuous and the integral is taken in the sense of Pettis.

The Cauchy kernel for cones

Sławomir Michalik (2004)

Studia Mathematica

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A new representation of the Cauchy kernel $_{Γ}$ for an arbitrary acute convex cone Γ in ℝⁿ is found. The domain of holomorphy of $_{Γ}$ is described. An estimation of the growth of $_{Γ}$ near the singularities is given.

Bounded Solutions for the Degasperis-Procesi Equation

Giuseppe Maria Coclite, Kenneth H. Karlsen (2008)

Bollettino dell'Unione Matematica Italiana

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This paper deals with the well-posedness in $L^{1}\cap L^{\infty}$ of the Cauchy problem for the Degasperis-Procesi equation. This is a third order nonlinear dispersive equation in one spatial variable and describes the dynamics of shallow water waves.

(Non-)amenability of ℬ(E)

Volker Runde (2010)

Banach Center Publications

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In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ℬ(E) of all bounded linear operators on a Banach space E could ever be amenable if dim E = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros-Haydon result that solves the “scalar plus compact problem”: there is an infinite-dimensional Banach space E, the dual of which is ℓ¹, such that $ℬ (E) = (E) + ℂ i d_{E}$ . Still, ℬ(ℓ²) is...

A note on a class of homeomorphisms between Banach spaces

Piotr Fijałkowski (2005)

Colloquium Mathematicae

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This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form $F (x) : = F ̃ x^{(2 n + 1)}$ where $F ̃ : X^{2 n + 1} \to Y$ is a continuous (2n+1)-linear operator.

On Ordinary and Standard Lebesgue Measures on $ℝ^{\infty}$

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on $ℝ^{\infty}$ are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on $ℝ^{\infty}$ for α = (1,1,...).

Local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation in Besov spaces

Gang Wu, Jia Yuan (2007)

Applicationes Mathematicae

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We study local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation $\partial_{t} u - \partial ³_{t x x} u + 2 κ \partial_{x} u + \partial_{x} [g (u) / 2] = γ (2 \partial_{x} u \partial ²_{x x} u + u \partial ³_{x x x} u)$ for the initial data u₀(x) in the Besov space $B_{p, r}^{s} (ℝ)$ with max(3/2,1 + 1/p) < s ≤ m and (p,r) ∈ [1,∞]², where g:ℝ → ℝ is a given $C^{m}$ -function (m ≥ 4) with g(0)=g’(0)=0, and κ ≥ 0 and γ ∈ ℝ are fixed constants. Using estimates for the transport equation in the framework of Besov spaces, compactness arguments and Littlewood-Paley theory, we get a local well-posedness result.

Osgood type conditions for an m th-order differential equation

Stanisaw Szufla (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We present a new theorem on the differential inequality $u^{(m)} \leq w (u)$ . Next, we apply this result to obtain existence theorems for the equation $x^{(m)} = f (t, x)$ .

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...

Lineability and spaceability on vector-measure spaces

Giuseppina Barbieri, Francisco J. García-Pacheco, Daniele Puglisi (2013)

Studia Mathematica

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It is proved that if X is infinite-dimensional, then there exists an infinite-dimensional space of X-valued measures which have infinite variation on sets of positive Lebesgue measure. In term of spaceability, it is also shown that $c a (ℬ, λ, X) ∖ M_{σ}$ , the measures with non-σ-finite variation, contains a closed subspace. Other considerations concern the space of vector measures whose range is neither closed nor convex. All of those results extend in some sense theorems of Muñoz Fernández et al. [Linear...

On vector valued measure spaces of bounded $Φ$ -variation containing copies of $ℓ_{\infty}$

María J. Rivera (2001)

Czechoslovak Mathematical Journal

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Given a Young function $Φ$ , we study the existence of copies of $c_{0}$ and $ℓ_{\infty}$ in ${c a b v}_{Φ} (μ, X)$ and in ${c a b s v}_{Φ} (μ, X)$ , the countably additive, $μ$ -continuous, and $X$ -valued measure spaces of bounded $Φ$ -variation and bounded $Φ$ -semivariation, respectively.

Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

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Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and $ℓ^{\infty}$ embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then $ℓ^{\infty}$ embeds in L(X,Y), and ℓ¹ embeds complementably in $X \otimes_{γ} Y *$ . Applications to embeddings of c₀ in various spaces of operators are given.

Solvability of the functional equation f = (T-I)h for vector-valued functions

Ryotaro Sato (2004)

Colloquium Mathematicae

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Let X be a reflexive Banach space and (Ω,,μ) be a probability measure space. Let T: M(μ;X) → M(μ;X) be a linear operator, where M(μ;X) is the space of all X-valued strongly measurable functions on (Ω,,μ). We assume that T is continuous in the sense that if (fₙ) is a sequence in M(μ;X) and $l i m_{n \to \infty} f ₙ = f$ in measure for some f ∈ M(μ;X), then also $l i m_{n \to \infty} T f ₙ = T f$ in measure. Then we consider the functional equation f = (T-I)h, where f ∈ M(μ;X) is given. We obtain several conditions for the existence of h ∈ M(μ;X)...

On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

Artur Michalak (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function $f : {[0, 1]}^{m} \to X$ is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function $f : {[0, 1]}^{m} \to X$ with respect to any norming subset there exists a separately increasing function $g : {[0, 1]}^{m} \to ℝ$ such that the sets of...