Solving variational inclusions by a multipoint iteration method under center-Hölder continuity conditions

Catherine Cabuzel; Alain Pietrus

Displaying similar documents to “Solving variational inclusions by a multipoint iteration method under center-Hölder continuity conditions”

Hukuhara's differentiable iteration semigroups of linear set-valued functions

Andrzej Smajdor (2004)

Annales Polonici Mathematici

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Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. A family $F^{t} : t \geq 0$ of continuous linear set-valued functions $F^{t} : K \to c c (K)$ is a differentiable iteration semigroup with F⁰(x) = x for x ∈ K if and only if the set-valued function $Φ (t, x) = F^{t} (x)$ is a solution of the problem $D_{t} Φ (t, x) = Φ (t, G (x)) : = ⋃ Φ (t, y) : y \in G (x)$ , Φ(0,x) = x, for x ∈ K and t ≥ 0, where $D_{t} Φ (t, x)$ denotes the Hukuhara derivative of Φ(t,x) with respect to t and $G (x) : = l i m_{s \to 0 +} (F^{s} (x) - x) / s$ for x ∈ K.

$P^{λ}$ -spaces and $L^{λ}$ -spaces

Nguyen To Nhu (1979)

Colloquium Mathematicae

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Sequentially Right Banach spaces of order $p$

Mahdi Dehghani, Mohammad B. Dehghani, Mohammad S. Moshtaghioun (2020)

Commentationes Mathematicae Universitatis Carolinae

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We introduce and study two new classes of Banach spaces, the so-called sequentially Right Banach spaces of order $p$ , and those defined by the dual property, the sequentially Right $^{*}$ Banach spaces of order $p$ for $1 \leq p \leq \infty$ . These classes of Banach spaces are characterized by the notions of $L_{p}$ -limited sets in the corresponding dual space and $R_{p}^{*}$ subsets of the involved Banach space, respectively. In particular, we investigate whether the injective tensor product of a Banach space $X$ and a reflexive Banach...

On Banach spaces C(K) isomorphic to c₀(Γ)

Witold Marciszewski (2003)

Studia Mathematica

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We give a characterization of compact spaces K such that the Banach space C(K) is isomorphic to the space c₀(Γ) for some set Γ. As an application we show that there exists an Eberlein compact space K of weight $ω_{ω}$ and with the third derived set $K^{(3)}$ empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and $c ₀ (ω_{ω})$ are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.

Generalized notions of amenability for a class of matrix algebras

Amir Sahami (2019)

Commentationes Mathematicae Universitatis Carolinae

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We investigate the amenability and its related homological notions for a class of $I \times I$ -upper triangular matrix algebra, say $UP (I, A)$ , where $A$ is a Banach algebra equipped with a nonzero character. We show that $UP (I, A)$ is pseudo-contractible (amenable) if and only if $I$ is singleton and $A$ is pseudo-contractible (amenable), respectively. We also study pseudo-amenability and approximate biprojectivity of $UP (I, A)$ .

Some remarks on descriptive characterizations of the strong McShane integral

Sokol Bush Kaliaj (2019)

Mathematica Bohemica

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We present the full descriptive characterizations of the strong McShane integral (or the variational McShane integral) of a Banach space valued function $f : W \to X$ defined on a non-degenerate closed subinterval $W$ of $ℝ^{m}$ in terms of strong absolute continuity or, equivalently, in terms of McShane variational measure $V_{ℳ} F$ generated by the primitive $F : ℐ_{W} \to X$ of $f$ , where $ℐ_{W}$ is the family of all closed non-degenerate subintervals of $W$ .

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...

Local analysis of a cubically convergent method for variational inclusions

Steeve Burnet, Alain Pietrus (2011)

Applicationes Mathematicae

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This paper deals with variational inclusions of the form 0 ∈ φ(x) + F(x) where φ is a single-valued function admitting a second order Fréchet derivative and F is a set-valued map from $ℝ^{q}$ to the closed subsets of $ℝ^{q}$ . When a solution z̅ of the previous inclusion satisfies some semistability properties, we obtain local superquadratic or cubic convergent sequences.

On the Aronszajn property for integral equations in Banach space

Stanisław Szufla (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$ , belonging to the class $L^{p}(J,E)$ , $p>1$ . Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

Variational Henstock integrability of Banach space valued functions

Luisa Di Piazza, Valeria Marraffa, Kazimierz Musiał (2016)

Mathematica Bohemica

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We study the integrability of Banach space valued strongly measurable functions defined on $[0, 1]$ . In the case of functions $f$ given by $\sum_{n = 1}^{\infty} x_{n} χ_{E_{n}}$ , where $x_{n}$ are points of a Banach space and the sets $E_{n}$ are Lebesgue measurable and pairwise disjoint subsets of $[0, 1]$ , there are well known characterizations for Bochner and Pettis integrability of $f$ . The function $f$ is Bochner integrable if and only if the series $\sum_{n = 1}^{\infty} x_{n} | E_{n} |$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability...

An extension of Mazur's theorem on Gateaux differentiability to the class of strongly α (·)-paraconvex functions

S. Rolewicz (2006)

Studia Mathematica

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Let (X,||·||) be a separable real Banach space. Let f be a real-valued strongly α(·)-paraconvex function defined on an open convex subset Ω ⊂ X, i.e. such that $f (t x + (1 - t) y) \leq t f (x) + (1 - t) f (y) + m i n [t, (1 - t)] α (| | x - y | |)$ . Then there is a dense $G_{δ}$ -set $A_{G} \subset Ω$ such that f is Gateaux differentiable at every point of $A_{G}$ .

On the Aronszajn property for integral equations in Banach space

Stanisław Szufla (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The new iteration methods for solving absolute value equations

Rashid Ali, Kejia Pan (2023)

Applications of Mathematics

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Many problems in operations research, management science, and engineering fields lead to the solution of absolute value equations. In this study, we propose two new iteration methods for solving absolute value equations $A x - | x | = b$ , where $A \in ℝ^{n \times n}$ is an $M$ -matrix or strictly diagonally dominant matrix, $b \in ℝ^{n}$ and $x \in ℝ^{n}$ is an unknown solution vector. Furthermore, we discuss the convergence of the proposed two methods under suitable assumptions. Numerical experiments are given to verify the feasibility, robustness...

A characterization of complex $L_{1}$ -preduals via a complex barycentric mapping

Petr Petráček, Jiří Spurný (2016)

Commentationes Mathematicae Universitatis Carolinae

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We provide a complex version of a theorem due to Bednar and Lacey characterizing real $L_{1}$ -preduals. Hence we prove a characterization of complex $L_{1}$ -preduals via a complex barycentric mapping.

Generalized gradients for locally Lipschitz integral functionals on non- $L^{p}$ -type spaces of measurable functions

Hôǹg Thái Nguyêñ, Dariusz Pączka (2008)

Banach Center Publications

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Let (Ω,μ) be a measure space, E be an arbitrary separable Banach space, $E *_{ω *}$ be the dual equipped with the weak* topology, and g:Ω × E → ℝ be a Carathéodory function which is Lipschitz continuous on each ball of E for almost all s ∈ Ω. Put $G (x) : = \int_{Ω} g (s, x (s)) d μ (s)$ . Consider the integral functional G defined on some non- $L^{p}$ -type Banach space X of measurable functions x: Ω → E. We present several general theorems on sufficient conditions under which any element γ ∈ X* of Clarke’s generalized gradient (multivalued...

Fréchet differentiability via partial Fréchet differentiability

Luděk Zajíček (2023)

Commentationes Mathematicae Universitatis Carolinae

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Let $X_{1}, \dots, X_{n}$ be Banach spaces and $f$ a real function on $X = X_{1} \times \dots \times X_{n}$ . Let $A_{f}$ be the set of all points $x \in X$ at which $f$ is partially Fréchet differentiable but is not Fréchet differentiable. Our results imply that if $X_{1}, \dots, X_{n - 1}$ are Asplund spaces and $f$ is continuous (respectively Lipschitz) on $X$ , then $A_{f}$ is a first category set (respectively a $σ$ -upper porous set). We also prove that if $X$ , $Y$ are separable Banach spaces and $f : X \to Y$ is a Lipschitz mapping, then there exists a $σ$ -upper porous set $A \subset X$ such that $f$ is Fréchet differentiable...

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...

On some properties of generalized Marcinkiewicz spaces

Evgeniy Pustylnik (2001)

Studia Mathematica

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We give a full solution of the following problems concerning the spaces $M_{φ} (X ⃗)$ : (i) to what extent two functions φ and ψ should be different in order to ensure that $M_{φ} (X ⃗) \neq M_{ψ} (X ⃗)$ for any nontrivial Banach couple X⃗; (ii) when an embedding $M_{φ} (X ⃗) ⊊ M_{ψ} (X ⃗)$ can (or cannot) be dense; (iii) which Banach space can be regarded as an $M_{φ} (X ⃗)$ -space for some (unknown beforehand) Banach couple X⃗.