Local analysis of a cubically convergent method for variational inclusions

Steeve Burnet; Alain Pietrus

Displaying similar documents to “Local analysis of a cubically convergent method for variational inclusions”

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

Catherine Cabuzel, Alain Pietrus (2007)

Applicationes Mathematicae

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We prove the existence of a sequence $(x_{k})$ satisfying $0 \in f (x_{k}) + \sum_{i = 1}^{M} a_{i} \nabla f (x_{k} + β_{i} (x_{k + 1} - x_{k})) (x_{k + 1} - x_{k}) + F (x_{k + 1})$ , where f is a function whose second order Fréchet derivative ∇²f satifies a center-Hölder condition and F is a set-valued map from a Banach space X to the subsets of a Banach space Y. We show that the convergence of this method is superquadratic.

$α$ -continuous multivalued maps

J. Eweret (1989)

Matematički Vesnik

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A Short Proof of the Variational Principle for a $ℤ_{+}^{N}$ Action on a Compact Space

Michal Misiurewicz (1975)

Publications mathématiques et informatique de Rennes

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A regularity theory for scalar local minimizers of splitting-type variational integrals

Michael Bildhauer, Martin Fuchs, Xiao Zhong (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $(p, q)$ -growth with exponents $p \leq q < \infty$ and show for the scalar case that locally bounded local minimizers are of class $C^{1, μ}$ . Note that to our knowledge the only $C^{1, μ}$ -results without imposing a relation between $p$ and $q$ concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

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...

Filippov Lemma for matrix fourth order differential inclusions

Grzegorz Bartuzel, Andrzej Fryszkowski (2014)

Banach Center Publications

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In the paper we give an analogue of the Filippov Lemma for the fourth order differential inclusions y = y”” - (A² + B²)y” + A²B²y ∈ F(t,y), (*) with the initial conditions y(0) = y’(0) = y”(0) = y”’(0) = 0, (**) where the matrices $A, B \in ℝ^{d \times d}$ are commutative and the multifunction $F : [0, 1] \times ℝ^{d} ⇝ c l (ℝ^{d})$ is Lipschitz continuous in y with a t-independent constant l < ||A||²||B||². Main theorem. Assume that $F : [0, 1] \times ℝ^{d} ⇝ c l (ℝ^{d}) i s m e a s u r a b l e i n t a n d i n t e g r a b l y b o u n d e d . L e t$ y₀ ∈ W4,1 $b e a n a r b i t r a r y f u n c t i o n s a t i s f y i n g (* *) a n d s u c h t h a t$ $d_{H} (y ₀ (t), F (t, y ₀ (t))) \leq p ₀ (t)$ a.e. in [0,1], where p₀ ∈ L¹[0,1]. Then there exists a solution y ∈ W4,1 of (*)...

Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity

Djairo Guedes de Figueiredo, Jean-Pierre Gossez, Pedro Ubilla (2006)

Journal of the European Mathematical Society

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We study the existence, nonexistence and multiplicity of positive solutions for the family of problems $- Δ u = f_{λ} (x, u)$ , $u \in H_{0}^{1} (Ω)$ , where $Ω$ is a bounded domain in $ℝ^{N}$ , $N \geq 3$ and $λ > 0$ is a parameter. The results include the well-known nonlinearities of the Ambrosetti–Brezis–Cerami type in a more general form, namely $λ a (x) u^{q} + b (x) u^{p}$ , where $0 \leq q < 1 < p \leq 2^{*} - 1$ . The coefficient $a (x)$ is assumed to be nonnegative but $b (x)$ is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential...

Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps

Piotr Niemiec (2009)

Studia Mathematica

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It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure $C F L (_{r})$ of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space $_{r}$ of diameter r, is (isometrically if r = +∞) isomorphic to the space $L I P (_{r})$ of equivalence classes of all real-valued Lipschitz maps on $_{r}$ . The space of all signed (real-valued) Borel measures on $_{r}$ is isometrically embedded in the dual space of $C F L (_{r})$ and it is shown that the image...

Some remarks providing discontinuous maps on some $C_{p} (X)$ spaces

S. Moll (2008)

Banach Center Publications

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Let X be a completely regular Hausdorff topological space and $C_{p} (X)$ the space of continuous real-valued maps on X endowed with the pointwise topology. A simple and natural argument is presented to show how to construct on the space $C_{p} (X)$ , if X contains a homeomorphic copy of the closed interval [0,1], real-valued maps which are everywhere discontinuous but continuous on all compact subsets of $C_{p} (X)$ .

Weighted local Orlicz-Hardy spaces with applications to pseudo-differential operators

Dachun Yang, Sibei Yang

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Let Φ be a concave function on (0,∞) of strictly critical lower type index $p_{Φ} \in (0, 1]$ and $ω \in A_{\infty}^{l o c} (ℝ ⁿ)$ (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space $h_{ω}^{Φ} (ℝ ⁿ)$ via the local grand maximal function. Let $ρ (t) \equiv t^{- 1} / Φ^{- 1} (t^{- 1})$ for all t ∈ (0,∞). We also introduce the BMO-type space $b m o_{ρ, ω} (ℝ ⁿ)$ and establish the duality between $h_{ω}^{Φ} (ℝ ⁿ)$ and $b m o_{ρ, ω} (ℝ ⁿ)$ . Characterizations of $h_{ω}^{Φ} (ℝ ⁿ)$ , including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are...

Pressure and recurrence

Véronique Maume-Deschamps, Bernard Schmitt, Mariusz Urbański, Anna Zdunik (2003)

Fundamenta Mathematicae

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We deal with a subshift of finite type and an equilibrium state μ for a Hölder continuous function. Let αⁿ be the partition into cylinders of length n. We compute (in particular we show the existence of the limit) $l i m_{n \to \infty} n^{- 1} l o g \sum_{j = 0}^{τ ₙ (x)} μ (α ⁿ (T^{j} (x)))$ , where $α ⁿ (T^{j} (x))$ is the element of the partition containing $T^{j} (x)$ and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).

A new function space and applications

Jean Bourgain, Haïm Brezis, Petru Mironescu (2015)

Journal of the European Mathematical Society

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We define a new function space $B$ , which contains in particular BMO, BV, and $W^{1 / p, p}$ , $1 < p < \infty$ . We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving $L^{p}$ norms of integer-valued functions in $B$ . We introduce a significant closed subspace, $B_{0}$ , of $B$ , containing in particular VMO and $W^{1 / p, p}$ , $1 \leq p < \infty$ . The above mentioned estimates imply in particular that integer-valued functions belonging to $B_{0}$ are necessarily constant. This framework provides a “common roof”...

Lipschitz extensions of convex-valued maps

Alberto Bressan, Agostino Cortesi (1986)

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

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Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz $M$ , definita su un sottoinsieme di uno spazio di Hilbert $H$ a valori compatti e convessi in $\mathbb{R}^{n}$ , può essere estesa su tutto $H$ ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz $M$ .

On two-to-one continuous functions

J. Mioduszewski

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CONTENTSIntroduction................................................................................................................................................................................3I. General properties of k-to-one functions on locally compact spaces1. Multi-valued functions Ф and ψ......................................................................................................................................... 62. The proof of (I.11)..................................................................................................................................................................

On the Rockafellar theorem for $Φ^{γ (\cdot, \cdot)}$ -monotone multifunctions

S. Rolewicz (2006)

Studia Mathematica

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Let X be an arbitrary set, and γ: X × X → ℝ any function. Let Φ be a family of real-valued functions defined on X. Let $Γ : X \to 2^{Φ}$ be a cyclic $Φ^{γ (\cdot, \cdot)}$ -monotone multifunction with non-empty values. It is shown that the following generalization of the Rockafellar theorem holds. There is a function f: X → ℝ such that Γ is contained in the $Φ^{γ (\cdot, \cdot)}$ -subdifferential of f, $Γ (x) \subset \partial_{Φ}^{γ (\cdot, \cdot)} {f |}_{x}$ .

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 $A_{G}$ .

Finite element variational crimes in the case of semiregular elements

Alexander Ženíšek (1996)

Applications of Mathematics

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The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain $Ω$ whose boundary $\partial Ω$ is formed by two circles $Γ_{1}$ , $Γ_{2}$ with the same center $S_{0}$ and radii $R_{1}$ , $R_{2} = R_{1} + ϱ$ , where $ϱ ≪ R_{1}$ . On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for $u = 0$ are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus...

Integral equalities for functions of unbounded spectral operators in Banach spaces

Benedetto Silvestri

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The work is dedicated to investigating a limiting procedure for extending “local” integral operator equalities to “global” ones in the sense explained below, and to applying it to obtaining generalizations of the Newton-Leibniz formula for operator-valued functions for a wide class of unbounded operators. The integral equalities considered have the form $g (R_{F}) \int f_{x} (R_{F}) d μ (x) = h (R_{F})$ . (1) They involve functions of the kind $X ∋ x \mapsto f_{x} (R_{F}) \in B (F)$ , where X is a general locally compact space, F runs over a suitable class of Banach subspaces...

Generalized α-variation and Lebesgue equivalence to differentiable functions

Jakub Duda (2009)

Fundamenta Mathematicae

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We find conditions on a real function f:[a,b] → ℝ equivalent to being Lebesgue equivalent to an n-times differentiable function (n ≥ 2); a simple solution in the case n = 2 appeared in an earlier paper. For that purpose, we introduce the notions of $C B V G_{1 / n}$ and $S B V G_{1 / n}$ functions, which play analogous rôles for the nth order differentiability to the classical notion of a VBG⁎ function for the first order differentiability, and the classes $C B V_{1 / n}$ and $S B V_{1 / n}$ (introduced by Preiss and Laczkovich) for Cⁿ smoothness....

On compactness and connectedness of the paratingent

Wojciech Zygmunt (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this note we shall prove that for a continuous function $ϕ : Δ \to ℝ^{n}$ , where $Δ \subset ℝ$ , the paratingent of $ϕ$ at $a \in Δ$ is a non-empty and compact set in $ℝ^{n}$ if and only if $ϕ$ satisfies Lipschitz condition in a neighbourhood of $a$ . Moreover, in this case the paratingent is a connected set.

On the structure of sequences with forbidden zero-sum subsequences

W. D. Gao, R. Thangadurai (2003)

Colloquium Mathematicae

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We study the structure of longest sequences in $ℤ ₙ^{d}$ which have no zero-sum subsequence of length n (or less). We prove, among other results, that for $n = 2^{a}$ and d arbitrary, or $n = 3^{a}$ and d = 3, every sequence of c(n,d)(n-1) elements in $ℤ ₙ^{d}$ which has no zero-sum subsequence of length n consists of c(n,d) distinct elements each appearing n-1 times, where $c (2^{a}, d) = 2^{d}$ and $c (3^{a}, 3) = 9$ .

Some theorems of Korovkin type

Tomoko Hachiro, Takateru Okayasu (2003)

Studia Mathematica

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We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ} (X)$ ) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ} (Y)$ ), and $ϕ_{\infty}$ a linear isometry from M into C(Y) (resp., $C_{ℝ} (Y)$ ). We show, under the assumption that $Π_{N} \subset Π_{T}$ , where $Π_{N}$ is...

An interpolatory estimate for the UMD-valued directional Haar projection

Richard Lechner

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We prove an interpolatory estimate linking the directional Haar projection $P^{(ε)}$ to the Riesz transform in the context of Bochner-Lebesgue spaces $L^{p} (ℝ ⁿ; X)$ , 1 < p < ∞, provided X is a UMD-space. If $ε_{i ₀} = 1$ , the result is the inequality $| | P^{(ε)} {u | |}_{L^{p} (ℝ ⁿ; X)} {\leq C | | u | |}_{L^{p} (ℝ ⁿ; X)}^{1 /} | | R_{i ₀} u {| |}_{L^{p} (ℝ ⁿ; X)}^{1 - 1 /}$ , (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of $L^{p} (ℝ ⁿ; X)$ . In order to obtain the interpolatory result (1) we analyze stripe operators $S_{λ}$ , λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....