Local convergence of a one parameter fourth-order Jarratt-type method in Banach spaces

I. K. Argyros; D. González; S. K. Khattri

Displaying similar documents to “Local convergence of a one parameter fourth-order Jarratt-type method in Banach spaces”

A correction to “Some mean convergence and complete convergence theorems for sequences of $m$ -linearly negative quadrant dependent random variables”

Yongfeng Wu, Andrew Rosalsky, Andrei Volodin (2017)

Applications of Mathematics

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The authors provide a correction to “Some mean convergence and complete convergence theorems for sequences of $m$ -linearly negative quadrant dependent random variables”.

A note on necessary and sufficient conditions for convergence of the finite element method

Kučera, Václav

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In this short note, we present several ideas and observations concerning finite element convergence and the role of the maximum angle condition. Based on previous work, we formulate a hypothesis concerning a necessary condition for $O (h)$ convergence and show a simple relation to classical problems in measure theory and differential geometry which could lead to new insights in the area.

Convergence of sequences of iterates of random-valued vector functions

Rafał Kapica (2003)

Colloquium Mathematicae

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Given a probability space (Ω,, P) and a closed subset X of a Banach lattice, we consider functions f: X × Ω → X and their iterates $f ⁿ : X \times Ω^{ℕ} \to X$ defined by f¹(x,ω) = f(x,ω₁), $f^{n + 1} (x, ω) = f (f ⁿ (x, ω), ω_{n + 1})$ , and obtain theorems on the convergence (a.s. and in L¹) of the sequence (fⁿ(x,·)).

Inexact Newton-type method for solving large-scale absolute value equation $A x - | x | = b$

Jingyong Tang (2024)

Applications of Mathematics

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Newton-type methods have been successfully applied to solve the absolute value equation $A x - | x | = b$ (denoted by AVE). This class of methods usually solves a system of linear equations exactly in each iteration. However, for large-scale AVEs, solving the corresponding system exactly may be expensive. In this paper, we propose an inexact Newton-type method for solving the AVE. In each iteration, the proposed method solves the corresponding system only approximately. Moreover, it adopts a new line...

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

J. T. Marti (1971)

Colloquium Mathematicae

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Pointwise convergence of nonconventional averages

I. Assani (2005)

Colloquium Mathematicae

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We answer a question of H. Furstenberg on the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} U ⁿ (f \cdot R ⁿ (g))$ , where U and R are positive operators. We also study the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} f (S ⁿ x) g (R ⁿ x)$ when T and S are measure preserving transformations.

Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅²₊

Péter Kórus, Ferenc Móricz (2010)

Studia Mathematica

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We investigate the convergence behavior of the family of double sine integrals of the form $\int_{0}^{\infty} \int_{0}^{\infty} f (x, y) s i n u x s i n v y d x d y$ , where (u,v) ∈ ℝ²₊:= ℝ₊ × ℝ₊, ℝ₊:= (0,∞), and f: ℝ²₊ → ℂ is a locally absolutely continuous function satisfying certain generalized monotonicity conditions. We give sufficient conditions for the uniform convergence of the remainder integrals $\int_{a ₁}^{b ₁} \int_{a ₂}^{b ₂}$ to zero in (u,v) ∈ ℝ²₊ as maxa₁,a₂ → ∞ and $b_{j} > a_{j} \geq 0$ , j = 1,2 (called uniform convergence in the regular sense). This implies the uniform convergence of the partial...

On the randomized complexity of Banach space valued integration

Stefan Heinrich, Aicke Hinrichs (2014)

Studia Mathematica

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We study the complexity of Banach space valued integration in the randomized setting. We are concerned with r times continuously differentiable functions on the d-dimensional unit cube Q, with values in a Banach space X, and investigate the relation of the optimal convergence rate to the geometry of X. It turns out that the nth minimal errors are bounded by $c n^{- r / d - 1 + 1 / p}$ if and only if X is of equal norm type p.

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 the rate of convergence of difference schemes for the heat transfer equation on the solutions from $w_{2}^{s, s / 2}$

L. D. Ivanović, B. S. Jovanović, E. E. Süli (1984)

Matematički Vesnik

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Linearized plasticity is the evolutionary $Γ$ -limit of finite plasticity

Alexander Mielke, Ulisse Stefanelli (2013)

Journal of the European Mathematical Society

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We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via $Γ$ -convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity.

Local convergence comparison between two novel sixth order methods for solving equations

Santhosh George, Ioannis K. Argyros (2019)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples...

Convergence in the dual of certain $K M_{p}$ -spaces

Larry Kitchens, Charles Swartz (1974)

Colloquium Mathematicae

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On the completeness of $ℒ_{p}$ -spaces over a charge

Sreela Gangopadhyay (1990)

Colloquium Mathematicae

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Estimation of the Szlenk index of Banach spaces via Schreier spaces

Ryan Causey (2013)

Studia Mathematica

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For each ordinal α < ω₁, we prove the existence of a Banach space with a basis and Szlenk index $ω^{α + 1}$ which is universal for the class of separable Banach spaces with Szlenk index not exceeding $ω^{α}$ . Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.

Nilakantha's accelerated series for π

David Brink (2015)

Acta Arithmetica

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We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula $π = \sum_{n = 0}^{\infty} ((5 n + 3) n! (2 n)!) / (2^{n - 1} (3 n + 2)!)$ with convergence as $13 . 5^{- n}$ , in much the same way as the Euler transformation gives $π = \sum_{n = 0}^{\infty} (2^{n + 1} n! n!) / (2 n + 1)!$ with convergence as $2^{- n}$ . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in...

On the convergence of sequences of iterates of random-valued vector functions

Rafał Kapica (2007)

Annales Polonici Mathematici

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Given a probability space (Ω,,P) and a subset X of a normed space we consider functions f:X × Ω → X and investigate the speed of convergence of the sequence (fⁿ(x,·)) of the iterates $f ⁿ : X \times Ω^{ℕ} \to X$ defined by f¹(x,ω ) = f(x,ω₁), $f^{n + 1} (x, ω) = f (f ⁿ (x, ω), ω_{n + 1})$ .

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...

On the existence of non-linear frames

Shah Jahan, Varinder Kumar, S.K. Kaushik (2017)

Archivum Mathematicum

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A stronger version of the notion of frame in Banach space called Strong Retro Banach frame (SRBF) is defined and studied. It has been proved that if $𝒳$ is a Banach space such that $𝒳^{*}$ has a SRBF, then $𝒳$ has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space $𝒳$ has an approximative Schauder frame, then $𝒳^{*}$ has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.

On ultra-weak convergence in $L^{p}$

Donald E. Myers (1976)

Annales Polonici Mathematici

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$L_{\infty}$ -Convergence of Finite Element Approximation

J. A. Nitsche (1975)

Publications mathématiques et informatique de Rennes

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Remarks and examples concerning distance ellipsoids

Dirk Praetorius (2002)

Colloquium Mathematicae

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We provide for every 2 ≤ k ≤ n an n-dimensional Banach space E with a unique distance ellipsoid such that there are precisely k linearly independent contact points between and $B_{E}$ . The corresponding result holds for spaces with non-unique distance ellipsoids as well. We construct n-dimensional Banach spaces E such that one distance ellipsoid has precisely k linearly independent contact points and all other distance ellipsoids have less than k-1 such points.

(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...

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.

Geometry of Banach spaces and biorthogonal systems

S. Dilworth, Maria Girardi, W. Johnson (2000)

Studia Mathematica

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A separable Banach space X contains $ℓ_{1}$ isomorphically if and only if X has a bounded fundamental total $w c_{0} *$ -stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total $w c_{0} *$ -biorthogonal system.

The Dual of a Non-reflexive L-embedded Banach Space Contains $l^{\infty}$ Isometrically

Hermann Pfitzner (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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A Banach space is said to be L-embedded if it is complemented in its bidual in such a way that the norm between the two complementary subspaces is additive. We prove that the dual of a non-reflexive L-embedded Banach space contains $l^{\infty}$ isometrically.

Banach spaces of bounded Szlenk index II

D. Freeman, E. Odell, Th. Schlumprecht, A. Zsák (2009)

Fundamenta Mathematicae

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For every α < ω₁ we establish the existence of a separable Banach space whose Szlenk index is $ω^{α ω + 1}$ and which is universal for all separable Banach spaces whose Szlenk index does not exceed $ω^{α ω}$ . In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with Tsirelson type upper estimates.