scholarly journals On Functions of Bounded (φ, k)-Variation

2019 ◽  
Vol 74 (1) ◽  
pp. 91-116
Author(s):  
Hugo Leiva ◽  
Nelson Merentes ◽  
Sergio T. Rivas ◽  
José Sánchez ◽  
Małgorzata Wróbel
Keyword(s):  
Banach Space ◽  
Real Function ◽  
Compact Interval ◽  
Second Variation ◽  
Absolutely Continuous ◽  
Orlicz Class ◽  
Class Of Functions ◽  
De La Vallée Poussin

Abstract Given a φ-function φ and k ∈ ℕ, we introduce and study the concept of (φ, k)-variation in the sense of Riesz of a real function on a compact interval. We show that a function u :[a, b] → ℝ has a bounded (φ, k)-variation if and only if u(k−1) is absolutely continuous on [a, b]and u(k) belongs to the Orlicz class L φ[a, b]. We also show that the space generated by this class of functions is a Banach space. Our approach simultaneously generalizes the concepts of the Riesz φ-variation, the de la Vallée Poussin second-variation and the Popoviciu kth variation.

scholarly journals On Functions of Bounded(p,k)-Variation

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
N. Merentes ◽  
S. Rivas ◽  
J. L. Sanchez
Keyword(s):  
Sobolev Space ◽  
Real Function ◽  
Compact Interval ◽  
Absolutely Continuous ◽  
Alternative Characterization ◽  
The One

We introduce and study the concept of(p,k)-variation (1<p<∞,k∈N)of a real function on a compact interval. In particular, we prove that a functionu:[a,b]→Rhas bounded(p,k)-variation if and only ifu(k-1)is absolutely continuous on[a,b]andu(k)belongs toLp[a,b]. Moreover, an explicit connection between the(p,k)-variation ofuand theLp-norm ofu(k)is given which is parallel to the classical Riesz formula characterizing functions in the spacesRVp[a,b]andAp[a,b]. This may also be considered as an alternative characterization of the one variable Sobolev spaceWpk[a,b].

scholarly journals The spectral theorem for well-bounded operators

1993 ◽  
Vol 54 (3) ◽  
pp. 334-351 ◽  
Author(s):  
Ian Doust ◽  
Qiu Bozhou
Keyword(s):  
Functional Calculus ◽  
Continuous Functions ◽  
Weak Compactness ◽  
Compact Interval ◽  
Spectral Theorem ◽  
Type B ◽  
Bounded Operators ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous

AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

scholarly journals The Bi-dimensional space of Korenblum and composition operator

2015 ◽  
Vol 62 (1) ◽  
pp. 1-12
Author(s):  
José A. Guerrero ◽  
Nelson Merentes ◽  
José L. Sánchez
Keyword(s):  
Banach Space ◽  
Composition Operator ◽  
Real Function ◽  
Dimensional Space ◽  
Similar Type ◽  
Functions Of Two Variables ◽  
Real Functions ◽  
Uniformly Continuous ◽  
Uniformly Bounded

Abstract In this paper we present the concept of total κ-variation in the sense of Hardy-Vitali-Korenblum for a real function defined in the rectangle Iab⊂R2. We show that the space κBV(Iab, R) of real functions of two variables with finite total κ-variation is a Banach space endowed with the norm ||f||κ = |f (a)| + κTV( f, Iab). Also, we characterize the Nemytskij composition operator H that maps the space of functions of two real variables of bounded κ-variation κBV(Iab, R) into another space of a similar type and is uniformly bounded (or Lipschitzian or uniformly continuous).

scholarly journals Local Convergence and Attraction Basins of Higher Order, Jarratt-Like Iterations

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1203
Author(s):  
Janak Raj Sharma ◽  
Deepak Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Local Convergence ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Geometrical Approach ◽  
Stable Convergence ◽  
Numerical Testing ◽  
Taylor Expansions ◽  
Attraction Basins ◽  
Class Of Functions

We studied the local convergence of a family of sixth order Jarratt-like methods in Banach space setting. The procedure so applied provides the radius of convergence and bounds on errors under the conditions based on the first Fréchet-derivative only. Such estimates are not proposed in the approaches using Taylor expansions of higher order derivatives which may be nonexistent or costly to compute. In this sense we can extend usage of the methods considered, since the methods can be applied to a wider class of functions. Numerical testing on examples show that the present results can be applied to the cases where earlier results are not applicable. Finally, the convergence domains are assessed by means of a geometrical approach; namely, the basins of attraction that allow us to find members of family with stable convergence behavior and with unstable behavior.

scholarly journals The Nemitskij operator on Lipk-type and BVk-type spaces

2013 ◽  
Vol 46 (3) ◽  
Author(s):  
José Giménez ◽  
Lorena López ◽  
N. Merentes
Keyword(s):  
Bounded Variation ◽  
Continuous Functions ◽  
Function Spaces ◽  
Compact Interval ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Type Spaces

AbstractIn this paper, we discuss and present various results about acting and boundedness conditions of the autonomous Nemitskij operator on certain function spaces related to the space of all real valued Lipschitz (of bounded variation, absolutely continuous) functions defined on a compact interval of ℝ. We obtain a result concerning the integrability of products of the form

scholarly journals Smooth variational principles in Radon-Nikodým spaces

1999 ◽  
Vol 60 (1) ◽  
pp. 109-118 ◽  
Author(s):  
Robert Deville ◽  
Abdelhakim Maaden
Keyword(s):  
Banach Space ◽  
Real Function ◽  
Variational Principles ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Weakly Continuous ◽  
Fréchet Differentiable

We prove that if f is a real valued lower semicontinuous function on a Banach space X, for which there exist a > 0 and b ∈ ℝ such that f(x) ≥ 2a∥x∥ + b, x ∈ X, and if X has the Radon-Nikody´m property, then for every Ε > 0 there exists a real function φ X such that φ is Fréchet differentiable, ∥φ∥∞ < Ε, ∥φ′∥∞ < Ε, φ′ is weakly continuous and f + φ attains a minimum on X. In addition, if we assume that the norm in X is β-smooth, we can take the function φ = g1 + g2 where g1 is radial and β-smooth, g2 is Fréchet differentiable, ∥g1∥∞ < Ε, ∥g2∥∞ < Ε, ∥g′1∥∞ < Ε, ∥g′1∥∞ < Ε, g′2 is weakly continuous and f + g1 + g2 attains a minimum on X.

scholarly journals A qualitative study on generalized Caputo fractional integro-differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed D. Kassim ◽  
Thabet Abdeljawad ◽  
Wasfi Shatanawi ◽  
Saeed M. Ali ◽  
Mohammed S. Abdo
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Continuous Functions ◽  
Fractional Operator ◽  
Stability Of Solutions ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Banach’S Fixed Point Theorem

AbstractThe aim of this article is to discuss the uniqueness and Ulam–Hyers stability of solutions for a nonlinear fractional integro-differential equation involving a generalized Caputo fractional operator. The used fractional operator is generated by iterating a local integral of the form $(I_{a}^{\rho }f)(t)=\int _{a}^{t}f(s)s^{\rho -1}\,ds$ ( I a ρ f ) ( t ) = ∫ a t f ( s ) s ρ − 1 d s . Our reported results are obtained in the Banach space of absolutely continuous functions that rely on Babenko’s technique and Banach’s fixed point theorem. Besides, our main findings are illustrated by some examples.

scholarly journals Local Convergence of an Efficient Multipoint Iterative Method in Banach Space

Algorithms ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 25
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Iterative Method ◽  
Local Convergence ◽  
Numerical Experiments ◽  
Order Method ◽  
Eighth Order ◽  
First Derivative ◽  
Taylor Expansions ◽  
Derivative Free ◽  
Class Of Functions

We discuss the local convergence of a derivative-free eighth order method in a Banach space setting. The present study provides the radius of convergence and bounds on errors under the hypothesis based on the first Fréchet-derivative only. The approaches of using Taylor expansions, containing higher order derivatives, do not provide such estimates since the derivatives may be nonexistent or costly to compute. By using only first derivative, the method can be applied to a wider class of functions and hence its applications are expanded. Numerical experiments show that the present results are applicable to the cases wherein previous results cannot be applied.

scholarly journals Approximation by double Walsh polynomials

1992 ◽  
Vol 15 (2) ◽  
pp. 209-220 ◽  
Author(s):  
Ferenc Móricz
Keyword(s):  
Banach Space ◽  
Fourier Series ◽  
Uniform Convergence ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Partial Sums ◽  
Lipschitz Classes ◽  
Sufficient Conditions For Convergence ◽  
By Products ◽  
De La Vallée Poussin

We study the rate of approximation by rectangular partial sums, Cesàro means, and de la Vallée Poussin means of double Walsh-Fourier series of a function in a homogeneous Banach spaceX. In particular,Xmay beLp(I2), where1≦p<∞andI2=[0,1)×[0,1), orCW(I2), the latter being the collection of uniformlyW-continuous functions onI2. We extend the results by Watari, Fine, Yano, Jastrebova, Bljumin, Esfahanizadeh and Siddiqi from univariate to multivariate cases. As by-products, we deduce sufficient conditions for convergence inLp(I2)-norm and uniform convergence onI2as well as characterizations of Lipschitz classes of functions. At the end, we raise three problems.

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