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On Bi-dimensional Second Variation

100%

Ereú J., Giménez J., Merentes N.

Commentationes Mathematicae

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2012

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tom 52

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nr 1

EN

In this paper we present the concept of bounded second variation of a real valued function defined on a rectangle in $\mathbb{R}^2$. We use Hardy-Vitali type technics in the plane in order to extend the classical notion of function of bounded second variation on intervals of $\mathbb{R}$. We introduce the class $BV^2(I_a^b )$, of all functions of bounded second variation on a rectangle $I_a^b \subset \mathbb{R}^2$, and show that this class can be equipped with a norm with respect to which it is a Banach space. Finally, we present two results that show that integrals of functions of first bounded variation (on $I_a^b$) are in $BV^2 (I_a^b)$.

2

On Korenblum convex functions

100%

Lopez L. M., Ereú J., Merentes N.

Commentationes Mathematicae

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2017

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tom 57

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nr 1

EN

We introduce a new class of generalized convex functions called the $\kappa$-convex functions, based on Korenblum's concept of $\kappa$-decreasing functions, where $\kappa$ is an entropy (distortion) function. We study continuity and differentiability properties of these functions, and we discuss a special subclass which is a counterpart of the class of so-called d.c. functions. We characterize this subclass in terms of the space of functions of bounded second $\kappa$-variation, extending a result of F. Riesz. We also present a formal structural decomposition result for the $\kappa$-convex functions.

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Remarks on strongly Wright-convex functions

100%

Merentes N., Nikodem K., Rivas S.

Annales Polonici Mathematici

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2011

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tom 102

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nr 3

271-278

EN

Some properties of strongly Wright-convex functions are presented. In particular it is shown that a function f:D → ℝ, where D is an open convex subset of an inner product space X, is strongly Wright-convex with modulus c if and only if it can be represented in the form f(x) = g(x)+a(x)+c||x||², x ∈ D, where g:D → ℝ is a convex function and a:X → ℝ is an additive function. A characterization of inner product spaces by strongly Wright-convex functions is also given.

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Solutions of the Hammerstein equations in $BV_\varphi(I_{a}^{b},\, \mathbb{R})

100%

Aziz W., Guerrero J., Azócar L., Merentes N.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2016

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tom 36

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nr 2

207-229

EN

In this paper we study existence and uniqueness of solutions for the Hammerstein equation \[ u(x)= v(x) + \lambda \int_{I_{a}^{b}}K(x,y)f(y,u(y))dy \] \noi in the space of function of bounded total $\varphi$-variation in the sense of Hardy-Vitali-Tonelli, where $\lambda\in \mathbb{R}$, $K:I_a^b\times I_a^b \longrightarrow \mathbb{R}$ and $f:I_a^b\times \mathbb{R} \longrightarrow \mathbb{R}$ are suitable functions. The existence and uniqueness of solutions are proved by means of the Leray-Schauder nonlinear alternative and the Banach contraction mapping principle.

5

Some remarks on the algebra of functions of two variables with bounded total $\Phi$-variation in Schramm sense

100%

Ereú T., Merentes N., Sánchez J. L.

Commentationes Mathematicae

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2010

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tom 50

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nr 1

EN

This paper is devoted to discuss some generalizations of the bounded total $\Phi$-variation in the sense of Schramm. This concept was defined by W. Schramm for functions of one real variable. In the paper we generalize the concept in question for the case of functions of of two variables defined on certain rectangle in the plane. The main result obtained in the paper asserts that the set of all functions having bounded total $\Phi$-variation in Schramm sense has the structure of a Banach algebra.

6

Superposition Operators in the Space of Functions of Waterman-Shiba Bounded Variation

100%

Gimenez J., Merentes N., Rodriguez L.

Commentationes Mathematicae

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2014

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tom 54

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nr 1

EN

In this paper we present a necessary condition for an autonomous superposition operator to act in the space of functions of Waterman-Shiba bounded variation. We also show that if a (general) superposition operator applies such space into itself and it is uniformly bounded, then its generating function satisfies a weak Matkowski condition.

7

Functions of bounded variations on compact subsets of $\mathbb{C}$

100%

Gimenez J., Merentes N., Vivas M.

Commentationes Mathematicae

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2014

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tom 54

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nr 1

EN

In this paper we introduce the concept of bounded variation for functions defined on compact subsets of the complex plane $\mathbb{C}$, based on the notion of variation along a curve as defined by Ashton and Doust; We describe in detail the space so generated and show that it can be equipped, in a natural way, with the structure of a Banach algebra. We also present a necessary condition for a composition operator $C_\varphi$ to act between two such spaces.

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A Representation Theorem for $\varphi$-Bounded Variation of Functions in the Sense of Riesz

100%

Aziz W., Leiva H., Merentes N., Rzepka B.

Commentationes Mathematicae

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2010

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tom 50

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nr 2

EN

In this paper we extend the well known Riesz lemma to the class of bounded $\varphi$-variation functions in the sense of Riesz defined on a rectangle $I_a^b\subset \mathbb{R}^2$. This concept was introduced in [2], where the authors proved that the space $BV_\varphi^R (I_a^b;\mathbb{R}$ of such functions is a Banach Algebra. Moreover, they characterized also the Nemytskii operator acting in this space. Thus our result creates a continuation of the paper [2].

9

A Generalization of m-Convexity and a Sandwich Theorem

88%

Lara T., Matkowski J., Merentes N., Quintero R., Wróbel M.

Annales Mathematicae Silesianae

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2017

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tom 31

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nr 1

107-126

EN

Functional inequalities generalizing m-convexity are considered. A result of a sandwich type is proved. Some applications are indicated.

Ograniczanie wyników

On Bi-dimensional Second Variation

On Korenblum convex functions

Remarks on strongly Wright-convex functions

Solutions of the Hammerstein equations in $BV_\varphi(I_{a}^{b},\, \mathbb{R})

Some remarks on the algebra of functions of two variables with bounded total \(\Phi\)-variation in Schramm sense

Superposition Operators in the Space of Functions of Waterman-Shiba Bounded Variation

Functions of bounded variations on compact subsets of \(\mathbb{C}\)

A Representation Theorem for \(\varphi\)-Bounded Variation of Functions in the Sense of Riesz

A Generalization of m-Convexity and a Sandwich Theorem