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1
Content available remote On Korenblum convex functions
Ereú J., López L., Merentes N.
Commentationes Mathematicae
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2017
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Vol 57, No. 1
25--44
EN
We introduce a new class of generalized convex functions called the K-convex functions, based on Korenblum’s concept of k-decreasing functions, where K 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 k-variation, extending a result of F. Riesz. We also present a formal structural decomposition result for the K-convex functions.
2
Content available Characterizations and decomposition of strongly wright-convex functions of higher order
Gilanyi A., Merentes N., Nikodem K., Pales Z.
Opuscula Mathematica
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2015
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Vol. 35, no. 1
37--46
EN
Motivated by results on strongly convex and strongly Jensen-convex functions by R. Ger and K. Nikodem in [Strongly convex functions of higher order, Nonlinear Anal. 74 (2011), 661-665] we investigate strongly Wright-convex functions of higher order and we prove decomposition and characterization theorems for them. Our decomposition theorem states that a function / is strongly Wright-convex of order n if and only if it is of the form [formula], where g is a (continuous) n-convex function and p is a polynomial function of degree n. This is a counterpart of Ng's decomposition theorem for Wright-convex functions. We also characterize higher order strongly Wright-convex functions via generalized derivatives.
3
Content available Approximate controllability of the impulsive semilinear heat equation
Leiva H., Merentes N.
Journal of Mathematics and Applications
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2015
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Vol. 38
85--104
EN
In this paper we apply Rothe's Fixed Point Theorem to prove the interior approximate controllability of the following semilinear impulsive Heat Equation [...] where k = 1, 2, . . . , p, Ω is a bounded domain in [...] is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to [...]. Under this condition we prove the following statement: For all open nonempty subsets ω of Ω the system is approximately controllable on [0, τ]. Moreover, we could exhibit a sequence of controls steering the nonlinear system from an initial state z0 to an ϵ neighborhood of the nal state z1 at time τ > 0.
4
Content available Uniformly bounded set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener
Guerrero J. A., Matkowski J., Merentes N., Wróbel M.
Journal of Applied Mathematics and Computational Mechanics
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2015
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Vol. 14, nr 4
41--51
EN
We show that the one-sided regularizations of the generator of any uniformly bounded set-valued Nemytskij composition operator mapping the space of bounded variation functions in the sense of Wiener into the space of bounded variation functions with closed bounded convex values (in the sense of Wiener) are affine functions.
5
Content available remote Uniformly continuous superposition operators in the space of functions of bounded n-dimensional Φ-variation
Bracamonte M., Ereú J., Giménez J., Merentes N.
Demonstratio Mathematica
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2014
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Vol. 47, nr 1
56--68
EN
We prove that if a superposition operator maps a subset of the space of all metric-vector-space-valued-functions of bounded n-dimensional Φ-variation into another such space, and is uniformly continuous, then the generating function of the operator is an affine function in the functional variable.
6
Content available remote On bi-dimensional second μ-variation
Ereú J., Giménez J., Merentes N.
Demonstratio Mathematica
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2014
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Vol. 47, nr 4
910--932
EN
In this paper, we present a generalization of the notion of bounded slope variation for functions defined on a rectangle Iba in R2. Given a strictly increasing function μ, defined in a closed real interval, we introduce the class BVμ,2 (Iba), of functions of bounded second μ-variation on Iba ; and show that this class can be equipped with a norm with respect to which it is a Banach space. We also deal with the important case of factorizable functions in BVμ,2 (Iba) and finally we exhibit a relation between this class and the one of double Riemann–Stieltjes integrals of functions of bi-dimensional bounded variation.
7
Content available remote Superposition Operators in the Space of Functions of Waterman-Shiba Bounded Variation
Giménez J., Merentes N., Rodriguez L.
Commentationes Mathematicae
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2014
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Vol 54, No. 1
79--93
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.
8
Content available remote Functions of bounded variations on compact subsets of C
Gimenez J., Merentes N., Vivas M.
Commentationes Mathematicae
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2014
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Vol 54, No. 1
3--19
EN
In this paper we introduce the concept of bounded variation for functions defined on compact subsets of the complex plane 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 Cφ to act between two such spaces.
9
Content available Remarks on uniformly bounded composition operator acting between Banach spaces of functions of two variables of bounded schramm Φ-variation
Ereú T., Merentes N., Wróbel M.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2013
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Vol. 18
19--27
EN
In this paper we prove that if the composition operator H of generator h : Ib a × C → Y (X is a real normed space, Y is a real Banach space, C is a convex cone in X and Ib a ⸦ R2) maps Φ1 BV (Ib a, C) into Φ2 BV (Ib a, Y) and is uniformly bounded, then the left-left regularization h* of h is an affine function in the third variable.
10
Content available remote On Nemytskii operator in the space of set-valued functions of bounded p-variation in the sense of Riesz with respect to the weight function
Aziz W., Guerrero J. A., Merentes N.
Fasciculi Mathematici
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2013
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Nr 50
17--32
EN
In this paper we consider the Nemytskii operator (Hf) (t) = h(t, f (t)), generated by a given set-valued function h is considered. It is shown that if H is globally Lipschitzian and maps the space of functions of bounded p-variation (with respect to a weight function α) into the space of set-valued functions of bounded q-variation (with respect to α) ) 1 < q < p, then H is of the form (Hϕ)(t) = A(t)ϕ(t) + B(t). On the other hand, if 1 < p < q, then H is constant. It generalizes many earlier results of this type due to Chistyakov, Matkowski, Merentes-Nikodem, Merentes-Rivas, Smajdor-Smajdor and Zawadzka.
11
Content available remote The Nemitskij operator on Lipk-type and BVk-type spaces
Giménez J., López L., Merentes N.
Demonstratio Mathematica
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2013
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Vol. 46, nr 3
543--558
EN
In 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 (…) and a generalized version of the chain rule for functions a.e differentiable, in the sense of Lebesgue. As an application, we get a generalization of a theorem due to V. I. Burenkov for the case of functions of bounded Riesz-p-variation.
12
Content available Uniformly continuous composition operator in the space of functions of two variables of Bounded ɸ-variation in the sense of Schramm
Aziz W., Ereú T., Merentes N., Sanchez J. L., Wróbel M.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2012
|
Vol. 17
7-16
EN
We prove in this paper that if the composition operator H, generated by a function h : I b a x C(Iba) Y , maps ɸBV (Iba ,C) into ɸ2 BV (Iba , Y ) and is uniformly continuous, then the left-left regularization h* of h is an affine function with respect to the third variable.
13
Content available Uniformly bounded set-valued composition operators in the spaces of functions of bounded variation in the sense of Schramm
Ereú T., Merentes N., Sanchez J. L., Wróbel M.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2012
|
Vol. 17
37-47
EN
We prove that, under some general assumptions, the one-sided regularizations of the generator of any uniformly bounded set-valued composition operator, acting in the spaces of functions of bounded variation in the sense of Schramm with nonempty bounded closed and convex values is an affine function. As a special case, we obtain an earlier result ([15]).
14
Content available Uniformly continuous composition operators in the space of bounded Φ-variation functions in the Schramm sense
Ereu T., Merentes N., Sanchez J. L., Wróbel M.
Opuscula Mathematica
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2012
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Vol. 32, no. 2
239-247
EN
We prove that any uniformly continuous Nemytskii composition operator in the space of functions of bounded generalized Φ-variation in the Schramm sense is affine. A composition operator is locally defined. We show that every locally defined operator mapping the space of continuous functions of bounded (in the sense of Jordan) variation into the space of continous monotonic functions is constant.
15
Content available Integral representation of functions of bounded second Φ-variation in the sense of Schramm
Gimenez J., Merentes N., Rivas S.
Opuscula Mathematica
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2012
|
Vol. 32, no. 1
137-151
EN
In this article we introduce the concept of second Φ--variation in the sense of Schramm for normed-space valued functions defined on an interval [a; b] ⊂ R. To that end we combine the notion of second variation due to de la Vallée Poussin and the concept of φ-variation in the sense of Schramm for real valued functions. In particular, when the normed space is complete we present a characterization of the functions of the introduced class by means of an integral representation. Indeed, we show that a function [formula] (where X is a reflexive Banach space) is of bounded second Φ-variation in the sense of Schramm if and only if it can be expressed as the Bochner integral of a function of (first) bounded variation in the sense of Schramm.
16
Content available remote Controllability of the semilinear Benjamin-Bona-Mahony equation
Leiva H., Merentes N., Sanchez J. L.
Journal of Mathematics and Applications
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2012
|
Vol. 35
39--51
EN
In this paper we prove the interior approximate controllability of the following Generalized Semilinear Benjamin-Bona-Mahony type equation (BBM) with homogeneous Dirichlet boundary conditions [formula/wzor] where a ≥ 0 and b > 0 are constans, Ω is a domain in IRN, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to L2(0, τ;L2(Ω)) and the nonlinear function ƒ:[0, τ] x IR x IR → IR is smooth enough and there are c, d, e ∈ IR, with c ≠ -1, ea + b > 0 such that [formula/wzor] where Qr = [0, τ] x IR x IR. We prove that for all τ > 0 and any nonempty open subset ω of Ω the system the system is approximately controllable on [0, τ]. Moreover, we exhibit a sequence of controls steering the system from an initial state z0 to an ε-neighborhood of the final state z1 on time > 0. As a consequence of this result we obtain the interior approximate controllability of the semilinear heat equation by putting a = 0 and b = 1.
17
Content available remote On Bi-dimensional Second Variation
Ereu J., Gimémez J., Merentes N.
Commentationes Mathematicae
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2012
|
Vol. 52, [Z] 1
39-59
EN
In this paper we present the concept of bounded second variation of a real valued function defined on a rectangle in R2. 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 R. We introduce the class [formula] of all functions of bounded second variation on a rectangle [formula] 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 are in [formula].
18
Content available Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Schramm
Ereú T., Sánchez J. L., Merentes N., Wróbel M.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2011
|
Vol. 16
23--32
EN
We show that the one-sided regularizations of the generator of any uniformly continuous set-valued Nemytskij operator, acting between the spaces of functions of bounded variation in the sense of Schramm, is an affine function. Results along these lines extend the study [1].
19
Content available Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener
Azocar A., Guerrero J. A., Matkowski J., Merentes N.
Opuscula Mathematica
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2010
|
Vol. 30, no. 1
53-60
EN
We show that the one-sided regularizations of the generator of any uniformly continuous and convex compact valued composition operator, acting in the spaces of functions of bounded variation in the sense of Wiener, is an affine function.
20
Content available Uniformly continuous set-valued composition operators in the space of total phi-bidimensional variation in the sense of Riesz
Aziz W., Gimenez J., Merentes N., Sanchez J. L.
Opuscula Mathematica
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2010
|
Vol. 30, no. 3
241-248
EN
In this paper we prove that if a Nemytskij composition operator, generated by a function of three variables in which the third variable is a function one, maps a suitable large subset of the space of functions of bounded total φ-bidimensional variation in the sense of Riesz, into another such space, and is uniformly continuous, then its generator is an affine function in the function variable. This extends some previous results in the one-dimensional setting.
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