Wyniki wyszukiwania - BazTech

1

A variant of Jensen’s functional equation on semigroups

Fadli B., Zeglami D., Kabbaj S.

Demonstratio Mathematica

|

2016

|

Vol. 49, nr 4

414--420

EN

We determine the solutions f : S → H of the following functional equation f(xy) + f(σ(y)x) = 2f(x), x,y∈S, and the solutions f1, f2, f3 : M → H of the functional equation f1(xy) + f2(σ(y)x) = 2f3(x), x,y∈M, where S is a semigroup, M is a monoid, H is an abelian group 2-torsion free, and σ is an involutive automorphism.

2

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

|

2015

|

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.

3

On the Nemytskii operator in the space of functions of bounded (p, 2, α)-variation with respect to the weight function

Aziz W.

Demonstratio Mathematica

|

2014

|

Vol. 47, nr 4

933--948

EN

In this paper, we consider the Nemytskii operator (H f) (t) = h(t, f(t)), generated by a given function h. It is shown that if H is globally Lipschitzian and maps the space of functions of bounded (p, 2, α)-variation (with respect to a weight function α) into the space of functions of bounded (q, 2, α)-variation (with respect to α) 1 < q < p, then H is of the form (H f) (t) = A(t)f(t) + B(t). On the other hand, if 1 < p < q then H is constant. It generalize several earlier results of this type due to Matkowski–Merentes and Merentes. Also, we will prove that if a uniformly continuous Nemytskii operator maps a space of bounded variation with weight function in the sense of Merentes into another space of the same type, its generator function is an affine function.

4

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

|

2013

|

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.

5

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

|

2012

|

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.

6

On Nemytskij operator in the space of absolutely continuous set-valued functions

Ludew J. J.

Journal of Applied Analysis

|

2011

|

Vol. 17, nr 2

277-290

EN

We consider the Nemytskij operator, defined by (Nφ)(x) ? G(x, φ(x)), where G is a given set-valued function. It is shown that if N maps AC(I, C), the space of all absolutely continuous functions on the interval I ? [0, 1] with values in a cone C in a reflexive Banach space, into AC(I, K), the space of all absolutely continuous set-valued functions on I with values in the set K, consisting of all compact intervals (including degenerate ones) on the real line R, and N is uniformly continuous, then the generator G is of the form G(x, y) = A(x)(y) + B(x), where the function A(x) is additive and uniformly continuous for every x ∈ I and, moreover, the functions x ? A(x)(y) and B are absolutely continuous. Moreover, a condition, under which the Nemytskij operator maps the space AC(I, C) into AC(I, K) and is Lipschitzian, is given.

7

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

|

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.

8

Uniformly continuous composition operator in the space of φ-variation functions in the sense of Riesz

Acosta A., Aziz W., Matkowski J., Merentes N.

Fasciculi Mathematici

|

2010

|

Nr 43

5-11

EN

Assuming that a Nemytskii operator maps a subset of the space of bounded variation functions in the sense of Riesz into another space of the same type, and is uniformly continuous, we prove that the generator of the operator is an affine function.

9

Uniformly continuous composition operators in the space of functions of two variables of bounded φ-variation in the sense of Wiener

Guerrero J.A., Matkowski J., Merentes N.

Commentationes Mathematicae

|

2010

|

Vol. 50, [Z] 1

41-48

EN

Assume that the generator of a Nemytskii composition operator is a function of three variables: the first two real and third in a closed convex subset of a normed space, with values in a real Banach space. We prove that if this operator maps a certain subset of the Banach space of functions of two real variables of bounded Wiener φ-variation into another Banach space of a similar type, and is uniformly continuous, then the one-sided regularizations of the generator are affine in the third variable.

10

Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Riesz

Aziz W., Guerrero J., Merentes N.

Bulletin of the Polish Academy of Sciences. Mathematics

|

2010

|

Vol. 58, no 1

39-45

EN

We show that any uniformly continuous and convex compact valued Nemytskii composition operator acting in the spaces of functions of bounded φ-variation in the sense of Riesz is generated by an affine function.

11

On Nemytskij operator of substitution in the C1 space of set-valued functions

Ludew J.

Demonstratio Mathematica

|

2008

|

Vol. 41, nr 2

403-414

EN

We consider the Nemytskij operator, i. e., the operator of substitution, defined by (N[...]x) := G(x,<[...](x)), where G is a given multifunction. It is shown that N maps C1 (I, C), the space of all continuously differentiable functions on the interval I with values in a cone C in a Banach space, into C1 (I, cc(Z)), the space of all continuously differentiable set-functions on I with compact and convex values in a Banach space Z and N fulfils the Lipschitz condition if and only if the generator G is of the form G(x,y)=A(x,y) + B(x) where A(x, •) is continuous, linear function, A(.,y) and B are continuously differentiable and the function x— > A(x, •) is Lipschitzian.

12

On Lipschitzian operators of substitution generated by set-valued functions

Ludew J. J.

Opuscula Mathematica

|

2007

|

Vol. 27, no. 1

13-24

EN

We consider the Nemytskii operator, i.e., the operator of substitution, defined by (Nφ)(x) := G(x,φ(x)), where G is a given multifunction. It is shown that if N maps a Hölder space Hα into Hβ and N fulfils the Lipschitz condition then G(x,y) = A(x,y) + B(x), where A(x,·) is linear and A(·,y), B ∈ Hβ. Moreover, some conditions are given under which the Nemytskii operator generated by (1) maps Hα into Hβ and is Lipschitzian.

13

Characterization of globally Lipschitz Nemytskii operators between spaces of set-valued functions of bounded [phi]- variation in the sense of Riesz

Merentes N., Hernandez S.

Bulletin of the Polish Academy of Sciences. Mathematics

|

2004

|

Vol. 52, nr 4

417-430

EN

Let (X, || . ||) and [Y, || . ||] be two normed spaces and K be a convex cone in X. Let CC(Y) be the family of all non-empty convex compact subsets of Y. We consider the Nemytskii operators, i.e. the composition operators defined by [Nu)(t) = H(t,u[t)), where H is a given set-valued function. It is shown that if the operator N maps the space RV[phi]1 ([a, b]; K) into RW[phi]2([a, b]; CC[Y)) (both are spaces of functions of bounded [phi]- variation in the sense of Riesz), and if it is globally Lipschitz, then it has to be of the form H(t,u[t)) = A(t]u(t)+B(t), where A(t) is a linear continuous set-valued function and B is a set-valued function of bounded [phi]2-variation in the sense of Riesz. This generalizes results of G. Zawadzka [12], A. Smajdor and W. Smajdor [II], N. Merentes and K. Nikodem [5], and N. Merentes and S. Rivas [7].

14

On conditional Jensen equation

Ziółkowski M.

Demonstratio Mathematica

|

2001

|

Vol. 34, nr 4

810-818

15

Note on Jensen and Pexider functional equations

Smajdor W.

Demonstratio Mathematica

|

1999

|

Vol. 32, nr 2

363-376

EN

We determine the general solutions of the Jensen functional equation 2f (x+y):2=f(x) + f(y), x,y zawiera się M and the Pexider functional equation f(x+y)=g(x)+h(y), x,y zawiera się M , for f, g, h : M --+ S , where M is an Abelian semigroup with the division by 2 and S is an abstract convex cone satisfying the cancellation law. Some applications to set-valued versions of these equations are given.