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On nonlinear differential equations in generalized Musielak-Orlicz spaces

Kozlowski W. M.

Commentationes Mathematicae

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2013

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Vol. 53, No. 2

13--33

EN

We consider ordinary differential equations u′(t)+(I−T)u(t)=0, where an unknown function takes its values in a given modular function space being a generalization of Musielak-Orlicz spaces, and T is nonlinear mapping which is nonexpansive in the modular sense. We demonstrate that under certain natural assumptions the Cauchy problem related to this equation can be solved. We also show a process for the construction of such a solution. This result is then linked to the recent results of the fixed point theory in modular function spaces.

2

Statistical approximation in multivariate modular function spaces

Belen C., Yildirim M.

Commentationes Mathematicae

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2011

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Vol. 51, [Z] 1

39-53

EN

In this paper, using the concept of A-statistical convergence we prove a Korovkin type approximation theorem in multivariate modular function spaces. Furthermore, giving an example via bivariate operators of Kantorovich type, it is shown that our theorem is stronger than its classical case.

3

On the Existence of Common Fixed Points for Semigroups of Nonlinear Mappings in Modular Function Spaces

Kozłowski W. M.

Commentationes Mathematicae

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2011

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Vol. 51, [Z] 1

81-98

EN

Let C be a ρ-bounded, ρ -closed, convex subset of a modular function space Lρ. We investigate the existence of common fixed points for semigroups of nonlinear mappings Tt : C → C, i.e. a family such that T0(x) = x, Ts+t = Ts(Tt(x)), where each Tt is either ρ -contraction or ρ -nonexpansive. We also briefly discuss existence of such semigroups and touch upon applications to differential equations.

4

On Cauchy Conservative Families of Nonlinear Integral Operators

Musielak A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2005

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Vol. 45, [Z] 1

1--11

EN

Let [formula] be nonlinear integral operators of the from (2). There is estimated the expresion [formula] where ρ is a modular on the space [formula]. This is applied in order to obtain a theorem concerning modular conservatitivy of a family [formula] of operators T of the form (2).

5

Characterization of the k-convexity property in the Besicovitch-Orlicz space of almost periodic functions

Bedouhene F., Morsli M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2005

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Vol. 45, [Z] 1

13--23

EN

In the present paper, we give criteria for the k-convexity of the Besicovitch-Orlicz space of almost periodic functions. Namely,it is shown that k-convexity is equivalent to strict convexity and reflexivity of this space in the case of Luxemburg norm.

6

Modular function spaces and control functions of almost everywhere convergence

Kita H., Miyamoto T., Yoneda K.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2001

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[Z] 41

99-133

EN

The modular spaces have been studied by many authors. In this paper we consider the modular function spaces which provide a generalization of Banach function spaces, and discuss some relations between properties of these spaces and control functions of almost everywhrere convergence of functions in modular function spaces. We give a necessary and sufficient condition for a control function to be in the same modular function space as functions appearing in almost everywhere convergence.

7

On one-sided trigonometric approximation in modular function spaces. II

Musielak A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2000

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[Z] 40

139-145

EN

There is proved the converse approximation theorem on one-sided trigonometric approximation of functions f from the modular space L^p_2Pi.

8

On one-sided trigonometric approximation in modular function spaces. I

Musielak A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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1999

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[Z] 39

127-138

EN

The purpose of this paper is to obtain the direct approximation theorems on one-sided approximation of 2(pi)-periodic functions f from a modular function space of 2(pi)- periodic functions by means of trigonometric polynomials of a given degree in the sense of the Luxemburg norm || || generated by a discretely convex modular .