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Content available remote On nonlinear differential equations in generalized Musielak-Orlicz spaces
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.
Content available remote Statistical approximation in multivariate modular function spaces
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.
Let C be a ρ-bounded, ρ-closed, convex subset of a modular function space L_ρ. We investigate the existence of common fixed points for semigroups of nonlinear mappings Tt : C→ C, i.e. a family such that To(x) = x, Ts+t = Ts(Tt(x)), where each T_t is either ρ-contraction or -nonexpansive. We also briefly discuss existence of such semigroups and touch upon applications to differential equations.
Content available remote On Cauchy Conservative Families of Nonlinear Integral Operators
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).
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.
Content available remote Modular function spaces and control functions of almost everywhere convergence
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.
Content available remote On one-sided trigonometric approximation in modular function spaces. II
There is proved the converse approximation theorem on one-sided trigonometric approximation of functions f from the modular space L^p_2Pi.
Content available remote On one-sided trigonometric approximation in modular function spaces. I
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 .
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