### A contribution to the theory of countably modulared spaces of double sequences

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Let G be a locally compact Hausdorff group with Haar measure, and let L⁰(G) be the space of extended real-valued measurable functions on G, finite a.e. Let ϱ and η be modulars on L⁰(G). The error of approximation ϱ(a(Tf - f)) of a function $f\in {\left(L\u2070\left(G\right)\right)}_{\varrho +\eta}\cap DomT$ is estimated, where $\left(Tf\right)\left(s\right)={\int}_{G}K(t-s,f\left(t\right))dt$ and K satisfies a generalized Lipschitz condition with respect to the second variable.

Let X denote the space of all real, bounded double sequences, and let Φ, φ, Γ be φ-functions. Moreover, let Ψ be an increasing, continuous function for u ≥ 0 such that Ψ(0) = 0.In this paper we consider some spaces of double sequences provided with two-modular structure given by generalized variations and the translation operator (...).

We obtain modular convergence theorems in modular spaces for nets of operators of the form $\left({T}_{w}f\right)\left(s\right)={\int}_{H}{K}_{w}(s-{h}_{w}\left(t\right),f\left({h}_{w}\left(t\right)\right))d{\mu}_{H}\left(t\right)$, w > 0, s ∈ G, where G and H are topological groups and ${{h}_{w}}_{w>0}$ is a family of homeomorphisms ${h}_{w}:H\to {h}_{w}\left(H\right)\subset G.$ Such operators contain, in particular, a nonlinear version of the generalized sampling operators, which have many applications in the theory of signal processing.

We investigate conditions under which the projective and the injective topologies coincide on the tensor product of two Köthe echelon or coechelon spaces. A major tool in the proof is the characterization of the επ-continuity of the tensor product of two diagonal operators from ${l}_{p}$ to ${l}_{q}$. Several sharp forms of this result are also included.

In this paper, we extend several concepts from geometry of Banach spaces to modular spaces. With a careful generalization, we can cover all corresponding results in the former setting. Main result we prove says that if $\rho $ is a convex, $\rho $-complete modular space satisfying the Fatou property and ${\rho}_{r}$-uniformly convex for all $r>0$, C a convex, $\rho $-closed, $\rho $-bounded subset of ${X}_{\rho}$, $T:C\to C$ a $\rho $-nonexpansive mapping, then $T$ has a fixed point.

Locally solid topologies on vector valued function spaces are studied. The relationship between the solid and topological structures of such spaces is examined.

In this paper we show that the ball-measure of non-compactness of a norm bounded subset of an Orlicz modular space L-Psi is equal to the limit of its n-widths. We also obtain several inequalities between the measures of non-compactness and the limit of the n-widths for modular bounded subsets of L-Psi which do not have Delta-2-condition. Minimum conditions on Psi to have such results are specified and an example of such a function Psi is provided.

We give results about embeddings, approximation and convergence theorems for a class of general nonlinear operators of integral type in abstract modular function spaces. Thus we extend some previous result on the matter.

The aim of this paper, is to introduce the convex structure (specially, Takahashi convex structure) on modular spaces. Moreover, we are interested in proving some common fixed point theorems for non-self mappings in modular space.

We introduce the spaces ${M}_{Y,\varphi}^{1}$, ${M}_{Y,\varphi}^{o,n}$, ${\tilde{M}}_{Y,\varphi}^{o}$ and ${M}_{Y,\mathbf{d},\varphi}^{o}$ of multifunctions. We prove that the spaces ${M}_{Y,\varphi}^{1}$ and ${M}_{Y,\mathbf{d},\varphi}^{o}$ are complete. Also, we get some convergence theorems.

We will present relationships between the modular ρ* and the norm in the dual spaces $\left({L}_{\Phi}\right)*$ in the case when a Musielak-Orlicz space ${L}_{\Phi}$ is equipped with the Orlicz norm. Moreover, criteria for extreme points of the unit sphere of the dual space $\left(L{\u2070}_{\Phi}\right)*$ will be presented.

The space of all order continuous linear functionals on an Orlicz space ${L}^{\varphi}$ defined by an arbitrary (not necessarily convex) Orlicz function $\varphi $ is described.