Let lϕ be a Musielak-Orlicz sequence space. Let X1ϕ and Xϕ be the modular spaces of multifunctions generated by lϕ. Let Kw,j: R→R for j = 0,1,2,..., w∈W, where W is an abstract set of indices. Assuming certain singularity assumption on the nonlinear kernel Kw,j and setting Tw(F)=(Tw(F)(i))i=0∞ with (Tw(F))(i) = {Σj=0iKw,j(f(j)) : f(j)∈F(j)}, convergence theorems Tw(F)→ϕ F in X1ϕ and Tw(F)→d,ϕ, F in Xϕ are obtained.
This paper presents a constructive proof of the existence of a regular non-atomic strictly-positive measure on any second-countable non-atomic locally compact Hausdorff space. This construction involves a sequence of finitely-additive set functions defined recursively on an ascending sequence of rings of subsets with a set function limit that is extendable to a measure with the desired properties. Non-atomicity of the space provides a meticulous way to ensure that the set function limit is σ-additive.
The sequence spaces ruℓ∞(O, ∇q), ruℓp(O, ∇q), ruc(O, ∇q), ruc0(O, ∇q), rumφ(O, ∇q, p), runφ(O, ∇q, p), rumφ(O, ∇q), runφ(O, ∇q) are defined by the Orlicz function in this article. We examine all of its characteristics, including symmetry, solidity, and completeness. A few geometric properties on convexity on the space rumφ(O, ∇q, p) are also examined in this article.
In this short paper, I recall the history of dealing with the lack of compactness of a sequence in the case of an unbounded domain and prove the vanishing Lions-type result for a sequence of Lebesgue-measurable functions. This lemma generalizes some results for a class of Orlicz-Sobolev spaces. What matters here is the behavior of the integral, not the space.
Two dual sequence functions describing some kind of local convexity and dimension of subspaces of linear metric spaces are introduced. It is shown that the functions give a useful tool in the investigations of fixed point properties of the Schauder type.
The object of this paper is to obtain necessary and sufficient conditions to characterize the matrices in classes (l∞y(p,s),l∞(q)), (c0y(p,s),l∞(q)),(l∞y(p,s),c0(q)), and(c0y(p,s),c0(q)) which will fill up a gap in the existing literature.
2003-01-30
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English