Nonsquareness in Musielak-Orlicz-Bochner Function Spaces

Nonsquareness in Musielak-Orlicz-Bochner Function Spaces

and Applied Analysis 3 Proposition 1.2. Function σ t is μ-measurable. Proof. Pick a dense set {ri}i 1 in 0,∞ and set Bk { t ∈ T : M ( t, 1 2 rk ) 1 2 M t, rk } , qk t rkχBk t k ∈ N . 1.7 It is easy to see that for all k ∈ N, σ t ≥ qk t μ-a.e on T . Hence, supk≥1qk t ≤ σ t . For μ-a.e t ∈ T , arbitrarily choose ε ∈ 0, σ t . Then, there exists rk ∈ σ t − ε, σ t such that M t, 1/2 rk 1/2 M t, rk ,...

Nonsquareness and Locally Uniform Nonsquareness in Orlicz-Bochner Function Spaces Endowed with Luxemburg Norm

Extreme Points and Rotundity in Musielak-Orlicz-Bochner Function Spaces Endowed with Orlicz Norm

and Applied Analysis 3 Put LM X { u ∈ XT : ρM λu < ∞ for some λ > 0 } . 1.5 Then the Musielak-Orlicz-Bochner function space ‖u‖ inf k>0 1 k [ 1 ρM ku ] 1.6 is Banach space. If X R, LM R is said to be Musielak-Orlicz function space. Set K u { k > 0 : 1 k ( 1 ρM ku ) ‖u‖ } . 1.7 In particular, the set K u can be nonempty. To show that, we give a proposition. Proposition 1.1. If limu→∞ M t, u /u ∞...

Some Approximation Properties in Musielak-Orlicz-Sobolev Spaces

Some approximation theorems involving the modular convergence, which improve known density results of interest in the existence theory for strongly nonlinear boundary value problems are presented.

SOME MULTIPLIER SEQUENCE SPACES DEFINED BY A MUSIELAK-ORLICZ FUNCTION IN n-NORMED SPACES

In the present paper we introduced some multiplier sequence spaces defined by a Musielak-Orlicz function M = (Mk) in n-normed spaces and examine some topological properties and some inclusion relation between the resulting sequence spaces.