and Applied Analysis 3 the fixed point theory and that a Banach space X has property L if and only if it is reflexive and has the uniform Opial property. For a real vector space X, a function ρ : X → 0,∞ is called amodular if it satisfies the following conditions: i ρ x 0 if and only if x 0; ii ρ αx ρ x for all scalar α with |α| 1; iii ρ αx βy ≤ ρ x ρ y , for all x, y ∈ X and all α, β ≥ 0 with ...

We obtain lacunary recurrent relations with gaps of length four for the Bernoulli and Euler polynomials and, as a consequence, known new numbers.

In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between Besov defined by differences and Fourier-analytical decompositions as well Sobolev/Triebel-Lizorkin spaces; Various new characterizations for norms in terms different K-functionals. For instance, derive...

The rational self-assembly of mono-lacunary Keggin clusters with 4d and 4f metal salts via a conventional method has yielded two novel polyoxometalate-based 4d-4f heterometallic compounds containing lacunary Keggin anions connected to ten metal atoms: {[Ag{Ag2(H2O)4}{Ln(H2O)6}2H ⊂ {SiW11Ln(H2O)4O39}2]·nH2O (Ln = Ce and n = 7 for 1, Ln = Pr and n = 3 for 2). Their structures were determined by s...

Based on the concept of lacunary statistical convergence sequences fuzzy numbers, convergence, uniformly and equi-lacunary double fuzzy-valued functions are defined investigated in this paper. The relationship among functions, their representations ? -level cuts discussed. In addition, we obtain form Egorov’s theorem for...