The main goal of the paper is to formulate some new properties of the Ishlinskii hysteresis operator $F$, which characterizes e.g. the relation between the deformation and the stress in a non-perfectly elastic (elastico-plastic) material. We introduce two energy functionals and derive the energy inequalities. As an example we investigate the equation $u^{\text{'}\text{'}}+F\left(u\right)=0$ describing the motion of a mass point at the extremity of an elastico-plastic spring.

We provide a complete isomorphic classification of the Banach spaces of continuous functions on the compact spaces $2^{}\oplus [0,\alpha ]$, the topological sums of Cantor cubes $2^{}$, with smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. In particular, we prove that it is relatively consistent with ZFC that the only isomorphism classes of $C\left(2^{}\oplus [0,\alpha ]\right)$ spaces with ≥ ℵ₀ and α ≥ ω₁ are the trivial ones. This result leads to some elementary questions on large cardinals.

We prove that if Köthe F-spaces X and Y on finite atomless measure spaces (ΩX; ΣX, µX) and (ΩY; ΣY; µY), respectively, with absolute continuous norms are isomorphic and have the property $\underset{\mu \left(A\right)\to 0}{lim}∥\mu {\left(A\right)}^{-1}{1}_A∥=0$ (for µ = µX and µ = µY, respectively) then the measure spaces (ΩX; ΣX; µX) and (ΩY; ΣY; µY) are isomorphic, up to some positive multiples. This theorem extends a result of A. Plichko and M. Popov concerning isomorphic classification of L p(µ)-spaces for 0 < p < 1. We also provide a new class of F-spaces...

It is shown that Jackson type inequality fails in the Orlicz classes φ(L) if φ(x) differs essentially from a power function of any order.

Some relations between the James (or non-square) constant J(X) and the Jordan-von Neumann constant $C_{NJ}\left(X\right)$, and the normal structure coefficient N(X) of Banach spaces X are investigated. Relations between J(X) and J(X*) are given as an answer to a problem of Gao and Lau [16]. Connections between $C_{NJ}\left(X\right)$ and J(X) are also shown. The normal structure coefficient of a Banach space is estimated by the $C_{NJ}\left(X\right)$-constant, which implies that a Banach space with $C_{NJ}\left(X\right)$-constant less than 5/4 has the fixed point property.

In this article, we formalized L1 space formed by complexvalued partial functions [11], [15]. The real-valued case was formalized in [22] and this article is its generalization.

This article contains some definitions and properties refering to function spaces formed by partial functions defined over a measurable space. We formalized a function space, the so-called L1 space and proved that the space turns out to be a normed space. The formalization of a real function space was given in [16]. The set of all function forms additive group. Here addition is defined by point-wise addition of two functions. However it is not true for partial functions. The set of partial functions...

This article is the continuation of [31]. We define the set of Lp integrable functions - the set of all partial functions whose absolute value raised to the p-th power is integrable. We show that Lp integrable functions form the Lp space. We also prove Minkowski's inequality, Hölder's inequality and that Lp space is Banach space ([15], [27]).

We study the spaces $H_{\mu }\left(\Omega \right)=f:\Omega \to ℂholomorphic:{\int }_0^R{\int }_0^{2\pi }\left|f\left(r{e}^{i\phi }\right)\right|d\phi d\mu \left(r\right)<\infty$ where Ω is a disc with radius R and μ is a given probability measure on [0,R[. We show that, depending on μ, $H_{\mu }\left(\Omega \right)$ is either isomorphic to l₁ or to ${\left(\sum \oplus Aₙ\right)}_{\left(1\right)}$. Here Aₙ is the space of all polynomials of degree ≤ n endowed with the L₁-norm on the unit sphere.