On Rapidly Varying Sequences

Mathematica Moravica, Vol. **8–2** (2004), 1–3.

doi: http://dx.doi.org/10.5937/MatMor0402001D

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Abstract and keywords

**Abstract.** In this paper relation between rapidly varying sequence $(c_{n})$ and
its generated function $f(x) = c_{[c]}$, $x\geq 1$, is considered. That relation is expressed by the concept of the Bojanić-Seneta
proposition type for rapidly variability.

**Keywords.** Slow variability, rapidly varying.

A Common Fixed Point Theorem on Transversal Upper Intervally Spaces

Mathematica Moravica, Vol. **8–2** (2004), 5–10.

doi: http://dx.doi.org/10.5937/MatMor0402005J

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Abstract and keywords

**Abstract.** This paper is to present a common fixed point theorem for family of commuting mappings
defined on transversal upper intervally spaces. This result extends results of M. Tasković [5].

**Keywords.** Fixed point, transversal upper intervally spaces, intervally upper contraction, upper bisection functions.

Classes of $d(\psi)$-functions, $d(L^{\psi})$-Spaces, and Orlitz Spaces

Mathematica Moravica, Vol. **8–2** (2004), 11–28.

doi: http://dx.doi.org/10.5937/MatMor0402011T

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Abstract and keywords

**Abstract.** The main purpose of this paper is to give an exposition on fundamental facts and
basic notions, and further on some key-results, on new classes of $d(\psi)$-functions and new classes of $d(L^{\psi})$-spaces.
This facts and results are directly in connection with Orlicz spaces.

**Keywords.** $d(\psi)$-function, $\varphi$-function, $d(L^{\psi})$-spaces, $L^{\varphi}$-spaces, Orlicz class,
Orlicz spaces, $d(\psi)$-class, transversal upper and lower modular spaces, transversal upper and lower modulars, complementary
$M$-function, $d(M)$-class, $\Delta_{2}$-condition, $d(\Delta_{2})$-condition, quasilinear representation.

Geometric Fixed Point Theorem on Transversal Spaces

Mathematica Moravica, Vol. **8–2** (2004), 29–52.

doi: http://dx.doi.org/10.5937/MatMor0402029T

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Abstract and keywords

**Abstract.** This paper presents new fixed point theorems on lower and upper transversal spaces.
The following main result is proved that if $T$ is a self-map on an orbitally DS-complete transversal lower space $(X,\rho)$,
if there exists an upper semicontinuous function $G : X \to \mathbb{R}$ such that
\[\rho[x,Tx] \geq G(Tx) − G(x) \geq 0\]
for every $x\in X$ and if $G(T^{n}a) \to +\infty$ as $n\to\infty$ for some $a\in X$, then $T$ has a fixed point in $X$.
For the lower transversal spaces are essential the mappings $T: X \to X$ which are unbounded variation, i.e.,
if there exists a function $A: X \times X \to \mathbb{R}_{+}^{0}$ such that
\[\sum_{n=0}^{\infty} A(T^{n}x, T^{n+1}x) = +\infty\]
for arbitrary $x\in X$. On the other hand, for upper transversal spaces are essential the mappings T: X \to X$ which are bounded variation.

**Keywords.** Fixed point theorems, lower and upper transversal spaces, orbitally DS-complete and CS-complete spaces,
functions of unbounded variation, function of bounded variation, geometric fixed point theorems.

Transversal Sets

Mathematica Moravica, Vol. **8–2** (2004), 53–93.

doi: http://dx.doi.org/10.5937/MatMor0402053T

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Abstract and keywords

**Abstract.** This paper presents a new concept of sets which we call it transversal
(upper, lower or medial) sets. We introduce this concept as a natural extension of ordinary, fuzzy and transversal fuzzy sets.
Transversal sets are a new way in the nonlinear analysis. Applications in minimax theory, algebra, topology, analysis, games theory,
algebraic equations theory, ideal theory of BCC-algebras, theory of measures and integration, and convex analysis are
considered. First time in history of the theory sets, we give a technology of an arbitrary set, in the sense that every
set has “three sides” which are invisible but they de facto existing.

**Keywords.** Transversal set, transversal (upper, lower or medial) fuzzy set, grade of membership, fuzzy set,
minimax equality, minimax principle, statement of separation, games theory, convex transversal (upper, lower or medial) set,
transversal (upper, lower or medial) BCC-ideal, BCC-algebra, Borsuk-Ulam theorem, transversal integrals, transversal points,
zeros of algebraic equations, transversal logic, transversal topological spaces.

On Congruences of Super-Associative Algebras with $n$-quasigroup Operations, $n\leq 3$

Mathematica Moravica, Vol. **8–2** (2004), 95–99.

doi: http://dx.doi.org/10.5937/MatMor0402095U

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Abstract and keywords

**Abstract.** In this paper $Con(Q,\Sigma)$, where $(Q,\Sigma)$ is a super-associative algebras with
$n$-quasigroup operations, $n\geq 3$, is described.

**Keywords.** $n$-group, super-associative algebra with $n$-quasigroup operations.