# Mathematica Moravica, Vol. 8–1 (2004)

New Generalizations of Caristi's Fixed Point Theorem via Brézis-Browder Principle
Mathematica Moravica, Vol. 8–1 (2004), 1–5.

Abstract. In this paper, some generalizations of Caristi's fixed point theorem are obtained via the Brézis-Browder principle: Theorems 2.1, 2.2, 3.1, and 3.2.
Keywords. Fixed point, Caristi's theorem, Brézis-Browder principle, orbit, lower semicontinuous functions.

Some Results on Commutativity of Rings
Mathematica Moravica, Vol. 8–1 (2004), 7–10.

Abstract. Some new results on commutativity theorems of torsion free unital rings have been obtained.
Keywords. Commutativity of rings, anticommutator, unital ring, torsion free ring.

A Note on $\lambda_2$ and $\lambda_n$ of a Graph
Mathematica Moravica, Vol. 8–1 (2004), 11–14.

Abstract. Using the eigenvalues and eigenvectors of a graph $G$, it was established the upper bound for the second eigenvalue $\lambda_2$ and the least eigenvalue $\lambda_n$ [1]. In this work using only the eigenvalues of G we obtain the upper bound for $\lambda_2$ and $\lambda_n$.
Keywords. Geometric inequalities, fundamental inequalities of triangle, Gerretsen's inequalities.

Q-analog of Determinant of a Kind of Binomial Coefficient Matrices
Mathematica Moravica, Vol. 8–1 (2004), 15–24.

Abstract. Recently, Ratliff and Rush gave the explanation of a kind of determinant whose entries are zero and either binomial coefficients of their negative. Such work can be considered in some generalized Pascal matrices. The purpose of this paper is to establish a $q$-analog of the result of Ratliff and Rush. Also, a $q$-analogs about problem 269 in [5] are given.
Keywords. Gaussian binomial coefficient; matrix, determinant, Vandermonde determinant, upper triangular matrix.

Fixed Points of Some Classes of Nonexpansive Mappings
Mathematica Moravica, Vol. 8–1 (2004), 25–31.

Abstract. In this paper we proves the convergence of a convex sequence $x_{n} = \lambda x_{n-1} + (1-\lambda)ƒ(x_{n-1})$, $\lambda \in (0,1)$, to a fixed point of the nonexpansive completely continuous operator in the normed $ƒ_{\lambda}$-orbitally complete spaces with $\lambda$-uniformly convex sphere. Further we shall prove some fixed point theorems of the star-shaped sets.
Keywords. Normed $ƒ_{\lambda}$-orbitally complete spaces with $\lambda$-uniformly convex sphere, nonexpansive operator, $\lambda$-orbit, fixed point, extremal point, star-shaped sets.

Stationary Points for Multifunctions on Two Complete Metric Spaces
Mathematica Moravica, Vol. 8–1 (2004), 33–38.

Abstract. In this paper we prove a general fixed point theorem for multi-functions on two complete metric spaces which generalizes the main results from [2] and [5].
Keywords. Fixed point, multifunction, implicit relation, complete metric space.

Axiom of Choice – 100th Next
Mathematica Moravica, Vol. 8–1 (2004), 39–62.

Abstract. In this survey, first, we give a short history oversee of the Axiom of Choice; and, second, in the connection with this we give oversee our main results which are new equivalents of he Axiom of Choice, i.e., Zorn's lemma. These statement are of fixed apex type and fixed point type theorems. The survey includes comments about these theorems and some historical facts.

On Hyperquasigroups
Mathematica Moravica, Vol. 8–1 (2004), 63–71.

Abstract. In the paper we define and study hyperquasigroups of the rang $m(\in N)$.
Keywords. Quasigroups, hypergroupoids, hyperquasigroups.

Two Characteriyations of $(n,m$)-groups for $n\leq 3$
Mathematica Moravica, Vol. 8–1 (2004), 73–78.

Abstract. In this paper two characterization of $(n,m)$-groups for $n\geq 3m$ are proved. (The case $m = 1$ is proved in [5].).
Keywords. $n$-group, $(n,m)$-groupoid, $(n,m)$-group, $\{1,n - m + 1\}$ neutral operation of the $(n,m)$-groupoid.

Mathematica Moravica, Vol. 8–1 (2004), 79–93.

Abstract. In this article several propositions of near-$P$-polyagroups are proved.
Keywords. $n$-group, $\{1,n\}$-neutral operation of the $n$-groupoid, near-$P$-polyagroup of the type $(s,n - 1)$.