Matching Theorems on Topological Spaces

Mathematica Moravica, Vol. **6** (2002), 1–7.

doi: http://dx.doi.org/10.5937/MatMor0206001A

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**Abstract.** In this paper we give one general matching theorem and some its applications in the fixed point theory.
Our results generalize earlier theorems obtained by Horvath, Chang - Zhang and Chang - Ma.

**Keywords.** Matching theorem, fixed point.

A Note on Weakly Separable Spaces

Mathematica Moravica, Vol. **6** (2002), 9–19.

doi: http://dx.doi.org/10.5937/MatMor0206009B

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**Abstract.** We study the notion of a weak separable space.
Some counterexamples, which show a strictness of previous results of the author, are given.
The weak separability of hyperspaces and function spaces is investigated.

**Keywords.** Weakly separable space, compact space, hyperspace, function space.

The Strengthening and Weakening Instrument: Comparability of Topological Spaces

Mathematica Moravica, Vol. **6** (2002), 21–64.

doi: http://dx.doi.org/10.5937/MatMor0206021D

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**Abstract.** Along with noncom parable topologies, the paper concentrates on situations,
where in a bitopological space one topology is finer than the other, which is frequently encountered in applications.
In this context, different families of sets are considered and the bitopological modification of the Cantor-Bendixson theorem is proved.
The three operators are defined, which characterize the degrees of nearness of the four boundaries of any set, tangency of topologies,
$S$-, $C$- and $N$-relations, and thus make it possible to compare small inductive dimensions at some special point.
Furthermore, different properties of pair wise small and pair wise large inductive dimensions are studied.
In the final part, the conditions are given, under which a bitopological space preserves the property
to be an $(i, j)$-Baire space to the image and preimage. Relations between pair wise small and large inductive dimensions
of the domain and the range of a $d$-closed and $d$-continuous function are investigated.

**Keywords.** Indicator of nearness, $(i,j)$-dense in itself set, $(i,j)$-perfect set, $(i,j)$-scattered set,
$(i,j)$-Baire space; almost, $(i,j)$-Baire space, $(i,j)$-small inductive dimension, $(i,j)$-large inductive dimension.

Quasi-Almost Convergence in a Normed Space

Mathematica Moravica, Vol. **6** (2002), 65–70.

doi: http://dx.doi.org/10.5937/MatMor0206065H

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**Abstract.** **1.** In [1] was shown the existence of the functionals of the kind of Banach
limits defined on the real vector space $\mathbf{m}$ of all bounded sequences in a real normed space $X$.
In [2] by these functionals was defined the almost convergence of a sequence $(x_{i})\in \mathbf{m}$ and shown that $(x_{i})$ almost converges
to $s\in X$ iff
\[\biggl\| \frac{1}{p}\sum_{i=0}^{p-1} X_{k+i} - s\biggr\|_{X} \to 0\quad\text{as } p\to \infty\]
uniformly in $k=(0,1,2,\dots)$.

**2.** The basic idea in this paper is to obtain a new method of sum ability of the vector sequences $(x_{i})\in \mathbf{m}$.
The paper is organized as follows. First, we will show the existence of an another family of functionals (of the kind of Banach limits)
defined on the space $\mathbf{m}$. Next, we define, by these functionals, a new method of sum ability of sequences $(x_{i})\in \mathbf{m}$
which will be called quasi almost convergence. Further, we will show a theorem which contains a necessary and
sufficient condition for a sequence $(x_{i})\in \mathbf{m}$ to be quasi almost convergent.
Next, we shall prove two theorems which shows that the class of quasi almost convergent sequences lies between the class
of almost convergent sequences and the class of $C$-sum able sequences.

**Keywords.** Functional, Banach limits, $C$-sum able sequences.

A Common Fixed Point on Transversal Probabilistic Spaces

Mathematica Moravica, Vol. **6** (2002), 71–76.

doi: http://dx.doi.org/10.5937/MatMor0206071J

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**Abstract.** In this paper we shall prove a common fixed point theorem for
family of commuting mappings defined on transversal probabilistic spaces.
This result extends some previous results.

**Keywords.** Convergence criteria of series, fixed point, lower transversal probabilistic spaces,
probabilistic contraction, bisection functions, Menger spaces.

Common Fixed Point Theorems of Gregus Type in Convex Metric Spaces

Mathematica Moravica, Vol. **6** (2002), 77–85.

doi: http://dx.doi.org/10.5937/MatMor0206077S

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**Abstract.** In this paper, we prove common fixed point theorems of Gregus type
for three mappings in convex metric spaces. We extend and generalize some well known results by many authors.

**Keywords.** Common fixed point, compatible mapping, convex metric space, $W$-affine mapping.

Fixed Point Theorems for Mappings in *d*-Complete Topological Spaces

Mathematica Moravica, Vol. **6** (2002), 87–92.

doi: http://dx.doi.org/10.5937/MatMor0206087P

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**Abstract.** Fixed point theorems are given for pairs of mappings satisfying an implicit
relation defined on $d$-complete topological spaces.

**Keywords.** $d$-topological space, fixed point

A General Fixed Point Theorem for Mappings in Pseudocompact Tichonoff Spaces

Mathematica Moravica, Vol. **6** (2002), 93–96.

doi: http://dx.doi.org/10.5937/MatMor0206093P

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**Abstract.** In [2] Jain and Dixit have obtained some interesting results on fixed points in pseudocompact Tichonoff spaces.
In this paper we present a general fixed point theorem in pseudocompact Tichonoff spaces for mappings satisfying an implicit relation
which generalize Theorems 1 and 2 from [2] and some results from [3] and [4].

**Keywords.** Fixed point, pseudocompact spaces, Tichonoff space, implicit relation.

Characterization of Reflexive Banach Spaces with Normal Structure

Mathematica Moravica, Vol. **6** (2002), 97–102.

doi: http://dx.doi.org/10.5937/MatMor0206097T

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**Abstract.** This paper presents a characterization of reflexive Banach spaces with normal
structure in term of fixed point for diametral contractive mappings.

**Keywords.** Reflexive Banach spaces, normal structure, diametral sequences, Brodskij-Milman theorem,
Šmulian property, characterization of normal structure, diametral contractive mappings, fixed points.

Diametral Contractive Mappings in Reflexive Banach Spaces

Mathematica Moravica, Vol. **6** (2002), 103–108.

doi: http://dx.doi.org/10.5937/MatMor0206103T

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**Abstract.** In this paper it is proved that if $K$ is a nonempty-bounded closed convex subset of a
reflexive Banach space $X$ and if $K$ has normal structure, then any diametral contractive mapping $T$ on $K$ into itself has a fixed point.

**Keywords.** Reflexive Banach spaces, normal structure, Šmulian property, Browder-Gohde-Kirk theorem, diametral contractive mappings, fixed points.

Extension of Theorems by Krasnoselskij, Stečenko, Dugundji, Granas, Kiventidis, Romaguera, Caristi and Kirk

Mathematica Moravica, Vol. **6** (2002), 109–118.

doi: http://dx.doi.org/10.5937/MatMor0206109T

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**Abstract.** In this paper we describe a class of conditions sufficient for the existence of
a furcate point which generalize several known results of fixed point.

**Keywords.** Fixed point theorems, topological spaces, weakly contractive mappings, general monotone principle of fixed point,
furcate points, monotone and forks principle.

On Schauder's 54th Problem in Scottish Book Revisited

Mathematica Moravica, Vol. **6** (2002), 119–126.

doi: http://dx.doi.org/10.5937/MatMor0206119T

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**Abstract.** The most famous of many problems in nonlinear analysis is Schauder's
problem (*Scottish book*, problem 54) of the following form, that if $C$ is a nonempty convex compact subset of a linear topological
space does every continuous mapping $f: C\to C$ has a fixed point? The answer we give in this paper is yes.

In this paper we prove that if $C$ is a nonempty convex compact subset of a linear topological space,
then every continuous mapping $f: C \to C$ has a fixed point.

On the other hand, in this sense, we extend and connected former results of Brouwer, Schauder, Tychonoff, Markoff,
Kakutani, Darbo, Sadovskij, Browder, Krasnoselskij, Ky Fan, Reinermann, Hukuhara, Mazur, Hahn, Ryll-Nardzewski, Day, Riedrich,
Jahn, Eisenack-Fenske, Idzik, Kirk, Ghöde, Caristi, Granas, Dugundji, Klee and some others.

**Keywords.** Fixed point theorems, Brouwer's theorem, Schauder's theorem, Tychonof's theorem,
Markoff's theorem, Kakutani's theorem, Sadovskij's theorem.

One Characterization of Near-*P*-Polyagroup

Mathematica Moravica, Vol. **6** (2002), 127–130.

doi: http://dx.doi.org/10.5937/MatMor0206127U

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**Abstract.** In the present paper the following proposition is proved.
Let $k>l$, $s>1$, $n = k \cdot s + 1$ and let $(Q, A)$ be an $n$-groupoid.
Then, $(Q, A)$ is an near-$P$-polyagroup (briefly: $NP$-polyagroup) of the type $(s,n-1)$ iff for some $i\in\bigl\{t\cdot s+l\mid t\in\{1,\dots,k-1\}\bigr\}$
the following conditions hold: (a) the $\langle i-s, i\rangle$ - associative law holds in $(Q, A)$;
(b) the $\langle i,i+s\rangle$ - associative law holds in $(Q, A)$; and (c) for every $a_{1}^{n}\in Q$ there is exactly one $x\in Q$
such that the following equality holds $A(a_{1}^{i-1},x,a_{i}^{n-1}) = a_{n}$.

**Keywords.** $n$-groupoid, $n$-semigroup, $n$-quasigroup, $Ps$-associative $n$-groupoid, $P$-polyagroup, $NP$-polyagroup.

Note on Congruence Classes of *n*-Groups

Mathematica Moravica, Vol. **6** (2002), 131–135.

doi: http://dx.doi.org/10.5937/MatMor0206131U

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**Abstract.** In the paper the following proposition is proved.
Let $(Q,A)$ be an $n$-group, $|Q|\in N\setminus\{1\}$, and let $n\geq 3$. Further on, let $\Theta$ be an arbitrary congruence of the $n$-group $(Q, A)$ and
let $C_{t}$ be an arbitrary class from the set $Q/\Theta$. Then there is a $k\in N$ such that
the pair $(C_{t},\overset{k}{A})$ is a $(k(n-1)+l)$-subgroup of the $(k(n-1)+l)$-group $(A,\overset{k}{A})$.

**Keywords.** $n$-semigroup, $n$-quasigroup, $n$-group.

Note on *n*-Groups

Mathematica Moravica, Vol. **6** (2002), 137–144.

doi: http://dx.doi.org/10.5937/MatMor0206137U

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**Abstract.** Among the results of the paper is the following proposition.
Let $n\geq 3$ and let $(Q,A)$ be an $n$-grupoid. Then: $(Q,A)$ is an $n$-group iff there are mappings $\alpha$ and $\beta$, respectively,
of the sets $Q^{n-2}$ and $Q$ into the set $Q$ such that the laws
\[A(A(x_{1}^{n}),x_{n+1}^{2n-1}) = A(x_{1},A(x_{2}^{n+1}),x_{n+2}^{2n-1})\]
\[\beta A(x_{1}^{n}) = A(x_{1}^{n-1},\beta(x_{n})) = A(x_{1}^{n-2},\beta(x_{n-1}),x_{n}),\]
\[A(x,a_{1}^{n-2},\alpha(a_{1}^{n-2})) = A(b_{1}^{n-2},\alpha(b_{1}^{n-2}),x),\text{ and}\]
\[\beta A(x,c_{1}^{n-2},\alpha(c_{1}^{n-2})) = x\]
hold in the algebra $(Q,\{A,\alpha,\beta\})$ [:3.1].

**Keywords.** $n$-semigroup, $n$-quasigroup, $n$-group, $\{1,n\}$-neutral operation on $n$-groupoid, central operation on $n$-group.

Note on Polyagroups

Mathematica Moravica, Vol. **6** (2002), 145–149.

doi: http://dx.doi.org/10.5937/MatMor0206145U

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**Abstract.** In the paper the following proposition is proved.
Let $k>1$, $s>1$, $n=k-s+1$ and let $(Q,A)$ be an $n$-groupoid. Then, $(Q,A)$ is a polyagroup of the type $(s,n-1)$ iff
the following statements hold: (i) $(Q,A)$ is an $\langle i, s+i\rangle$-associative $n$-groupoid for
all $i\in\{1,\dots,s\}$; $\langle l,n\rangle$-associative $n$-groupoid;
(iii) for every $a_{1}^{n}\in Q$ there is at least one $x\in Q$ and at least one $y\in Q$ such that the following equalities hold
$A(x,a_{1}^{n-1})=a_{n}$ and $A(a_{1}^{n-1}) = a_{n}$; and (iv) for every $a_{1}^{n}\in Q$ and for all $i\in \{2,\dots,s\}\cup \{(k-1)\cdot s+2,\dots,k\cdot s\}$
there is exactly one $X_{i}\in Q$ such that the following equality holds $A(a_{1}^{i-1},x_{i},a_{i}^{n-1}) = a_{n}$.
[The case $s=1$ (: (i) – (iii)) is discribed in [4].]

**Keywords.** $n$-groupoid, $n$-semigroup, $n$-quasigroup, $iPs$-associative $n$-groupoid, polyagroup, near-$P$-polyagroup.