Mathematica Moravica, Vol. 18-2 (2014)
The Fuzzy Stability of a Pexiderized Functional Equation
Mathematica Moravica, Vol. 18-2 (2014), 1–14.
doi: http://dx.doi.org/10.5937/MatMor1402001K
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Abstract and keywords
Abstract. In this paper, Hyers-Ulam-Rassias Stability of the Pexiderized functional
Equation $f(x+y) = g(x) + h(y)$ is concerned in fuzzy Banach spaces.
Keywords. Fuzzy norm, fuzzy Banach space, Pexiderized functional equation, Hyers-Ulam-Rassias stability.
On Decompositions of Continuity and α-Continuity
Mathematica Moravica, Vol. 18-2 (2014), 15–20.
doi: http://dx.doi.org/10.5937/MatMor1402015D
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Abstract and keywords
Abstract. Several results concerning a decomposition of $\alpha$-continuous,
continuous and complete continuous functions are offered.
Keywords. Preopen set, $\alpha$-open set, semi-open set, $\alpha$-precontinuity, continuity, complete continuity.
A Quadruple Fixed Point Theorem for Contractive Type Condition by Using ICS Mapping and Application to Integral Equation
Mathematica Moravica, Vol. 18-2 (2014), 21–34.
doi: http://dx.doi.org/10.5937/MatMor1402021R
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Abstract and keywords
Abstract. In this paper, we obtain a Quadruple fixed point theorem for
$\psi-\phi$ contractive condition in partially ordered partial metric spaces by using ICS mapping.
We are also given an example and an application to integral equation which supports our main theorem.
Keywords. Partial metric space, Quadruple fixed point, ICS mapping, mixed monotone property.
On s-Topological Groups
Mathematica Moravica, Vol. 18-2 (2014), 35–44.
doi: http://dx.doi.org/10.5937/MatMor1402035B
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Abstract and keywords
Abstract. In this paper we study the class of $s$-topological groups and a
wider class of $S$-topological groups which are defined by using semi-open sets and semi-continuity introduced by N. Levine.
It is shown that these groups form a generalization of topological groups, and that they are different from several distinct
notions of semitopological groups which appear in the literature. Counterexamples are given to strengthen these concepts.
Some basic results and applications of $s$- and $S$-topological groups are presented. Similarities with and differences from
topological groups are investigated.
Keywords. Semi-open set, semi-continuity, semi-homeomorphism, $s$-regular space, $s$-topological group, $S$-topological group.
Coupled Common Fixed Point Theorems in Partially Ordered $G$-metric Spaces for Nonlinear Contractions
Mathematica Moravica, Vol. 18-2 (2014), 45–62.
doi: http://dx.doi.org/10.5937/MatMor1402045J
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Abstract and keywords
Abstract. The aim of this paper is to prove coupled coincidence and coupled common
fixed point theorems for a mixed $ɡ$-monotone mapping satisfying nonlinear contractive conditions in the setting of partially
ordered $G$-metric spaces. Present theorems are true generalizations of the recent results of Choudhury and Maity
[Math Comput. Modelling 54 (2011), 73-79], and Luong and Thuan [Math. Comput. Modelling 55 (2012), 1601-1609].
Keywords. Partially ordered set, $G$-metric space, coupled coincidence point, coupled common fixed point,
mixed monotone mappings.
A Common Fixed Point Theorem for Weakly Compatible Multi-Valued Mappings Satisfying Strongly Tangential Property
Mathematica Moravica, Vol. 18-2 (2014), 63–72.
doi: http://dx.doi.org/10.5937/MatMor1402063B
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Abstract and keywords
Abstract. In this paper we prove a common fixed point theorem for two weakly
compatible pairs of single and set-valued mappings which satisfying contractive condition of integral type in metric space
by using the concept of strongly tangential property, our results generalize and extend some previous results.
Keywords. Common fixed point, strongly tangential, weakly compatible, multi-valued maps.
Quadruple Coincidence and Common Quadruple Fixed Point for Hybrid Pair of Mappings Under New Contractive Condition
Mathematica Moravica, Vol. 18-2 (2014), 73–90.
doi: http://dx.doi.org/10.5937/MatMor1402073D
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Abstract and keywords
Abstract. We establish a quadruple coincidence and common quadruple fixed point theorem
for hybrid pair of mappings satisfying new contractive condition. It is to be noted that to find quadruple coincidence points,
we do not use the condition of continuity of any mapping involved. An example supporting to our result has also been cited.
We improve, extend and generalize several known results.
Keywords. Quadruple fixed point, quadruple coincidence point, $w$-compatible mappings, F-weakly commuting.
Stability in Nonlinear Neutral Differential Equations with Infinite Delay
Mathematica Moravica, Vol. 18-2 (2014), 91–103.
doi: http://dx.doi.org/10.5937/MatMor1402091A
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Abstract and keywords
Abstract. In this paper we use the contraction mapping theorem to obtain asymptotic
stability results of the nonlinear neutral differential equation with infinite delay
\[ \frac{\mathrm{d}}{\mathrm{d}t}x(t) = -a(t) x(t-\tau_{1}(t)) + \frac{\mathrm{d}}{\mathrm{d}t} Q(t, x(t-\tau_{2}(t))) +
\int_{-\infty}^{t} D(t,s) f(x(s))\mathrm{d}s.\]
An asymptotic stability theorem with a necessary and sufficient condition is proved, which improves and generalizes
some results due to Burton [6], Zhang [17], Althubiti, Makhzoum, Raffoul [1].
Keywords. Fixed points, Stability, Neutral differential equation, Integral equation, Infinite delay.
Transversal Functional Analysis
Mathematica Moravica, Vol. 18-2 (2014), 105–181.
doi: http://dx.doi.org/10.5937/MatMor1402105T
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Abstract and keywords
Abstract. This paper provides an introduction to the ideas and methods of
transversal functional analysis based on the transversal sets theory. A unifying concept that lies at the heart of
transversal functional analysis is that of a transversal normed linear space. I have developed the theory far enough
to include facts of have called the three new basic principles of linear analysis as: Form of Hahn-Banach theorem,
Form of Principle of Uniform Boundedness (= Form of Banach-Steinhaus theorem), and Form of Open Mapping theorem.
In the classical functional analysis fundamental fact is Riesz lemma. In transversal functional analysis
(on lower transversal normed spaces) its role play so-called Geometrical lemma! This paper presents applications
of the Axiom of Infinite Choice.
Keywords. Transversal (upper, lower and middle) normed spaces, Transversal seminorms, Form of Hahn-Banach theorem,
Form of Principle of Uniform Boundedness, Form of Banach-Steinhaus theorem, Form of Open Mapping theorem, Form of Riesz lemma,
Geometrical lemma, Form of onvergence principle, Lower compactness, Lower total continuous operators, Lower locally compactness,
Lower Fredholm alternative, Extension Leray-Schauder principle, Peano’s equation, Lower compact operators.