Mathematica Moravica, Vol. 27, No. 1 (2023)
Homotopy extension property for multi-valued functions
Mathematica Moravica, Vol. 27, No. 1 (2023), 1–12.
doi: http://dx.doi.org/10.5937/MatMor2301001K
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Abstract and keywords
Abstract. In this study, we introduce some well-known definitions
and properties for multi-valued functions. We present new definitions such as, m-retraction, m-section,
m-homeomorphism, m-HEP and reducible function. We give a new result on the relation between multi-homotopy groups
and m-homeomorphism. We also deal with some properties of m-HEP.
Keywords. Multi-valued function, m-retraction, m-homeomorphism, homotopy extension property.
Convergence analysis for Suzuki's generalized nonexpansive mappings
Mathematica Moravica, Vol. 27, No. 1 (2023), 13–22.
doi: http://dx.doi.org/10.5937/MatMor2301013G
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Abstract and keywords
Abstract. In this paper, we study the Picard-Mann hybrid iteration process
to approximate fixed points of Suzuki's generalized nonexpansive mappings.
We establish some weak and strong convergence theorems for such mappings in uniformly convex Banach space.
Keywords. Nonexpansive mapping, fixed points, Suzuki's generalized nonexpansive mappings, iterative methods.
The Tribonacci-type balancing numbers and their applications
Mathematica Moravica, Vol. 27, No. 1 (2023), 23–36.
doi: http://dx.doi.org/10.5937/MatMor2301023H
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Abstract and keywords
Abstract. In this paper,
we define the Tribonacci-type balancing numbers via a
Diophantine equation with a complex variable and then give their
miscellaneous properties. Also, we study the Tribonacci-type balancing
sequence modulo $m$ and then obtain some interesting results concerning the
periods of the Tribonacci-type balancing sequences for any $m$. Furthermore,
we produce the cyclic groups using the multiplicative orders of the
generating matrices of the Tribonacci-type balancing numbers when read
modulo $m$. Then give the connections between the periods of the
Tribonacci-type balancing sequences modulo $m$ and the orders of the cyclic
groups produced. Finally, we expand the Tribonacci-type balancing sequences
to groups and give the definition of the Tribonacci-type balancing sequences
in the $3$-generator groups and also, investigate these sequences in the
non-abelian finite groups in detail. In addition, we obtain the periods of
the Tribonacci-type balancing sequences in the polyhedral groups
$(2,2,n)$, $(2,n,2)$, $(n,2,2)$, $(2,3,3)$, $(2,3,4)$, $(2,3,5)$.
Keywords. The Tribonacci-type balancing sequence, Matrix, Group, Presentation, Period, Rank.
New fixed figure results with the notion of $k$-ellipse
Mathematica Moravica, Vol. 27, No. 1 (2023), 37–52.
doi: http://dx.doi.org/10.5937/MatMor2301037A
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Abstract and keywords
Abstract. In this paper,
as a geometric approach to the fixed-point theory, we prove
new fixed-figure results using the notion of $k$-ellipse on a metric space.
For this purpose, we are inspired by the Caristi type mapping, Kannan type
contraction, Chatterjea type contraction and Ćirić type contraction.
After that, we give some existence and uniqueness theorems of a fixed $k$-ellipse.
We also support our obtained results with illustrative examples.
Finally, we present a new application to the $S$-Shaped Rectified Linear
Activation Unit ($SReLU$) to show the importance of our theoretical results.
Keywords. Fixed figure, fixed $k$-ellipse, metric space, activation function.
Frames generated by double sequences in Hilbert spaces
Mathematica Moravica, Vol. 27, No. 1 (2023), 53–72.
doi: http://dx.doi.org/10.5937/MatMor2301053B
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Abstract and keywords
Abstract. In this paper,
we introduce frames generated by double sequences ($d$-frame)
in Hilbert spaces and describe some of their properties.
Furthermore, we discuss frame operators, alternate dual frames and stability for $d$-frames.
Keywords. Frame, Double sequence, $d$-frame, Alternate dual $d$-frame, Frame operator.
New fixed-circle results on fuzzy metric spaces with an application to dynamic market equilibrium
Mathematica Moravica, Vol. 27, No. 1 (2023), 73–83.
doi: http://dx.doi.org/10.5937/MatMor2301073K
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Abstract and keywords
Abstract. In this study, the fixed point theory on fuzzy metric spaces has been generalized to the fixed-circle
theory by making a geometric interpretation. The necessary conditions to exist the fixed circles of a
self-mapping have been investigated and the uniqueness of the circle is examined under suitable conditions.
We present some illustrative examples of obtained results and also offer an application to confirm the utility
of our established result for finding the unique solution of an integral equation appearing in the dynamic
market equilibrium aspects of economics.
Keywords. Fixed circle, Fuzzy metric, Dynamic market equilibrium.
Fixed point theorems in complex valued $b$-metric spaces
Mathematica Moravica, Vol. 27, No. 1 (2023), 85–96.
doi: http://dx.doi.org/10.5937/MatMor2301085B
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Abstract and keywords
Abstract. In this paper, we have proved common fixed point theorems
using Hardy and Rogers type contraction condition
in complex-valued $b$-metric spaces The results of the paper extend the results proved in S. Ali [1].
Keywords. Complex valued $b$-metric space.
On approximation properties of functions by means of Fourier and Faber series in weighted Lebesgue spaces with variable exponent
Mathematica Moravica, Vol. 27, No. 1 (2023), 97–112.
doi: http://dx.doi.org/10.5937/MatMor2301097J
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Abstract and keywords
Abstract. In this paper the approximation of functions by linear means of Fourier
series in weighted variable exponent Lebesgue spaces was studied. This
result was applied to the approximation of the functions by linear means of
Faber series in Smirnov classes with variable exponent defined on simply
connected domain of the complex plane.
Keywords. Trigonometric approximation, Muckenhoupt weight, Lebesgue space
with variable exponent, weighted modulus of smoothness, best approximation.
Ideal convergence and ideal Cauchy sequences in intuitionistic fuzzy metric spaces
Mathematica Moravica, Vol. 27, No. 1 (2023), 113–128.
doi: http://dx.doi.org/10.5937/MatMor2301113O
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Abstract and keywords
Abstract. The present study introduces the concepts of
ideal convergence (${I}$–convergence), ideal Cauchy (${I}$–Cauchy) sequences,
$I^*$–convergence, and ${I^*}$–Cauchy sequences in intuitionistic fuzzy metric spaces.
It defines ${I}$–limit and ${I}$–cluster points as a sequence in these spaces.
Afterward, it examines some of their basic properties.
Lastly, the paper discusses whether phenomena should be further investigated.
Keywords. Ideal convergence, ideal Cauchy sequences,
cluster points, limit points, intuitionistic fuzzy metric spaces.