Homotopy extension property for multi-valued functions
Mathematica Moravica, Vol. 27, No. 1 (2023), 1–12.
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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.
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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.
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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.
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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.
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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), 73–82.
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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.