Mathematica Moravica, Vol. 28, No. 1 (2024)

Deepak Khantwal, Rajendra Pant
Suzuki-type fixed point theorems in relational metric spaces with applications
Mathematica Moravica, Vol. 28, No. 1 (2024), 1–15.
Download PDF file: (510kB) | Abstract and keywords
Abstract. In this paper, we establish a relation-theoretic version of the results presented by Kim et al. (Journal of Nonlinear and Convex Analysis, 16 (9) (2015), 1779-1786). To showcase the versatility of our results, we furnish some illustrative examples. Furthermore, we exhibit an application of our results to establish sufficient conditions for the existence of a positive definite common solution to a pair of nonlinear matrix equations.
Keywords. Suzuki contraction, binary relation, continuity, completeness.

Samira Hamani
Perturbed functional fractional differential equation of Caputo-Hadamard order
Mathematica Moravica, Vol. 28, No. 1 (2024), 17–28.
Download PDF file: (469kB) | Abstract and keywords
Abstract. In this paper, we investigate the existence of solution and extremal solutions for a initial-value problem of perturbed functional fractional differential equations with Caputo-Hadamard derivative.
Our analysis relies on the fixed point theorem of Burton and Kirk and the concept of upper and lower solutions combined with a fixed point theorem in ordered Banach space established by Dhage and Henderson.
Keywords. Fractional differential equation, Caputo-Hadamard fractional derivatives, Fixed point, Extremal solutions.

Michal Jánoši, Beáta Stehlíková
Accuracy of analytical approximation formula for bond prices in a three-factor convergence model of interest rates
Mathematica Moravica, Vol. 28, No. 1 (2024), 29–38.
Download PDF file: (431kB) | Abstract and keywords
Abstract. We consider a convergence model of interest rates, in which the behaviour of the domestic instantaneous interest rate (so called short rate) depends on the short rate in a monetary union that the country is going to join. The short rate in the monetary union is modelled by a two-factor model, which leads to a three-factor model for the domestic rate. In this setting, term structures of interest rates are computed from bond prices, which are obtained as solutions to a parabolic partial differential equation. A closed-form solution is known only in special cases. An analytical approximation formula for the domestic bond prices has been proposed, with the error estimate only for certain parameter values, when the solution has a separable form. In this paper, we derive the order of accuracy in the general case. We also study a special case, which makes it possible to model the phenomenon of negative interest rates that were observed in the previous years. It turns out that it leads to a higher accuracy than the one achieved in the general case without restriction on parameters.
Keywords. Short rate, convergence model, bond-pricing partial differential equation, approximate analytical solution, order of accuracy.

Silvestru Sever Dragomir
Operator upper bounds for Davis-Choi-Jensen's difference in Hilbert spaces
Mathematica Moravica, Vol. 28, No. 1 (2024), 39–51.
Download PDF file: (479kB) | Abstract and keywords
Abstract. In this paper we obtain several operator inequalities providing upper bounds for the Davis-Choi-Jensen's Difference $\Phi \left( f\left( A\right) \right) -f\left( \Phi \left( A\right) \right)$ for any convex function $f:I\rightarrow \mathbb{R}$, any selfadjoint operator $A$ in $H$ with the spectrum $\mathrm{Sp}\left( A\right) \subset I$ and any linear, positive and normalized map $\Phi :\mathcal{B}\left( H\right) \rightarrow \mathcal{B}\left(K\right) ,$ where $H$ and $K$ are Hilbert spaces. Some examples for convex and operator convex functions are also provided.
Keywords. Selfadjoint bounded linear operators, Functions of operators, Operator convex functions, Jensen's operator inequality, Linear, positive and normalized map.

Bekbolot Kanetov, Dinara Kanetova, Anara Baidzhuranova
About uniformly Menger spaces
Mathematica Moravica, Vol. 28, No. 1 (2024), 53–61.
Download PDF file: (458kB) | Abstract and keywords
Abstract. Precompact type properties – precompactness (=totally precompactness), $\alpha$-precompactness, pre-Lindelöfness, (=$\aleph _{0}$-bounded\-ness), $\tau $-boundedness – belong to the basic important invariants studied in the uniform topology.
The theory of these invariants is widely and goes on to develop. However, in a sense, the class of uniformly Menger spaces escaped the attention of researchers.
Lj.D.R. Kočicinac was the first who introduced and studied the class of uniformly Menger spaces in [3, 4]. It immediately follows from the definition that the class of uniformly Menger spaces lies between the class of precompact uniform spaces and the class of pre-Lindelöf uniform spaces and should therefore have many good properties.
In this paper some important properties of the uniformly Menger spaces are investigated. In particular, it is established that under uniformly perfect mappings the uniformly Menger property is preserved both in the image and the preimage direction.
Keywords. Uniform space, uniform Menger space, uniformly continuous mapping, uniformly perfect mapping.