Suzuki-type fixed point theorems in relational metric spaces with applications

Mathematica Moravica, Vol. **28**, No. **1** (2024), 1–15.

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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.

Perturbed functional fractional differential equation of Caputo-Hadamard order

Mathematica Moravica, Vol. **28**, No. **1** (2024), 17–28.

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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.

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.

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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.

Operator upper bounds for Davis-Choi-Jensen's difference in Hilbert spaces

Mathematica Moravica, Vol. **28**, No. **1** (2024), 39–51.

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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.

About uniformly Menger spaces

Mathematica Moravica, Vol. **28**, No. **1** (2024), 53–61.

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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.