Mathematica Moravica, Vol. 28, No. 1 (2024)
Suzuki-type fixed point theorems in relational metric spaces with applications
Mathematica Moravica, Vol. 28, No. 1 (2024), 1–15.
doi: https://doi.org/10.5937/MatMor2401001K
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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.
doi: https://doi.org/10.5937/MatMor2401017H
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Abstract and keywords
Abstract.
In this paper, we investigate the existence of solution and extremal solutions for an
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.
doi: https://doi.org/10.5937/MatMor2401029J
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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.
doi: https://doi.org/10.5937/MatMor2401039S
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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 $\Phi(f(A)) - f(\Phi(A))$
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 $\Phi :\mathcal{B}(H) \rightarrow \mathcal{B}(K)$,
where $H$ and $K$ are Hilbert spaces. Some examples of 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.
doi: https://doi.org/10.5937/MatMor2401053K
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Abstract and keywords
Abstract.
Precompact type properties – precompactness (=totally
precompactness), $\alpha$-precompactness, pre-Lindelöfness,
(=$\aleph _{0}$-boundedness), $\tau$-boundedness – belong to the
basic important invariants studied in the uniform topology.
The theory of these invariants is wide and continues 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. Therefore,
we expect it to 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.
Some Bullen-Simpson type inequalities for differentiable s-convex functions
Mathematica Moravica, Vol. 28, No. 1 (2024), 63–85.
doi: https://doi.org/10.5937/MatMor2401063M
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Abstract and keywords
Abstract.
Convexity is one of the fundamental principles of analysis.
Over the past few decades, many important inequalities have been established
for different classes of convex functions.
In this paper, some Bullen-Simpson type integral inequalities for functions
whose first derivatives are s-convex in the second sense are established.
The cases where the first derivatives are bounded as well as Hölderian are also provided.
Some applications to numerical integration and inequalities involving means are given.
Keywords. Bullen-Simpson's inequality, s-convex functions, Hölder inequality, power mean inequality.
Connectedness in graded ditopological texture spaces
Mathematica Moravica, Vol. 28, No. 1 (2024), 87–96.
doi: https://doi.org/10.5937/MatMor2401087E
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Abstract and keywords
Abstract.
The aim of this paper is to introduce two different types of connectedness notions for graded ditopological texture spaces:
the connectedness function which gives the grade of connectedness of a set and the connectedness spectrum by means of spectrum idea.
Also, the properties of these connectedness notions and their relationships with the connectedness notion in ditopological
case are investigated.
Further, the relation between these two different types of connectedness notions is studied.
Keywords. Texture, ditopology, connectedness.
Some new results for the generalized bivariate Fibonacci and Lucas polynomials
Mathematica Moravica, Vol. 28, No. 1 (2024), 97–108.
doi: https://doi.org/10.5937/MatMor2401097Y
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Abstract and keywords
Abstract.
In this paper, new identities are obtained by using the generalized bivariate Fibonacci and Lucas polynomials.
Firstly, several binomial summations and the closed formulas for summation of powers are investigated for these polynomials.
Also, general summation formulas, different generating functions, and relations of these polynomials are presented.
Keywords. Binomial, Fibonacci polynomials, generating function, Lucas polynomials.
On a generalization of the Gadovan numbers
Mathematica Moravica, Vol. 28, No. 1 (2024), 109–119.
doi: https://doi.org/10.5937/MatMor2401109O
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Abstract and keywords
Abstract.
In this paper, we define $(k,l)-$Gadovan numbers. We give the Binet-like formula, the generating functions,
the exponential generating function of the $(k,l)-$ Gadovan numbers. Also, we derive Cassini-like identity,
Catalan-like identity, Vajda-like identity, Honsberger-like identity and D'ocagne-like identity for the $(k,l)-$Gadovan numbers.
Keywords. Gadovan numbers, $k-$Gadovan numbers, Binet formula, Generating functions, Cassini identity, Catalan identity.
$\mathcal{I}$-asymptotically lacunary statistical equivalent sequences in partial metric spaces
Mathematica Moravica, Vol. 28, No. 1 (2024), 121–134.
doi: https://doi.org/10.5937/MatMor2401121O
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Abstract and keywords
Abstract.
The present study deals with $\mathcal{I}$-asymptotically equivalent sequences in partial metric spaces.
We define the notions of strongly $\mathcal{I}$-asymptotically lacunary equivalent,
$\mathcal{I}$-asymptotically statistical equivalent, and $\mathcal{I}$-asymptotically lacunary statistical equivalent.
We theoretically contribute to these notions and investigate some of their basic properties.
Keywords. $\mathcal{I}$-asymptotically statistical equivalent, Strong $\mathcal{I}$-asymptotically lacunary equivalent,
$\mathcal{I}$-asymptotically lacunary statistical equivalent, Partial metric space.