Mathematica Moravica, Vol. 28, No. 2 (2024)
IN MEMORIAM: Prof. dr Mališa Žižović (1948–2024)
Mathematica Moravica, Vol. 28, No. 2 (2024).
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Some coupled fixed point results for multi-valued nonlinear contractions in metric spaces using $w$-distance
Mathematica Moravica, Vol. 28, No. 2 (2024), 1–16.
doi: https://doi.org/10.5937/MatMor2402001R
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Abstract and keywords
Abstract.
This study employs contractive criteria related to $w$-distances to design and address a non-linear multivalued fixed point problem.
The results presented in this study are substantiated by the inclusion of three illustrative examples.
In this paper, a dedicated portion is allocated to the examination of the use of $w$-distances.
Specifically, it explores how the utilization of $w$-distances in the current context expands upon the findings derived from metric distances.
Keywords. Multi-valued contraction, $w$-distance, ceiling distance, coupled fixed point.
Opial inequalities for a conformable $\Delta$-fractional calculus on time scales
Mathematica Moravica, Vol. 28, No. 2 (2024), 17–32.
doi: https://doi.org/10.5937/MatMor2402017M
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Abstract and keywords
Abstract.
In this paper, an Opial-type inequality is introduced on time scale for a conformal $\Delta$-fractional
differentiable function of order $\alpha$, $\alpha\in (0,1]$.
In the case where the certain weight functions are included, one generalization of the Opial inequality
is proved using conformal $\Delta$-fractional calculus on time scales.
Moreover, for $n$ times conformal $\Delta$-fractional differentiable function on time scale,
$n \in\mathbb N$, an Opial inequality is obtained.
In particular, through examples, the main results from the paper are compared
with classical ones on generalized time scales.
At the end of the paper, we indicate possible applications of the obtained Opial-type inequalities
in the consideration of stochastic dynamical equations where conformal $\Delta$-fractional calculus
on time scales is included, which requires further research.
Keywords. Opial inequality, conformable $\Delta$-fractional differentiable function, time scale.
Criteria for the asymptotic behaviour of solutions to certain third-order nonlinear differential equations
Mathematica Moravica, Vol. 28, No. 2 (2024), 33–44.
doi: https://doi.org/10.5937/MatMor2402033A
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Abstract and keywords
Abstract.
We investigate and provide sufficient criteria for ultimate boundedness of solutions
as well as the asymptotic stability of the trivial solution to certain nonlinear differential equation of order three.
Using the Lyapunov second method and the Yoshizawa limit point approach, we establish our results.
The equation considered is new and more general. Hence, our results are new, generalized and improved
on some earlier established results. In order to verify the correctness of our obtained results,
a numerical example is provided with the trajectories of the solutions.
Keywords. Stability, boundedness, asymptotic behaviour, Lyapunov function.
Trigonometric approximation of periodic functions in Morrey spaces using matrix means of their Fourier series
Mathematica Moravica, Vol. 28, No. 2 (2024), 45–56.
doi: https://doi.org/10.5937/MatMor2402045J
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Abstract and keywords
Abstract.
We examine the generalized methods of summability of Fourier series of functions belonging to Morrey spaces
$L^{p,\lambda}$, $0 < \lambda \leq 2$, $1 < p < \infty$.
In this study, the approximation of functions by matrix means in terms of the continuity modulus in Morrey spaces
$L^{p,\lambda}$, $0 < \lambda \leq 2$, $1 < p < \infty$, is investigated.
Keywords. Morrey spaces, trigonometric polynomials, Fourier series, summablity of Fourier series, modulus of continuity.
Porous-elastic system with boundary dissipation of fractional derivative type
Mathematica Moravica, Vol. 28, No. 2 (2024), 57–84.
doi: https://doi.org/10.5937/MatMor2402057O
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Abstract and keywords
Abstract.
This paper deals with the solution and asymptotic analysis for a porous-elastic system
with two dynamic control boundary conditions of fractional derivative type.
We consider an augmented model. The energy function is presented, and the dissipative property of the system is established.
We use the semigroup theory. The existence and uniqueness of the solution are obtained by applying the well-known Lumer-Phillips Theorem.
We present two results for the asymptotic behavior: Strong stability of the $C_0$-semigroup associated with the system
using Arendt-Batty and Lyubich-Vũ's general criterion and polynomial stability applying Borichev-Tomilov's Theorem.
Keywords. Porous-elastic system, dynamic boundary dissipation of fractional derivative type, existence of solution, asymptotic analysis
Clairaut slant submersion from almost Hermitian manifolds
Mathematica Moravica, Vol. 28, No. 2 (2024), 85–102.
doi: https://doi.org/10.5937/MatMor2402085K
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Abstract and keywords
Abstract.
Our main aim is to introduce Clairaut slant submersions in complex geometry.
We give the notion of Clairaut slant submersions from almost Hermitian manifolds
onto Riemannian manifold in this article.
We obtain some basic results on discussed submersions.
Furthermore, we provide some examples to explore the geometry of Clairaut slant submersions.
Keywords. Kähler manifold, Riemannian submersions, Clairaut slant submersions.
Functional impulsive fractional differential inclusions involving the Caputo-Hadamard derivative
Mathematica Moravica, Vol. 28, No. 2 (2024), 103–118.
doi: https://doi.org/10.5937/MatMor2402103I
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Abstract and keywords
Abstract.
This paper establishes sufficient conditions for the existence of solutions to fractional impulsive
functional differential inclusions, utilizing fixed-point theorems for multivalued mappings.
Keywords. Fractional differential functional inclusion,
Caputo-Hadamard derivative, convex, Fixed point, nonconvex.
Fixed points of $k$-nonexpansive mappings
Mathematica Moravica, Vol. 28, No. 2 (2024), 119–128.
doi: https://doi.org/10.5937/MatMor2402119K
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Abstract and keywords
Abstract.
In this paper, we generalizes the notion of fundamentally nonexpansive mappings introduced in
[Fixed Point Theory and Applications, 2014 (2014), Paper No.~76] to $k$-nonexpansive mappings.
We investigate various properties associated with these mappings and demonstrate
some existence and convergence results for $k$-nonexpansive mappings in the setting of Banach spaces.
Keywords. Fixed point, nonexpansive mapping, fundamentally nonexpansive mappings,
Suzuki type nonexpansive mappings.
Inequalities for the normalized determinant of positive operators in
Hilbert spaces via some inequalities in terms of Kantorovich ratio
Mathematica Moravica, Vol. 28, No. 2 (2024), 129–140.
doi: https://doi.org/10.5937/MatMor2402129D
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Abstract and keywords
Abstract.
For positive invertible operators $A$ on a Hilbert space $H$ and a fixed
unit vector $x\in H$, define the normalized determinant by
$\Delta_{x}(A):=\exp \left\langle \ln Ax,x\right\rangle$.
In this paper we prove among others that, if $0 < mI\leq A\leq MI$, then
\begin{align*}
1 &\leq K\left(\frac{M}{m}\right) ^{\left[ \frac{1}{2}-\frac{1}{M-m}
\left\langle \left\vert A-\frac{1}{2}\left( m+M\right) I\right\vert x,x\right\rangle\right]} \\
&\leq \frac{\Delta_{x}(A)}{m^{\frac{M-\left\langle Ax,x\right\rangle }{M-m}}
M^{\frac{\left\langle Ax,x\right\rangle -m}{M-m}}} \\
&\leq \left[ K\left( \frac{M}{m}\right) \right] ^{\left[ \frac{1}{2} + \frac{1}{M-m}\left\langle \left\vert A
-\frac{1}{2}\left( m+M\right) I\right\vert x,x\right\rangle \right] }\leq K\left( \frac{M}{m}\right),
\end{align*}
for $x\in H$, $\Vert x\Vert =1$, where $K(\cdot)$ is Kantorovich's ratio.
Keywords. Positive operators, Normalized determinants, Inequalities.