Mathematica Moravica, Vol. 24, No. 1 (2020)
Stability and boundedness of nonautonomous neutral differential equation with
delay
Mathematica Moravica, Vol. 24, No. 1 (2020),
1–16.
doi: http://dx.doi.org/10.5937/MatMor2001001R
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(477kB) |
Abstract and keywords
Abstract. We consider the nonautonomous neutral
differential
equation with delay
\begin{gather*}
&\left[ p(t)\Big(q(t)\big(x(t)+\beta_{1} x(t-r_{1})\big)'\Big)'\right] ^{ \prime }+a(t)\big(
x^{\prime \prime}(t)+
\beta_{2} x''(t-r_{2})\big)\\
&+b(t)\big(x^{\prime }(t)+\beta_{3} x'(t-r_{3})\big)+c(t)f(x(t-\sigma))=e(t, x, x', x'').
\end{gather*}
Using the method of Lyapunov, we give
conditions for the uniform asymptotic stability and uniform boundedness and square integrability of
solutions for the considered system. Our theorems generalize
and extend some related results known in the literature. Example is given to show our results.
Keywords. Uniform ultimate boundedness, square integrability, Lyapunov functional,
neutral
differential equation of third order.
Differential sandwich results for Wanas operator of analytic functions
Mathematica Moravica, Vol. 24, No. 1 (2020),
17–28.
doi: http://dx.doi.org/10.5937/MatMor2001017K
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Abstract and keywords
Abstract. In the present article, we determine some
subordination and superordination results
involving Wanas operator for certain normalized analytic functions defined in the unit disk
$\mathbb{U}$.
These results are applied to establish sandwich results.
Our results extend corresponding previously known results.
Keywords. Analytic function, differential subordination, differential
superoordination, dominant, subordinant, Wanas operator.
Some common fixed point theorems on partial metric spaces satisfying implicit
relation
Mathematica Moravica, Vol. 24, No. 1 (2020),
29–43.
doi: http://dx.doi.org/10.5937/MatMor2001029S
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Abstract and keywords
Abstract. The aim of this article is to prove some fixed
point and
common fixed point theorems on partial metric spaces satisfying implicit relation.
Our results extend and generalize several results from the existing literature.
Keywords. Fixed point, common fixed point, implicit relation, partial metric space.
On exponentially $(h_{1},h_{2})$-convex functions and fractional integral
inequalities related
Mathematica Moravica, Vol. 24, No. 1 (2020),
45–62.
doi: http://dx.doi.org/10.5937/MatMor2001045V
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Abstract and keywords
Abstract. In this work the concept of exponentially $
(h_{1},h_{2})$-convex function is introduced and using it,
the Hermite-Hadamard inequality and some bounds for the right side of this inequality,
via Raina's fractional integral operator and generalized convex functions, are established.
Keywords. Exponentially $ (h_{1},h_{2})$-convex function, Raina's fractional
integral operator, fractional integral inequalities.
On $G$-transitive version of perfectly meager sets
Mathematica Moravica, Vol. 24, No. 1 (2020),
63–70.
doi: http://dx.doi.org/10.5937/MatMor2001063N
Download PDF file: (536kB) |
Abstract and keywords
Abstract. We study the $G-$ invariant version of
perfectly meager
sets (a generalization of the notion of $\mathrm{AFC}'$ sets).
We find the necessary and sufficient conditions for the inclusion
$\mathrm{AFC}'_{G} \subseteq \mathcal{I}$. In particular, we partially characterize for which groups
$G$ of automorphisms of the Cantor space every $\mathrm{AFC}'_{G}$ set is Lebesgue null.
Keywords. Perfectly meager sets, strongly meager sets, $\mathrm{AFC}'$ sets.
Some inequalities for Heinz operator mean
Mathematica Moravica, Vol. 24, No. 1 (2020),
71–82.
doi: http://dx.doi.org/10.5937/MatMor2001071S
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Abstract and keywords
Abstract. In this paper we obtain some new inequalities for Heinz operator mean.
Keywords. Young's Inequality, Convex functions, Arithmetic mean-Geometric
mean inequality, Heinz means.
On relationships between $q$-product identities and combinatorial partition identities
Mathematica Moravica, Vol. 24, No. 1 (2020),
83–91.
doi: http://dx.doi.org/10.5937/MatMor2001083C
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Abstract and keywords
Abstract. Andrews et al. [2] discussed about the combinatorial partition
identities. We aim to present some relationships between $q$-product identities and combinatorial partition identities,
by using and combining known formulas.
Keywords. Combinatorial partition identities, Jacobi’s triple-product
identity, Ramanujan’s theta functions, $q$-Product identities, Combinatorial partition
identities, Euler’s Pentagonal Number Theorem, Rogers-Ramanujan continued fraction,
Rogers-Ramanujan identities.
Inverse-$C$-class function on weak semi compatibility and fixed point theorems for expansive mappings in $G$-metric spaces
Mathematica Moravica, Vol. 24, No. 1 (2020),
93–108.
doi: http://dx.doi.org/10.5937/MatMor2001093H
Download PDF file: (462kB) |
Abstract and keywords
Abstract. In this paper we introduce the concept of inverse $C$-class
function in $G$-metric setting and established some fixed point theorems.
We also put some examples in support of proved fixed point results.
Keywords. Common fixed point, inverse-$C$-class function.
Existence and Ulam stability results for nonlinear hybrid implicit Caputo fractional differential equations
Mathematica Moravica, Vol. 24, No. 1 (2020),
109–122.
doi: http://dx.doi.org/10.5937/MatMor2001109L
Download PDF file: (462kB) |
Abstract and keywords
Abstract. In this paper, we study the existence, uniqueness and estimate
of solutions for nonlinear hybrid implicit Caputo fractional differential
equations by using the contraction mapping principle and the
generalization of Gronwall’s inequality. After that, we also establish the
Ulam stability for the problem at hand. Finally, an example is given to
illustrate this work.
Keywords. Implicit fractional differential equations, Caputo fractional
derivatives, fixed point theorems, existence, uniqueness, Ulam stability.