# Mathematica Moravica, Vol. 23, No. 2 (2019)

A note on q-analogue of Hermite-poly-Bernoulli numbers and polynomials
Mathematica Moravica, Vol. 23, No. 2 (2019), 1–16.

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Abstract. In this paper, we introduce the Hermite-based poly-Bernoulli numbers and polynomials with q-parameter and give some of their basic properties including not only addition property, but also derivative properties and integral representations. We also define the Hermitebased -Stirling polynomials of the second kind and then provide some relations, identities of these polynomials related to the Stirling numbers of the second kind. We derive some symmetric identities for these families of special functions by applying the generating functions.
Keywords. Hermite polynomials, q-analogue of poly-Bernoulli polynomials, q-analogue of Hermite poly-Bernoulli polynomials, Stirling numbers of the second kind, q-polylogarithm function, Symmetric identities.

Existence of positive periodic solutions for third-order nonlinear delay differential equations with variable coefficients
Mathematica Moravica, Vol. 23, No. 2 (2019), 17–28.

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Abstract. In this paper, the following third-order nonlinear delay differential equation with periodic coefficients \begin{gather*} x'''(t) + p(t)x''(t)+q(t)x'(t)+r(t)x(t) \\ = f\bigl(t,x(t),x(t-\tau(t))\bigr) + \frac{d}{dt}g\bigl(t,x(t-\tau(t))\bigr), \end{gather*} is considered. By employing Green's function and Krasnoselskii's fixed point theorem, we state and prove the existence of positive periodic solutions to the third-order nonlinear delay differential equation.
Keywords. Fixed point, positive periodic solutions, third-order delay differential equations.

A common fixed point result for two pairs of maps in b-metric spaces without (E.A.)-property
Mathematica Moravica, Vol. 23, No. 2 (2019), 29–44.

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Abstract. In this paper, we investigate a common fixed point problem for two pairs $\{f,S\}$ and $\{g,T\}$ of weakly compatible selfmaps of a complete b-metric $(X,d;s)$, satisfying a contractive condition of Ćirić type. This contraction and some of its variants were used in the paper [29] published in 2016 by V. Ozturk and S. Radenović, requiring the (E.A.)-property for the pairs $\{f,S\}$ and $\{g,T\}$. The aim of this paper is to provide some improvements to the main result of [29]. Our main theorem will improve certain results published in 2015, by V. Ozturk and D. Turkoglu (see [30] and [31]). We improve also results from other related papers (see the references herin). Indeed, we remove the (E.A.)-property and weaken certain assumptions imposed in these papers. So, our work aims to extend and unify, in one go, several common fixed point results known in a recent literature. We furnish two illustrative examples and we prove that the fixed point problem, considered here, for the pairs $\{f,S\}$ and $\{g,T\}$ is well-posed. We compare our main result with a recent result obtained in 2018 by N. Hussain, Z. D. Mitrović and S. Radenović in [19].
Keywords. b-metric spaces, common fixed point for four maps, weakly compatible maps, compatible maps, property (E.A.), well-posedness.

Strong Differential Sandwich Results of $\lambda$-Pseudo-Starlike Functions with Respect to Symmetrical Points
Mathematica Moravica, Vol. 23, No. 2 (2019), 45–58.

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Abstract. In the present investigation, by considering suitable classes of admissible functions, we establish strong differential subordination and superordination properties for $\lambda$-pseudo-starlike functions with respect to symmetrical points in the open unit disk $U$. These results are applied to obtain strong differential sandwich results.
Keywords. Strong differential subordination, Strong differential superordination, $\lambda$-pseudo-starlike functions, symmetrical points, Admissible functions.

$Z_s$-Contractive Mappings and Weak Compatibility in Fuzzy Metric Space
Mathematica Moravica, Vol. 23, No. 2 (2019), 59–68.

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Abstract. The aim of this paper is to introduce $Z_s$-contractive condition for a pair of self maps in a fuzzy metric space, which enlarges and unifies the existing fuzzy contractions (by Gregori and Sapena [4]), $\psi$-contraction (by Mihet [8]), $Z$-contractions (by Shukla [15]) and Tirado contraction ([16]), which are for only one self map. Using this, we establish a unique common fixed point theorem for two self maps satisfying condition (S), which was introduced by Shukla et al. in [15] through weak compatibility. The article includes an example, which shows the validity of our results.
Keywords. Fuzzy metric space, $t$-norm, $M$-cauchy sequence, common fixed points, $Z_s$-contraction, weak compatibility.

Some new generalizations for $m-$convexity via new conformable fractional integral operators
Mathematica Moravica, Vol. 23, No. 2 (2019), 69–77.

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Abstract. In this paper, some new generalizations for $m-$convex functions have been given by using an integral identity via new conformable fractional integrals and some further properties. It is pointed out that special cases of our findings gave some earlier inequalities involving Riemann-Liouville fractional integrals.
Keywords. New Conformable fractional integrals, $m-$convexity, Euler Beta function.

Derivations satisfying certain algebraic identities on Lie ideals
Mathematica Moravica, Vol. 23, No. 2 (2019), 79–86.

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Abstract. Let $d$ be a derivation of a semiprime ring $R$ and $L$ a nonzero Lie ideal of $R$. In this note, it is proved that every noncentral square-closed Lie ideal of $R$ contains a nonzero ideal of $R$. Further, we use this result to characterize the conditions: $d(xy)=d(x)d(y)$, $d(xy)=d(y)d(x)$ on $L$. With this, a theorem of Ali et al. [14] can be deduced.
Keywords. Semiprime ring, Lie ideals, Derivation.

Refinements of Hermite-Hadamard inequality for trigonometrically $\rho$-convex functions
Mathematica Moravica, Vol. 23, No. 2 (2019), 87–96. Retracted
doi: http://dx.doi.org/10.5937/MatMor1902087B
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Impulsive fractional differential inclusions with state-dependent delay
Mathematica Moravica, Vol. 23, No. 2 (2019), 97–113.

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Abstract. In this paper, we prove the existence of mild solution for impulsive fractional inclusions with state-dependent delay in Banach spaces. Our study is based on the nonlinear alternative of Leray-Schauder type for multivalued maps due to Martelli. An example is provided to illustrate the main result.
Keywords. Impulsive fractional differential inclusions, $\alpha$-resolvent family, solution operator, Caputo fractional derivative, mild solution, multivalued map, fixed point, Banach spaces.