# Mathematica Moravica, Vol. 19-1 (2015)

On the Non-Commutative Neutrix Convolution of $\tan^{-1}x$ and $x^{r}$
Mathematica Moravica, Vol. 19-1 (2015), 1–17.

Abstract. We define regular distributions $\tan_{+}^{-1}x$ and $\tan_{-}^{-1}x$ from the function $\tan^{-1}x$. We then evaluate some convolutions and neutrix convolutions of these distributions and the functions $x_{+}^{r}$, $x_{-}^{r}$ and $x^{r}$.
Keywords. Distribution, convolution, neutrix, neutrix convolution.

Convergence of Modified $S$-iteration Process for Two Generalized Asymptotically Quasi-nonexpansive Mappings in CAT(0) Spaces
Mathematica Moravica, Vol. 19-1 (2015), 19–31.

Abstract. In this paper, we give the sufficient condition of newly defined modified S-iteration process to converge to common fixed point for two generalized asymptotically quasi-nonexpansive mappings in the framework of CAT(0) spaces. Also we establish some strong convergence theorems of the said iteration process and mappings under suitable conditions. Our results extend and improve many known results from the existing literature.
Keywords. Generalized asymptotically quasi-nonexpansive mapping, strong convergence, modified $S$-iteration process, common fixed point, CAT(0) space.

Convergence to Common Fixed Point for Two Asymptotically Quasi-nonexpansive Mappings in the Intermediate Sense in Banach Spaces
Mathematica Moravica, Vol. 19-1 (2015), 33–48.

Abstract. Suppose $K$ is a nonempty closed convex subset of a real uniformly convex Banach space $E$. Let $S,T: K \to K$ be two asymptotically quasi-nonexpansive mappings in the intermediate sense such that $F = F(S) \cap F(T) = \{x\in K : Sx = Tx = x\} \neq \emptyset$. Suppose $\{x_{n}\}$ is generated iteratively by $x_{1}\in K$, $x_{n+1} = (1 - \alpha_{n})T^{n}x_{n} + \alpha_{n}S^{n}y_{n}$, $y_{n} = (1-\beta_{n})x_{n} + \beta_{n}T^{n}x_{n}$, $n \geq 1$, where $\{\alpha_{n}\}$ and $\{\beta_{n}\}$ are real sequences in $[a,b]$ for some $a,b\in (0,1)$. If $S$ and $T$ satisfy condition (B) or either $S$ or $T$ is semi-compact, then the sequence $\{x_{n}\}$ converges strongly to some $q\in F$ and if $E$ satisfying the Opial's condition, then the sequence $\{x_{n}\}$ converges weakly to some $q\in F.$
Keywords. Asymptotically quasi-nonexpansive mapping in the intermediate sense, modified Ishikawa type iteration process, common fixed point, strong convergence, uniformly convex Banach space, weak convergence.

Generalized Open Sets of Minkowski Space
Mathematica Moravica, Vol. 19-1 (2015), 49–56.

Abstract. We consider the Euclidean topology and $s$-topology on $n$-dimensional Minkowski space and investigate the interior and closure of the space cone and perforated space cone with respect to these topologies. In these regards, we determine that whether the space cone and perforated space cone are generalized open set or not. Moreover, we compare the generalized Euclidean topologies with the generalized $s$-topologies.
Keywords. Generalized topology, Minkowski space.

Periodic Solutions for a System of Nonlinear Neutral Functional Difference Equations with Two Functional Delays
Mathematica Moravica, Vol. 19-1 (2015), 57–71.

Abstract. In this paper, we study the existence and uniqueness of periodic solutions of the system of nonlinear neutral difference equations $\Delta x(n) = A(n) x(n-\tau(n)) + \Delta Q(n, x(n-g(n))) + G(n, x(n), x(n-g(n))).$ By using Krasnoselski's fixed point theorem we obtain the existence of periodic solution and by contraction mapping principle we obtain the uniqueness. An example is given to illustrate our result. Our results extend and generalize the work [13].
Keywords. Rasnoselskii's theorem, contraction, neutral difference equation, periodic solution, fundamental matrix solution.

Irresolute-Topological Groups
Mathematica Moravica, Vol. 19-1 (2015), 73–80.

Abstract. We introduce and study two classes of topologized groups, which are called here irresolute-topological and Irr-topological, investigate their properties, and establish their differences from topological groups. For this purpose we employ irresoluteness of the group operations instead of continuity in the case of topological groups and their variations.
Keywords. Semi-open set, semi-closed set, irresolute mapping, semi-homeomorphism, semi-regular space, irresolute-topological group, Irr-topological group.

On the Theorem of Wan for $K$-Quasiconformal Hyperbolic Harmonic Self Mappings of the Unit Disk
Mathematica Moravica, Vol. 19-1 (2015), 81–85.

Abstract. We give a new glance to the theorem of Wan (Theorem 1.1) which is related to the hyperbolic bi-Lipschicity of the $K$-quasiconformal, $K\geq 1$, hyperbolic harmonic mappings of the unit disk $\mathbb{D}$ onto itself. Especially, if $f$ is such a mapping and $f(0) = 0$, we obtained that the following double inequality is valid $2|z|/(K+1) \leq |f(z)| \leq \sqrt{K|z|}$, whenever $z\in\mathbb{D}$.
Keywords. Hyperbolic metric, harmonic mappings, quasiconformal mappings.

Spinor Darboux Equations of Curves in Euclidean 3-Space
Mathematica Moravica, Vol. 19-1 (2015), 87–93.

Abstract. In this paper, the spinor formulation of Darboux frame on an oriented surface is given. Also, the relation between spinor formulation of Frenet frame and Darboux frame is obtained.
Keywords. Frenet equations, Darboux equations, spinors.

Two-Step Iteration Scheme for Nonexpansive Mappings in Banach Space
Mathematica Moravica, Vol. 19-1 (2015), 95–105.

Abstract. In this paper, we introduce a new two-step iteration process to approximate common fixed points of two nonexpansive mappings in Banach spaces and established strong and weak convergence results of this iterative scheme. We also shows that our iteration process converges faster than of Mann, S-iterative and modified Ishikawa processes. Our result also illustrated with help of an example with numerical calculation. The results obtained in this paper is generalizations of Sahu [7].
Keywords. Two-step iteration process, nonexpansive mappings, condition (A'), Opial's condition, common fixed point.

Inequalities of Jensen Type for $h$-Convex Functions on Linear Spaces
Mathematica Moravica, Vol. 19-1 (2015), 107–121.

Abstract. Some inequalities of Jensen type for $h$-convex functions defined on convex subsets in real or complex linear spaces are given. Applications for norm inequalities are provided as well.
Keywords. Convex functions, integral inequalities, $h$-convex functions.

On Fuzzy Differential Subordination
Mathematica Moravica, Vol. 19-1 (2015), 123–129.

$p^{\ast}$-Closure Operator and $p^{\ast}$-Regularity in Fuzzy Setting
Abstract. In this paper a new type of fuzzy regularity, viz. fuzzy $p^{\ast}$- regularity has been introduced and studied by a newly defined closure operator, viz., fuzzy $p^{\ast}$-closure operator. Also we have found the mutual relationship of this closure operator among other closure operators defined earlier. In $p^{\ast}$-regular space, $p^{\ast}$-closure operator is an idempotent operator. In the last section, $p^{\ast}$-closure operator has been characterized via $p^{\ast}$-convergence of a fuzzy net.
Keywords. Fuzzy $p^{\ast}$-closure operator, fuzzy $p^{\ast}$-closed set, fuzzy $p^{\ast}$-regular space, $p^{\ast}$-convergence of a fuzzy net.