# Mathematica Moravica, Vol. 20:1 (2016)

Obituary in memory of Slavik Jablan (1952-2015)
Mathematica Moravica, Vol. 20:1 (2016), 1–6.

On the logarithmic integral and the convolution
Mathematica Moravica, Vol. 20:1 (2016), 7–16.

Abstract. The logaritmic integral $\mathrm{li}(x)$ and its associated functions $\mathrm{li}_{+}(x)$ and $\mathrm{li}_{-}(x)$ are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of these functions and other functions are then found.
Keywords. Logarithmic integral, distribution, convolution, neutrix, neutrix convolution.

Periodic solutions for neutral nonlinear difference equations with functional delay
Mathematica Moravica, Vol. 20:1 (2016), 17–29.

Abstract. We use a variant of Krasnoselskii's fixed point theorem to show that the nonlinear difference equation with functional delay $\Delta x(t) = -a(t) g(x(t)) + c(t)\Delta x(t -\tau(t)) + q(t, x(t), x(t-\tau(t))),$ has periodic solutions. For that end, we invert this equation to construct a fixed point mapping written as a sum of a completely continuous map and a large contraction which is suitable for the application of Krasnoselskii-Burton's theorem.
Keywords. Krasnoselskii theorem, fixed point, periodic solutions, large contraction, difference equations.

Solvability of boundary value problems for second order impulsive differential equations on whole line with a non-Carathédory nonlinearity
Mathematica Moravica, Vol. 20:1 (2016), 31–49.

Abstract. We study a class of boundary value problems of the impulsive differential equations on whole lines with a non-Carathéodory nonlinearity. Sufficient conditions to guarantee the existence of solutions are established. A new Banach function space $X$ and its relatively compact property of subset of $X$ is proved. An example is given to illustrate the main results.
Keywords. Second order impulsive differential equation, boundary value problem, non-Carathéodory nonlinearity, fixed point theorem.

Some results on the q-Beta function
Mathematica Moravica, Vol. 20:1 (2016), 51–57.

Abstract. In this article we find some results on the $q$-analogue of the beta function via using the concepts of neutrix and neutrix limit.
Keywords. Neutrix, neutrix limit, beta Function, $q$-Beta function, $q$-integral, $q$-integration, $q$-derivative.

Asymmetric maximal and minimal open sets
Mathematica Moravica, Vol. 20:1 (2016), 59–67.

Abstract. We introduce the notions of maximal and minimal open sets in bitopological spaces and obtain some properties of them. In contrary to maximal and minimal open sets in topological spaces, we observe that maximal and minimal open sets in bitopological spaces behave differently. The maximal and minimal open sets in a bitopological space under the operations of union and intersection respectively sometimes become slightly different types of maximal and minimal open sets in that bitopological space. We also obtain results concerning an asymmetric minimal open set on a subspace of a bitopological space.
Keywords. $(\mathscr{P}_{i}, \mathscr{P}_{j})$maximal open set, $(\mathscr{P}_{i}, \mathscr{P}_{j})$minimal open set, pairwise maximal open set, pairwise minimal open set, disconnected bitopological space.

Strong convergence theorem for generalized mixed equilibrium problems and Bregman nonexpansive mapping in Banach spaces
Mathematica Moravica, Vol. 20:1 (2016), 69–87.

Abstract. In this paper, we study an iterative method for a common fixed point of a Bregman strongly nonexpansive mapping in the frame work of reflexive real Banach spaces. Moreover, we prove the strong convergence theorem for finding common fixed points with the solutions of a generalized mixed equilibrium problem.
Keywords. Banach space, Bregman projection, Bregman distance, Bregman strongly nonexpansive mapping, fixed point, generalized mixed equilibrium problem.

Slant helices, Darboux helices and similar curves in dual space $\mathbb{D}^{3}$
Mathematica Moravica, Vol. 20:1 (2016), 89–103.

Abstract. In this paper, we give definitions and characterizations of slant helices, normalized Darboux helices and similar curves in dual space $\mathbb{D}^{3}$. First, we define dual slant helices and dual normalized Darboux helices and show that dual slant helices are also dual normalized Darboux helices. Then, we introduce the concept of dual similar curves and obtain that the family of dual slant helices forms a family of dual similar curves.
Keywords. Dual space, dual slant helix, dual Darboux helix, dual similar curves.

Characterization of triple $\chi^{3}$ sequence spaces via Orlicz functions
Mathematica Moravica, Vol. 20:1 (2016), 105–114.

Abstract. In this paper we study of the characterization and general properties of triple gai sequence via Orlicz space of $\chi_{M}^{3}$ of $\chi^{3}$ establishing some inclusion relations.
Keywords. Gai sequence, analytic sequence, triple sequence, dual space, Orlicz space.

The strongly generalized triple difference $\Gamma^{3}$ sequence spaces defined by a modulus
Mathematica Moravica, Vol. 20:1 (2016), 115–123.

Abstract. In this paper we introduce the strongly generalized difference sequence spaces using non-negative four dimensional matrix of complex numbers. We also give natural relationship between strongly generalized difference ${V_{3_{\Gamma^{3}}}}^{\lambda_{3}}[A,\Delta^{m},p,f]$ — summable sequences with respect to $f$. We examine some topological properties of ${V_{3_{\Gamma^{3}}}}^{\lambda_{3}}[A,\Delta^{m},p,f]$ — spaces and investigate some inclusion relations between these spaces.
Keywords. Entire sequence, analytic sequence, triple sequence, difference sequence.

Weak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings
Mathematica Moravica, Vol. 20:1 (2016), 125–144.

Abstract. This paper presents the Axiom of Infinite Choice Given any set $P$, there exist at least countable choice functions or there exist at least finite choice functions. The author continues herein with the further study of two papers of the Axiom of Choice in order by E. Zermelo [Neuer Beweis für die Möglichkeit einer Wohlordung, Math. Annalen, 65 (1908), 107-128; translated in van Heijenoort 1967, 183-198], and by M. Tasković [The axiom of choice, fixed point theorems, and inductive ordered sets, Proc. Amer. Math. Soc., 116 (1992), 897-904]. Fredholm and Leray-Schauder alternatives are two direct consequences of the Axiom of Infinite Choice! This paper presents applications of the Axiom of Infinite Choice to the Fredholm and Leray-Schauder theory. In this sense, I give a solution and some extensions of Schauder's problem (in Scottish book, problem 54). This paper presents some new mathematical n-person games. In the theory of n-person games, there have been some further developments in the direction of transversal games and mathematical alternative theory.
Keywords. The axiom of infinite choice, Zermelo's axiom of choice, lemma of infinite maximality, Zorn's lemma, foundation of the fixed point theory, geometry of the axiom of infinite choice, axioms of infinite choice for points and apices, general Brouwer theorem, general Schauder theorem, extension of Schauder’s 54th problem in Scottish book, transversal $n$-person games, Nash’s theorem, Peano’s theorem.