# Mathematica Moravica, Vol. 19-2 (2015)

Stability for Nonlinear Neutral Integro-Differential Equations with Variable Delay
Mathematica Moravica, Vol. 19-2 (2015), 1–18.

Abstract. In this paper we use the contraction mapping principle to obtain asymptotic stability results of a nonlinear neutral integro-differential equation with variable delay. An asymptotic stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some previous results due to Burton [7], Becker and Burton [6] and Jin and Luo [17]. In the end we provide an example to illustrate our claim.
Keywords. Fixed points, stability, integro-differential equation, variable delay.

On a General Class of $q$-Rational Type Operators
Mathematica Moravica, Vol. 19-2 (2015), 19–33.

Abstract. In this study, we define a general class of rational type operators based on $q$-calculus and investigate the weighted approximation properties of these operators by using $A$-statistical convergence. We also estimate the rates of $A$-statistical convergence of these operators by modulus of continuity and Petree's $K$-functional. The operators to be introduced, include some well known $q$-operators so our results are true in a large spectrum of these operators.
Keywords. $q$-calculus, rational type operators, $A$-statistical convergence, weighted spaces, linear positive operators.

An Introduction to Fuzzy Soft Graph
Mathematica Moravica, Vol. 19-2 (2015), 35–48.

Abstract. The notions of fuzzy soft graph, union, intersection of two fuzzy soft graphs are introduced in this paper and a few properties relating to finite union and intersection of fuzzy soft graphs are established here.
Keywords. Soft set, fuzzy soft set, fuzzy graph, fuzzy soft graph.

On Weak and Strong Convergence Theorems for a Finite Family of Nonself $I$-asymptotically Nonexpansive Mappings
Mathematica Moravica, Vol. 19-2 (2015), 49–64.

Abstract. We prove the weak and strong convergence of S iterative scheme to a common fixed point of a family of nonself asymptotically $I$-nonexpansive mappings $\{T_{i}\}_{i}^{N}$ and a family of nonself asymptotically nonexpansive mappings $\{I_{i}\}_{i}^{N}$ defined on a nonempty closed convex subset of a Banach space. Our scheme converges faster than Mann and Ishikawa iteration for contractions. Our weak convergence theorem is proved under more general setup of space as different from weak convergence theorems proved in previously.
Keywords. Nonself asymptotically $I$-nonexpansive mappings, Iterative scheme, Kadec-Klee property, condition (B), common fixed point.

Convolutions Involving the Exponential Function and the Exponential Integral
Mathematica Moravica, Vol. 19-2 (2015), 65–73.

Abstract. The exponential integral $\mathrm{ei}(\lambda x)$ and its associated functions $\mathrm{ei}_{+}(\lambda x)$ and $\mathrm{ei}_{-}(\lambda x)$ are defined as locally summable functions on the real line and their derivatives are found as distributions. The convolutions $x^{r}\mathrm{ei}_{+}(x) \ast x^{s}e_{+}^{x}$ and $x^{r}\mathrm{ei}_{+}(x) \ast x^{s}e^{x}$ are evaluated.
Keywords. Exponential integral, convolution.

On a New Subclass of Harmonic Univalent Functions Defined by Multiplier Transformation
Mathematica Moravica, Vol. 19-2 (2015), 75–87.

Abstract. The purpose of the present paper is to introduce a new subclass of harmonic univalent functions by using Multiplier transformation. Coefficient estimates, distortion bounds, extreme points, convolution condition and convex combination for functions belonging to this class are determined. The results obtained for the class reduce to the corresponding several known results are briefly indicated.
Keywords. Harmonic, univalent function, multiplier transformation.

Some New Integral Inequalities via Variant of Pompeiu’s Mean Value Theorem
Mathematica Moravica, Vol. 19-2 (2015), 89–95.

Abstract. The main of this paper is to establish an inequality providing some better bounds for integral mean by using a mean value theorem. Our results generalize the results of Ahmad et. al in [8].
Keywords. Ostrowski inequality, $p$-norm, mean value theorem.

Some Fixed Point Theorems for $(CAB)$-contractive Mappings and Related Results
Mathematica Moravica, Vol. 19-2 (2015), 97–112.

Abstract. In this paper, we introduced the concept of $(CAB)$-contractive mappings and provide sufficient conditions for the existence and uniqueness of a fixed point for such class of generalized nonlinear contractive mappings in metric spaces and several interesting corollaries are deduced. Also, as application, we obtain some results on coupled fixed points, fixed point on metric spaces endowed with $N$-transitive binary relation and fixed point for cyclic mappings. The proved results generalize and extend various well-known results in the literature.
Keywords. Fixed point, $(CAB)$-contraction mappings, binary relations, metric space.

Monotone Principle of Forked Points and Its Consequences
Mathematica Moravica, Vol. 19-2 (2015), 113–124.

Abstract. This paper presents applications of 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]. Monotone principle of forked points is a direct consequence of the Axiom of Infinite Choice, i.e., of the Lemma of Infinite Maximality! Brouwer and Schauder theorems are two direct censequences of the monotone principle od forked points.