# Mathematica Moravica, Vol. 21, No. 2 (2017)

In Memoriam - Roman Jan Dwilewicz
Mathematica Moravica, Vol. 21, No. 2 (2017)

On topological BE-algebras
Mathematica Moravica, Vol. 21, No. 2 (2017), 1–13.

Abstract. In this paper, we study some properties of uniform topology and topological BE-algebras and compare this topologies.
Keywords. BE-algebra, filter, uniform topology, topological BE-algebra.

A coincidence and common fixed point theorem for subsequentially continuous hybrid pairs of maps satisfying an implicit relation
Mathematica Moravica, Vol. 21, No. 2 (2017), 15–25.

Abstract. In this paper, we introduce the notion of subsequential continuity for a hybrid pair of maps and combine this concept with compatibility, to establish a coincidence and common fixed point theorem for a hybrid quadruple of maps. Our main result also demonstrates that several fixed point theorems can be unified using implicit relations. We also give two examples in support our results.
Keywords. Common fixed point, subsequentially continuous, compatible mappings, implicit relation.

Extended biorthogonal matrix polynomials
Mathematica Moravica, Vol. 21, No. 2 (2017), 27–50.

Abstract. The pair of biorthogonal matrix polynomials for commutative matrices were first introduced by Varma and Tasdelen in [22]. The main aim of this paper is to extend the properties of the pair of biorthogonal matrix polynomials of Varma and Tasdelen and certain generating matrix functions, finite series, some matrix recurrence relations, several important properties of matrix differential recurrence relations, biorthogonality relations and matrix differential equation for the pair of biorthogonal matrix polynomials $J_{n}^{(A,B)} (x;k)$ and $K_{n}^{(A,B)} (x;k)$ are discussed. For the matrix polynomials $J_{n}^{(A,B)} (x;k)$, various families of bilinear and bilateral generating matrix functions are constructed in the sequel.
Keywords. Matrix functional calculus, biorthogonal matrix polynomials, Jacobi matrix polynomials, generating matrix functions, differential operator.

Common Fixed Points for Faintly Compatible Mappings
Mathematica Moravica, Vol. 21, No. 2 (2017), 51–59.

Abstract. In this paper, we obtain a generalized common fixed point theorem for four mappings using the conditions of non-compatibility and faint compatibility.
Keywords. Common fixed points, compatibility, faint compatibility.

On the solutions of a second order difference equation

Abstract. In this paper, we discuss the global behavior of all solutions of the difference equation $x_{n+1} = \frac{x_{n}x_{n-1}}{ax_{n} + bx_{n-1}},\quad n\in\mathbb{N}_{0},$ where $a,b$ are real numbers and the initial conditions $x_{-1},x_{0}$ are real numbers. We determine the forbidden set and give an explicit formula for the solutions. We show the existence of periodic solutions, under certain conditions.
Keywords. Difference equation, forbidden set, periodic solution, unbounded solution.

General integral formulas involving Humbert hypergeometric functions of two variables
Mathematica Moravica, Vol. 21, No. 2 (2017), 75–84.

Abstract. In this paper, we have established two general integral formulas involving Humbert hypergeometric functions of two variables $\Phi_{2}$ and $\Psi_{2}$. The results are obtained with the help of a generalization of classical Kummer's summation theorem on the sum of the series ${}_{}F_{1}(-1)$ due to Lavoie et al. [5]. Some interesting applications are also presented.
Keywords. Humbert functions, Exton functions, Kampé de Férietfunction, Integral formulas, Kummer's theorem.

Best proximity points for generalized $\alpha-\eta-\psi$-Geraghty proximal contraction mappings
Mathematica Moravica, Vol. 21, No. 2 (2017), 85–102.

Abstract. In this paper, we introduce the new notion of generalized $\alpha-\eta-\psi$-Geraghty proximal contraction mappings and prove the existence of the best proximity point for such mappings in $\alpha-\eta$ complete metric spaces. we give an example to illustrate our result. Our result extends some of the results in the literature.
Keywords. Best proximty point, $\alpha$-triangular proximal admissble withrespect to $\eta$, generalized $\alpha-\eta-\psi$-Geraghty proximal contraction.
Abstract. The purpose of this paper is to extend Theorem 3.2 [16] for two hybrid pairs of mappings satisfying a mixed implicit relation and a new type of common limit range property without weak compatibility. As applications, some fixed point results for pairs of mappings satisfying contractive conditions of integral type and $\varphi$-contractive maps are obtained.
Keywords. Fixed point, hybrid pairs of mappings, common limit range property, integral type, $\varphi$-contractive maps, mixed implicit relation.