# Mathematica Moravica, Vol. 16-2 (2012)

Hybrid Pairs of Mappings with Some Weaker Conditions in Consideration of Common Fixed Point on 2-Metric Spaces
Mathematica Moravica, Vol. 16-2 (2012), 1–12.

Abstract. In this paper, we define weak commutativity of type $(KB)$ for hybrid pairs of mappings in 2-metric spaces. We prove a common fixed point theorem for two hybrid pairs of mappings by using weak commutativity of type $(KB)$ in 2-metric spaces. We give an example to throw light on our claim. We improve, extend and generalize several known results.
Keywords. Coincidence point, common fixed point, compatible maps, weak commutativity of type $(KB)$.

Second-order method for Parabolic Volterra Integral Equations with Crank-Nicolson Method
Mathematica Moravica, Vol. 16-2 (2012), 13–20.

Abstract. In this paper using Crank-Nicolson method we investigate convergence properties of time discretization of Parabolic Volterra integral equations are studied. The rectangle, the trapezoidal and Simpson’s rules applied for these equations. The integral is approximated in each case by the quadrature rule. The main idea of this paper quadrature rules based a fewer points, reducing the number of time levels at which the data need to be saved. We consider time step methods based on Crank-Nicolson method and combined with the appropriate quadrature rules.
Keywords. Parabolic Volterra integro-differential(integral) equation, Crank-Nicolson (CN) scheme, mixed rule, quadrature rule.

On ACVF of a Regime Switching AR(1) Process
Mathematica Moravica, Vol. 16-2 (2012), 21–24.

Abstract. This paper considers the auto-covariance function (ACVF) of a regime switching AR(1) process. Two independent Markov chains governs on auto-regressive coefficient and standard deviation of white noise process. Our approach to solve this problem is to obtain the ACVF of a AR(1) model with time varying parameters and then extend this result to regime switching case. Finally, an application of our formulae in model selection is proposed.
Keywords. Auto-covariance function, Auto-regressive model, Markov chain, Non-stationary process, Regime switching.

Fixed Point for Completely Norm Space and Map $T_{\alpha}$
Mathematica Moravica, Vol. 16-2 (2012), 25–35.

Abstract. We consider completely norm space $X$, and define $\varepsilon$-fixed point for $T_{\alpha}$ maps, and obtain some sufficient and necessary condi­tions on that, we also obtain some sufficient and necessary theorems on $\varepsilon$-fixed point for two maps.
Keywords. Fixed point, Completely norm space, Convex norm space.

Fixed Point Theorems for Monotone Mappings on Partial $D^{\ast}$-metric Spaces
Mathematica Moravica, Vol. 16-2 (2012), 37–52.

Abstract. In this paper, we introduce the concept of partial $D^{\ast}$-metric on a nonempty set $X$. In the present paper, we give some fixed point results on these interesting spaces.
Keywords. Fixed point, partial metric.

A Remark on One Iterative Process for\\ Finding the Roots of Equations
Mathematica Moravica, Vol. 16-2 (2012), 53–57.

Abstract. n this paper we consider the convergence of one iterative formula for finding the roots of equations.
Keywords. Iteration formulas, approximate solutions of equations.

On the Location of Zeros of Some Polynomials
Mathematica Moravica, Vol. 16-2 (2012), 59–61.

Abstract. In this paper we determine the regions in the complex plane containing zeros of some polynomials.
Keywords. Zeros of the polynomials, region of zeros.

Acceleration of Convergence of One Iterative Method for Finding the Roots of Equations
Mathematica Moravica, Vol. 16-2 (2012), 63–68.

Abstract. In this paper we consider two iterative methods which ac­celerate the finding of the roots of equations.
Keywords. Iteration formulas, approximate solutions of equations.

Some Fixed Point Theorems for Certain Contractive Mappings on Metric and Generalized Metric Spaces
Mathematica Moravica, Vol. 16-2 (2012), 69–77.

Abstract. In the present paper we obtain sufficient conditions for the existence of a unique fixed point of Reich and Rhoades type contractive conditions on generalized, complete, metric spaces dependent on another function. Our results generalize and extend some well-known previous results.
Keywords. Fixed point, contractive mapping, sequently convergent, sub­sequently convergent.

General System of Nonconvex Variational Inequalities and Parallel Projection Method
Mathematica Moravica, Vol. 16-2 (2012), 79–87.

Abstract. Using the prox-regularity notion, we introduce and study a system of general nonconvex variational inequalities. Using the parallel projection technique, we suggest and analyze a three-step iterative method for this system. We establish a convergence result for the proposed iteration method. We obtain some known results as a particular case.
Keywords. System of nonconvex general variational inequality, fixed point problem, parallel algorithm, proximal normal cone, relaxed cocoercive mapping.

Numerical Solution of Mixed Problem of Parabolic Equation with an Integral Conditions by using Finite Difference and Orthogonal Function Approximation
Mathematica Moravica, Vol. 16-2 (2012), 89–98.