# Mathematica Moravica, Vol. 14-2 (2010)

A Procedure for Obtaining a Family of Iterative Formulas for Finding Zeros of Function
Mathematica Moravica, Vol. 14-2 (2010), 1–5.

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Abstract. In this paper a family of iterative formulas for finding zeros of functions is obtained. The family includes the Laguerre method. All the methods of the family are cubically convergent for a simple zero. The superior behavior of Laguerre method when starting from the point $x_{k}$ for which $|x_{k}|$ is large, is also explained.
Keywords. Iterative formulas, approximative solutions of equations.

A remark on the Upper Bounds of the Moduli of the Roots of Algebraic Equations
Mathematica Moravica, Vol. 14-2 (2010), 7–9.

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Abstract. In this paper we obtain one upper bound of the moduli of the roots of the algebraic equations.
Keywords. Roots of algebraic equations, upper bounds for roots moduli.

Fixed Points on Tasković's Upper Transversal Intervally Spaces
Mathematica Moravica, Vol. 14-2 (2010), 11–18.

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Abstract. The main mathematical object of this paper is Taskovic's transversal space. There is a new class of spaces among various kinds of spaces with a possibility of big and significant uses. That includes upper transversal intervally spaces as well as the existence of fixed points and the uniqueness with a large number of specific features and can be a model for various uses in various fields of human creation.
Keywords. TCS-convergence, topological spaces, metric spaces, nonlinear conditions of contractions, fixed apex, nonnumerical transverses, completeness, fixed points, forked points, controlling function, principle of transpose, transversal spaces.

Transversal Theory of Fixed Point, Fixed Apices, and Forked Points
Mathematica Moravica, Vol. 14-2 (2010), 19–97.

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Abstract. In this paper on topological spaces we formulate new monotone principles of fixed point, forked point and fixed apex. This text continues the further study of the paper by M. R. Tasković [A monotone principle of fixed points, Proc. Amer. Math. Soc., 94 (1985), 427-432, Lemma 2 and Theorem 2]. New monotone principles to include some recent results of author, which contains, as special cases, some results of S. Banach, J. Dugundji and A. Granas, F. Browder, D. W. Boyd and J. S. Wong, J. Caristi, T. L. Hicks and B. E. Rhoades, B. Fisher, S. Massa, Đ. Kurepa, M. Kwapisz, W. Kirk, S. Park, M. Krasnoselskij, V. J. Stecenko, T. Kiventidis, I. Rus, K. Iséki, J. Walter, J. Daneš, A. Meir and E. Keeler, L. Collatz, J. Istraµescu, A. Miczko, and B. Palczewski, C. S. Wong, and many others.
Keywords. TCS-convergence, topological spaces, metric spaces, nonlinear conditions of contractions, fixed apex, nonnumerical transverses, completeness, fixed points, forked points, controlling function, principle of transpose, transversal spaces.

Transversal Spring Spaces, the Equation $x = T(x,\dots,x)$ and Applications
Mathematica Moravica, Vol. 14-2 (2010), 99–124.

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Abstract. This paper continues the study of the transversal spaces. In this sense we formulate a new structure of spaces which we call it transversal (upper, lower, or middle) spring spaces. Also, we consider problems of the fixed point theory on transversal spring spaces. In connection with this, we give some solutions for the equation $x = T(x,\dots,x)$. This paper presents an extended asymptotic fixed point theory.
Keywords. General ecart, distance, Fréchet's spaces, Kurepa's spaces, Menger's spaces, transversal spaces, transversal intervally spaces, middle transversal intervally spaces, transverse, bisection functions, fixed points, transversal chaos spaces, asymptotic fixed point.

New Solutions of Peano’s Differential Equation
Mathematica Moravica, Vol. 14-2 (2010), 125–143.

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Abstract. This paper gives sufficient conditions for new solutions of Peano's differential equation in the class of all lower continuous mappings. In this sense, this paper presents new fixed point theorems of Schauder type on lower transversal spaces. For the lower transversal space $(X,\rho)$ are essential the mappings $T: X\to X$ which are unbounded variation, i.e., if $\sum_{n=0}^{\infty} (T^{n}x, T^{n+1}x)=+\infty$ for arbitrary $x\in X$. On the other hand, for upper transversal spaces are essential the mappings $T: X\to X$ which are bounded variation.
Keywords. Fixed points, diametral $\varphi$-contractions, complete metric spaces, nonlinear conditions for fixed points, optimization.