# Mathematica Moravica, Vol. 15-2 (2011)

Well-Posedness of Fixed Point Problem for a Multifunction Satisfying an Implicit Relation
Mathematica Moravica, Vol. 15-2 (2011), 1–9.

Download PDF file: (216kB) | Abstract and keywords
Abstract. The notion of well-posedness of a fixed point problem for a single valued mapping has generated much interest to a several mathematicians, for examples, F.S. De Blassi and J. Myjak (1989), S. Reich and A. J. Zaslavski (2001), B.K. Lahiri and P. Das (2005) and V. Popa (2006 and 2008). In this paper we extend the notion of well-posedness known for single valued mappings to the case of multifunctions. We establish the well-posedness of fixed point problem for a multifunction satisfying an implicit relation in orbitally complete metric spaces.
Keywords. Well-posedness of fixed point problem for a multifunction, strict fixed points, implicit relations, orbit ally complete metric spaces.

Related Fixed Point Theorems for Three Metric Spaces, II
Mathematica Moravica, Vol. 15-2 (2011), 11–17.

Download PDF file: (205kB) | Abstract and keywords
Abstract. In this paper, we have proved some related fixed point theorems for three metric spaces which improve the results of Jain, Sahu and Fisher .
Keywords. Complete metric space, compact metric space, related fixed point.

Continued Fractions Expansion of $\sqrt{D}$ and Pell Equation $x^{2}-Dy^{2}=1$
Mathematica Moravica, Vol. 15-2 (2011), 19–27.

Download PDF file: (239kB) | Abstract and keywords
Abstract. Let $D\neq 1$ be a positive non-square integer. In the first section, we give some preliminaries from Pell equations and simple continued fraction expansion. In the second section, we give a formula for the continued fraction expansion of $\sqrt{D}$ for some specific values of $D$ and then we consider the integer solutions of Pell equations $x^{2}-Dy^{2} = 1$ for these values of $D$ including recurrence relations on the integer solutions of it.
Keywords. Pell equation, solutions of the Pell equation, continued fractions.

Some Results for Fuzzy Maps Under Nonexpansive Type Condition
Mathematica Moravica, Vol. 15-2 (2011), 29–39.

Download PDF file: (205kB) | Abstract and keywords
Abstract. In this paper, we have proved some results for fuzzy maps satisfying non-expansive type condition.
Keywords. Fuzzy maps, common fixed point, non-expansive map.

The $Thy$-Angle and $g$-Angle in a Quasi-Inner Product Space
Mathematica Moravica, Vol. 15-2 (2011), 41–46.

Download PDF file: (208kB) | Abstract and keywords
Abstract. In this note we prove that in a so-called quasi-inner product spaces, introduced a new angle ($Thy$-angle) and the so-called $g$-angle (previously defined) have many common characteristics. Important statements about parallelograms that apply to the Euclidean angles in the Euclidean space are also valid for the angles in a q.i.p. space (see Theorem 1).
Keywords. Quasi-inner product space, $Thy$-angle, $g$-angle.

A Note on the Zeros of One Form of Composite Polynomials
Mathematica Moravica, Vol. 15-2 (2011), 47–49.

Download PDF file: (196kB) | Abstract and keywords
Abstract. In this paper we consider one form of composite polynomials. Several relations concerning their zeros are obtained.
Keywords. Roots of algebraic equations, upper bounds for roots moduli.

A Remark on One Family of Iterative Formulas
Mathematica Moravica, Vol. 15-2 (2011), 51–53.

Download PDF file: (195kB) | Abstract and keywords
Abstract. In this paper we obtain one family of iterative formulas of the second order for finding zeros of a given function $F(x)$.
Keywords. Iteration formulas, approximate solutions of equations.

Principles of Transpose in the Fixed Point Theory for Cone Metric Spaces
Mathematica Moravica, Vol. 15-2 (2011), 55–63.

Download PDF file: (231kB) | Abstract and keywords
Abstract.This paper presents new principles of transpose in the fixed point theory as for example: Let $X$ be a nonempty set and let $\mathfrak{C}$ be an arbitrary formula which contains terms $x,y \in X$, $\leq$, $+$, $\preccurlyeq$, $\oplus$, $T: X \to X$, and $\rho$. Then, as assertion of the form: For every $T$ and for every $\rho(x,y)\in \mathbb{R}_{+}^{0} := [0,+\infty)$ the following fact
(A) $\qquad \mathfrak{C}(x,y\in X,\leq, +, T, \rho)$ implies $T$ has a fixed point
is a theorem if and only if the assertion of the form: For every $T$ and for every $\rho(x,y)\in C$, where $C$ is a cone of the set $G$ of all cones, the following fact in the form
(TA)$\qquad \mathfrak{C}(x,y\in X, \preccurlyeq, T, \rho)$ implies $T$ has a fixed point
is a theorem. Applications of the principles of transpose in nonlinear functional analysis and fixed point theory are numerous.
Keywords. Coincidence points, common fixed points, cone metric spaces, principles of transpose, Banach's contraction principle, numerical and non numerical distances, characterizations of contractive mappings, Banach's mappings, nonnumerical transversals.

On a Statement by I. Aranđelović for Asymptotic Contractions in Appl. Anal. Discrete Math.
Mathematica Moravica, Vol. 15-2 (2011), 65–68.

Download PDF file: (206kB) | Abstract and keywords
Abstract. We prove that main result of asymptotic contractions by I. Aranđelović [Appl. Anal. Disrete Math. 1 (2007), 211-216, Theorem 1, p. 212] has been for the first time proved 21 years ago in Tasković [Fundamental elements of the fixed point theory, ZUNS- 1986, Theorem 4, p. 170]. But, the author (and next other authors) this historical fact is to neglect and to ignore.
Keywords. Metric and topological spaces, TCS-convergence, complete spaces, contraction, asymptotic contraction, nonlinear conditions for fixed points, Kirk'stheorem for asymptotic contractions, Tasković's characterizations of asymptotic conditions for fixed points.

A Question of Priority Regarding a Fixed Point Theorem in a Cartesian Product of Metric Spaces
Mathematica Moravica, Vol. 15-2 (2011), 69–71.

Download PDF file: (207kB) | Abstract and keywords
Abstract. We prove that a result of Ćirić and Prešić [Acta Math. Univ. Comenianae, 76 (2007), 143-147, Theorem 2, p. 144] has been for the first time proved before 31 years in Tasković [Publ. Inst. Math., 34 (1976), 231-242, Theorem 3, p. 238]. But the authors neglected and ignored this historical fact.
Keywords. Kuratowski's problem, fixed points, Cartesian product, complete metric spaces, Cauchy's sequence, Banach's principle of contraction.

Lower Normal Topological Spaces and Lower Continuity
Mathematica Moravica, Vol. 15-2 (2011), 73–86.

Download PDF file: (204kB) | Abstract and keywords
Abstract. In this paper we formulate a new structure of topological spaces which we call it lower normal spaces. This concept of spaces is directly and nature connection with the lower transversal continuous mappings on topological spaces. In this sense, we shall study spaces in which it is possible in the same way to separate two disjoint closed sets by a lower continuous real valued function. Applications in nonlinear functional analysis are considered. The concept of lower normal spaces is closely connected with the concept of normal topological spaces and the results of Alexandroff, Urysohn, Tietze, Lebesgue, Dieudonné, Tychonoff, Lefschetz, and Vietoris.
Keywords. Topological spaces, lower normal topological spaces, lower continuity, extension of one lower continuous real-valued function, lower continuous partitions of unity.