Well-Posedness of Fixed Point Problem for a Multifunction Satisfying an Implicit Relation

Mathematica Moravica, Vol. **15-2** (2011), 1–9.

doi: http://dx.doi.org/10.5937/MatMor1102001A

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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.

doi: http://dx.doi.org/10.5937/MatMor1102011N

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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 [2].

**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.

doi: http://dx.doi.org/10.5937/MatMor1102019T

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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.

doi: http://dx.doi.org/10.5937/MatMor1102029M

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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.

doi: http://dx.doi.org/10.5937/MatMor1102041M

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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.

doi: http://dx.doi.org/10.5937/MatMor1102047S

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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.

doi: http://dx.doi.org/10.5937/MatMor1102051S

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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.

doi: http://dx.doi.org/10.5937/MatMor1102055T

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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.

doi: http://dx.doi.org/10.5937/MatMor1102065T

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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.

doi: http://dx.doi.org/10.5937/MatMor1102069T

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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.

doi: http://dx.doi.org/10.5937/MatMor1102073T

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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.