Mathematica Moravica, Vol. 15-1 (2011)

Perturbation of Farthest Points in Weakly Compact Sets
Mathematica Moravica, Vol. 15-1 (2011), 1–6.

Abstract. If $f$ is a real valued weakly lower semi-continuous function on a Banach space $X$ and $C$ a weakly compact subset of $X$, we show that the set of $x\in X$ such that $z \mapsto\|x-z\| -f(z)$ attains its supremum on $C$ is dense in $X$. We also construct a counter example showing that the set of $x\in X$ such that $z\mapsto\|x-z\| + \|z\|$ attains its supremum on $C$ is not always dense in $X$.
Keywords: Normed space, weakly compact set, farthest points.

Mappings with a Common Fixed Point in Generalized $D^{\ast}$-Metric Spaces
Mathematica Moravica, Vol. 15-1 (2011), 7–16.

Abstract. The purpose of this paper is to establish a common fixed point theorem in a generalized $D^{\ast}$-metric space. Our results unify, generalize and complement the comparable results from the current literature.
Keywords: Eneralized $D^{\ast}$-metric space, normal cones, fixed point.

Convexity and Hausdorff-Pompeiu Distance
Mathematica Moravica, Vol. 15-1 (2011), 17–23.

Abstract. The aim of this paper is to realize a decomposition of the usual convexity structures on metric spaces. Thus, a metric space is totally convex if and only if it satisfies the conditions (A) and (B) (Proposition 2). Also, it is totally externally convex if and only if both conditions (A') and (B') are satisfied (Proposition 4). Some connections between the convexity conditions (A) and (A0) and the Hausdorff-Pompeiu metric are investigated (see, for example, Corollary 3).
Keywords: Convexity condition (A), convexity condition (B), totally convex, Hausdorff-Pompeiu metric.

Fixed Points of Occasionally Weakly Compatible Maps Satisfying General Contractive Conditions of Integral Type
Mathematica Moravica, Vol. 15-1 (2011), 25–29.

Abstract. In this paper, two common fixed point theorems for four occasionally weakly compatible maps satisfying a contractive condition of integral type are obtained. Our results improve some results especially Theorem 2.1 of [3] and Theorem 1 of [1].
Keywords: Weakly compatible maps, occasionally weakly compatible maps, contractive condition, integral type, common fixed point theorems.

Common Fixed Points for Nonexpansive Type Mappings
Mathematica Moravica, Vol. 15-1 (2011), 31–39.

Abstract. he aim of the present paper is to obtain common fixed point theorems for three mappings satisfying nonexpansive type condition. For this purpose we use the notion of pointwise $R$-weak commutativity or $R$-weak commutativity of type $(A_{g})$ but without assuming the completeness of the space or continuity of the mappings involved. We further generalize the results obtained in first three theorems by replacing the condition of noncompatibility of maps with the property (E.A). In Theorem 5, we show that if the aspect of noncompatibility is taken in place of the property (E.A), the maps become discontinuous at their common fixed point. We are, thus, able to provide one more answer to the problem posed by Rhoades [13] regarding the existence of contractive definition which is strong enough to generate fixed point but does not forces the maps to become continuous. In Theorem 6, we use the notion of conditionally commuting maps recently introduced by Pant and Pant [12] and prove a common fixed point theorem under minimal commutative condition.
Keywords: Nonexpansive conditions, nocompatible mappings, pointwise $R$-weak commutativity, $R$-weak commutativity, contractive conditions, property (E.A), conditionally commuting maps.

Some Presic Type Generalizations of the Banach Contraction Principle
Mathematica Moravica, Vol. 15-1 (2011), 41–47.

Abstract. In this paper, we extend and generalize Presic Type theorems for a pair of maps and Jungck type maps.
Keywords: Weakly $k$-compatible maps, Jungck type maps, commonfixed point.

Singer Orthogonality and James Orthogonality in the So-Called Quasi-Inner Product Space
Mathematica Moravica, Vol. 15-1 (2011), 49–52.

Abstract. In this note we prove that, in a quasi-inner product space, $S$-orthogonality and $J$-orthogonality can be defined with the best approximations.
Keywords: Singer orthogonality, James orthogonality, quasi-inner product space.

On a Result of T. Suzuki for Generalized Distance and Fixed Points
Mathematica Moravica, Vol. 15-1 (2011), 53–57.

Abstract. We prove that a main result of T. Suzuki [J. Math. Anal. Appl. 253 (2001), 440-458, Theorem 1, p. 451] has been for the first time proved 20 years ago by Tasković [Proc. Amer. Math. Soc. 94 (1985), 427-432, Theorem 2, p. 430], and second time proved by Tasković [Math. Japonica, 35 (1990), 645-666, Theorem 2, p. 654] as a very special case of the so-called Localization Monotone Principle.
Keywords: Fixed points, $\tau$-distance, complete metric spaces, topological spaces, localization monotone principle, TCS-convergence, nonlinear conditions for fixed points, Tasković's localization monotone principle of fixed point.

Diametral $\varphi$-Contraction on Topological Spaces
Mathematica Moravica, Vol. 15-1 (2011), 59–63.

Keywords: Fixed point theorems, diametral $\varphi$-Contraction, topological spaces, topological cauchy sequence, topological orbital completeness, topological diameter.
Abstract. In this paper we consider a concept of lower compact operators in lower transversal normed spaces. In this sense we obtain the basic statements for lower compact operators. Applications in nonlinear functional analysis are considered. 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 $\sum_{n=0}^{\infty} \rho(T^{n}x, T^{n+1}x) = +\infty$ 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.