Mathematica Moravica, Vol. 18-2 (2014)

Nabin Chandra Kayal, Pratap Mondal, T.K. Samanta
The Fuzzy Stability of a Pexiderized Functional Equation
Mathematica Moravica, Vol. 18-2 (2014), 1–14.
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Abstract. In this paper, Hyers-Ulam-Rassias Stability of the Pexiderized functional Equation $f(x+y) = g(x) + h(y)$ is concerned in fuzzy Banach spaces.
Keywords. Fuzzy norm, fuzzy Banach space, Pexiderized functional equation, Hyers-Ulam-Rassias stability.

Zbigniew Duszyński
On Decompositions of Continuity and α-Continuity
Mathematica Moravica, Vol. 18-2 (2014), 15–20.
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Abstract. Several results concerning a decomposition of $\alpha$-continuous, continuous and complete continuous functions are offered.
Keywords. Preopen set, $\alpha$-open set, semi-open set, $\alpha$-precontinuity, continuity, complete continuity.

K.P.R. Rao, G.N.V. Kishore, K.V. Siva Parvathi
A Quadruple Fixed Point Theorem for Contractive Type Condition by Using ICS Mapping and Application to Integral Equation
Mathematica Moravica, Vol. 18-2 (2014), 21–34.
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Abstract. In this paper, we obtain a Quadruple fixed point theorem for $\psi-\phi$ contractive condition in partially ordered partial metric spaces by using ICS mapping. We are also given an example and an application to integral equation which supports our main theorem.
Keywords. Partial metric space, Quadruple fixed point, ICS mapping, mixed monotone property.

Muhammad Siddique Bosan, Moiz ud Din Khan, Ljubiša D.R. Kočinac
On s-Topological Groups
Mathematica Moravica, Vol. 18-2 (2014), 35–44.
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Abstract. In this paper we study the class of $s$-topological groups and a wider class of $S$-topological groups which are defined by using semi-open sets and semi-continuity introduced by N. Levine. It is shown that these groups form a generalization of topological groups, and that they are different from several distinct notions of semitopological groups which appear in the literature. Counterexamples are given to strengthen these concepts. Some basic results and applications of $s$- and $S$-topological groups are presented. Similarities with and differences from topological groups are investigated.
Keywords. Semi-open set, semi-continuity, semi-homeomorphism, $s$-regular space, $s$-topological group, $S$-topological group.

Manish Jain, Calogero Vetro, Neetu Gupta, Sanjay Kumar
Coupled Common Fixed Point Theorems in Partially Ordered $G$-metric Spaces for Nonlinear Contractions
Mathematica Moravica, Vol. 18-2 (2014), 45–62.
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Abstract. The aim of this paper is to prove coupled coincidence and coupled common fixed point theorems for a mixed $ɡ$-monotone mapping satisfying nonlinear contractive conditions in the setting of partially ordered $G$-metric spaces. Present theorems are true generalizations of the recent results of Choudhury and Maity [Math Comput. Modelling 54 (2011), 73-79], and Luong and Thuan [Math. Comput. Modelling 55 (2012), 1601-1609].
Keywords. Partially ordered set, $G$-metric space, coupled coincidence point, coupled common fixed point, mixed monotone mappings.

Said Beloul
A Common Fixed Point Theorem for Weakly Compatible Multi-Valued Mappings Satisfying Strongly Tangential Property
Mathematica Moravica, Vol. 18-2 (2014), 63–72.
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Abstract. In this paper we prove a common fixed point theorem for two weakly compatible pairs of single and set-valued mappings which satisfying contractive condition of integral type in metric space by using the concept of strongly tangential property, our results generalize and extend some previous results.
Keywords. Common fixed point, strongly tangential, weakly compatible, multi-valued maps.

Bhavana Deshpande, Amrish Handa
Quadruple Coincidence and Common Quadruple Fixed Point for Hybrid Pair of Mappings Under New Contractive Condition
Mathematica Moravica, Vol. 18-2 (2014), 73–90.
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Abstract. We establish a quadruple coincidence and common quadruple fixed point theorem for hybrid pair of mappings satisfying new contractive condition. It is to be noted that to find quadruple coincidence points, we do not use the condition of continuity of any mapping involved. An example supporting to our result has also been cited. We improve, extend and generalize several known results.
Keywords. Quadruple fixed point, quadruple coincidence point, $w$-compatible mappings, F-weakly commuting.

Abdelouaheb Ardjouni, Ahcene Djoudi
Stability in Nonlinear Neutral Differential Equations with Infinite Delay
Mathematica Moravica, Vol. 18-2 (2014), 91–103.
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Abstract. In this paper we use the contraction mapping theorem to obtain asymptotic stability results of the nonlinear neutral differential equation with infinite delay \[ \frac{\mathrm{d}}{\mathrm{d}t}x(t) = -a(t) x(t-\tau_{1}(t)) + \frac{\mathrm{d}}{\mathrm{d}t} Q(t, x(t-\tau_{2}(t))) + \int_{-\infty}^{t} D(t,s) f(x(s))\mathrm{d}s.\] An asymptotic stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton [6], Zhang [17], Althubiti, Makhzoum, Raffoul [1].
Keywords. Fixed points, Stability, Neutral differential equation, Integral equation, Infinite delay.

Milan R. Tasković
Transversal Functional Analysis
Mathematica Moravica, Vol. 18-2 (2014), 105–181.
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Abstract. This paper provides an introduction to the ideas and methods of transversal functional analysis based on the transversal sets theory. A unifying concept that lies at the heart of transversal functional analysis is that of a transversal normed linear space. I have developed the theory far enough to include facts of have called the three new basic principles of linear analysis as: Form of Hahn-Banach theorem, Form of Principle of Uniform Boundedness (= Form of Banach-Steinhaus theorem), and Form of Open Mapping theorem. In the classical functional analysis fundamental fact is Riesz lemma. In transversal functional analysis (on lower transversal normed spaces) its role play so-called Geometrical lemma! This paper presents applications of the Axiom of Infinite Choice.
Keywords. Transversal (upper, lower and middle) normed spaces, Transversal seminorms, Form of Hahn-Banach theorem, Form of Principle of Uniform Boundedness, Form of Banach-Steinhaus theorem, Form of Open Mapping theorem, Form of Riesz lemma, Geometrical lemma, Form of onvergence principle, Lower compactness, Lower total continuous operators, Lower locally compactness, Lower Fredholm alternative, Extension Leray-Schauder principle, Peano’s equation, Lower compact operators.

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