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1

(ε, δ) contractive condition and common fixed points

Pant V., Jha K.

Fasciculi Mathematici

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2009

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Nr 42

73-84

EN

In the present paper we prove a common fixed point theorem (Theorem 1) for four mappings under the (ε, δ) contractive condition, however, without either imposing any additional restriction on δ or assuming the ∅-contractive condition together with. While proving the theorem, neither the completeness of the metric space is assumed nor any of the mappings is required to be continuous. Thus we also provide one more answer to the problem of Rhoades [24] which ensures the existence of common fixed point, however, does not force the maps to be continuous at the common fixed point. Theorem 2 generalizes further the result obtained in Theorem 1.

2

Common fixed points of mappings not satisfying contractive condition

Pant V., Padaliya S., Pant R. P.

Fasciculi Mathematici

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2008

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Nr 40

71-77

EN

The aim of this paper is to consider a new approach for obtaining common fixed point theorems in metric spaces by subjecting the triangle inequality to a Lipschitz type condition. For values of the Lipschitz constant k < 1/3 the condition reduces to a Banach type contractive condition and we get the results known so far. However, values of k ≥ 1/3 yield new result. It may be observed that in the setting of metric spaces k ≥ 1/3 generally does not ensure the existence of fixed points and there is no known method for dealing these cases. In Theorem 1 and Theorem 2 we provide results under a new condition. In the last section of this paper (Theorem 3 and Theorem 4) by using the (E.A) property introduced by Aamri and Moutawakil [2] we extend the results obtained in Theorem 1 and Theorem 2.

3

Common fixed point theorems for hybrid pairs of mappings with some weaker conditions

Sharma S., Deshpande B., Pathak R.

Fasciculi Mathematici

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2008

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Nr 39

71-86

EN

In this paper, we prove a common fixed point theorem for hybrid pairs of set and single valued mappings without assuming compatibility and continuity of any mapping on noncomplete metric spaces. To prove the theorem, we use a noncompatible condition, that is, weak commutativity of type (KB). We show that completeness of the whole space is not necessary for the existence of common fixed point. Our result improves, extends and generalizes the results of Fisher [5], Sastry and Naidu [18]. We give an example to validate our result. We also prove a common fixed point theorem on compact metric spaces. At the end, we improve our theorem by omitting the assumption of compactness. We also improve and generalize the results of Ahmed [2] and Fisher [5].

4

Common of fixed point of multivalued mappings without continuity

Kubiaczyk I., Deshpande B.

Fasciculi Mathematici

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2007

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Nr 37

19-26

EN

In this paper, we prove a common fixed point theorem for single-valued and multivalued mappings on a metric space using the minimal type commutativity condition. We show that continuity of any mapping is not necessary for the existence of a common fixed point.

5

Fixed point theorems for set nd single valued maps witout continuity and compatibility

Sharma S., Deshpande B.

Demonstratio Mathematica

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2007

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Vol. 40, nr 3

649-658

EN

The new concept of weak commutativity of type (KB) is used to prove some fixed point theorems for set and single-valued mappings. We show that continuity of any mapping is not necessary for the existence of common fixed point. We also show that completeness of the whole space can be replaced by a weaker condition.