A fixed point principle in ordered metric spaces and applications to rational type contractions

I. Eroǧlu; E. Güner; H. Aygün; O. Valero; I. Eroǧlu; E. Güner; H. Aygün; O. Valero

doi:10.3934/math.2022750

AIMS Mathematics

2022, Volume 7, Issue 7: 13573-13594. doi: 10.3934/math.2022750

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Research article

A fixed point principle in ordered metric spaces and applications to rational type contractions

1.
Ordu University, Department of Mathematics, Altinordu, 52200, Ordu, Turkey
2.
Kocaeli University, Kocaeli University, Department of Mathematics, Umuttepe Campus, 41380, Kocaeli, Turkey
3.
Balearic Islands University, Department of Mathematics and Computer Science, Palma, 07122, Baleares, Spain
4.
Health Research Institute of the Balearic Islands (IdISBa), Hospital Universitari Son Espases, Palma, 07120, Baleares, Spain

Received: 08 February 2022 Revised: 03 May 2022 Accepted: 12 May 2022 Published: 20 May 2022
MSC : 47H10, 54E50, 54E55, 54F05, 54H25

Fixed points results for rational type contractions in metric spaces have been widely studied in the literature. In the last years, many of these results are obtained in the context of partially ordered metric spaces. In this paper, we introduce a fixed point principle for a class of mappings between partially ordered metric spaces that we call orbitally order continuous. We show that the hypotheses in the statement of such a principle are not redundant and, in addition, that they cannot be weakened in order to guarantee the existence of a fixed point. Moreover, the relationship between this kind of mappings and those that are continuous and orbitally continuous is discussed. As an application, we extend many fixed point theorems for continuous contractions of rational type to the framework of those that are only orbitally order continuous. Furthermore, we get extensions of the aforementioned metric fixed point results to the framework of partial metrics. This is achieved thanks to the fact that each partial metric induces in a natural way a metric in such a way that our new principle is applicable. In both approaches, the metric and the partial metric, we show that there are orbitally order continuous mappings that satisfy all assumptions in our new fixed point principle but that they are not contractions of rational type. The explored theory is illustrated by means of appropriate examples.
- partial order,
- metric space,
- partial metric space,
- fixed point principle,
- rational contraction
Citation: I. Eroǧlu, E. Güner, H. Aygün, O. Valero. A fixed point principle in ordered metric spaces and applications to rational type contractions[J]. AIMS Mathematics, 2022, 7(7): 13573-13594. doi: 10.3934/math.2022750

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Abstract

Fixed points results for rational type contractions in metric spaces have been widely studied in the literature. In the last years, many of these results are obtained in the context of partially ordered metric spaces. In this paper, we introduce a fixed point principle for a class of mappings between partially ordered metric spaces that we call orbitally order continuous. We show that the hypotheses in the statement of such a principle are not redundant and, in addition, that they cannot be weakened in order to guarantee the existence of a fixed point. Moreover, the relationship between this kind of mappings and those that are continuous and orbitally continuous is discussed. As an application, we extend many fixed point theorems for continuous contractions of rational type to the framework of those that are only orbitally order continuous. Furthermore, we get extensions of the aforementioned metric fixed point results to the framework of partial metrics. This is achieved thanks to the fact that each partial metric induces in a natural way a metric in such a way that our new principle is applicable. In both approaches, the metric and the partial metric, we show that there are orbitally order continuous mappings that satisfy all assumptions in our new fixed point principle but that they are not contractions of rational type. The explored theory is illustrated by means of appropriate examples.

References

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