Abstract
In this article, we propose two Banach-type fixed point theorems on bipolar metric spaces. More specifically, we look at covariant maps between bipolar metric spaces and consider iterates of the map involved. We also propose a generalization of the Banach fixed point result via Caristi-type arguments.
Highlights
Introduction and PreliminariesMetric fixed point theory deals with the existence of a point x∗ such that Tx∗ = x∗ where T is a self-mapping defined on a metric space ðX, mÞ
One of the most recent is that of bipolar metric space which was introduced by Mutlu and Gürdal [1]
They explored the link between metric spaces and bipolar metric spaces and proved some well-known fixed point theorems in that new setting
Summary
Metric fixed point theory deals with the existence of a point x∗ such that Tx∗ = x∗ where T is a self-mapping defined on a metric space ðX, mÞ. One of the most recent is that of bipolar metric space which was introduced by Mutlu and Gürdal [1] They explored the link between metric spaces and bipolar metric spaces and proved some well-known fixed point theorems in that new setting. Mutlu et al [2] proved coupled fixed point theorems in the setting of bipolar metric spaces. Recent work in fixed-point characterization of completeness in bipolar metric spaces can be read in [15]. A bipolar metric space is called complete, if every Cauchy bisequence is convergent, biconvergent. A covariant or a contravariant map f from the bipolar metric space ðX1, Y1, d1Þ to the bipolar metric space ðX2, Y2, d2Þ is continuous, if and only if ðunÞ ⟶ v on ðX1, Y1, d1Þ implies ð f ðunÞÞ ⟶ f ðvÞ on ðX2, Y2, d2Þ
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