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A fixed point theorem in Menger space through weak compatibility. (English) Zbl 1068.54044

A Menger space is a probabilistic generalization of a metric space \(d\), in which, to every two points \(x\) and \(y\), we assign a cumulative distribution function \(F_{xy}(t)\) whose intended meaning is \(\text{Prob}(d(x,y)\leq t)\). Several theorems about fixed points and joint fixed points have been generalized from metric spaces to Menger spaces. These fixed point theorems use different ideas and techniques. The authors succeed in combining many of these results into a single new joint fixed point theorem for six self-maps in a complete Menger space. This theorem includes, as particular cases, many known results about fixed points in Menger spaces and in metric spaces.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54E70 Probabilistic metric spaces
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