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Probabilistic Metric Spaces and Some Contraction Mappings


Parvin Azhdari


Vol. 17  No. 2  pp. 56-60


There exist many types of contraction mappings in the case of single valued and multi- valued that researchers are interested in proving fixed point theorems. Now, I consider two types of contraction: (φ ? k) ? B contraction and bn-contraction . In this paper, after presenting the definition of (φ ? k) ? B contraction, I prove a lemma about continuity of (φ ? k) ? B contraction and by using it I prove a unique single valued fixed point theorem for (φ ? k) ? B contraction with t-norm which is φ-convergent in probabilistic metric space. Then multi-valued bn-contraction definition is illustrated. I obtain first multi-valued theorem with new assumptions. Finally, I prove another fixed point theorem for multi-valued case by the definition of a large class of mappings called weakly demicompact mappings.


Probabilistic metric space, bn-contraction, (φ ? k) ? B contraction, fixed point.