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.


54H25 Fixed-point and coincidence theorems (topological aspects)
54E70 Probabilistic metric spaces
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[1] Dhage, B.C., On common fixed point of coincidentally commuting mappings in D-metric spaces, Indian J. pure appl. math., 30, 395-406, (1999) · Zbl 0957.54023
[2] Hadgic, O., Common fixed point theorems for families of mapping in complete metric space, Math. japon., 29, 127-134, (1984)
[3] Jungck, G., Compatible mappings and common fixed points, Internat. J. math. math. sci., 771-779, (1986) · Zbl 0613.54029
[4] Jungck, G., Compatible mappings and common fixed points (2), Internat. J. math. math. sci., 285-288, (1988) · Zbl 0647.54035
[5] Jungck, G.; Rhoades, B.E., Fixed points for set valued functions without continuity, Indian J. pure appl. math., 29, 227-238, (1998) · Zbl 0904.54034
[6] Menger, K., Statistical metrics, Proc. nat. acad. sci. USA, 28, 535-537, (1942) · Zbl 0063.03886
[7] Mishra, S.N., Common fixed points of compatible mappings in PM-spaces, Math. japon., 36, 283-289, (1991) · Zbl 0731.54037
[8] Sessa, S., On a weak commutative condition in fixed point consideration, Publ. inst. math. (beograd), 32, 146-153, (1982)
[9] Schweizer, B.; Sklar, A., Statistical metric spaces, Pacific J. math., 10, 313-334, (1960) · Zbl 0091.29801
[10] Sehgal, V.M.; Bharucha-Reid, A.T., Fixed points of contraction mappings in PM-spaces, Math. system theory, 6, 97-102, (1972) · Zbl 0244.60004
[11] Singh, S.L.; Pant, B.D., Common fixed point theorems in PM spaces and extension to uniform spaces, Honam math. J., 6, 1-12, (1984) · Zbl 0945.54502
[12] Xieping, D., A common fixed point theorem of commuting mappings in PM-spaces, Kexue tongbao, 29, 147-150, (1984) · Zbl 0522.54040
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