Discontinuity and fixed points. (English) Zbl 0938.54040

The author gives a fixed point theorem for a self-map of a metric space \((X,d)\) under a contractivity condition which does not force the map to be continuous at the fixed point. He also gives a common fixed point theorem for two self-mappings \(f,g\) which satisfy a contractivity condition and moreover are “weakly commuting”, i.e. there exists some positive real number \(R\) such that \(d(ffx, gfx)\leq R\cdot d(fx,gx)\).
Reviewer: D.Roux (Milano)


54H25 Fixed-point and coincidence theorems (topological aspects)
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[1] Jachymski, J., Common fixed point theorems for some families of maps, Indian J. pure appl. math., 25, 925-937, (1994) · Zbl 0811.54034
[2] Jungck, G., Compatible mappings and common fixed points, Internat. J. math. math. sci., 9, 771-779, (1986) · Zbl 0613.54029
[3] Kannan, R., Some results on fixed points, Bull. Calcutta math. soc., 60, 71-76, (1968) · Zbl 0209.27104
[4] Pant, R.P., Common fixed points of noncommuting mappings, J. math. anal. appl., 188, 436-440, (1994) · Zbl 0830.54031
[5] Pant, R.P., Common fixed point theorems for contractive maps, J. math. anal. appl., 226, 251-258, (1998) · Zbl 0916.54027
[6] Pathak, H.K.; Cho, Y.J.; Kang, S.M., Remarks on R-weakly commuting mappings and common fixed point theorems, Bull. Korean math. soc., 34, 247-257, (1997) · Zbl 0878.54032
[7] Rhoades, B.E., Contractive definitions and continuity, Contemp. math., 72, 233-245, (1982)
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