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Remarks on some recent metrical common fixed point theorems. (English) Zbl 1296.54065

Summary: In an interesting article, H. Bouhadjera and C. Godet-Thobie [Acta Math. Vietnam. 36, No. 1, 1–17 (2011; Zbl 1231.54019)] introduced notions of subcompatibility and subsequential continuity, and utilized them to prove several common fixed point theorems. The results of the aforementioned article contain flaws, and they are not correct in their present form. But these results can be recovered in two ways by strengthening either of the newly introduced definitions. The results, in their corrected form, still bring about noted improvements over a multitude of relevant fixed point theorems of the existing literature, substantiating the utility of these (two) new notions.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces

Citations:

Zbl 1231.54019
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References:

[1] Sessa, S., On a weak commutativity condition in fixed point considerations, Publ. inst. math. (beograd) (NS), 34, 46, 149-153, (1982) · Zbl 0523.54030
[2] Jungck, G., Compatible mappings and common fixed points, Int. J. math. math. sci., 9, 771-779, (1986) · Zbl 0613.54029
[3] Murthy, P.P., Important tools and possible applications of metric fixed point theory, Nonlinear anal., 47, 3479-3490, (2001) · Zbl 1042.54507
[4] Al-Thagafi, M.A.; Shahzad, Naseer, Generalized \(I\)-nonexpansive selfmaps and invariant approximations, Acta math. sin. (engl. ser.), 24, 867-876, (2008) · Zbl 1175.41026
[5] Pant, R.P., Common fixed points of four mappings, Bull. Calcutta math. soc., 90, 281-286, (1998) · Zbl 0936.54043
[6] H. Bouhadjera, C. Godet-Thobie, Common fixed theorems for pairs of subcompatible maps, 17 June 2009. arXiv:0906.3159v1 [math.FA]. · Zbl 1231.54019
[7] Jungck, G.; Rhoades, B.E., Fixed point theorems for occasionally weakly compatible mappings, Fixed point theory, 7, 2, 287-296, (2006) · Zbl 1118.47045
[8] Djoudi, A.; Nisse, L., Gregus type fixed points for weakly compatible maps, Bull. belg. math. soc. Simon stevin, 10, 3, 369-378, (2003) · Zbl 1040.54019
[9] Pathak, H.K.; Cho, Y.J.; Kang, S.M.; Madharia, B., Compatible mappings of type (C) and common fixed point theorems of Gregus type, Demonstratio math., 31, 3, 499-518, (1998) · Zbl 0922.54036
[10] Ali, Javid; Imdad, M., An implicit function implies several contraction conditions, Sarajevo J. math., 4, 17, 269-285, (2008) · Zbl 1180.54052
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