A comparison of various definitions of contractive mappings

Rhoades, B. E.

doi:10.1090/S0002-9947-1977-0433430-4

A comparison of various definitions of contractive mappings
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Trans. Amer. Math. Soc. 226 (1977), 257-290 Request permission

Abstract:

A number of authors have defined contractive type mappings on a complete metric space X which are generalizations of the well-known Banach contraction, and which have the property that each such mapping has a unique fixed point. The fixed point can always be found by using Picard iteration, beginning with some initial choice ${x_0} \in X$. In this paper we compare this multitude of definitions. X denotes a complete metric space with distance function d, and f a function mapping X into itself.

References

D. F. Bailey, Some theorems on contractive mappings, J. London Math. Soc. 41 (1966), 101–106. MR 189007, DOI 10.1112/jlms/s1-41.1.101
L. P. Belluce and W. A. Kirk, Fixed-point theorems for certain classes of nonexpansive mappings, Proc. Amer. Math. Soc. 20 (1969), 141–146. MR 233341, DOI 10.1090/S0002-9939-1969-0233341-4
Rosa Maria Tiberio Bianchini, Su un problema di S. Reich riguardante la teoria dei punti fissi, Boll. Un. Mat. Ital. (4) 5 (1972), 103–108 (Italian, with English summary). MR 0308875
D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458–464. MR 239559, DOI 10.1090/S0002-9939-1969-0239559-9
S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci. 25 (1972), 727–730. MR 324493
S. K. Chatterjea, Fixed point theorems for a sequence of mappings with contractive iterates, Publ. Inst. Math. (Beograd) (N.S.) 14(28) (1972), 15–18. MR 397700
Ljubomir B. Ćirić, Generalized contractions and fixed-point theorems, Publ. Inst. Math. (Beograd) (N.S.) 12(26) (1971), 19–26. MR 309092
Lj. B. Ćirić, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 45 (1974), 267–273. MR 356011, DOI 10.1090/S0002-9939-1974-0356011-2
Michael Edelstein, An extension of Banach’s contraction principle, Proc. Amer. Math. Soc. 12 (1961), 7–10. MR 120625, DOI 10.1090/S0002-9939-1961-0120625-6
M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74–79. MR 133102, DOI 10.1112/jlms/s1-37.1.74
K. Goebel, W. A. Kirk, and Tawfik N. Shimi, A fixed point theorem in uniformly convex spaces, Boll. Un. Mat. Ital. (4) 7 (1973), 67–75 (English, with Italian summary). MR 0320834
Pramila Srivastava and Vijay Kumar Gupta, A note on common fixed points, Yokohama Math. J. 19 (1971), no. 2, 91–95. MR 410720
Vijay Kumar Gupta and Pramila Srivastava, On common fixed points, Rev. Roumaine Math. Pures Appl. 17 (1972), 531–538. MR 305157
L. F. Guseman Jr., Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc. 26 (1970), 615–618. MR 266010, DOI 10.1090/S0002-9939-1970-0266010-3
G. E. Hardy and T. D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull. 16 (1973), 201–206. MR 324495, DOI 10.4153/CMB-1973-036-0
Kiyoshi Iséki, On common fixed points of mappings, Bull. Austral. Math. Soc. 10 (1974), 365–370. MR 356014, DOI 10.1017/S0004972700041058
R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71–76. MR 257837
R. Kannan, Some results on fixed points. II, Amer. Math. Monthly 76 (1969), 405–408. MR 257838, DOI 10.2307/2316437
Silvio Massa, Un’osservazione su un teorema di Browder-Roux-Soardi, Boll. Un. Mat. Ital. (4) 7 (1973), 151–155 (Italian, with English summary). MR 0315531
Nicolae Mureşan, Families of mappings and fixed points, Studia Univ. Babeş-Bolyai Ser. Math.-Mech. 19 (1974), no. 1, 13–15 (Romanian, with English and Russian summaries). MR 341463
E. Rakotch, A note on contractive mappings, Proc. Amer. Math. Soc. 13 (1962), 459–465. MR 148046, DOI 10.1090/S0002-9939-1962-0148046-1
Baidyanath Ray, Some results on fixed points and their continuity, Colloq. Math. 27 (1973), 41–48. MR 336728, DOI 10.4064/cm-27-1-41-48
Barada K. Ray, On non-expansive mappings in a metric space, Nanta Math. 7 (1974), no. 1, 86–92. MR 383165
Simeon Reich, Some remarks concerning contraction mappings, Canad. Math. Bull. 14 (1971), 121–124. MR 292057, DOI 10.4153/CMB-1971-024-9
Simeon Reich, Kannan’s fixed point theorem, Boll. Un. Mat. Ital. (4) 4 (1971), 1–11. MR 0305163
Simeon Reich, Fixed points of contractive functions, Boll. Un. Mat. Ital. (4) 5 (1972), 26–42 (English, with Italian summary). MR 0309095
Simeon Reich, Remarks on fixed points, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 52 (1972), 689–697 (English, with Italian summary). MR 331139
Delfina Roux and Paolo Soardi, Alcune generalizzazioni del teorema di Browder-Göhde-Kirk, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 52 (1972), 682–688 (Italian, with English summary). MR 326512
Ioan A. Rus, On common fixed points, Studia Univ. Babeş-Bolyai Ser. Math.-Mech. 18 (1973), no. 1, 31–33 (English, with Romanian and Russian summaries). MR 336729

Some common fixed point theorems for compact metric spaces

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V. M. Sehgal, On fixed and periodic points for a class of mappings, J. London Math. Soc. (2) 5 (1972), 571–576. MR 319176, DOI 10.1112/jlms/s2-5.3.571
S. P. Singh, Some results on fixed point theorems, Yokohama Math. J. 17 (1969), 61–64. MR 257839
Paolo Soardi, Su un problema di punto unito di S. Reich, Boll. Un. Mat. Ital. (4) 4 (1971), 841–845 (Italian, with English summary). MR 0301583
Chi Song Wong, Common fixed points of two mappings, Pacific J. Math. 48 (1973), 299–312. MR 328903, DOI 10.2140/pjm.1973.48.299
Chi Lin Yen, Remark on common fixed points, Tamkang J. Math. 3 (1972), 95–96. MR 322845
Tudor Zamfirescu, A theorem on fixed points, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 52 (1972), 832–834 (1973) (English, with Italian summary). MR 328904
Tudor Zamfirescu, Fix point theorems in metric spaces, Arch. Math. (Basel) 23 (1972), 292–298. MR 310859, DOI 10.1007/BF01304884