A generalized Banach contraction principle that characterizes metric completeness

Suzuki, Tomonari

doi:10.1090/S0002-9939-07-09055-7

A generalized Banach contraction principle that characterizes metric completeness
HTML articles powered by AMS MathViewer

by Tomonari Suzuki PDF

Proc. Amer. Math. Soc. 136 (2008), 1861-1869 Request permission

Abstract:

We prove a fixed point theorem that is a very simple generalization of the Banach contraction principle and characterizes the metric completeness of the underlying space. We also discuss the Meir-Keeler fixed point theorem.

References

S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181.
James Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241–251. MR 394329, DOI 10.1090/S0002-9947-1976-0394329-4
James Caristi and William A. Kirk, Geometric fixed point theory and inwardness conditions, The geometry of metric and linear spaces (Proc. Conf., Michigan State Univ., East Lansing, Mich., 1974) Lecture Notes in Math., Vol. 490, Springer, Berlin, 1975, pp. 74–83. MR 0399968
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
Ljubomir Ćirić, A new fixed-point theorem for contractive mappings, Publ. Inst. Math. (Beograd) (N.S.) 30(44) (1981), 25–27. MR 672538
E. H. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc. 10 (1959), 974–979. MR 110093, DOI 10.1090/S0002-9939-1959-0110093-3
J. Dugundji, Positive definite functions and coincidences, Fund. Math. 90 (1975/76), no. 2, 131–142. MR 400192, DOI 10.4064/fm-90-2-131-142
I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. MR 346619, DOI 10.1016/0022-247X(74)90025-0
Ivar Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 3, 443–474. MR 526967, DOI 10.1090/S0273-0979-1979-14595-6
T. K. Hu, On a fixed-point theorem for metric spaces, Amer. Math. Monthly 74 (1967), 436–437. MR 210107, DOI 10.2307/2314587
R. Kannan, Some results on fixed points. II, Amer. Math. Monthly 76 (1969), 405–408. MR 257838, DOI 10.2307/2316437
Jacek Jachymski, Equivalent conditions and the Meir-Keeler type theorems, J. Math. Anal. Appl. 194 (1995), no. 1, 293–303. MR 1353081, DOI 10.1006/jmaa.1995.1299
W. A. Kirk, Caristi’s fixed point theorem and metric convexity, Colloq. Math. 36 (1976), no. 1, 81–86. MR 436111, DOI 10.4064/cm-36-1-81-86
W. A. Kirk, Contraction mappings and extensions, Handbook of metric fixed point theory, Kluwer Acad. Publ., Dordrecht, 2001, pp. 1–34. MR 1904272
W. A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003), no. 2, 645–650. MR 1961251, DOI 10.1016/S0022-247X(02)00612-1
Marek Kuczma, Bogdan Choczewski, and Roman Ger, Iterative functional equations, Encyclopedia of Mathematics and its Applications, vol. 32, Cambridge University Press, Cambridge, 1990. MR 1067720, DOI 10.1017/CBO9781139086639
A. Meir and Emmett Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326–329. MR 250291, DOI 10.1016/0022-247X(69)90031-6
Sam B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475–488. MR 254828
Sehie Park, Characterizations of metric completeness, Colloq. Math. 49 (1984), no. 1, 21–26. MR 774845, DOI 10.4064/cm-49-1-21-26
Simeon Reich, Kannan’s fixed point theorem, Boll. Un. Mat. Ital. (4) 4 (1971), 1–11. MR 0305163
P. V. Subrahmanyam, Remarks on some fixed-point theorems related to Banach’s contraction principle, J. Mathematical and Physical Sci. 8 (1974), 445–457; errata, ibid. 9 (1975), 195. MR 358749
P. V. Subrahmanyam, Completeness and fixed-points, Monatsh. Math. 80 (1975), no. 4, 325–330. MR 391065, DOI 10.1007/BF01472580
Tomonari Suzuki, Generalized distance and existence theorems in complete metric spaces, J. Math. Anal. Appl. 253 (2001), no. 2, 440–458. MR 1808147, DOI 10.1006/jmaa.2000.7151
Tomonari Suzuki, Several fixed point theorems concerning $\tau$-distance, Fixed Point Theory Appl. 3 (2004), 195–209. MR 2096951, DOI 10.1155/S168718200431003X
Tomonari Suzuki, Contractive mappings are Kannan mappings, and Kannan mappings are contractive mappings in some sense, Comment. Math. (Prace Mat.) 45 (2005), no. 1, 45–58. MR 2199893
J. D. Weston, A characterization of metric completeness, Proc. Amer. Math. Soc. 64 (1977), no. 1, 186–188. MR 458359, DOI 10.1090/S0002-9939-1977-0458359-2