A fixed-point theorem for asymptotically contractive mappings

Penot, Jean-Paul

doi:10.1090/S0002-9939-03-06999-5

A fixed-point theorem for asymptotically contractive mappings
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Proc. Amer. Math. Soc. 131 (2003), 2371-2377 Request permission

Abstract:

We present fixed point theorems for a nonexpansive mapping from a closed convex subset of a uniformly convex Banach space into itself under some asymptotic contraction assumptions. They generalize results valid for bounded convex sets or asymptotically compact sets.

References

A. AGADI and J.-P. PENOT, Asymptotic approximation of sets with application in mathematical programming, preprint, Univ. of Pau, February 1996.
Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. MR 187120, DOI 10.1073/pnas.54.4.1041
Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1976, pp. 1–308. MR 0405188
Jean-Pierre Dedieu, Cône asymptote d’un ensemble non convexe. Application à l’optimisation, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 7, A501–A503 (French, with English summary). MR 458117
Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258 (German). MR 190718, DOI 10.1002/mana.19650300312
Vasile I. Istrăţescu, Fixed point theory, Mathematics and its Applications, vol. 7, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. An introduction; With a preface by Michiel Hazewinkel. MR 620639
W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. MR 189009, DOI 10.2307/2313345
W. A. Kirk, Nonexpansive mappings and asymptotic regularity, Nonlinear Anal. 40 (2000), no. 1-8, Ser. A: Theory Methods, 323–332. Lakshmikantham’s legacy: a tribute on his 75th birthday. MR 1768416, DOI 10.1016/S0362-546X(00)85019-1
W. A. Kirk, The fixed point property and mappings which are eventually nonexpansive, Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Appl. Math., vol. 178, Dekker, New York, 1996, pp. 141–147. MR 1386675
W. A. Kirk, Carlos Martinez Yañez, and Sang Sik Shin, Asymptotically nonexpansive mappings, Nonlinear Anal. 33 (1998), no. 1, 1–12. MR 1623029, DOI 10.1016/S0362-546X(97)00541-5
M. A. Krasnosel′skiĭ, Positive solutions of operator equations, P. Noordhoff Ltd., Groningen, 1964. Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron. MR 0181881
D. T. Luc, Recession maps and applications, Optimization 27 (1993), no. 1-2, 1–15. MR 1275544, DOI 10.1080/02331939308843868
D. T. LUC, Recessively compact sets: uses and properties, Set-Valued Anal. 10 (2002), 15-35.
Dinh The Luc and Jean-Paul Penot, Convergence of asymptotic directions, Trans. Amer. Math. Soc. 353 (2001), no. 10, 4095–4121. MR 1837222, DOI 10.1090/S0002-9947-01-02664-2
Jean-Paul Penot, Fixed point theorems without convexity, Bull. Soc. Math. France Mém. 60 (1979), 129–152. Analyse non convexe (Proc. Colloq., Pau, 1977). MR 562261
Jean-Paul Penot, Compact nets, filters, and relations, J. Math. Anal. Appl. 93 (1983), no. 2, 400–417. MR 700155, DOI 10.1016/0022-247X(83)90184-1
Jean-Paul Penot, What is quasiconvex analysis?, Optimization 47 (2000), no. 1-2, 35–110. CODE Workshop on Generalized Convexity and Monotonicity in Economic Modeling (Bellaterra, 1998). MR 1755580, DOI 10.1080/02331930008844469
J.-P. PENOT, A metric approach to asymptotic analysis, preprint, Univ. of Pau, June 2001.
J.-P. PENOT and C. ZALINESCU, Continuity of usual operations and variational convergences, preprint, Univ. of Pau, 2000 and 2001, to appear in Set-Valued Analysis.
Behzad Djafari Rouhani and W. A. Kirk, Asymptotic properties of nonexpansive iterations in reflexive spaces, J. Math. Anal. Appl. 236 (1999), no. 2, 281–289. MR 1704583, DOI 10.1006/jmaa.1999.6423
Sankatha Singh, Bruce Watson, and Pramila Srivastava, Fixed point theory and best approximation: the KKM-map principle, Mathematics and its Applications, vol. 424, Kluwer Academic Publishers, Dordrecht, 1997. MR 1483076, DOI 10.1007/978-94-015-8822-5
C. Zălinescu, Recession cones and asymptotically compact sets, J. Optim. Theory Appl. 77 (1993), no. 1, 209–220. MR 1222792, DOI 10.1007/BF00940787
Eberhard Zeidler, Nonlinear functional analysis and its applications. I, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated from the German by Peter R. Wadsack. MR 816732, DOI 10.1007/978-1-4612-4838-5