Zeroes of accretive operators and the asymptotic behavior of nonlinear semigroups

Houston Journal of Mathematics

Volumn 32, Issue 4, 2006, Pages 1197-1225

  García Falset, Jesús ^a   Reich, Simeon ^b  

^a UNIVERSITY OF VALENCIA   (Spain)

^b TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Accretive operator; Asymptotic behavior; Nonexpansive semigroup; Stationary point

Indexed keywords

EID: 33846695700     PISSN: 03621588     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (40)

1. D.E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981), 423-424.
2. F. Andreu, C. Ballester, V. Caselles, J.M. Mazón, Minimizing total variation flow, Differential and Integral equations 14 (2001), 321-360.
3. Ph. Bénilan, Equations d'évolution dans un espace de Banach quelconque et applications, Thesis, Orsay, 1972.
4. Ph. Bénilan, M.G. Crandall, Regularizing effects of homogeneous evolution equations, Contributions to Analysis and Geometry, D.N. Clark et al., eds., Johns Hopkins University Press, Baltimore, 1981, 23-39.
5. Ph. Bénilan, M.G. Crandall, Completely accretive operators, Semigroup Theory and Evolution Equations, Ph. Clément, E. Mitidieri and Ben de Payter, eds., LN in Pure Applied Math. 135, Marcel Dekker, New York, 1991, 41-76.
6. 1, C. R. Acad. Sci. Paris 311 (1990), 243-246.
7. H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, 101-156.
8. H. Brézis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486-490.
9. F. E. Browder, Nonlinear accretive operators in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 470-476.
10. J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241-251.
11. Y.-Q. Chen, On the ranges of nonlinear set-valued accretive operators in Banach spaces, J. Math. Anal. Appl. 233 (1999), 827-842.
12. I. Cioranescu, Geometry of Banach spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, 1990.
13. M.G. Crandall, T. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265-298.
14. T. Domínguez Benavides, J. García Falset, M. Japon Pineda, The τ-fixed point property for nonexpansive mappings, Abstract and Applied Analysis 3 (1998), 343-362.
15. J. García Falset, Fixed point property in Banach lattices with the Banach-Saks property, Nonlinear Anal. 22 (1994), 1097-1103.
16. J. García Falset, W. Kaczor, T. Kuczumow, S. Reich, Weak convergence theorems for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 43 (2001), 377-401.
17. J. García Falset, Fixed points for mappings with the range type condition, Houston J. Math. 26 (2002), 143-158
18. K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.
19. E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York, 1975.
20. A.G. Kartsatos, Sets in the range of nonlinear accretive operators in Banach spaces, Studia Math. 114 (1995), 261-273.
21. T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508-520.
22. M.A. Khamsi, On the weak-star fixed point property, Contemp. Math. 85 (1989), 325-337.
23. M.A. Khamsi, T.H. Turpin, Fixed points of nonexpansive mappings in Banach lattices, Proc. Amer. Math. Soc. 105 (1989), 102-110.
24. M.A. Khamsi, On uniform Opial condition and uniform Kadec-Klee property in Banach and metric spaces, Nonlinear Anal. 26 (1996), 1733-1748.
25. K. Kobayashi, I. Miyadera, On the asymptotic behavior of almost-orbits of nonlinear con-traction semigroups in Banach spaces, Nonlinear Anal. 6 (1982), 349-365.
26. T. Kuczumow, Weak convergence theorems for nonexpansive mappings and semigroups in Banach spaces with Opial's property, Proc. Amer. Math. Soc. 93 (1985), 430-432.
27. 1, Studia Math. 100 (1991), 95-108.
28. R.H. Martin Jr, Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc. 179 (1973), 399-414.
29. C. Morales, Nonlinear equations involving m-accretive operators, J. Math. Anal. Appl. 97 (1983), 329-336.
30. C. Morales, Multivalued pseudo-contractive mappings defined on unbounded sets in Banach spaces, Comment. Math. Univ. Carolinae 33 (1992), 625-630.
31. O. Nevanlinna, S. Reich, Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces, Israel J. Math. 32 (1979), 44-58.
32. Z. Opial, Weak convergence of the sequence of succesive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 595-597.
33. A. Pazy, On the asymptotic behavior of semigroups of nonlinear contractions in Hilbert spaces, J. Func. Anal. 27 (1978), 292-307.
34. A. Pazy, Strong convergence of semigroups of nonlinear contractions in Hilbert space, J. d'Analyse Math. 34 (1978), 1-35.
35. A. Pazy, The Lyapunov method for semigroups of nonlinear contractions in Banach spaces, J. d'Analyse Math. 40 (1981), 239-262.
36. S. Reich, On fixed point theorems obtained from existence theorems for differential equations, J. Math. Anal. Appl. 54 (1976), 26-36.
37. S. Reich, Iterative methods for accretive sets, Nonlinear Equations in Abstract Spaces, Academic Press, New York, 1978, 317-326.
38. S. Reich, R. Torrejón, Zeros of accretive operators, Commen. Math. Univ. Carolinae 21 (1980), 619-625.
39. B. Sims, The weak-star Karlovitz lemma for dual lattices, Bull. Austral. Math. Soc. 45 (1992), 171-176.
40. H.K. Xu, Strong asymptotic behavior of almost-orbits of nonlinear semigroups, Nonlinear Anal. 46 (2001), 135-151.