Kết quả hội tụ cho bài toán tìm điểm không và các điểm cố định của các nhóm không mở rộng và các phép biến đổi giả nghiêm ngặt

Springer Science and Business Media LLC - Tập 2012 - Trang 1-16 - 2012
Prasit Cholamjiak1
1School of Science, University of Phayao, Phayao, Thailand

Tóm tắt

Trong công trình này, chúng tôi thiết lập các định lý hội tụ mạnh cho việc giải quyết bài toán điểm cố định của các nhóm không mở rộng và các phép biến đổi giả nghiêm ngặt, cũng như bài toán tìm zero của các phép toán đơn điệu cực đại trong không gian Hilbert. Chúng tôi cũng áp dụng kết quả của mình cho bài toán tối ưu hóa lồi và các nhóm giao hoán.

Từ khóa

#Hội tụ #điểm cố định #nhóm không mở rộng #phép biến đổi giả nghiêm ngặt #không gian Hilbert #tối ưu hóa lồi.

Tài liệu tham khảo

Acedo GL, Xu HK: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. TMA 2007, 67: 2258–2271. 10.1016/j.na.2006.08.036 Atsushiba S, Takahashi W: Approximation common fixed points of nonexpansive semigroups by the Mann iteration process. Ann. Univ. Mariae Curie-Skl̄odowska, Sect. A 1997, 51: 1–16. Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6 Browder FE: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 1965, 54: 1041–1044. 10.1073/pnas.54.4.1041 Bruck RE: On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces. Isr. J. Math. 1981, 38: 304–314. 10.1007/BF02762776 Chen R, He H: Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space. Appl. Math. Lett. 2007, 20: 751–757. 10.1016/j.aml.2006.09.003 Chen R, Song Y: Convergence to common fixed point of nonexpansive semigroups. J. Comput. Appl. Math. 2007, 200: 566–575. 10.1016/j.cam.2006.01.009 Cho YJ, Kang SM, Zhou H: Approximate proximal point algorithms for finding zeroes of maximal monotone operators in Hilbert spaces. J. Inequal. Appl. 2008., 2008: Article ID 598191 Cholamjiak P, Suantai S: Viscosity approximation methods for a nonexpansive semigroup in Banach spaces with gauge functions. J. Glob. Optim. 2011. doi:10.1007/s10898–011–9756–4 Cholamjiak P, Cho YJ, Suantai S: Composite iterative schemes for maximal monotone operators in reflexive Banach spaces. Fixed Point Theory Appl. 2011, 2011: 7. 10.1186/1687-1812-2011-7 Day MM: Amenable semigroup. Ill. J. Math. 1957, 1: 509–544. Goebel K, Kirk WA Cambridge Studies in Advanced Mathematics 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge, UK; 1990. Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0 Kamimura S, Takahashi W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 2000, 106: 226–240. 10.1006/jath.2000.3493 Kohsaka F, Takahashi W: Proximal point algorithms with Bregman functions in Banach spaces. J. Nonlinear Convex Anal. 2005, 6: 505–523. Lau AT-M: Invariant means on almost periodic functions and fixed point properties. Rocky Mt. J. Math. 1973, 3: 69–76. 10.1216/RMJ-1973-3-1-69 Lau AT-M: Invariant means and fixed point properties of semigroup of nonexpansive mappings. Taiwan. J. Math. 2008, 12: 1525–1542. Lau AT-M, Takahashi W: Invariant means and fixed point properties for nonexpansive representations of topological semigroups. Topol. Methods Nonlinear Anal. 1995, 5: 39–57. Lau AT-M, Miyake H, Takahashi W: Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces. Nonlinear Anal. TMA 2007, 67: 1211–1225. 10.1016/j.na.2006.07.008 Lau AT-M, Shioji N, Takahashi W: Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces. J. Funct. Anal. 1999, 161: 62–75. 10.1006/jfan.1998.3352 Mann WR: Mean value methods in iterations. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3 Marino G, Xu HK: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028 Marino G, Xu HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 2007, 329: 336–346. 10.1016/j.jmaa.2006.06.055 Martinet B: Régularisation d’inéquations variationelles par approximations successives. Rev. Francaise d’Informatique et de Recherche Opérationelle 1970, 4: 154–159. Moudafi A: Viscosity approximation methods for fixed point problems. J. Math. Anal. Appl. 2000, 241: 46–55. 10.1006/jmaa.1999.6615 Petruşel A, Yao J-C: Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings. Nonlinear Anal. TMA 2008, 69: 1100–1111. 10.1016/j.na.2007.06.016 Qin X, Kang SM, Cho YJ: Approximating zeros of monotone operators by proximal point algorithms. J. Glob. Optim. 2010, 46: 75–87. 10.1007/s10898-009-9410-6 Qin X, Shang M, Kang SM: Strong convergence theorems of modified Mann iterative process for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. TMA 2009, 70: 1257–1264. 10.1016/j.na.2008.02.009 Rockafellar RT: On the maximality of suns of nonlinear monotone operators. Trans. Am. Math. Soc. 1970, 149: 75–88. 10.1090/S0002-9947-1970-0282272-5 Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056 Saeidi S: Existence of ergodic retractions for semigroups in Banach spaces. Nonlinear Anal. TMA 2008, 69: 3417–3422. 10.1016/j.na.2007.09.031 Saeidi S: Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings. Nonlinear Anal. TMA 2009, 70: 4195–4208. 10.1016/j.na.2008.09.009 Shioji N, Takahashi W: Strong convergence of average approximants for asymptotically nonexpansive mappings in Banach spaces. J. Approx. Theory 1999, 97: 53–64. 10.1006/jath.1996.3251 Solodov MV, Svaiter BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 2000, 87: 189–202. Song Y, Xu S: Strong convergence theorems for nonexpansive semigroup in Banach spaces. J. Math. Anal. Appl. 2008, 338: 152–161. 10.1016/j.jmaa.2007.05.021 Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017 Takahashi W: Nonlinear Function Analysis. Yokohama Publishers, Yokohama; 2000. Takahashi W: Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces. Nonlinear Anal. TMA 2009, 70: 719–734. 10.1016/j.na.2008.01.005 Takahashi W: A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space. Proc. Am. Math. Soc. 1981, 81: 253–256. 10.1090/S0002-9939-1981-0593468-X Wang S, Wang F: On relaxed and contraction-proximal point algorithms in Hilbert spaces. J. Inequal. Appl. 2011, 2011: 41. 10.1186/1029-242X-2011-41 Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332 Xu HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059 Xu HK: A strong convergence theorem for contraction semigroups in Banach spaces. Bull. Aust. Math. Soc. 2005, 72: 371–379. 10.1017/S000497270003519X Zhou H: Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 2008, 343: 546–556. 10.1016/j.jmaa.2008.01.045