Ergodic and fixed point theorems for sequences and nonlinear mappings in a Hilbert space

• Autorzy: Rouhani B. D.

Identyfikatory

DOI

10.1515/dema-2018-0005

• Języki publikacji: EN

Abstrakty

EN

In this paper, we introduce the notion of 2-generalized hybrid sequences, extending the notion of nonexpansive and hybrid sequences introduced and studied in our previous work [Djafari Rouhani B., Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. thesis, Yale University, 1981; and other published in J. Math. Anal. Appl., 1990, 2002, and 2014; Nonlinear Anal., 1997, 2002, and 2004], and prove ergodic and convergence theorems for such sequences in a Hilbert space H. Subsequently, we apply our results to prove new fixed point theorems for 2-generalized hybrid mappings, first introduced in [Maruyama T., Takahashi W., Yao M., Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces, J. Nonlinear Convex Anal., 2011, 12, 185-197] and further studied in [Lin L.-J., Takahashi W., Attractive point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces, Taiwanese J. Math., 2012, 16, 1763-1779], defined on arbitrary nonempty subsets of H.

Słowa kluczowe

EN

ergodic theorem; asymptotically regular; weak convergence theorem; generalized hybrid sequence; fixed point; absolute fixed point; Hilbert space

PL

twierdzenie ergodyczne; słaba zbieżność; punkt stały; przestrzeń Hilberta

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2018
• Tom: Vol. 51, nr 1
• Strony: 27--36
• Opis fizyczny: Bibliogr. 27 poz.

Twórcy

autor

Rouhani B. D.

• Department of Mathematical Sciences, University of Texas at El Paso, 500 W. University Avenue, El Paso, Texas 79968 U.S.A.

Bibliografia

• [1] Goebel K., Kirk W. A., Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, United Kingdom, 1990
• [2] Goebel K., Reich S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics, Dekker, New York, Basel, 1984, 83
• [3] Takahashi W., Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000
• [4] Baillon J. B., Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert, C. R. Acad. Sci. Paris Sér., 1975, A-B 280, 1511-1514
• [5] Reich S., Almost convergence and nonlinear ergodic theorems, J. Approximation Theory, 1978, 24, 269-272
• [6] Bruck R. E., Reich S., Accretive operators, Banach limits, and dual ergodic theorems, Bull. Aca. Polon. Sci., 1981, 29, 585-589
• [7] Djafari Rouhani B., Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. thesis, Yale University, Part I (1981), 1-76
• [8] Djafari Rouhani B., Asymptotic behaviour of quasi-autonomous dissipative systems in Hilber spaces, J. Math. Anal. Appl., 1990, 147, 465-476
• [9] Djafari Rouhani B., Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl., 1990, 151, 226-235
• [10] Djafari Rouhani B., On the fixed point property for nonexpansive mappings and semigroups, Nonlinear Anal., 1997, 30, 389-396
• [11] Djafari Rouhani B., Remarks on asymptotically nonexpansive mappings in Hilbert space, Nonlinear Anal., 2002, 49, 1099-1104
• [12] Djafari Rouhani B., Asymptotic behavior of uniformly asymptotically almost nonexpansive curves in a Hilbert space, Nonlinear Anal., 2004, 58, 143-157
• [13] Djafari Rouhani B., Kim J. K., Asymptotic behavior for almost-orbits of a re versible semigroup of non-Lipschitzian mappings in a metric space, J. Math. Anal. Appl., 2002, 276, 422-431
• [14] Kohsaka F., Takahashi W., Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math., 2008, 91, 166-177
• [15] Takahashi W., Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal., 2010, 11, 79-88
• [16] Iemoto S., Takahashi W., Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space, Nonlinear Anal., 2009, 71, e2082-e2089
• [17] Kohsaka F., Takahashi W., Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach sapces, SIAM J. Optim., 2008, 19, 824-835
• [18] Kocourek P., Takahashi W., Yao J.-C., Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces, Taiwanese J. Math., 2010, 14, 2497-2511
• [19] Takahashi W., Takeuchi Y., Nonlinear ergodic theorem without convexity for generalized hybrid mappings in a Hilbert space, J. Nonlinear Convex Anal., 2011, 12, 399-406
• [20] Maruyama T., Takahashi W., Yao M., Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces, J. Nonlinear Convex Anal., 2011, 12, 185-197
• [21] Lin L.-J., Takahashi W., Attractive point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces, Taiwanese J. Math., 2012, 16, 1763-1779
• [22] Djafari Rouhani B., Ergodic theorems for hybrid sequences in a Hilbert space with applications, J. Math. Anal. Appl., 2014, 409, 205-211
• [23] Goebel K., Schöneberg R., Moons, bridges, birds... and nonexpansive mappings in Hilbert space, Bull. Austral. Math. Soc., 1977, 17, 463-466
• [24] Schoenberg I. J., On a theorem of Kirszbraun and Valentine, Amer. Math. Monthly, 1953, 60, 620-622
• [25] Reich S., Simons S., Fenchel duality, Fitzpatrick functions and the Kirszbraun-Valentine extension theorem, Proc. Amer. Math. Soc., 2005, 133, 2657-2660
• [26] Edelstein M., The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc. , 1972, 78, 206-208
• [27] Takahashi W., Toyoda M., Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 2003, 118, 417-428

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).