318
Views
0
CrossRef citations to date
0
Altmetric
Research Articles

Rates of Convergence and Metastability for Chidume’s Algorithm for the Approximation of Zeros of Accretive Operators in Banach Spaces

& ORCID Icon
Pages 216-233 | Received 09 Apr 2023, Accepted 09 Feb 2024, Published online: 28 Feb 2024

References

  • Martinet, B. (1970). Régularisation d’inéquations variationnelles par approximations successives. Rev. Française Informat. Recherche Opérationnelle 4:154–158. DOI: 10.1051/m2an/197004R301541.
  • Rockafellar, R. T. (1976). Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5):877–898. DOI: 10.1137/0314056.
  • Bruck, R. E., Reich, S. (1977). Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3:459–470.
  • Güler, O. (1991). On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29(2):403–419. DOI: 10.1137/0329022.
  • Bauschke, H. H., Matoušková, E., Reich, S. (2004). Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal. 56:715–738. DOI: 10.1016/j.na.2003.10.010.
  • Kamimura, S., Takahashi, W. (2000). Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106(2):226–240. DOI: 10.1006/jath.2000.3493.
  • Xu, H.-K. (2002). Iterative algorithms for nonlinear operators. J. London Math. Soc. 66(1):240–256. DOI: 10.1112/S0024610702003332.
  • Aoyama, K., Toyoda, M. (2017). Approximation of zeros of accretive operators in a Banach space. Israel J. Math. 220:803–816. DOI: 10.1007/s11856-017-1511-1.
  • Bruck, R. E. (1974). A strongly convergent iterative method for the solution of 0∈U(x) for a maximal monotone operator U in Hilbert space. J. Math. Anal. Appl. 48:114–126. DOI: 10.1016/0022-247X(74)90219-4.
  • Reich, S. (1977). Extension problems for accretive sets in Banach spaces. J. Funct. Ana. 26:378–395. DOI: 10.1016/0022-1236(77)90022-2.
  • Chidume, C. E., Djitte, N. (2012). Strong convergence theorems for zeros of bounded maximal monotone nonlinear operators. Abstr. Appl. Anal. 2012:681348. DOI: 10.1155/2012/681348.
  • Chidume, C. E. (2016). Strong convergence theorems for bounded accretive operators in uniformly smooth Banach spaces. Contemp. Math. 659:31–41. DOI: 10.1090/conm/659.
  • Bruck, R. E., Reich, S. (1981). Accretive operators, Banach limits, and dual ergodic theorems. Bull. Acad. Polon. Sci. Sér. Math. 29:585–589.
  • Reich, S. (1980). Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75(1):287–292. DOI: 10.1016/0022-247X(80)90323-6.
  • Tao, T. (2008). Soft analysis, hard analysis, and the finite convergence principle. In: Tao, T., ed. Structure and Randomness: Pages from Year One of a Mathematical Blog. Providence, RI: American Mathematical Society, pp. 298.
  • Tao, T. (2008). Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Theory Dyn. Syst. 28(2):657–688. DOI: 10.1017/S0143385708000011.
  • Sipos, A. (2024). On quantitative metastability for accretive operators. Z. Anal. Anwend., to appear. DOI: 10.4171/zaa/1741.
  • Kohlenbach, U., Sipoş, A. (2021). The finitary content of sunny nonexpansive retractions. Commun. Contemp. Math. 23(01):1950093. DOI: 10.1142/S0219199719500937.
  • Takahashi, W. (2000). Nonlinear Functional Analysis. Fixed Point Theory and its Applications. Yokohama: Yokohama Publishers, iv + 276pp.
  • Cioranescu, I. (1990). Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Mathematics and its Applications, 62. Dordrecht: Kluwer Academic Publishers Group, xiv + 260pp.
  • Reich, S. (1992). Review of [20]. Bull. Amer. Math. Soc. 26:367–370. DOI: 10.1090/S0273-0979-1992-00287-2.
  • Kato, T. (1967). Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 19(4):508–520. DOI: 10.2969/jmsj/01940508.
  • Chidume, C. E. (2009). Geometric Properties of Banach Spaces and Nonlinear Iterations. Lecture Notes in Mathematics, Vol. 1965. London: Springer.
  • Kohlenbach, U., Leuştean, L. (2012). On the computational content of convergence proofs via Banach limits. Philos. Trans. Royal Soc. A: Math. Phys. Eng. Sci. 370:3449–3463. DOI: 10.1098/rsta.2011.0329.
  • Pischke, N. Logical metatheorems for accretive and (generalized) monotone set-valued operators. J. Math. Logic, to appear DOI: 10.1142/S0219061323500083.
  • Lindenstrauss, J., Tzafriri, L. (Reprint of 2013). Classical Banach Spaces II: Function Spaces, Vol. 97. Berlin: Springer 1979.
  • Xu, Z.-B., Roach, G. F. (1991). Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157(1):189–210. DOI: 10.1016/0022-247X(91)90144-O.
  • Petryshyn, W. V. (1970). A characterization of strict convexity of Banach spaces and other uses of duality mappings. J. Funct. Anal. 6(2):282–291. DOI: 10.1016/0022-1236(70)90061-3.
  • Kohlenbach, U., Leuştean, L. (2012). Effective metastability of Halpern iterates in CAT(0) spaces. Adv. Math. 231:2526–2556. DOI: 10.1016/j.aim.2012.06.028.
  • Körnlein, D. (2015). Quantitative results for Halpern iterations of nonexpansive mappings. J. Math. Anal. Appl. 428:1161–1172. DOI: 10.1016/j.jmaa.2015.03.020.
  • Körnlein, D., Kohlenbach, U. (2014). Rate of Metastability for Bruck’s iteration of pseudocontractive mappings in Hilbert space. Numer. Funct. Anal. Optim. 35:20–31. DOI: 10.1080/01630563.2013.809361.
  • Körnlein, D., Kohlenbach, U. (2011). Effective rates of convergence for Lipschitzian pseudocontractive mappings in general Banach spaces. Nonlinear Anal 74:5253–5267. DOI: 10.1016/j.na.2011.04.028.
  • Findling, R. (2023). Logische Analyse von C.E. Chidumes Algorithmus zur Berechnung von Nullstellen akkretiver Operatoren. Bachelor Thesis, TU Darmstadt.