Abstract
In this paper, in order to solve a variational inequality problem over the fixed point set of a nonexpansive mapping on uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm, we investigate an explicit iteration method, based on the steepest-descent and Krasnosel’skii–Mann algorithms. We also show that some modifications of the last and Halpern-type algorithms are special cases of our result.
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References
Agarwal R. P., O’Regan D., Sahu D. R.: Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Springer, New York (2009)
Browder E. F., Petryshin W. V.: Constructions of fixed points of nonlinear mappings. J. Math. Anal. Appl. 20, 197–228 (1967)
Buong Ng., Duong L. Th.: An explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 151, 513–524 (2011)
Byrne C.: A unified treatment of some iterative algorithm in signal processing and image reconstruction. Inverse Problems 20, 103–120 (2004)
Ceng L. C., Ansari Q. H., Yao J. Ch.: Mann-type steepest-descent and modified hybrid steepest descent methods for variational inequalities in Banach spaces. Numer. Funct. Anal. Optim. 29, 987–1033 (2008)
Cioranescu I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Acad. Publ., Dordrecht (1990)
Goebel K., Reich S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)
Halpern B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 957–961 (1967)
Iiduka I.: Fixed point optimization algorithm and its application to power control in CDMA data networks. Math. Program. 133, 227–242 (2012)
Iiduka I.: An ergodic algorithm for the power-control games for CDMA data networks. J. Math. Model. Algorithms 8, 1–18 (2009)
Iiduka I.: Fixed point optimization algorithm and its application to network bandwidth allocation. J. Comput. Appl. Math. 236, 1733–1742 (2012)
Iiduka I.: Iterative algorithm for triple-hierarchical constrained nonconvex optimization problem and its application to network bandwidth allocation. SIAM J. Optim. 22, 862–878 (2012)
Iiduka I.: Fixed point optimization algorithms for distributed optimization in network systems. SIAM J. Optim. 23, 1–26 (2013)
Ishikawa S.: Fixed points by a new iteration method. Proc. Amer. Math. Soc. 44, 147–150 (1974)
L. V. Kantorovich and G. P. Akilov, Functional Analysis. 2nd ed., Nauka, Moscow, 1977 (in Russian).
Kim T. H., Xu H. K.: Strong convergence of modified Mann iterations. Nonlinear Anal. 61, 51–60 (2005)
Krasnosel’skiĭ M. A.: Two remarks on the method of successive approximations. Uspekhi Mat. Nauk 10, 123–127 (1955)
Mann W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)
Moudafi A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000)
Reich S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979)
Reich S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75, 287–292 (1980)
Reich S.: Review of: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, by Ioana Cioranescu [6]. Bull. Amer. Math. Soc. 26, 367–370 (1992)
Shehu Y.: Modified Krasnoselskii-Mann iterative algorithm for nonexpansive mappings in Banach spaces. Arab J. Math. (Springer) 2, 209–219 (2013)
Suzuki T.: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 103–123 (2005)
Suzuki T.: A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings. Proc. Amer. Math. Soc. 135, 99–106 (2007)
Takahashi W., Ueda Y.: On Reich’s strong convergence theorems for resolvents of accretive operators. J. Math. Anal. Appl. 104, 546–553 (1984)
Vainberg M.M.: Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations. Wiley, New York (1973)
Xu H.-K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. (2) 66, 240–256 (2002)
Xu Z., Roach G.F.: A necessary and sufficient condition for convergence of steepest descent approximation to accretive operator equations. J. Math. Anal. Appl. 167, 340–354 (1992)
I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (D. Butnariu, Y. Censor and S. Reich, eds.), North–Holland, Amsterdam, 2001, 473–504.
Yao Y., Zhou H., Liou Y.-Ch.: Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings. J. Appl. Math. Comput. 29, 383–389 (2009)
Zhou H.: A characteristic condition for convergence of steepest descent approximation to accretive operator equations. J. Math. Anal. Appl. 271, 1–6 (2002)
Zhou H., Wang P.: A simpler explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 161, 716–727 (2014)
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Buong, N., Quynh, V.X. & Thuy, N.T.T. A steepest-descent Krasnosel’skii–Mann algorithm for a class of variational inequalities in Banach spaces. J. Fixed Point Theory Appl. 18, 519–532 (2016). https://doi.org/10.1007/s11784-016-0290-3
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DOI: https://doi.org/10.1007/s11784-016-0290-3