Viscosity regularization iterative methods and convergence analysis
- School of Information Engineering, North China University of Water Resources and Electric Power, Zhengzhou 450011, China.
- School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China.
In this paper, a Moudafi's type viscosity regularization iterative method is introduced and investigated for an \(m\)-accretive mapping and a nonexpansive mapping. Strong convergence of the regularization iterative method is obtained in the framework of real uniformly smooth Banach spaces. Some subresults are also provided as applications of the main results.
- Accretive mapping
- regularization iteration
- uniform smoothness
- operator equation
I. K. Argyros, S. George, S. M. Erappa, Expanding the applicability of the generalized Newton method for generalized equations, Commun. Optim. Theory, 2017 (2017), 12 pages.
V. Barbu , Nonlinear semigroups and differential equations in Banach spaces, Translated from the Romanian, Editura Academiei Republicii Socialiste Romania, Bucharest; Noordhoff International Publishing, Leiden (1976)
F. E. Browder , Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1272–1276.
F. E. Browder, Existence and approximation of solutions of nonlinear variational inequalities, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1080–1086.
L.-C. Ceng, C.-F. Wen, Y.-H. Yao, Iteration approaches to hierarchical variational inequalities for infinite nonexpansive mappings and finding zero points of m-accretive operators, J. Nonlinear Var. Anal., 1 (2017), 213–235.
O. Chadli, A. Koukkous, A. Saidi , Existence of anti-periodic solutions for nonlinear implicit evolution equations with time dependent pseudomonotone operators, J. Nonlinear Var. Aanl., 1 (2017), 71–88.
S.-S. Chang, Some problems and results in the study of nonlinear analysis, Proceedings of the Second World Congress of Nonlinear Analysts, Part 7, Athens, (1996), Nonlinear Anal., 30 (1997), 4197–4208.
S.-S. Chang, H. W. J. Lee, C. K. Chan, Strong convergence theorems by viscosity approximation methods for accretive mappings and nonexpansive mappings, J. Appl. Math. Inform., 27 (2009), 59–68.
S. Y. Cho, B. A. Bin Dehaish, X.-L. Qin, Weak convergence of a splitting algorithm in Hilbert spaces, J. Appl. Anal. Comput., 7 (2017), 427–438.
S. Y. Cho, X.-L. Qin, L. Wang , Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., 2014 (2014), 15 pages.
T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19 (1967), 508–520.
T.-H. Kim, H.-K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal., 61 (2005), 51–60.
W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004–1006.
L. S. Liu , Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194 (1995), 114–125.
A. Moudafi , Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46–55.
M. A. Noor, R. Kamal, K. I. Noor , General variational inclusions and dynamical systems, Commun. Optim. Theory, 2017 (2017), 18 pages.
M. A. Noor, T. Rassias, Z.-Y. Huang, Three-step iterations for nonlinear accretive operator equations, J. Math. Anal. Appl., 274 (2002), 59–68.
X.-L. Qin, S. Y. Cho , Convergence analysis of a monotone projection algorithm in reflexive Banach spaces, Acta Math. Sci. Ser. B Engl. Ed., 37 (2017), 488–502.
X.-L. Qin, S. Y. Cho, L. Wang, Iterative algorithms with errors for zero points of m-accretive operators, Fixed Point Theory Appl., 2013 (2013), 17 pages.
X.-L. Qin, J.-C. Yao, Projection splitting algorithms for nonself operators, J. Nonlinear Convex Anal., 18 (2017), 925–935.
S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274–276.
S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75 (1980), 287–292.
R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), 97–116.
T. Suzuki , Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., 2005 (2005), 103–123.