On nonexpansive and accretive operators in Banach spaces

Li, Dongfeng¹

¹ School of Information Engineering, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 7, p. 3437-3446.

Résumé

The purpose of this article is to investigate common solutions of a zero point problem of a accretive operator and a fixed point problem of a nonexpansive mapping via a viscosity approximation method involving a $\tau$ -contractive mapping

Détail

DOI : 10.22436/jnsa.010.07.08

Classification : 47H05, 65J15
Keywords: Accretive operator, approximation solution, viscosity method, variational inequality.

Bibliographie

[1] Argyros, I. K.; George, S. Iterative regularization methods for nonlinear ill-posed operator equations with m-accretive mappings in Banach spaces, Acta Math. Sci. Ser. B Engl. Ed., Volume 35 (2015), pp. 1318-1324 | Zbl | DOI

[2] Argyros, I. K.; George, S. Extending the applicability of a new Newton-like method for nonlinear equations, Commun. Optim. Theory, Volume 2016 (2016), pp. 1-9

[3] Barbu, V. Nonlinear semigroups and differential equations in Banach spaces, Translated from the Romanian, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976

[4] Dehaish, B. A. Bin; Latif, A.; Bakodah, H. O.; Qin, X.-L. A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces, J. Inequal. Appl., Volume 2015 (2015), pp. 1-14 | DOI | Zbl

[5] Dehaish, B. A. Bin; Qin, X.-L.; Latif, A.; Bakodah, H. O. Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., Volume 16 (2015), pp. 1321-1336 | Zbl

[6] Browder, F. E. Existence and approximation of solutions of nonlinear variational inequalities, Proc. Nat. Acad. Sci. U.S.A., Volume 56 (1966), pp. 1080-1086 | DOI

[7] Bruck, R. E.; Jr. A strongly convergent iterative solution of \(0 \in U(x)\) for a maximal monotone operator U in Hilbert space, J. Math. Anal. Appl., Volume 48 (1974), pp. 114-126 | Zbl | DOI

[8] Chang, S.-S.; Lee, H. W. J.; Chan, C. K. Strong convergence theorems by viscosity approximation methods for accretive mappings and nonexpansive mappings, J. Appl. Math. Inform., Volume 27 (2009), pp. 59-68

[9] Chidume, C. E. Iterative solutions of nonlinear equations in smooth Banach spaces, Nonlinear Anal., Volume 26 (1996), pp. 1823-1834

[10] Cho, S. Y.; Dehaish, B. A. Bin; Qin, X.-L. Weak convergence of a splitting algorithm in Hilbert spaces, J. Appl. Anal. Comput., Volume 7 (2017), pp. 427-438

[11] Cho, S. Y.; Qin, X.-L.; Wang, L. Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., Volume 2014 (2014), pp. 1-15 | DOI | Zbl

[12] Jung, J. S.; Cho, Y. J.; Zhou, H.-Y. Iterative processes with mixed errors for nonlinear equations with perturbed m-accretive operators in Banach spaces, Appl. Math. Comput., Volume 133 (2002), pp. 389-406 | DOI | Zbl

[13] Kato, T. Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, Volume 19 (1967), pp. 508-520 | DOI

[14] Liu, L. S. Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., Volume 194 (1995), pp. 114-125 | DOI

[15] Moudafi, A. Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., Volume 241 (2000), pp. 46-55 | DOI

[16] Qin, X.-L.; Cho, S. Y. Convergence analysis of a monotone projection algorithm in reflexive Banach spaces, Acta Math. Sci. Ser. B Engl. Ed., Volume 37 (2017), pp. 488-502 | Zbl | DOI

[17] Qin, X.-L.; Cho, S. Y.; Kim, J. K. On the weak convergence of iterative sequences for generalized equilibrium problems and strictly pseudocontractive mappings, Optimization, Volume 61 (2012), pp. 805-821 | Zbl | DOI

[18] Qin, X.-L.; Cho, S. Y.; Wang, L. Iterative algorithms with errors for zero points of m-accretive operators, Fixed Point Theory Appl., Volume 2013 (2013), pp. 1-17 | DOI | Zbl

[19] Qin, X.-L.; Yao, J.-C. Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators, J. Inequal. Appl., Volume 2016 (2016), pp. 1-9 | Zbl | DOI

[20] Reich, S. On fixed point theorems obtained from existence theorems for differential equations, J. Math. Anal. Appl., Volume 54 (1976), pp. 26-36 | DOI

[21] Rockafellar, R. T. Monotone operators and the proximal point algorithm, SIAM J. Control Optim., Volume 14 (1976), pp. 877-898 | DOI

[22] Suzuki, T. Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., Volume 2005 (2005), pp. 1-21 | Zbl | DOI

[23] Zhou, H.-Y. A characteristic condition for convergence of steepest descent approximation to accretive operator equations, J. Math. Anal. Appl., Volume 271 (2002), pp. 1-6 | DOI | Zbl

[24] Zhou, H.-Y. Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces, Nonlinear Anal., Volume 70 (2009), pp. 4039-4046 | DOI