# On nonexpansive and accretive operators in Banach spaces

Volume 10, Issue 7, pp 3437--3446 Publication Date: July 21, 2017       Article History
• 461 Views

### Authors

Dongfeng Li - School of Information Engineering, North China University of Water Resources and Electric Power, Zhengzhou 450011, China.

### Abstract

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

### Keywords

• Accretive operator
• approximation solution
• viscosity method
• variational inequality.

•  47H05
•  65J15

### References

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

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

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

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

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

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

• [7] R. E. Bruck, Jr., A strongly convergent iterative solution of $0 \in U(x)$ for a maximal monotone operator U in Hilbert space, J. Math. Anal. Appl., 48 (1974), 114–126.

• [8] 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.

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

• [10] 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.

• [11] 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.

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

• [13] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19 (1967), 508–520.

• [14] 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.

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

• [16] 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.

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

• [18] 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.

• [19] X.-L. Qin, J.-C. Yao, Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators, J. Inequal. Appl., 2016 (2016), 9 pages.

• [20] S. Reich, On fixed point theorems obtained from existence theorems for differential equations, J. Math. Anal. Appl., 54 (1976), 26–36.

• [21] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877–898.

• [22] T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., 2005 (2005), 21 pages.

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

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