# zbMATH — the first resource for mathematics

Strong convergence of composite iterative schemes for zeros of $$m$$-accretive operators in Banach spaces. (English) Zbl 1226.47069
Summary: We introduce a new composite iterative scheme to approximate a zero of an $$m$$-accretive operator $$A$$ defined on uniform smooth Banach spaces and a reflexive Banach space having a weakly continuous duality map. It is shown that the iterative process in each case converges strongly to a zero of $$A$$. The results presented in this paper substantially improve and extend the results due to the first author and H.-K. Xu [Taiwanese J. Math. 11, No. 3, 661–682 (2007; Zbl 1219.47102)], T.-H. Kim and H.-K. Xu [Nonlinear Anal., Theory Methods Appl. 61, No. 1–2, A, 51–60 (2005; Zbl 1091.47055)] and H.-K. Xu [J. Math. Anal. Appl. 314, No. 2, 631–643 (2006; Zbl 1086.47060)]. Our work provides a new approach for the construction of a zero of $$m$$-accretive operators.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H06 Nonlinear accretive operators, dissipative operators, etc.
Full Text:
##### References:
 [1] Barbu, V., Nonlinear semigroups and differential equations in Banach space, (1976), Noordhoff [2] Browder, F.E., Fixed point theorems for noncompact mappings in Hilbert space, Proc. nat. acad. sci. USA, 53, 1272-1276, (1965) · Zbl 0125.35801 [3] Browder, F.E., Convergence theorems for sequences of nonlinear operators in Banach space, Math. Z., 100, 201-225, (1967) · Zbl 0149.36301 [4] Bruck, R.E., Nonexpansive projections on subsets of Banach spaces, Pacific J. math., 47, 341-355, (1973) · Zbl 0274.47030 [5] L.C. Ceng, H.K. Xu, J.C. Yao, Strong convergence of an iterative method with perturbed mapping for nonexpansive and accretive operators, Numer. Funct. Anal. Optim. (in press) · Zbl 1140.47050 [6] Ceng, L.C.; Xu, H.K., Strong convergence of a hybrid viscosity approximation method with perturbed mappings for nonexpansive and accretive operators, Taiwanese J. math., 11, 3, 661-682, (2007) · Zbl 1219.47102 [7] Geobel, K.; Reich, S., Uniform convexity, hyperbolic geometry, and nonexpansive mappings, (1984), Marcel Dekker New York [8] Kim, T.H.; Xu, H.K., Strong convergence of modified Mann iterations, Nonlinear anal., 61, 51-60, (2005) · Zbl 1091.47055 [9] Lim, T.C.; Xu, H.K., Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear anal., 22, 1345-1355, (1994) · Zbl 0812.47058 [10] Qin, X.L.; Su, Y.F., Approximation of a zero point of accretive operator in Banach spaces, J. math. anal. appl., 329, 415-424, (2007) · Zbl 1115.47055 [11] Reich, S., Asymptotic behavior of contractions in Banach spaces, J. math. anal. appl., 44, 57-70, (1973) · Zbl 0275.47034 [12] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. math. anal. appl., 75, 287-292, (1980) · Zbl 0437.47047 [13] Xu, H.K., Iterative algorithms for nonlinear operators, J. London math. soc., 66, 240-256, (2002) · Zbl 1013.47032 [14] Xu, H.K., An iterative approach to quadratic optimization, J. optim. theory appl., 116, 659-678, (2003) · Zbl 1043.90063 [15] Xu, H.K., Strong convergence of an iterative method for nonexpansive and accretive operators, J. math. anal. appl., 314, 631-643, (2006) · Zbl 1086.47060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.