Viscosity approximation methods for two accretive operators in Banach spaces. (English) Zbl 1470.47056

Chen, Jun-Min; Fan, Tie-Gang

Abstr. Appl. Anal. 2013, Article ID 670523, 9 p. (2013).

Summary: We introduced a viscosity iterative scheme for approximating the common zero of two accretive operators in a strictly convex Banach space which has a uniformly Gâteaux differentiable norm. Some strong convergence theorems are proved, which improve and extend the results of L. C. Ceng et al. [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 5, 1830–1840 (2009; Zbl 1226.47069)] and some others.

MSC:

47J25	Iterative procedures involving nonlinear operators
47H06	Nonlinear accretive operators, dissipative operators, etc.

Keywords:

viscosity iterative scheme; common zero; accretive operators; strictly convex Banach space; uniformly Gâteaux differentiable norm; strong convergence

Citations:

Zbl 1226.47069

References:

[1]	Diestel, J., Geometry of Banach Spaces. Geometry of Banach Spaces, Lecture Notes in Mathematics, 485 (1975), Berlin, Germany: Springer, Berlin, Germany · Zbl 0307.46009
[2]	Goebel, K.; Reich, S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics, 83 (1984), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0537.46001
[3]	Xu, H. K., Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications, 298, 1, 279-291 (2004) · Zbl 1061.47060
[4]	Bruck, R. E.; Passty, G. B., Almost convergence of the infinite product of resolvents in Banach spaces, Nonlinear Analysis, 3, 2, 279-282 (1979) · Zbl 0407.47035
[5]	Bruck, R. E.; Reich, S., Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston Journal of Mathematics, 3, 4, 459-470 (1977) · Zbl 0383.47035
[6]	Jung, J. S.; Takahashi, W., Dual convergence theorems for the infinite products of resolvents in Banach spaces, Kodai Mathematical Journal, 14, 3, 358-365 (1991) · Zbl 0755.47037
[7]	Jung, J. S.; Takahashi, W., On the asymptotic behavior of infinite products of resolvents in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 20, 5, 469-479 (1993) · Zbl 0781.47007
[8]	Reich, S., On infinite products of resolvents, Atti della Accademia Nazionale dei Lincei, 63, 5, 338-340 (1977) · Zbl 0407.47034
[9]	Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications, 67, 2, 274-276 (1979) · Zbl 0423.47026
[10]	Ceng, L. C.; Khan, A. R.; Ansari, Q. H.; Yao, J. C., Strong convergence of composite iterative schemes for zeros of m-accretive operators in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 70, 5, 1830-1840 (2009) · Zbl 1226.47069
[11]	Chen, R.; Liu, Y.; Shen, X., Iterative approximation of a zero of accretive operator in Banach space, Nonlinear Analysis: Theory, Methods & Applications, 71, 12, e346-e350 (2009) · Zbl 1238.47044
[12]	Jung, J. S., Convergence of composite iterative methods for finding zeros of accretive operators, Nonlinear Analysis: Theory, Methods & Applications, 71, 5-6, 1736-1746 (2009) · Zbl 1219.47118
[13]	Hu, L.; Liu, L., A new iterative algorithm for common solutions of a finite family of accretive operators, Nonlinear Analysis: Theory, Methods & Applications, 70, 6, 2344-2351 (2009) · Zbl 1218.47101
[14]	Zegeye, H.; Shahzad, N., Strong convergence theorems for a common zero of a countably infinite family of \(α\)-inverse strongly accretive mappings, Nonlinear Analysis: Theory, Methods & Applications, 71, 1-2, 531-538 (2009) · Zbl 1221.47137
[15]	Alber, Y. I.; Guerre-Delabriere, S., Principle of weakly contractive maps in Hilbert spaces, New Results in Operator Theory and Its Applications. New Results in Operator Theory and Its Applications, Operator Theory: Advances and Applications, 98, 7-22 (1997) · Zbl 0897.47044
[16]	Alber, Y. I.; Guerre-Delabriere, S.; Zelenko, L., The principle of weakly contractive mappings in metric spaces, Communications on Applied Nonlinear Analysis, 5, 1, 45-68 (1998) · Zbl 1108.54303
[17]	Rhodes, B. E., Some theorems on weakly contractive maps, Nonlinear Analysis: Theory, Methods & Applications, 47, 4, 2683-2693 (2001) · Zbl 1042.47521
[18]	Pascali, D.; Sburlan, S., Nonlinear Mappings of Monotone Type (1978), Alphen aan den Rijn, The Netherlands: Sijthoff and Noordholff International Publishers, Alphen aan den Rijn, The Netherlands · Zbl 0392.47026
[19]	Calvert, B. D.; Gupta, C. P., Nonlinear elliptic boundary value problems in \(L^p\)-spaces and sums of ranges of accretive operators, Nonlinear Analysis, 2, 1, 1-26 (1978) · Zbl 0369.47033