×

Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. (English) Zbl 1090.47059

Summary: Mann iterations for nonexpansive mappings in general display only weak convergence in a Hilbert space. We modify an iterative method of Mann’s type introduced by K. Nakajo and W. Takahashi [J. Math.Anal.Appl.279, No. 2, 372–379 (2003; Zbl 1035.47048)] for nonexpansive mappings and prove the strong convergence of our modified Mann iteration process for asymptotically nonexpansive mappings and semigroups.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H20 Semigroups of nonlinear operators
65J15 Numerical solutions to equations with nonlinear operators

Citations:

Zbl 1035.47048
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brézis, H.; Browder, F. E., Nonlinear ergodic theorems, Bull. Am. Math. Soc., 82, 959-961 (1976) · Zbl 0339.47029
[2] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20, 103-120 (2004) · Zbl 1051.65067
[3] Genel, A.; Lindenstrass, J., An example concerning fixed points, Israel J. Math., 22, 81-86 (1975) · Zbl 0314.47031
[4] Goebel, K.; Kirk, W. A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. Am. Math. Soc., 35, 171-174 (1972) · Zbl 0256.47045
[5] Kim, T. W.; Xu, H. K., Strong convergence of modified Mann iterations, Nonlinear Anal., 61, 51-60 (2005) · Zbl 1091.47055
[6] Lin, P. K.; Tan, K. K.; Xu, H. K., Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal., 24, 929-946 (1995) · Zbl 0865.47040
[7] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279, 372-379 (2003) · Zbl 1035.47048
[8] Podilchuk, C. I.; Mammone, R. J., Image recovery by convex projections using a least-squares constraint, J. Opt. Soc. Am. A, 7, 517-521 (1990)
[9] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67, 274-276 (1979) · Zbl 0423.47026
[10] Schu, J., Weak, strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43, 153-159 (1991) · Zbl 0709.47051
[11] Sezan, M. I.; Stark, H., Applications of convex projection theory to image recovery in tomography and related areas, (Stark, H., Image Recovery Theory and Applications (1987), Academic Press: Academic Press Orlando), 415-462
[12] Tan, K. K.; Xu, H. K., The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces, Proc. Am. Math. Soc., 114, 399-404 (1992) · Zbl 0781.47045
[13] Tan, K. K.; Xu, H. K., Fixed point theorems for Lipschitzian semigroups in Banach spaces, Nonlinear Anal., 20, 395-404 (1993) · Zbl 0781.47044
[14] Tan, K. K.; Xu, H. K., Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Am. Math. Soc., 122, 733-739 (1994) · Zbl 0820.47071
[15] Xu, H. K., Strong asymptotic behavior of almost-orbits of nonlinear semigroups, Nonlinear Anal., 46, 135-151 (2001) · Zbl 0993.47038
[16] Youla, D., Mathematical theory of image restoration by the method of convex projections, (Stark, H., Image Recovery Theory and Applications (1987), Academic Press: Academic Press Orlando), 29-77
[17] Youla, D., On deterministic convergence of iterations of relaxed projection operators, J. Visual Comm. Image Representation, 1, 12-20 (1990)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.