Strong convergence of a composite Halpern-type iteration for a family of nonexpansive mappings in CAT(0) spaces

Sajad Ranjbar

doi:10.1515/auom-2017-0044

.blurhash-client-img { display: none !important; }

Strong convergence of a composite Halpern-type iteration for a family of nonexpansive mappings in CAT(0) spaces

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 25 (2017): Issue 3 (December 2017)

By: Sajad Ranjbar

Open Access

|Mar 2018

Abstract

In this paper, we prove the strong convergence of the composite Halpern-type iteration for a family of nonexpansive mappings in CAT(0) spaces and compare our results with several recent results in this subject. Also, the inexact version of the Halpern iteration is studied in CAT(0) spaces.

References

[1] Aoyama, K., Kimura, Y., Takahashi, W., Toyoda, M.: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 67, (2007) 2350-2360.10.1016/j.na.2006.08.032
Search in Google Scholar Back to article
[2] Bridson, M., Haeiger, A.: Metric Spaces of Non-Positive Curvature.Fundamental Principles of Mathematical Sciences. Springer, Berlin 319, (1999).10.1007/978-3-662-12494-9
Search in Google Scholar Back to article
[3] Brown, K. S.: Buildings. Springer, New York, (1989).10.1007/978-1-4612-1019-1
Search in Google Scholar Back to article
[4] Chaoha, P., Phon-on, A.: A note on fixed point sets in CAT(0) spaces. J. Math. Anal. Appl. 320(2), (2006) 983-987.10.1016/j.jmaa.2005.08.006
Search in Google Scholar Back to article
[5] Chidume, C. E., Chidume, C. O.: Iterative approximation of fixed points of nonexpansive mappings. J. Math. Anal. Appl. 318, (2006) 288-295.10.1016/j.jmaa.2005.05.023
Search in Google Scholar Back to article
[6] Cuntavepanit, A., Panyanak, B.: Strong Convergence of Modi_ed Halpern Iterations in CAT(0) Spaces. Fixed Point Theory Appl. 869458, (2011) doi:10.1155/2011/869458.
Open DOI Search in Google Scholar Back to article
[7] Dhompongsa, S., Panyanak, B.: On _-convergence theorems in CAT(0) spaces. Comput. Math. Appl. 56(10), (2008) 2572-2579.10.1016/j.camwa.2008.05.036
Search in Google Scholar Back to article
[8] Goebel, K., Reich, S,: Uniform Convexity, hyperbolic Geometry, and nonexpansive Mappings. Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc, New York, 83, (1984).
Search in Google Scholar Back to article
[9] Halpern, B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, (1967) 957-961.10.1090/S0002-9904-1967-11864-0
Search in Google Scholar Back to article
[10] Kim, T. H., Xu, H. K.: Strong convergence of modi_ed Mann iterations. J. Math. Anal. Appl. 61, (2005) 51-60.10.1016/j.na.2004.11.011
Search in Google Scholar Back to article
[11] Kirk, W. A.: Fixed point theorems in CAT(0) spaces and R-trees. Fixed Point Theory Appl. 4, (2004) 309-316.
Search in Google Scholar Back to article
[12] Kirk, W. A.: Geodesic geometry and fixed point theory II. Paper pre- sented at the International Conference on Fixed Point Theory and Ap- plications, Yokohama Publ. (2004) 113-142 .10.1155/S1687182004406081
Search in Google Scholar Back to article
[13] Kirk, W. A.: Geodesic geometry and fixed point theory. Paper presented at the Seminar of Mathematical Analysis, University of Sevilla Secretry, sevilla, spain, 64, (2003) 195-225.
Search in Google Scholar Back to article
[14] Lions, P. L.: Approximation de points fixes de contractions. C.R. Acad Sci Paris Ser. A-B 284, (1977) A1357-A1359.
Search in Google Scholar Back to article
[15] Qin, X., Su. Y., Shang, M.: Strong convergence of the composite Halpern iteration. J. Math. Anal. Appl. 339, (2008) 996-1002.10.1016/j.jmaa.2007.07.062
Search in Google Scholar Back to article
[16] Reich, S.: Approximating fixed points of non-expansive mappings. Panam. Math. J. 4(2), (1994) 23-28.
Search in Google Scholar Back to article
[17] Saejung, S.: Halperns iteration in CAT(0) spaces. Fixed Point Theory Appl. 471781, (2010).10.1155/2010/471781
Search in Google Scholar Back to article
[18] Shioji, N., Takahashi, W.: Strong convergence of approximated sequences for non-expansive mappings in Banach spaces. Proc. Amer. Math. Soc. 125, (1997) 3641-3645.10.1090/S0002-9939-97-04033-1
Search in Google Scholar Back to article
[19] Suzuki, T.: A su_cient and necessary condition for Halpern-type strong convergence to the fixed point of nonexpansive mapping. Proc. Amer. Math. Soc. 135, (2007) 99-106.10.1090/S0002-9939-06-08435-8
Search in Google Scholar Back to article
[20] Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch Math. 58, (1992) 486-491.10.1007/BF01190119
Open DOI Search in Google Scholar Back to article
[21] Xu, H. K.: Another control condition in an iterative method for nonex- pansive mappings. Bull. Austral. Math. Soc. 65, (2002) 109-113.10.1017/S0004972700020116
Open DOI Search in Google Scholar Back to article
[22] Xu, H. K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66, (2002) 240-256.10.1112/S0024610702003332
Open DOI Search in Google Scholar Back to article