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Finding minimum norm fixed point of nonexpansive mappings and applications. (English) Zbl 1216.47102
Summary: We construct two new methods for finding the minimum norm fixed point of nonexpansive mappings in Hilbert spaces. Some applications are also included.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47N10Applications of operator theory in optimization, convex analysis, programming, economics
WorldCat.org
Full Text: DOI EuDML
References:
[1] F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, pp. 82-90, 1967. · Zbl 0148.13601 · doi:10.1007/BF00251595
[2] F. E. Browder, “Convergence theorems for sequences of nonlinear operators in Banach spaces,” Mathematische Zeitschrift, vol. 100, pp. 201-225, 1967. · Zbl 0149.36301 · doi:10.1007/BF01109805 · eudml:170809
[3] F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197-228, 1967. · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[4] B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957-961, 1967. · Zbl 0177.19101 · doi:10.1090/S0002-9904-1967-11864-0
[5] Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591-597, 1967. · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[6] P.-L. Lions, “Approximation de points fixes de contractions,” Comptes Rendus de l’Académie des Sciences. Série I. Mathématique, vol. 284, no. 21, pp. A1357-A1359, 1977. · Zbl 0349.47046
[7] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. · Zbl 0708.47031 · doi:10.1017/CBO9780511526152
[8] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984. · Zbl 0537.46001
[9] S. Reich, “Strong convergence theorems for resolvents of accretive operators in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287-292, 1980. · Zbl 0437.47047 · doi:10.1016/0022-247X(80)90323-6
[10] S. Reich and H.-K. Xu, “An iterative approach to a constrained least squares problem,” Abstract and Applied Analysis, vol. 2003, no. 8, pp. 503-512, 2003. · Zbl 1053.65041 · doi:10.1155/S1085337503212082 · eudml:51108
[11] S. Reich and A. J. Zaslavski, “Convergence of Krasnoselskii-Mann iterations of nonexpansive operators,” Mathematical and Computer Modelling, vol. 32, no. 11-13, pp. 1423-1431, 2000. · Zbl 0977.47046 · doi:10.1016/S0895-7177(00)00214-4
[12] A. T.-M. Lau and W. Takahashi, “Fixed point properties for semigroup of nonexpansive mappings on Fréchet spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 11, pp. 3837-3841, 2009. · Zbl 1219.47082 · doi:10.1016/j.na.2008.07.041
[13] E. M. Mazcuñán-Navarro, “Three-dimensional convexity and the fixed point property for nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 1-2, pp. 587-592, 2009. · Zbl 1175.47054 · doi:10.1016/j.na.2008.10.101
[14] A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46-55, 2000. · Zbl 0957.47039 · doi:10.1006/jmaa.1999.6615
[15] A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative least-squares and regularization,” IEEE Transactions on Signal Processing, vol. 46, no. 9, pp. 2345-2352, 1998. · Zbl 0978.93085 · doi:10.1109/78.709518
[16] Y. J. Cho and X. Qin, “Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Computational and Applied Mathematics, vol. 228, no. 1, pp. 458-465, 2009. · Zbl 1173.47047 · doi:10.1016/j.cam.2008.10.004
[17] Y.-L. Cui and X. Liu, “Notes on Browder’s and Halpern’s methods for nonexpansive mappings,” Fixed Point Theory, vol. 10, no. 1, pp. 89-98, 2009. · Zbl 1190.47068
[18] Y. Yao and H. K. Xu, “Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications,” Optimization. In press. · Zbl 05957102
[19] H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240-256, 2002. · Zbl 1013.47032 · doi:10.1112/S0024610702003332
[20] H.-K. Xu, “Another control condition in an iterative method for nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 65, no. 1, pp. 109-113, 2002. · Zbl 1030.47036 · doi:10.1017/S0004972700020116
[21] H.-K. Xu, “Iterative methods for constrained Tikhonov regularization,” Communications on Applied Nonlinear Analysis, vol. 10, no. 4, pp. 49-58, 2003. · Zbl 1047.65035
[22] H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185-201, 2003. · Zbl 1045.49018 · doi:10.1023/B:JOTA.0000005048.79379.b6
[23] H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279-291, 2004. · Zbl 1061.47060 · doi:10.1016/j.jmaa.2004.04.059
[24] T. Suzuki, “A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 135, no. 1, pp. 99-106, 2007. · Zbl 1117.47041 · doi:10.1090/S0002-9939-06-08435-8
[25] Y. Yao, Y. C. Liou, and G. Marino, “Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 279058, 7 pages, 2009. · Zbl 1186.47080 · doi:10.1155/2009/279058 · eudml:45677
[26] X. Liu and Y. Cui, “The common minimal-norm fixed point of a finite family of nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 1, pp. 76-83, 2010. · Zbl 1214.47050 · doi:10.1016/j.na.2010.02.041
[27] N. Shahzad, “Approximating fixed points of non-self nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 6, pp. 1031-1039, 2005. · Zbl 1089.47058 · doi:10.1016/j.na.2005.01.092
[28] H. Zegeye and N. Shahzad, “Viscosity methods of approximation for a common fixed point of a family of quasi-nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 7, pp. 2005-2012, 2008. · Zbl 1153.47059 · doi:10.1016/j.na.2007.01.027
[29] H. Zegeye and N. Shahzad, “Strong convergence theorems for a finite family of nonexpansive mappings and semigroups via the hybrid method,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 1, pp. 325-329, 2010. · Zbl 1225.47122 · doi:10.1016/j.na.2009.06.056
[30] H. Zegeye and N. Shahzad, “Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 155-163, 2007. · Zbl 1194.47089 · doi:10.1016/j.amc.2007.02.072
[31] L. Wang, “An iteration method for nonexpansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2007, Article ID 28619, 8 pages, 2007. · Zbl 1159.47052 · doi:10.1155/2007/28619 · eudml:55290
[32] L. Wang, Y.-J. Chen, and R.-C. Du, “Hybrid iteration method for common fixed points of a finite family of nonexpansive mappings in Banach spaces,” Mathematical Problems in Engineering, vol. 2009, Article ID 678519, 9 pages, 2009. · Zbl 1184.47052 · doi:10.1155/2009/678519 · eudml:45879
[33] K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,” Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387-404, 2001. · Zbl 0993.47037
[34] K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 8, pp. 2350-2360, 2007. · Zbl 1130.47045 · doi:10.1016/j.na.2006.08.032
[35] N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641-3645, 1997. · Zbl 0888.47034 · doi:10.1090/S0002-9939-97-04033-1
[36] W. Takahashi and K. Shimoji, “Convergence theorems for nonexpansive mappings and feasibility problems,” Mathematical and Computer Modelling, vol. 32, no. 11-13, pp. 1463-1471, 2000. · Zbl 0971.47040 · doi:10.1016/S0895-7177(00)00218-1
[37] K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372-379, 2003. · Zbl 1035.47048 · doi:10.1016/S0022-247X(02)00458-4
[38] M. Kikkawa and W. Takahashi, “Approximating fixed points of infinite nonexpansive mappings by the hybrid method,” Journal of Optimization Theory and Applications, vol. 117, no. 1, pp. 93-101, 2003. · Zbl 1033.65037 · doi:10.1023/A:1023652406878
[39] P. L. Combettes, “The convex feasibility problem in image recovery,” in Advances in Imaging and Electron Physics, P. Hawkes, Ed., vol. 95, pp. 155-270, Academic Press, New York, NY, USA, 1996.
[40] P. L. Combettes and T. Pennanen, “Generalized Mann iterates for constructing fixed points in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 275, no. 2, pp. 521-536, 2002. · Zbl 1032.47034 · doi:10.1016/S0022-247X(02)00221-4
[41] S. A. Hirstoaga, “Iterative selection methods for common fixed point problems,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 1020-1035, 2006. · Zbl 1106.47057 · doi:10.1016/j.jmaa.2005.12.064
[42] H. H. Bauschke, “The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 202, no. 1, pp. 150-159, 1996. · Zbl 0956.47024 · doi:10.1006/jmaa.1996.0308