Generalized Krasnoselskii-Mann Type Iterations for Two Nonexpansive Mappings in Real Hilbert Spaces
محورهای موضوعی : Application of Game Theory in Finance
Sirous Moradi
1
(Department of Mathematics, Faculty of science, Lorestan University, Khoramabad 68151-4-4316, Iran)
Najmeh Mohitazar
2
(Department of Mathematics, Faculty of science, Arak University, Arak
38156-8-8349, Iran)
کلید واژه: Hilbert space, Nonexpansive mapping, Maximal monotone operators, fixed point, Mann iterative,
چکیده مقاله :
In this paper, we discuss a new type of Mann iterative algorithmfor finding a common fixed point of two nonexpansive mappings in real Hilbertspaces, and we establish the weak convergence of this fixed point approachby considering new circumstances. We can also show the algorithm’s strongconvergence by taking into account an extra requirement. Our results extendthe previous results given by Kanzow and Shehu as well as by Cho at et. al. Also in the last section, some applications of the main results are shown, and by some examples, we show the generality of our main results.
[1] Bauschke, H.H., Combettes, P.L., Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York, 2011.
[2] Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H., (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optimization and Its Applications, 49 Springer, 2011.
[3] Berinde, V., Iterative Approximation of Fixed Points, Lecture Notes in Mathematics, vol. 1912. Springer, Berlin, 2007.
[4] Browder, F. E., Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc, 1968, 74, P. 661-665. Doi:10.1090/S0002-9904-1968-11979-2
[5] Cegielski, A., Iterative Methods for Fixed Point Problems in Hilbert Spaces: Lecture Notes in Mathematics, 2057. Springer, Berlin, 2012.
[6] Chang, S.S., Cho, Y.J., Zhou, H. (eds.), Iterative Methods for Nonlinear Operator Equations in Banach
Spaces, Nova Science, Huntington, NY, 2002.
[7] Chidume, C.E., Geometric Properties of Banach Spaces and Nonlinear Iterations, Lecture Notes in Mathematics, 1965. Springer, London, 2009.
[8] Chidume, C.E., Chidume, C.O., Iterative approximation of fixed points of nonexpansive mappings, J.
Math. Anal. Appl, 2006, 318, P. 288–295. Doi: 10.1016/j.jmaa.2005.05.023
[9] Cho, Y.J., Kang, S.M., Qin, X., Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces, Comput. Math. Appl, 2008, 56, P. 2058–2064.
Doi: 10.1016/j.camwa.2008.03.035
[10] Combettes, P.L., Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, , 2004, 53, P. 475–504. Doi: 10.1080/02331930412331327157
[11] Genel, A., Lindenstrauss, J., An example concerning fixed points Isr. J. Math, 1975, 22, P. 81–86. Doi:10.1007/BF02757276
[12] Hao, Y., Cho, S. Y., Qin, X., Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings, Fixed Point Theory Appl, Article ID 218573, 2010. Doi:10.1155/2010/218573
[13] Kanzow, C., Shehu, Y., Generalized Krasnoselskii–Mann type iterations for nonexpansive mappings in Hilbert spaces, Comput. Optim. Appl, 2017, 67, P. 595–620. Doi: 10.1007/s10589-017-9902-0
[14] Kim, T.-H., Xu, H.-K., Strong convergence of modified Mann iterations, Nonlinear Anal, 2005, 61, P. 51–60. Doi:10.1016/j.na.2004.11.011
[15] Krasnoselskii, M.A., Two remarks on the method of successive approximations, Uspekhi Mat. Nauk, 1955, 10, P. 123–127. mi.mathnet.ru/umn7954
[16] Liang, J., Fadili, J., Peyré, G., Convergence rates with inexact non-expansive operators, Math. Program, 2016, 159, P. 403–434. Doi: 10.1007/s10107-015-0964-4
[17] Mann, W.R., Mean value methods in iteration, Bull. Am. Math. Soc, 1953, 4, P. 506–510.
Doi: 10.1090/S0002-9939-1953-0054846-3
[18] Opial, Z., Weak convergence of the sequence of successive appproximations for nonexpansive mappings, Bull. Amer. Math. Soc, 1967, 73, P. 591-597. Doi: 10.1090/S0002-9904-1967-11761-0
[19] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl, 1979, 67, P. 274-276. Doi: 10.1016/0022-247X(79)90024-6
[20] Suzuki, T., A sufficient and necessary condition for Halpern-type strong convergence to fixed points
of nonexpansive mappings, Proc. Am. Math. Soc, , 2007, 135, P. 99–106. www.jstor.org/stable/20534551
[21] Tani, K. K., Xu, H. K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl, 1993, 178, P. 301-308. Doi: 10.1006/jmaa.1993.1309
[22] Xu, H. K., Ori, M. G., An implicit iterative process for nonexpansive mappings, Numer. Funct. Anal. Optim, 2001, 22, P. 767-773. Doi: 10.1081/NFA-100105317
[23] Nasr, N., Farhadi Sartangi, M., Madahi, Z., A Fuzzy Random Walk Technique to Forecasting Volatility of Iran Stock Exchange Index, Advances in Mathematical Finance and Applications, 2019, 4(1), P.15-30.
Doi: 10.22034/amfa.2019.583911.1172
