Common fixed points of a three-step iteration with errors of asymptotically quasi-nonexpansive nonself-mappings in the intermediate sense in Banach spaces

• Autorzy: Okeke, G. A. , Olaleru, J. O.
• Języki publikacji: EN

Abstrakty

EN

In this paper, we extend the results of Inprasit and Wattanataweekul [7] to the class of asymptotically quasi-nonexpansive nonself-mappings in the intermediate sense. We prove some strong convergence theorems for asymptotically quasi-nonexpansive nonself-mappings in the intermediate sense using a three-step iterative method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of common fixed points of a finite family of nonexpansive mappings in a real Hilbert space. Our results extends, improves, unifies and generalizes the results of [13], [25] and [27].

Słowa kluczowe

EN

nonexpansive mappings; quasi-nonexpansive mappings; asymptotically nonexpansive mappings; asymptotically quasi-nonexpansive mappings in the intermediate sense; nonself-mappings; Banach spaces

• Czasopismo: Fasciculi Mathematici
• Rocznik: 2014
• Tom: Nr 52
• Strony: 93--115
• Opis fizyczny: Bibliogr. 27 poz.

Twórcy

autor

Okeke, G. A.

• Department of Mathematics University of Lagos Lagos, Nigeria,

autor

Olaleru, J. O.

• Department of Mathematics University of Lagos Lagos, Nigeria,

Bibliografia

• [1] Bnouhachem A., Noor M.A., Rassias Th.M., Three-steps iterative algorithms for mixed variational inequalities, Appl. Math. Comput., 183(2006), 436-446.
• [2] Bruck R.E., Kuczumow T., Reich S., Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloquium Mathematicum, 65(2)(1993), 169-179.
• [3] Chidume C.E., Geometric properties of Banach spaces and nonlinear iterations, Springer, 2008.
• [4] Fukhar-Ud-Din H., Khan S.H., Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications, J. Math. Anal. Appl., 328(2007), 821-829.
• [5] Glowinski R., Le Tallec P., Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM, Philadelphia, 1989.
• [6] Goebel K., Kirk W., A fixed point theorem for asymptotically nonexpansive mappings, Proceedings of the American Mathematical Society, 35(1972), 171-174.
• [7] Inprasit U., Wattanataweekul H., Common fixed points of a new three-step iteration with errors of asymptotically quasi-nonexpansive non-self-mappings in Banach spaces, Journal of Nonlinear Analysis and Optimization, 1(1)(2010), 169-182.
• [8] Khan S.H., Takahashi W., Approximating common fixed points of two asymptotically nonexpansive mappings, Sci. Math. Jpn., 53(2001), 133-138.
• [9] Khan S.H., Fukhar-Ud-Din H., Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal., 8(2005), 1295-1301.
• [10] Kirk W.A., Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type, Israel Journal of Mathematics, 17(1974), 339-346.
• [11] Mogbademu A.A., Olaleru J.O., Modified Noor iterative methods for a family of strongly pseudocontractive maps, Bulletin of Mathematical Analysis and Applications, 3(4)(2011), 132-139.
• [12] Nammanee K., Noor M.A., Suantai S., Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings, J. Math. Anal. Appl., in press.
• [13] Nilsrakoo W., Saejung S., A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings, J. Appl. Math. Comput., 181(2006), 1026-1034.
• [14] Noor M.A., New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251(2000), 217-229.
• [15] Noor M.A., Three-step iterative algorithms for multivalued quasi variational inclusions, J. Math. Anal. Appl., 255(2001), 589-604.
• [16] Noor M.A., Some developments in general variational inequalities, Appl. Math. Comput., 152(2004), 199-277.
• [17] Noor M.A., Kassias T.M., Huang Z., Three-step iterations for nonlinear accretive operator equations, J. Math. Anal. Appl., 274(2001), 59-68.
• [18] Olaleru J.O., Mogbademu A.A., On the modified Noor iteration scheme for non-linear maps, Acta Math. Univ. Comenianae, LXXX, 2(2011), 221-228..
• [19] Olaleru J.O., Mogbademu A.A., Approximation of fixed points of strongly successively pseudocontractive maps in Banach space, International Journal of Computational and Applied Mathematics, 7(2)(2012), 121-132.
• [20] Opial Z., Weak convergence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73(1967), 591-597.
• [21] Qihou L., Iteration sequences for asymptotically quasi-nonexpansive map-ping with an error member of uniform convex Banach space, J. Math. Anal. Appl., 266(2002), 468-471.
• [22] Schu J., Iterative construction of fixed points of asymptotically nonexpansive mappings, Mathematical Analysis and Applications, 158(2)(1991), 407-413.
• [23] Senter H.F., Dotson W.G., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44(1974), 375-380.
• [24] Shahzad N., Udomene A., Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., (2006), article ID 18909, 10 pages.
• [25] Suantai S., Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 311(2005), 506-517.
• [26] Tan K.-K., Xu H.K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, Journal of Mathematical Analysis and Applications, 178(2)(1993), 301-308.
• [27] Xu B.L., Noor M.A., Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 267(2002), 444-453.