Weak convergence theorems for two nearly asymptotically nonexpansive

Gurucharan Singh Saluja

Abstract


In this paper, we establish some weak convergence results of
modified $S$-iteration process to converge to common fixed points involving two nearly asymptotically nonexpansive non-self mappings in the framework of uniformly convex Banach spaces under the following conditions (i) the Banach space $E$ satisfying Opial condition and (ii) the dual $E^{*}$ of $E$ has the Kadec-Klee property. Our results extend and improve many known results from the existing literature.

Full Text:

PDF

References


bibitem{AOS07} {sc R. P. Agarwal, Donal O'Regan, D. R. Sahu}, {rm Iterative construction of fixed points of nearly

asymptotically nonexpansive mappings}, Nonlinear Convex Anal. textbf{8(1)}(2007), 61--79.

bibitem{B76}{sc F. E. Browder}, {rm Nonlinear operators and nonlinear equations of evolution}, Proc. Amer. Math. Symp. Pure Math. XVII, Amer. Math. Soc., Providence, 1976.

bibitem{COZ03} {sc C. E. Chidume, E. U. Ofoedu, H. Zegeye}, {rm Strong and weak convergence theorems for asymptotically nonexpansive mappings}, J. Math. Anal. Appl. textbf{280}(2003), 364--374.

bibitem{CSZ04} {sc C.E. Chidume; N. Shahzad, H. Zegeye}, {rm Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense}, Numerical Functional and Optimization {bf 25} (2004), Nos. 3-4, 239--257.

bibitem{FKKR01} {sc J. Garcia Falset, W. Kaczor, T. Kuczumow, S. Reich}, {rm Weak convergence theorems for asymptotically nonexpansive mappings and semigroups}, Nonlinear Anal., TMA, textbf{43(3)}(2001), 377--401.

bibitem{GK72} {sc K. Goebel, W. A. Kirk}, {rm A fixed point theorem for asymptotically nonexpansive mappings}, Proc. Amer. Math. Soc. textbf{35(1)}(1972), 171--174.

bibitem{J89} {sc J. Gornicki}, {rm Weak convergence theorems for asymptotically mappings in uniformly convex Banach spaces}, Comment. Math. Univ. Carolinae textbf{30}(1989), 249--252.

bibitem{JST98} {sc J. S. Jung, D. R. Sahu, B. S. Thakur}, {rm Strong convergence theorems for asymptotically nonexpansive mappings in Banach spaces}, Comm. Appl. Nonlinear Anal. textbf{5}(1998), 53--69.

bibitem{KT01} {sc S. H. Khan, W. Takahashi}, {rm Approximating common fixed points of two asymptotically nonexpansive mappings}, Sci. Math. Jpn. textbf{53(1)}(2001), 143--148.

bibitem{K14} {sc S. H. Khan}, {rm Weak convergence for non-self nearly asymptotically nonexpansive mappings by iterations}, Demostr. Math. textbf{47(2)}(2014), 371--381.

bibitem{K65} {sc W. A. Kirk}, {rm A fixed point theorem for mappings which do not increase distances}, Amer. Math. Monthly textbf{72}(1965), 1004--1006.

bibitem{K74} {sc W. A. Kirk}, {rm Fixed point theorems for

non-Lipschitzian mappings of asymptotically nonexpansive type}, Israel J. Math. textbf{17}(1974), 339--346.

bibitem{KMK98} {sc W. A. Kirk, C. Martinez Yanez, S. S. Kim}, {rm Asymptotically nonexpansive mappings}, Nonlinear Anal. textbf{33}(1998), 1345--1365.

bibitem{LX94} {sc T. C. Lim, H. Xu}, {rm Fixed point theorems for mappings of asymptotically nonexpansive mappings}, Nonlinear Anal. textbf{22}(1994), 1345--1355.

bibitem{O67} {sc Z. Opial}, {rm Weak convergence of the sequence of successive approximations for nonexpansive mappings}, Bull. Amer. Math. Soc. textbf{73}(1967), 591--597.

bibitem{OA00} {sc M. O. Osilike, S. C. Aniagbosor}, {rm Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings}, Math. and Computer Modelling textbf{32}(2000), 1181--1191.

bibitem{R94} {sc B. E. Rhoades}, {rm Fixed point iteration for certain nonlinear mappings}, J. Math. Anal. Appl. textbf{183}(1994), 118--120.

bibitem{S05} {sc D. R. Sahu}, {rm Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces}, Comment. Math. Univ. Carolinae textbf{46(4)}(2005), 653--666.

bibitem{S08} {sc G. S. Saluja}, {rm Approximating common fixed points of finite family of asymptotically nonexpansive non-self mappings}, Filomat textbf{22(2)}(2008), 23--42.

bibitem{S191} {sc J. Schu}, {rm Weak and strong convergence to fixed points of asymptotically nonexpansive mappings}, Bull. Austral. Math. Soc. textbf{43(1)}(1991), 153--159.

bibitem{S291} {sc J. Schu}, {rm Iterative construction of fixed points of asymptotically nonexpansive mappings}, J. Math. Anal. Appl. textbf{158}(1991), 407--413.

bibitem{SS09} {sc K. Sitthikul, S. Saejung}, {rm Convergence theorems for a finite family of nonexpansive and asymptotically nonexpansive mappings}, Acta Univ. Palack. Olomuc. Math. textbf{48}(2009), 139--152.

bibitem{TX92} {sc K. K. Tan, H. K. Xu}, {rm The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach space}, Proc. Amer. Math. Soc. textbf{114}(1992), 399--404.

bibitem{TX93} {sc K. K. Tan, H. K. Xu}, {rm Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process}, J. Math. Anal. Appl. textbf{178}(1993), 301--308.

bibitem{TX94} {sc K. K. Tan and H. K. Xu}, {rm Fixed point iteration processes for asymptotically nonexpansive mappings}, Proc. Amer. Math. Soc. {bf 122}(1994), 733--739.

bibitem{W06} {sc L. Wang}, {rm Strong and weak convergence theorems for common fixed point of nonself asymptotically nonexpansive mappings}, J. Mth. Anal. Appl. textbf{323(1)}(2006), 550--557.


Refbacks

  • There are currently no refbacks.