Weak convergence theorems for two nearly asymptotically nonexpansive non-self mappings

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 space under the following conditions (i) the Banach space $E$ satisfying Opial condition and (ii) the dual $E^{\ast}$ 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} R. P. Agarwal, Donal O'Regan and D. R. Sahu,

{it Iterative construction of fixed points of nearly

asymptotically nonexpansive mappings}, Nonlinear Convex Anal. {bf

(1)}(2007), 61-79.

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

bibitem{COZ03} C. E. Chidume, E. U. Ofoedu and H. Zegeye, {it Strong

and weak convergence theorems for asymptotically nonexpansive

mappings}, J. Math. Anal. Appl. textbf{280}(2003), 364-374.

bibitem{CSZ04} C. E. Chidume, N. Shahzad and H. Zegeye,

{it Convergence theorems for mappings which are asymptotically

nonexpansive in the intermediate sense}, Numerical Functional and

Optimization {bf 25(3-4)} (2004), 239--257.

bibitem{CA07} C. E. Chidume and B. Ali, {it Weak and strong

convergence theorems for finite families of asymptotically

nonexpansive mappings in Banach spaces}, J. Math. Anal. Appl. {bf

}(2007), 377-387.

bibitem{FKKR01} J. Garcia Falset, W. Kaczor, T. Kuczumow and S.

Reich, {it Weak convergence theorems for asymptotically

nonexpansive mappings and semigroups}, Nonlinear Anal., TMA,

textbf{43(3)}(2001), 377-401.

bibitem{GL89} R. Glowinski and P. Le Tallec, {it "Augemented

Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics"}

Siam, Philadelphia, (1989).

bibitem{GK72} K. Goebel and W. A. Kirk, {it A fixed point theorem for

asymptotically nonexpansive mappings}, Proc. Amer. Math. Soc. {bf

(1)}(1972), 171-174.

bibitem{HNS98} S. Haubruge, V. H. Nguyen and J. J. Strodiot,

{it Convergence analysis and applications of the Glowinski Le

Tallec splitting method for finding a zero of the sum of two

maximal monotone operators}, J. Optim. Theory Appl. {bf

}(1998), 645-673.

bibitem{KCA} S. H. Khan, Y. J. Cho and M. Abbas, {it Convergence

to common fixed points by a modified iteration process}, J. Appl.

Math. Comput. doi:10.1007/s12190-010-0381-z.

bibitem{KOA14} S. H. Khan, M. Ozdemir and S. Akbulut, {it Some common fixed

fixed point results for non-self asymptotically quasi-nonexpansive mappings by a two-step iterative process}, Indian J. Math. {bf

(2)}(2014), 169-188.

bibitem{K14} S. H. Khan, {it Weak convergence for non-self nearly asymptotically nonexpansive mappings by iterations}, Demonstr. Math. {bf

(2)}(2014), 371-381.

bibitem{N2000} M. A. Noor, {it New approximation schemes for general

variational inequalities}, J. Math. Anal. Appl. {bf 251}(2000),

-229.

bibitem{N01} M. A. Noor, {it Three-step iterative algorithms for

multivalued quasi variational inclusions}, J. Math. Anal. Appl.

{bf 255}(2001), 589-604.

bibitem{O67} Z. Opial, {it Weak convergence of the sequence of

successive approximations for nonexpansive mappings}, Bull. Amer.

Math. Soc. {bf 73}(1967), 591-597.

bibitem{OA} M. O. Osilike and S. C. Aniagbosor, {it Weak and

strong convergence theorems for fixed points of asymptotically

nonexpansive mappings}, Math. Comput. Model. {bf 32}(2000),

-1191.

bibitem{R94} B. E. Rhoades, {it Fixed point iteration for certain nonlinear

mappings}, J. Math. Anal. Appl. {bf 183}(1994), 118-120.

bibitem{S05} D. R. Sahu, {it Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces}, Comment. Math. Univ. Carolinae {bf 46(4)}(2005),

-666.

bibitem{S14} G. S. Saluja, {it Convergence theorems for two asymptotically

nonexpansive non-self mappings in uniformly convex Banach spaces},

J. Indian Math. Soc. {bf 81(3-4)} (2014), 369-385.

bibitem{S191} J. Schu, {it Weak and strong convergence to fixed

points of asymptotically nonexpansive mappings}, Bull. Austral.

Math. Soc. {bf 43(1)}(1991), 153-159.

bibitem{S291} J. Schu, {it Iterative construction of fixed points of

asymptotically nonexpansive mappings}, J. Math. Anal. Appl. {bf

}(1991), 407-413.

bibitem{SS09} K. Sitthikul and S. Saejung, {it Convergence

theorems for a finite family of nonexpansive and asymptotically

nonexpansive mappings}, Acta Univ. Palack. Olomuc. Math.

textbf{48}(2009), 139-152.

bibitem{TX93} K. K. Tan and H. K. Xu, {it Approximating fixed points of

nonexpansive mappings by the Ishikawa iteration process}, J. Math.

Anal. Appl. {bf 178}(1993), 301-308.

bibitem{TX94} K. K. Tan and H. K. Xu, {it Fixed point iteration process

for asymptotically nonexpansive mappings}, Proc. Amer. Math. Soc.

{bf 122}(1994), 733-739.

bibitem{XN02} B. L. Xu and M. A. Noor, {it Fixed point iterations for

asymptotically nonexpansive mappings in Banach spaces}, J. Math.

Anal. Appl. {bf 267(2)}(2002), 444-453.

bibitem{ZCK07} H. Y. Zhou, Y. J. Cho and S. M. Kang, {it A new iterative algorithm for approximating common fixed points for asymptotically nonexpansive mappings}, Fixed Point Theory Appl. vol. {bf (2007)}, Article ID 64874, 10 pages.


Refbacks

  • There are currently no refbacks.