Servicios Personalizados
Articulo
Indicadores
- Citado por SciELO
Links relacionados
Cubo (Temuco)
versión On-line ISSN 0719-0646
Cubo vol.14 no.3 Temuco oct. 2012
http://dx.doi.org/10.4067/S0719-06462012000300010
CUBO A Mathematical Journal Vol.14, No 03, (143–166). October 2012
Weak and strong convergence theorems of a multistep iteration to a common fixed point of a family of nonself asymptotically nonexpansive mappings in banach spaces
Shrabani Banerjee and Binayak S.Choudhury
Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, India. email: banerjee.shrabani@yahoo.com, binayak12@yahoo.co.in
ABSTRACT
In this paper we have defined a multistep iterative scheme with errors involving a family of asymptotically nonexpansive nonself mappings in Banach spaces. A retraction has been used in the construction of theiteration. We prove here weak and strong convergences of the iteration to common fixed points of the family of asymptotically nonexpansive nonself mappings. We have used several concepts of Banach space geometry. Our results improve and extend some recent results.
Keywords and Phrases: Modified multistep iterative process with errors; nonself asymptotically nonexpansive mapping; retraction; Opial’s condition; uniformly convex Banach space; common fixed point; Kadec-klee property; Condition ; weak and strong convergence.
RESUMEN
En este artículo definimos un esquema de multi paso iterativo con errores que involucran una familia de aplicaciones no expansivas y no auto asintóticamente en espacios deBanach. Una retracción se ha usado en la construcción de la iteración. Probamos convergencias débiles y fuertes de las iteraciones a puntos fijos clásicos de la familia de aplicaciones no expansivas no auto asintóticamente. Hemos usado varios conceptos de geometría en espacios de Banach. Nuestro resultado mejora y extiende algunos resultados recientes.
2010 AMS Mathematics Subject Classification: 47H10
References
[1] R.E.Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J.Math. 32(1979),107-116. [ Links ]
[2] C.E.Chidume, E.U.Ofoedu, H.Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 280(2003)364-374. [ Links ]
[3] C.E.Chidume, Bashir Ali, Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 326(2007)960-973. [ Links ]
[4] C.E.Chidume, Bashir Ali, Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 330(2007)377-387. [ Links ]
[5] Y.J.Cho, H.Y.Zhou, G.Guo; Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl. 47(2004)707-717. [ Links ]
[6] J.G.Falset, W.Kaczor, T.Kuczumow, S.Reich, Weak convrgence theorems for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal.43(2001)377-401. [ Links ]
[7] K.Goebel and W.A.Kirk; A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35(1972), 171-174. [ Links ]
[8] S.Ishikawa; Fixed points by a new iteration, Proc. Amer. Math. Soc. 44(1974), 147-150. [ Links ]
[9] J.U.Jeong, S.H.Kim, Weak and strong convergence of the Ishikawa iteration process with errors for two asymptotically nonexpansive mappings, Appl. Math. Comp. 181(2006)1394-1401. [ Links ]
[10] W.Kaczor, Weak convergence of almost orbits of asymptotically nonexpansive mappings, J.Math.Anal.Appl. 272(2002)565-574. [ Links ]
[11] W.R.Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4(1953),506-510. [ Links ]
[12] Z.Opial, Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc. 73(1967)591-597. [ Links ]
[13] M.O.Osilike, A.Udomene, Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type, J. Math. Anal. Appl. 256(2001)431-445. [ Links ]
[14] H.F.Senter andW.G.Dotson, Jr., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44,375-380,(1974). [ Links ]
[15] N.Shahzad, Approximating fixed points of nonself nonexpansive mappings in Banach spaces, Nonlinear Anal.61(2005)1031-1039. [ Links ]
[16] S.Y.Matsushita, D.Kuroiwa, Strong convergence of averaging iteration of nonexpansive nonself-mappings, J. Math. Anal. Appl. 294(2004)206-214. [ Links ]
[17] W.Takahashi, G.E.Kim, Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces, Nonlinear Anal.3(32)(1998)447-454. [ Links ]
[18] K.K.Tan, H.K.Xu, Approximating fixed points of nonexpansive mapping by the Ishikawa iteration process, J. Math. Anal. Appl. 178(1993)301-308. [ Links ]
[19] L.Wang, Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl. 323(2006)550-557. [ Links ]
[20] L.Yang, Modified multistep iterative for some common fixed point of a finite family of nonself asymptotically nonexpansive mappings, Math. Comput. Modelling 45(2007)1157-1169. [ Links ]
[21] H.K.Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16(1991) 1127-1138. [ Links ]
Received: December 2011. Revised: September 2012.