Composite Implicit Iteration Process Research Articles

Stability and convergence of a new composite implicit iterative sequence in Banach spaces

The purpose of this paper is to study stability and strong convergence of asymptotically pseudocontractive mappings by using a new composite implicit iteration process in an arbitrary real Banach space. The results in this paper improve and extend the corresponding results in the literature.

Composite Implicit Iteration Process for Asymptotically Hemi-Pseudocontractive Mappings

In Banach space, the composite implicit iterative process for uniformly L- Lipschitzian asymptotically hemi-pesudocontractive mappings are studied, and the sufficient and necessary conditions of strong convergence for the composite implicit iterative process are obtained. AMS subject classifications: 47H09, 47J05, 47J25

General composite implicit iteration process for a finite family of asymptotically pseudo-contractive mappings

Abstract In this paper, a modified general composite implicit iteration process is used to study the convergence of a finite family of asymptotically nonexpansive mappings. Weak and strong convergence theorems have been proved, in the framework of a Banach space. MSC:47H09, 47H10.

Composite Implicit Iteration Process for Two Finite Families of Generalized Asymptotically Quasi-Nonexpansive Mappings

In this article, A demiclosed principle is proved for generalized asymptotically nonexpansive mapping in a Banach space. Additionally, it proves strong convergence of a composite implicit iteration scheme to a common fixed point for two finite families of generalized asymptotically quasi-nonexpansive mappings in a nonempty closed convex subset of a uniformly convex Banach space. We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. The results of this article improve and extend the corresponding results of Xu and Ori [23], Zhou and Chang [26], Chang et al. [3], Yang and Yu [25], Shahzad and Zegeye [20].

Convergence of perturbed composite implicit iteration process for a finite family of asymptotically nonexpansive mappings

In this paper, we introduce a perturbed composite implicit iterative process with errors for a finite family of asymptotically nonexpansive mappings. Under Opial’s condition, semicompact and conditions, respectively, we prove that this iterative scheme converges weakly or strongly to a common fixed point of a finite family of asymptotically nonexpansive mappings in uniformly convex Banach spaces. The results presented in this paper generalize and improve the corresponding results of Sun (J. Math. Anal. Appl. 286:351-358, 2003), Chang (J. Math. Anal. Appl. 313:273-283, 2006), Gu (J. Math. Anal. Appl. 329:766-776, 2007), Thakur (Appl. Math. Comput. 190:965-973, 007), Rafiq (Rostock. Math. Kolloqu. 62:21-39, 2007) and some others. MSC:47H9, 47H10.

Convergence of an implicit iteration process with errors for two asymptotically nonexpansive mappings

AbstractThe purpose of this paper is to introduce an implicit iterative process with errors for approximating common fixed point of two finite families of asymptotically nonexpansive mappings in the framework of Banach space. The results presented in this paper extend and generalize the corresponding results of Qin et al. [Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings, Appl. Math. Comp. 210 (2009), 542–550], Thakur [Weak and strong convergence of composite implicit iteration process, Appl. Math. Comp. 190 (2007), 965–973] and some others.

Strong convergence theorems for strictly asymptotically pseudocontractive mappings in Hilbert spaces

The purpose of this paper is by using CSQ method to study the strong convergence problem of iterative sequences for a pair of strictly asymptotically pseudocontractive mappings to approximate a common fixed point in a Hilbert space. Under suitable conditions some strong convergence theorems are proved. The results presented in the paper are new which extend and improve some recent results of Acedo and Xu [Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal., 67(7), 2258–2271 (2007)], Kim and Xu [Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal., 64, 1140–1152 (2006)], Martinez-Yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal., 64, 2400–2411 (2006)], Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl., 279, 372–379 (2003)], Marino and Xu [Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl., 329(1), 336–346 (2007)], Osilike et al. [Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps. J. Math. Anal. Appl., 326, 1334–1345 (2007)], Liu [Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal., 26(11), 1835–1842 (1996)], Osilike et al. [Fixed points of demi-contractive mappings in arbitrary Banach spaces. Panamer Math. J., 12 (2), 77–88 (2002)], Gu [The new composite implicit iteration process with errors for common fixed points of a finite family of strictly pseudocontractive mappings. J. Math. Anal. Appl., 329, 766–776 (2007)].

A Modified Averaging Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of k-Strictly Asymptotically Pseudocontractive Mappings

The composite implicit iteration process introduced by Su and Li [J. Math. Anal. Appl. 320 (2006) 882-891] is modified. A strong convergence theorem for approximation of common fixed points of finite family of k-strictly asymptotically pseudo-contractive mappings is proved in Banach spaces using the modified iteration process.

IMPLICIT ITERATION PROCESS FOR COMMON FIXED POINTS OF AN INFINITE FAMILY OF STRICTLY PSEUDOCONTRACTIVE MAPPINGS IN BANACH SPACES

Some convergence theorems for approximating to a common fixed point of an infinite family of strictly pseudocontractive mappings of Browder-Petryshyn type are proved in the setting of Banach spaces by using a new composite implicit iterative process with errors. The results presented in the paper generalize and improve the main results of Bai and Kim [1], Gu [4], Osilike [5], Su and Li [7], and Xu and Ori [8].

On the implicit iterative process for strictly pseudo-contractive mappings in Banach spaces

In this paper, we consider a composite implicit iterative process for an infinite family of strictly pseudo-contractive mappings. Strong convergence theorems are established in an arbitrary real Banach space. The results presented in this paper improve and extend the recent ones announced by many others.

Relaxed Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of Strictly Pseudocontractive Mappings

AbstractWe propose a relaxed composite implicit iteration process for finding approximate common fixed points of a finite family of strictly pseudocontractive mappings in Banach spaces. Several convergence results for this process are established.

A new composite implicit iterative algorithm for nonself asymptotically nonexpansive mappings

Let E be a real uniformly convex and smooth Banach space with P as a sunny nonexpansive retraction, K be a nonempty closed convex subset of E. Let {Si}i=1N, {Ti}i=1N:K→K be two finite families of weakly inward and asymptotically nonexpansive mappings with respect to P. It is proved that the composite implicit iteration process converges weakly and strongly to a common fixed point of {Si}i=1N, {Ti}i=1N under certain conditions. The results of this paper improve and extend some well known corresponding results.

Convergence of the new composite implicit iteration process with random errors

In this paper, strong convergence theorems for approximation of common fixed points of a finite family of asymptotically demicontractive mappings are proved in Banach spaces using the new composite implicit iteration scheme with errors. Our results of this paper improve and extend the corresponding results of Chen, Song, Zhou [R.D. Chen, Y.S. Song, H.Y. Zhou, Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings, J. Math. Anal. Appl. 314 (2006) 701–709], Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73–81], Gu [F. Gu, The new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings, J. Math. Anal. Appl. 329 (2007) 766–776] and Yang and Hu [L.P. Yang, G. Hu, Convergence of implicit iteration process with random errors, Acta Math. Sinica (Chin. Ser.) 51 (1) (2008) 11–22].

Weak and strong convergence of composite implicit iteration process

In the present paper, we propose a modified composite implicit iteration process for finite family of asymptotically nonexpansive mappings and prove some weak and strong convergence theorems for this family of mappings using proposed iteration process. Our results extend and generalize some well known results.

Strong Convergence of Modified Implicit Iteration Processes for Common Fixed Points of Nonexpansive Mappings

Strong convergence theorems are obtained by hybrid method for modified composite implicit iteration process of nonexpansive mappings in Hilbert spaces. The results presented in this paper generalize and improve the corresponding results of Nakajo and Takahashi (2003) and others.

The new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings

Convergence theorems for approximation of common fixed points of strictly pseudocontractive mappings of Browder–Petryshyn type are proved in Banach spaces using a new composite implicit iteration scheme with errors. The results presented in this paper generalize and improve the corresponding results of M.O. Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73–81].

A new composite implicit iterative process for a finite family of nonexpansive mappings in Banach spaces

AbstractThe purpose of this paper is to study the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Chang and Cho (2003), Xu and Ori (2001), and Zhou and Chang (2002).

Composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps

Let E be a real Banach space and let K be a nonempty closed convex subset of E. Let { T i } i = 1 N be N strictly pseudocontractive self-maps of K such that F = ⋂ i = 1 N F ( T i ) ≠ ∅ , where F ( T i ) = { x ∈ K : T i x = x } and let { α n } n = 1 ∞ , { β n } n = 1 ∞ ⊂ [ 0 , 1 ] be two real sequence satisfying the conditions: ( i ) ∑ n = 1 ∞ ( 1 − α n ) = + ∞ ; ( ii ) ∑ n = 1 ∞ ( 1 − α n ) 2 < + ∞ ; ( iii ) ∑ n = 1 ∞ ( 1 − β n ) < + ∞ ; ( iv ) ( 1 − α n ) ( 1 − β n ) L 2 < 1 , ∀ n ⩾ 1 , where L ⩾ 1 is common Lipschitz constant of { T i } i = 1 N . For x 0 ∈ K , let { x n } n = 1 ∞ be new implicit process defined by x n = α n x n − 1 + ( 1 − α n ) T n y n , y n = β n x n − 1 + ( 1 − β n ) T n x n where T n = T n mod N , then ( i ) lim n → ∞ ‖ x n − p ‖ exists, for all p ∈ F ; ( ii ) lim inf n → ∞ ‖ x n − T n x n ‖ = 0 . The results of this paper generalize and improve the results of Osilike in 2004. In this paper, the proof methods of the main results are also different from that of Osilike.